Notes on Stochastic Finance-Nicolas Privaut

Nicolas Privault

Notes on Stochastic Finance

This version: April 24, 2013

http://www.ntu.edu.sg/home/nprivault/indext.html



Preface

This text is an introduction to pricing and hedging in discrete and contin-uous time financial models without friction (i.e. without transaction costs),with an emphasis on the complementarity between analytical and probabilis-tic methods. Its contents are mostly mathematical, and also aim at makingthe reader aware of both the power and limitations of mathematical modelsin finance, by taking into account their conditions of applicability. The bookcovers a wide range of classical topics including Black-Scholes pricing, exoticand american options, term structure modeling and change of numeraire, aswell as models with jumps. It is targeted at the advanced undergraduate andgraduate level in applied mathematics, financial engineering, and economics.The point of view adopted is that of mainstream mathematical finance inwhich the computation of fair prices is based on the absence of arbitrage hy-pothesis, therefore excluding riskless profit based on arbitrage opportunitiesand basic (buying low/selling high) trading. Similarly, this document is notconcerned with any “prediction” of stock price behaviors that belong otherdomains such as technical analysis, which should not be confused with thestatistical modeling of asset prices. The text also includes 104 figures andsimulations, along with about 20 examples based on actual market data.

The descriptions of the asset model, self-financing portfolios, arbitrage andmarket completeness, are first given in Chapter 1 in a simple two time-stepsetting. These notions are then reformulated in discrete time in Chapter 2.Here, the impossibility to access future information is formulated using thenotion of adapted processes, which will play a central role in the constructionof stochastic calculus in continuous time.

In order to trade efficiently it would be useful to have a formula to esti-mate the “fair price” of a given risky asset, helping for example to determinewhether the asset is undervalued or overvalued at a given time. Althoughsuch a formula is not available, we can instead derive formulas for the pric-ing of options that can act as insurance contracts to protect their holdersagainst adverse changes in the prices of risky assets. The pricing and hedgingof options in discrete time, particularly in the fundamental example of the

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Cox-Ross-Rubinstein model, are considered in Chapter 3, with a descriptionof the passage from discrete to continuous time that prepares the transitionto the subsequent chapters.

A simplified presentation of Brownian motion, stochastic integrals andthe associated Ito formula, is given in Chapter 4. The Black-Scholes model ispresented from the angle of partial differential equation (PDE) methods inChapter 5, with the derivation of the Black-Scholes formula by transformingthe Black-Scholes PDE into the standard heat equation wich is then solvedby a heat kernel argument. The martingale approach to pricing and hedgingis then presented in Chapter 6, and complements the PDE approach of Chap-ter 5 by recovering the Black-Scholes formula via a probabilistic argument.An introduction to volatility estimation is given in Chapter 7, including his-torical, local, and implied volatilities. This chapter also contains a comparisonof the prices obtained by the Black-Scholes formula with option price marketdata.

Exotic options such as barrier, lookback, and Asian options in continuousasset models are treated in Chapter 8. Optimal stopping and exercise, withapplication to the pricing of American options, are considered in Chapter 9.The construction of forward measures by change of numeraire is given inChapter 10 and is applied to the pricing of interest rate derivatives in Chap-ter 12, after an introduction to the modeling of forward rates in Chapter 11,based on material from [60]. The pricing of defaultable bonds is consideredin Chapter 13.

Stochastic calculus with jumps is dealt with in Chapter 14 and is restrictedto compound Poisson processes which only have a finite number of jumps onany bounded interval. Those processes are used for option pricing and hedgingin jump models in Chapter 15, in which we mostly focus on risk minimiz-ing strategies as markets with jumps are generally incomplete. Chapter 16contains an elementary introduction to finite difference methods for the nu-merical solution of PDEs and stochastic differential equations, dealing withthe explicit and implicit finite difference schemes for the heat equations andthe Black-Scholes PDE, as well as the Euler and Milshtein schemes for SDEs.The text is completed with an appendix containing the needed probabilisticbackground.

The material in this book has been used for teaching in the Masters ofScience in Financial Engineering at City University of Hong Kong and at theNanyang Technological University in Singapore. The author thanks Ju-YiYen (University of Cincinnati) for several corrections and improvements.

The cover graph represents the time evolution of the HSBC stock pricefrom January to September 2009, plotted on the price surface of a European

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call option on that asset, expiring on October 05, 2009, cf. § 5.5.

This pdf file contains external links, and animated figures in Chapters 8,9, 11 and 14, that may require using Acrobat Reader for viewing on the com-plete pdf file. Clicking on an exercise number inside the solution section willsend to the original problem text inside the file. Conversely, clicking on theproblem number sends the reader to the corresponding solution, however thisfeature should not be misused.

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Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1 Assets, Portfolios and Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . 91.1 Definitions and Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2 Portfolio Allocation and Short-Selling . . . . . . . . . . . . . . . . . . . . . 101.3 Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4 Risk-Neutral Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.5 Hedging of Contingent Claims . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.6 Market Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.7 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2 Discrete-Time Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.1 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2 Portfolio Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.3 Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.4 Contingent Claims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.5 Martingales and Conditional Expectation . . . . . . . . . . . . . . . . . . 312.6 Risk-Neutral Probability Measures . . . . . . . . . . . . . . . . . . . . . . . . 362.7 Market Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.8 The Cox-Ross-Rubinstein (CRR) Market Model . . . . . . . . . . . . 38Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3 Pricing and hedging in discrete time . . . . . . . . . . . . . . . . . . . . . . 413.1 Pricing of Contingent Claims . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2 Hedging of Contingent Claims - Backward Induction . . . . . . . . 453.3 Pricing of Vanilla Options in the CRR Model . . . . . . . . . . . . . . 473.4 Hedging of Vanilla Options in the CRR model . . . . . . . . . . . . . 493.5 Hedging of Exotic Options in the CRR Model . . . . . . . . . . . . . . 523.6 Convergence of the CRR Model . . . . . . . . . . . . . . . . . . . . . . . . . . 59Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

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4 Brownian Motion and Stochastic Calculus . . . . . . . . . . . . . . . . 674.1 Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.2 Wiener Stochastic Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.3 Ito Stochastic Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.4 Deterministic Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.5 Stochastic Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.6 Geometric Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.7 Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . 87Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5 The Black-Scholes PDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.1 Continuous-Time Market Model . . . . . . . . . . . . . . . . . . . . . . . . . . 935.2 Self-Financing Portfolio Strategies . . . . . . . . . . . . . . . . . . . . . . . . 935.3 Arbitrage and Risk-Neutral Measures . . . . . . . . . . . . . . . . . . . . . 975.4 Market Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985.5 The Black-Scholes PDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.6 The Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085.7 Solution of the Black-Scholes PDE . . . . . . . . . . . . . . . . . . . . . . . . 109Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6 Martingale Approach to Pricing and Hedging . . . . . . . . . . . . . 1176.1 Martingale Property of the Ito Integral . . . . . . . . . . . . . . . . . . . . 1176.2 Risk-neutral Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196.3 Girsanov Theorem and Change of Measure . . . . . . . . . . . . . . . . 1226.4 Pricing by the Martingale Method . . . . . . . . . . . . . . . . . . . . . . . . 1236.5 Hedging Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

7 Estimation of Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1417.1 Historical Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1417.2 Implied Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1427.3 The Black-Scholes Formula vs Market Data . . . . . . . . . . . . . . . . 1437.4 Local Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

8 Exotic Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1518.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1518.2 The Reflexion Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1558.3 Barrier Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1638.4 Lookback Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1828.5 Asian Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

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9 American Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2279.1 Filtrations and Information Flow . . . . . . . . . . . . . . . . . . . . . . . . . 2279.2 Martingales, Submartingales, and Supermartingales . . . . . . . . . 2289.3 Stopping Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2309.4 Perpetual American Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2399.5 Finite Expiration American Options . . . . . . . . . . . . . . . . . . . . . . 252Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

10 Change of Numeraire and Forward Measures . . . . . . . . . . . . . 26910.1 Notion of Numeraire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26910.2 Change of Numeraire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27110.3 Foreign Exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27810.4 Pricing of Exchange Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28410.5 Self-Financing Hedging by Change of Numeraire . . . . . . . . . . . . 286Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

11 Forward Rate Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29511.1 Short Term Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29511.2 Zero-Coupon Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29711.3 Forward Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30511.4 The HJM Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31111.5 Forward Vasicek Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31511.6 Modeling Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32011.7 The BGM Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

12 Pricing of Interest Rate Derivatives . . . . . . . . . . . . . . . . . . . . . . 33712.1 Forward Measures and Tenor Structure . . . . . . . . . . . . . . . . . . . . 33712.2 Bond Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33912.3 Caplet Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34112.4 Forward Swap Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34412.5 Swaption Pricing on the LIBOR . . . . . . . . . . . . . . . . . . . . . . . . . . 345Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

13 Default Risk in Bond Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35913.1 Survival Probabilities and failure rate . . . . . . . . . . . . . . . . . . . . . 35913.2 Stochastic Default . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36113.3 Defaultable Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36313.4 Credit Default Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36413.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366

14 Stochastic Calculus for Jump Processes . . . . . . . . . . . . . . . . . . . 36914.1 The Poisson Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36914.2 Compound Poisson Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37514.3 Stochastic Integrals with Jumps . . . . . . . . . . . . . . . . . . . . . . . . . . 37814.4 Ito Formula with Jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380

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14.5 Stochastic Differential Equations with Jumps . . . . . . . . . . . . . . 38514.6 Girsanov Theorem for Jump Processes . . . . . . . . . . . . . . . . . . . . 389Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395

15 Pricing and Hedging in Jump Models . . . . . . . . . . . . . . . . . . . . . 39715.1 Risk-Neutral Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39715.2 Pricing in Jump Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39815.3 Black-Scholes PDE with Jumps . . . . . . . . . . . . . . . . . . . . . . . . . . 40015.4 Exponential Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40215.5 Self-Financing Hedging with Jumps . . . . . . . . . . . . . . . . . . . . . . . 404Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408

16 Basic Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41116.1 The Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41116.2 The Black-Scholes PDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41316.3 Euler Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41716.4 Milshtein Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418

Appendix: Background on Probability Theory . . . . . . . . . . . . . . . . 421Probability Spaces and Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421Probability Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425Conditional Probabilities and Independence . . . . . . . . . . . . . . . . . . . . 426Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427Probability Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429Expectation of a Random Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434Conditional Expectation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441Moment Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445

Exercise Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499Chapter 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505Chapter 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511Chapter 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525Chapter 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528Chapter 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532Background on Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535

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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541

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List of Figures

0.1 Graph of the Hang Seng index - holding a put option might be

useful here. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50.2 Sample price processes simulated by a geometric Brownian motion. . . 60.3 “Infogrames” stock price curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.1 Another example of absence of arbitrage. . . . . . . . . . . . . . . . . . . . . . . 12

2.1 Illustration of the self-financing condition (2.3). . . . . . . . . . . . . . . . . . 27

4.1 Sample paths of one-dimensional Brownian motion. . . . . . . . . . . . . . . 704.2 Two sample paths of a two-dimensional Brownian motion. . . . . . . . . 714.3 Sample paths of a three-dimensional Brownian motion. . . . . . . . . . . . 714.4 Step function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.1 Illustration of the self-financing condition (5.2). . . . . . . . . . . . . . . . . . 945.2 Graph of the Black-Scholes call price function with strike K = 100. . 1035.3 Graph of the stock price of HSBC Holdings. . . . . . . . . . . . . . . . . . . . 1035.4 Path of the Black-Scholes price for a call option on HSBC. . . . . . . . . 1045.5 Time evolution of the hedging portfolio for a call option on HSBC. . 1065.6 Graph of the Black-Scholes put price function with strike K = 100. . 1065.7 Path of the Black-Scholes price for a put option on HSBC. . . . . . . . . 1075.8 Time evolution of the hedging portfolio for a put option on HSBC. . 1075.9 Option price as a function of the volatility σ. . . . . . . . . . . . . . . . . . . . 113

6.1 Drifted Brownian path. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1206.2 Option price as a function of the underlying and of time to maturity 1366.3 Delta as a function of the underlying and of time to maturity . . . . . . 1376.4 Gamma as a function of the underlying and of time to maturity . . . . 1376.5 Option price as a function of the underlying and of time to maturity 1396.6 Delta as a function of the underlying and of time to maturity . . . . . . 1396.7 Gamma as a function of the underlying and of time to maturity . . . . 140

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7.1 Implied volatility of Asian options on light sweet crude oil futures.1 . 1437.2 Graph of the (market) stock price of Cheung Kong Holdings. . . . . . . 1447.3 Graph of the (market) call option price on Cheung Kong Holdings. . 1447.4 Graph of the Black-Scholes call option price on Cheung Kong Holdings.1457.5 Graph of the (market) stock price of HSBC Holdings. . . . . . . . . . . . . 1457.6 Graph of the (market) call option price on HSBC Holdings. . . . . . . . 1467.7 Graph of the Black-Scholes call option price on HSBC Holdings. . . . 1467.8 Graph of the (market) put option price on HSBC Holdings. . . . . . . . 1477.9 Graph of the Black-Scholes put option price on HSBC Holdings. . . . 147

8.1 Brownian motion Bt and its supremum Xt. . . . . . . . . . . . . . . . . . . . . 1538.2 Brownian motion Bt and its moving average. . . . . . . . . . . . . . . . . . . . 1548.3 Reflected Brownian motion with a = 1. . . . . . . . . . . . . . . . . . . . . . . . 1578.4 Probability density of the maximum of Brownian motion. . . . . . . . . . 1588.5 Joint probability density of B1 and its maximum over [0,1]. . . . . . . . . 1608.6 Probability density of the maximum of drifted Brownian motion. . . . . 1628.7 Graph of the up-and-out call option price. . . . . . . . . . . . . . . . . . . . . . . 1678.8 Graph of the up-and-out put option price with B > K. . . . . . . . . . . . 1738.9 Graph of the up-and-out put option price with K > B. . . . . . . . . . . . 1738.10 Graph of the down-and-out call option price with B < K. . . . . . . . . . 1758.11 Graph of the down-and-out call option price with K > B. . . . . . . . . . 1758.12 Graph of the down-and-out put option price with K > B. . . . . . . . . . 1768.13 Delta for the up-and-out option. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1808.14 Graph of the lookback put option price. . . . . . . . . . . . . . . . . . . . . . . . 1838.15 Graph of the normalized lookback put option price. . . . . . . . . . . . . . . 1928.16 Black-Scholes put price in the decomposition (8.29). . . . . . . . . . . . . . . 1938.17 Function hp(τ, z) in the decomposition (8.29). . . . . . . . . . . . . . . . . . . 1948.18 Graph of the lookback call option price. . . . . . . . . . . . . . . . . . . . . . . . 1958.19 Normalized lookback call option price. . . . . . . . . . . . . . . . . . . . . . . . . . 2008.20 Graph of the underlying asset price. . . . . . . . . . . . . . . . . . . . . . . . . . . 2008.21 Graph of the lookback call option price. . . . . . . . . . . . . . . . . . . . . . . . 2018.22 Running minimum of the underlying asset price. . . . . . . . . . . . . . . . . . 2018.23 Black-Scholes call price in the normalized lookback call price. . . . . . . 2028.24 Function hc(τ, z) in the normalized lookback call option price. . . . . . . 2038.25 Graph of the Asian option price with σ = 0.3, r = 0.1 and K = 90. . . 2108.26 Lognormal approximation to the Asian option price. . . . . . . . . . . . . . 211

9.1 Drifted Brownian path. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2299.2 Evolution of the fortune of a poker player vs number of games played. 2299.3 Stopped process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2329.4 Graphs of the option price by exercise at τL for several values of L. . 2439.5 Animated graph of the option price depending on the values of L. . . 2449.6 Option price as a function of L and of the underlying asset price. . . . 2449.7 Path of the American put option price on the HSBC stock. . . . . . . . 2459.8 Graphs of the option price by exercising at τL for several values of L. 250

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9.9 Graphs of the option prices parametrized by different values of L. . . . 2509.10 Expected Black-Scholes European call price vs (x, t) 7→ (x−K)+. . . . 2539.11 Black-Scholes put price function vs (x, t) 7→ (K − x)+. . . . . . . . . . . . . 2549.12 Optimal frontier for the exercise of a put option. . . . . . . . . . . . . . . . . 2549.13 Numerical values of the finite expiration American put price. . . . . . . . 2569.14 Longstaff-Schwartz algorithm for the American put price. . . . . . . . . . 2579.15 Comparison between Longstaff-Schwartz and finite differences. . . . . . 257

11.1 Graph of t 7−→ rt in the Vasicek model. . . . . . . . . . . . . . . . . . . . . . . . 29611.2 Graphs of t 7−→ P (t, T ) and t 7−→ e−r0(T−t). . . . . . . . . . . . . . . . . . . 30311.3 Graph of t 7−→ P (t, T ) for a bond with a 2.3% coupon. . . . . . . . . . . . 30311.4 Bond price graph with coupon rate 6.25%. . . . . . . . . . . . . . . . . . . . . . 30411.5 Graph of T 7−→ f(t, T, T + δ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30711.6 Stochastic process of forward curves. . . . . . . . . . . . . . . . . . . . . . . . . . 31211.7 Forward rate process t 7−→ f(t, T, S). . . . . . . . . . . . . . . . . . . . . . . . . . 31611.8 Instantaneous forward rate process t 7−→ f(t, T ). . . . . . . . . . . . . . . . . 31711.9 Forward instantaneous curve in the Vasicek model. . . . . . . . . . . . . . 31811.10 Forward instantaneous curve x 7−→ f(0, x) in the Vasicek model. . . . 31811.11 Short term interest rate curve t 7−→ rt in the Vasicek model. . . . . . . 31911.12 Market example of yield curves (11.22). . . . . . . . . . . . . . . . . . . . . . . 31911.13 Graph of x 7−→ g(x) in the Nelson-Siegel model. . . . . . . . . . . . . . . . . 32011.14 Graph of x 7−→ g(x) in the Svensson model. . . . . . . . . . . . . . . . . . . . 32111.15 Comparison of market data vs a Svensson curve. . . . . . . . . . . . . . . . 32111.16 Graphs of forward rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32211.17 Forward instantaneous curve in the Vasicek model. . . . . . . . . . . . . . . 32211.18 Graph of t 7−→ P (t, T1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32311.19 Graph of forward rates in a two-factor model. . . . . . . . . . . . . . . . . . 32611.20 Random evolution of forward rates in a two-factor model. . . . . . . . . 32611.21 Graph of stochastic interest rate modeling. . . . . . . . . . . . . . . . . . . . . 328

12.1 Forward rates arranged according to a tenor structure. . . . . . . . . . . . 337

14.1 Sample path of a Poisson process (Nt)t∈R+ . . . . . . . . . . . . . . . . . . 37014.2 Sample path of a compound Poisson process (Yt)t∈R+

. . . . . . . . 37614.3 Sample trajectories of a gamma process. . . . . . . . . . . . . . . . . . . . . 38314.4 Sample trajectories of a stable process. . . . . . . . . . . . . . . . . . . . . . 38314.5 Sample trajectories of a variance gamma process. . . . . . . . . . . . . 38414.6 Sample trajectories of an inverse Gaussian process. . . . . . . . . . . 38414.7 Sample trajectories of a negative inverse Gaussian process. . . . 38414.8 Geometric Poisson process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38614.9 Ranking data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38714.10Geometric compound Poisson process. . . . . . . . . . . . . . . . . . . . . . . . . 38814.11Geometric Brownian motion with compound Poisson jumps. . . . . . . . 389

16.1 Divergence of the explicit finite difference method. . . . . . . . . . . . . . . 415

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16.2 Stability of the implicit finite difference method. . . . . . . . . . . . . . . . . 41716.3 Average return by selling at the maximum vs selling at maturity T . . 46916.4 Graph of the down-and-in long forward contract price with

K < B = 80. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47216.5 Delta of the down-and-in long forward contract with K < B = 80. . . . 47316.6 Graph of the up-and-out long forward contract price with K < B = 80.47416.7 Graph of the up-and-out long forward contract price with K < B = 80.47516.8 Graph of the down-and-in long forward contract price with

K < B = 80. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47616.9 Delta of the down-and-in long forward contract with K < B = 80. . . . 47616.10Graph of the down-and-out long forward contract price with

K < B = 80. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47716.11Delta of the down-and-out long forward contract with K < B = 80. . 47816.12Lookback call option as a function of maturity time T . . . . . . . . . . . . . 479

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Introduction

Modern mathematical finance and quantitative analysis require a strongbackground in fields such as stochastic calculus, optimization, partial differ-ential equations (PDEs) and numerical methods, or even infinite dimensionalanalysis. In addition, the emergence of new complex financial instruments onthe markets makes it necessary to rely on increasingly sophisticated mathe-matical tools. Not all readers of this book will eventually work in quantitativefinancial analysis, nevertheless they may have to interact with quantitativeanalysts, and becoming familiar with the tools they employ be an advantage.In addition, despite the availability of ready made financial calculators it stillmakes sense to be able oneself to understand, design and implement suchfinancial algorithms. This can be particularly useful under different types ofconditions, including an eventual lack of trust in financial indicators, possibleunreliability of expert advice such as buy/sell recommendations, or other fac-tors such as market manipulation. To some extent we would like to have someform of control on the future behaviour of random (risky) assets, however,since knowledge of the future is not possible, the time evolution of the pricesof risky assets will be modelled by random variables and stochastic processes.

Historical Sketch

We start with a description of some of the main steps, ideas and individualsthat played an important role in the development of the field over the lastcentury.

Robert Brown, botanist, 1827

Brown observed the movement of pollen particles as described in his paper“A brief account of microscopical observations made in the months of June,July and August, 1827, on the particles contained in the pollen of plants; and

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on the general existence of active molecules in organic and inorganic bodies.”Phil. Mag. 4, 161-173, 1828.

Philosophical Magazine, first published in 1798, is a journal that “publishesarticles in the field of condensed matter describing original results, theoriesand concepts relating to the structure and properties of crystalline materials,ceramics, polymers, glasses, amorphous films, composites and soft matter.”

Louis Bachelier, mathematician, PhD 1900

Bachelier used Brownian motion for the modelling of stock prices in hisPhD thesis “Theorie de la speculation”, Annales Scientifiques de l’Ecole Nor-male Superieure 3 (17): 21-86, 1900.

Albert Einstein, physicist

Einstein received his 1921 Nobel Prize in part for investigations on thetheory of Brownian motion: “... in 1905 Einstein founded a kinetic theory toaccount for this movement”, presentation speech by S. Arrhenius, Chairmanof the Nobel Committee, Dec. 10, 1922.

Albert Einstein, Uber die von der molekularkinetischen Theorie der Warmegeforderte Bewegung von in ruhenden Flussigkeiten suspendierten Teilchen”,Annalen der Physik 17 (1905) 223.

Norbert Wiener, mathematician, founder of cybernetics

Wiener is credited, among other fundamental contributions, for the math-ematical foundation of Brownian motion, published in 1923. In particular heconstructed the Wiener space and Wiener measure on C0([0, 1]) (the space ofcontinuous functions from [0, 1] to R vanishing at 0).

Norbert Wiener, “Differential space”, Journal of Mathematics and Physics ofthe Massachusetts Institute of Technology, 2, 131-174, 1923.

Kiyoshi Ito, mathematician, Gauss prize 2006

Ito constructed the Ito integral with respect to Brownian motion, cf. Ito,Kiyosi, Stochastic integral. Proc. Imp. Acad. Tokyo 20, (1944). 519-524. Healso constructed the stochastic calculus with respect to Brownian motion,which laid the foundation for the development of calculus for random pro-cesses, cf. Ito, Kiyoshi, “On stochastic differential equations”, Mem. Amer.Math. Soc. 1951, (1951).

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“Renowned math wiz Ito, 93, dies.” (The Japan Times, Saturday, Nov. 15,2008).

Kiyoshi Ito, an internationally renowned mathematician and professoremeritus at Kyoto University died Monday of respiratory failure at a Ky-oto hospital, the university said Friday. He was 93. Ito was once dubbed“the most famous Japanese in Wall Street” thanks to his contributionto the founding of financial derivatives theory. He is known for his workon stochastic differential equations and the “Ito Formula”, which laid thefoundation for the Black-Scholes model, a key tool for financial engineer-ing. His theory is also widely used in fields like physics and biology.

Paul Samuelson, economist, Nobel Prize 1970

In 1965, Samuelson rediscovered Bachelier’s ideas and proposed geometricBrownian motion as a model for stock prices. In an interview he stated “Inthe early 1950s I was able to locate by chance this unknown [Bachelier’s]book, rotting in the library of the University of Paris, and when I opened itup it was as if a whole new world was laid out before me.” We refer to “Ra-tional theory of warrant pricing” by Paul Samuelson, Industrial ManagementReview, p. 13-32, 1965.

In recognition of Bachelier’s contribution, the Bachelier Finance Society wasstarted in 1996 and now holds the World Bachelier Finance Congress every2 years.

Robert Merton, Myron Scholes, economists

Robert Merton and Myron Scholes shared the 1997 Nobel Prize in eco-nomics: “In collaboration with Fisher Black, developed a pioneering formulafor the valuation of stock options ... paved the way for economic valuationsin many areas ... generated new types of financial instruments and facilitatedmore efficient risk management in society.”

Black, Fischer; Myron Scholes (1973). ”The Pricing of Options and Corpo-rate Liabilities”. Journal of Political Economy 81 (3): 637-654.

The development of options pricing tools contributed greatly to the expansionof option markets and led to development several ventures such as the “LongTerm Capital Management” (LTCM), founded in 1994. The fund yielded an-nualized returns of over 40% in its first years, but registered lost US$ 4.6billion in less than four months in 1998, which resulted into its closure inearly 2000.

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Oldrich Vasicek, economist, 1977

Interest rates behave differently from stock prices, notably due to the phe-nomenon of mean reversion, and for this reason they are difficult to modelusing geometric Brownian motion. Vasicek was the first to suggest a mean-reverting model for stochastic interest rates, based on the Ornstein-Uhlenbeckprocess, in “An equilibrium characterisation of the term structure”, Journalof Financial Economics 5: 177-188.

David Heath, Robert Jarrow, A. Morton

These authors proposed in 1987 a general framework to model the evolu-tion of (forward) interest rates, known as the HJM model, see their joint paper“Bond Pricing and the Term Structure of Interest Rates: A New Methodol-ogy for Contingent Claims Valuation”, Econometrica, (January 1992), Vol.60, No. 1, pp 77-105.

Alan Brace, Dariusz Gatarek, Marek Musiela (BGM)

The BGM model is actually based on geometric Brownian motion, and itis specially useful for the pricing of interest rate derivatives such as caps andswaptions on the LIBOR market, see “The Market Model of Interest RateDynamics”. Mathematical Finance Vol. 7, page 127. Blackwell 1997, by AlanBrace, Dariusz Gatarek, Marek Musiela.

European Call and Put Options

We close this introduction with a description of European call and put op-tions, which are at the basis of risk management. As mentioned above, animportant concern for the buyer of a stock at time t is whether its price STcan fall down at some future date T . The buyer of the stock may seek pro-tection from a market crash by purchasing a contract that allows him to sellhis asset at time T at a guaranteed price K fixed at time t. This contract iscalled a put option with strike price K and exercise date T .

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Fig. 0.1: Graph of the Hang Seng index - holding a put option might be useful here.

Definition 0.1. A (European) put option is a contract that gives its holderthe right (but not the obligation) to sell a quantity of assets at a predefinedprice K called the strike and at a predefined date T called the maturity.

In case the price ST falls down below the level K, exercising the contractwill give the holder of the option a gain equal to K − ST in comparison tothose who did not subscribe the option and sell the asset at the market priceST . In turn, the issuer of the option will register a loss also equal to K − ST(in the absence of transaction costs and other fees).

If ST is above K then the holder of the option will not exercise the optionas he may choose to sell at the price ST . In this case the profit derived fromthe option is 0.

In general, the payoff of a (so called European) put option will be of theform

φ(ST ) = (K − ST )+ =

K − ST , ST ≤ K,

0, ST ≥ K.Two possible scenarios (ST finishing above K or below K) are illustrated

in Figure 0.2.

On the other hand, if the trader aims at buying some stock or commodity,his interest will be in prices not going up and he might want to purchase acall option, which is a contract allowing him to buy the considered asset attime T at a price not higher than a level K fixed at time t.

Here, in the event that ST goes above K, the buyer of the option willregister a potential gain equal to ST −K in comparison to an agent who did

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0

1

2

3

4

5

6

7

8

9

10

0 0.2 0.4 0.6 0.8 1

S t

K=

S0=

T=t=0.62

|

Strike

ST-K>0

ST-K<0

Fig. 0.2: Sample price processes simulated by a geometric Brownian motion.

not subscribe to the call option.

Definition 0.2. A (European) call option is a contract that gives its holderthe right (but not the obligation) to buy a quantity of assets at a predefinedprice K called the strike and at a predefined date T called the maturity.

In general, a (European) call option is an option with payoff function

φ(ST ) = (ST −K)+ =

ST −K, ST ≥ K,

0, ST ≤ K.

In market practice, options are often divided into a certain number n of war-rants, the (possibly fractional) quantity n being called the entitlement ratio.

In order for an option contract to be fair, the buyer of the option shouldpay a fee (similar to an insurance fee) at the signature of the contract. Thecomputation of this fee is an important issue, which is known as optionpricing.

The second important issue is that of hedging, i.e. how to manage a givenportfolio in such a way that it contains the required random payoff (K−ST )+

(for a put option) or (ST −K)+ (for a call option) at the maturity date T .

The next figure illustrates a sharp increase and sharp drop in asset price,making it valuable to hold a call option during the first half of the graph,whereas holding a put option would be recommended during the second half.

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Fig. 0.3: “Infogrames” stock price curve.

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Chapter 1

Assets, Portfolios and Arbitrage

We consider a simplified financial model with only two time instants t = 0 andt = 1. In this simple setting we introduce the notions of portfolio, arbitrage,completeness, pricing and hedging using the notation of [25]. A binary assetprice model is considered as an example in Section 1.7.

1.1 Definitions and Formalism

We will use the following notation. An element x of Rd+1 is a vector

x = (x0, x1, . . . , xd)

made of d+1 components. The scalar product x · y of two vectors x, y ∈ Rd+1

is defined byx · y = x0y0 + x1y1 + · · ·xdyd.

The vectorπ =

(π(0), π(1), . . . , π(d)

)denotes the prices π(i) > 0 at time t = 0 of d + 1 assets numberedi = 0, 1, . . . , d.

The values S(i) > 0 at time t = 1 of assets i = 1, . . . , d are represented bythe random vector

S =(S(0), S(1), . . . , S(d)

)defined on a probability space (Ω,F ,P).

In addition we will assume that asset no 0 is a riskless asset (of savingsaccount type) that yields an interest rate r > 0, i.e. we have

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S(0) = (1 + r)π(0).

1.2 Portfolio Allocation and Short-Selling

A portfolio based on the assets 0, 1, 2, . . . , d is viewed as a vector ξ ∈ Rd+1

in which ξ(i) represents the (possibly fractional) quantity of asset no i ownedby an investor, i = 0, 1, . . . , d. The price of such a portfolio is given by

ξ · π =d∑i=0

ξ(i)π(i)

at time t = 0.

At time t = 1 the value of the portfolio has evolved into

ξ · S =d∑i=0

ξ(i)S(i).

If ξ(0) > 0, the investor puts the amount ξ(0)π(0) > 0 on a savings accountwith interest rate r, while if ξ(0) < 0 he borrows the amount −ξ(0)π(0) > 0with the same interest rate.

For i = 1, . . . , d, if ξ(i) > 0 then the investor buys a (possibly fractional)quantity ξ(i) > 0 of the asset no i, while if ξ(i) < 0 he borrows a quantity−ξ(i) > 0 of asset i and sells it to obtain the amount −ξ(i)π(i) > 0. In thelatter case one says that the investor short sells a quantity −ξ(i) > 0 of theasset no i.

Usually, profits are made by first buying at a lower price and then sellingat a higher price. Short-sellers apply the same rule but in the reverse timeorder: first sell high, and then buy low if possible, by applying the followingprocedure.

1. Borrow the asset no i.

2. At time t = 0, sell the asset no i on the market at the price π(i) andinvest the amount π(i) at the interest rate r > 0.

3. Buy back the asset no i at time t = 1 at the price S(i), with hopefullyS(i) < (1 + r)π(i).

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4. Return the asset to its owner, with possibly a (small) fee p > 0.1

At the end of the operation the profit made on share no i equals

(1 + r)π(i) − S(i) − p > 0,

which is positive provided S(i) < (1 + r)π(i) and p > 0 is sufficiently small.

1.3 Arbitrage

As stated in the next definition, an arbitrage opportunity is the possibilityto make a strictly positive amount of money starting from 0 or even from anegative amount. In a sense, an arbitrage opportunity can be seen as a wayto “beat” the market.

The short-selling procedure described in Section 1.2 represents a way torealize an arbitrage opportunity. One can proceed similarly by simply buyingan asset instead short-selling it.

1. Borrow the amount −ξ(0)π(0) > 0 on the riskless asset no 0.

2. Use the amount −ξ(0)π(0) > 0 to buy the risky asset no i at time t = 0and price π(i), for a quantity ξ(i) = −ξ(0)π(0)/π(i), i = 1, . . . , d.

3. At time t = 1, sell the risky asset no i at the price S(i), with hopefullyS(i) > π(i).

4. Refund the amount −(1 + r)ξ(0)π(0) > 0 with interest rate r > 0.

At the end of the operation the profit made is

S(i) − (1 + r)π(i) > 0,

which is positive provided S(i) > π(i) and r is sufficiently small.

Next we state a mathematical formulation of the concept of arbitrage.

Definition 1.1. A portfolio ξ ∈ Rd+1 constitutes an arbitrage opportunity ifthe three following conditions are satisfied:

i) ξ · π ≤ 0, [start from 0 or even with a debt]

1 The cost p of shortselling will not be taken into account in later calculations.

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ii) ξ · S ≥ 0, [finish with a non-negative amount]

iii) P(ξ · S > 0) > 0. [a profit is made with non-zero probability]

The are many real-life examples of situations where arbitrage opportunitiescan occur, such as:

- assets with different returns (finance),

- servers with different speeds (queueing, networking, computing),

- highway lanes with different speeds (driving).

In the latter two examples, the absence of arbitrage is consequence of thefact that switching to a faster lane or server may result into congestion, thusannihilating the potential benefit of the shift.

Fig. 1.1: Another example of absence of arbitrage.

In the table of Figure 1.1 the absence of arbitrage opportunities is material-ized by the fact that the price of each combination is found to be proportionalto its probability, thus making the game fair and disallowing any opportunityor arbitrage that would result of betting on a more profitable combination.

In the sequel we will work under the assumption that arbitrage oppor-tunities do not occur and we will rely on this hypothesis for the pricing offinancial instruments.

Let us give a market example of pricing by absence of arbitrage.

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From March 24 to 31, 2009, HSBC issued rights to buy shares at the priceof $28. This right actually behaves like a call option since it gives the right(with no obligation) to buy the stock at K = $28. On March 24 the HSBCstock price finished at $41.70.

The question is: how to value the price $R of the right to buy one share ?This question can be answered by looking for arbitrage opportunities. Indeed,there are two ways to buy the stock:

1. directly buy the stock on the market at the price of $41.70. Cost: $41.70,or:2. first purchase the right at price $R and then the stock at price $28.Total cost: $R+$28.

For an investor who owns no stock and no rights, arbitrage would be possiblein case $R + $28 < $41.70 by buying the right at a price $R, then the stockat price $28, and finally selling the stock at the market price of $41.70. Theprofit made by the investor would equal

$41.70− ($R+ $28) > 0.

On the other hand, for an investor who owns the rights, in case $R + $28 >$41.70, arbitrage would be possible by firt selling the right at price $R, andthen buying the stock on the market at $41.70. At time t = 1 the stock couldbe sold at around $28, and profit would equal

$R+ $28− $41.70 > 0.

In the absence of arbitrage opportunities, the above argument implies that$R should satisfy

$R+ $28− $41.70 = 0,

i.e. the arbitrage price of the right is given by the equation

$R = $41.70− $28 = $13.70. (1.1)

Interestingly, the market price of the right was $13.20 at the close of thesession on March 24. The difference of $0.50 can be explained by the presenceof various market factors such as transaction costs, the time value of money,or simply by the fact that asset prices are constantly fluctuating over time.It may also represent a small arbitrage opportunity, which cannot be at allexcluded. Nevertheless, the absence of arbitrage argument (1.1) prices theright at $13.70, which is quite close to its market value. Thus the absence ofarbitrage hypothesis appears as an accurate tool for pricing.

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N. Privault

1.4 Risk-Neutral Measures

In order to use absence of arbitrage in the general context of pricing financialderivatives, we will need the notion of risk-neutral measure.

The next definition says that under a risk-neutral (probability) measure,the risky assets no 1, . . . , d have same average rate of return as the risklessasset no 0.

Definition 1.2. A probability measure P∗ on Ω is called a risk-neutral mea-sure if

IE∗[S(i)] = (1 + r)π(i), i = 0, 1, . . . , d. (1.2)

Here, IE∗ denotes the expectation under the probability measure P∗.

In other words, P∗ is called “risk neutral” because taking risks under P∗by buying a stock S(i) has a neutral effect: on average the expected yield ofthe risky asset equals the riskless rate obtained by investing on the savingsaccount with interest rate r.

On the other hand, under a “risk premium” probability measure P#, theexpected return of the risky asset S(i) would be higher than r, i.e. we wouldhave

IE#[S(i)] > (1 + r)π(i), i = 1, . . . , d.

The following result can be used to check for the existence of arbitrage op-portunities, and is known as the first fundamental theorem of mathematicalfinance. In the sequel we will only consider probability measures P∗ that areequivalent to P in the sense that P∗(A) = 0 if and only if P(A) = 0 for allA ∈ F .

Theorem 1.1. A market is without arbitrage opportunity if and only if itadmits at least one equivalent risk-neutral measure P∗.

Proof. For the sufficiency, given P∗ a risk-neutral measure we have

ξ · π =

d∑i=0

ξ(i)π(i) =1

1 + r

d∑i=0

ξ(i) IE∗[S(i)] =1

1 + rIE∗[ξ · S] > 0,

because P∗(ξ · S > 0) > 0 as P(ξ · S > 0) > 0 and P∗ is equivalent to P,and this contradicts Definition 1.1-(i). The proof of necessity relies on thetheorem of separation of convex sets by hyperplanes, cf. Theorem 1.6 of [25].

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1.5 Hedging of Contingent Claims

In this section we consider the notion of contingent claim, according to thefollowing broad definition.

Definition 1.3. A contingent claim is any non-negative random variableC ≥ 0.

In practice the random variable C represents the payoff of an (option)contract at time t = 1.

Referring to Definition 0.2, a European call option with maturity t = 1 onthe asset no i is a contingent claim whose the payoff C is given by

C = (S(i) −K)+ =

S(i) −K if S(i) ≥ K,

0 if S(i) < K,

where K is called the strike price. The claim C is called “contingent” be-cause its value may depend on various market conditions, such as S(i) > K.A contingent claim is also called a “derivative” for the same reason.

Similarly, referring to Definition 0.1, a European put option with maturityt = 1 on the asset no i is a contingent claim with payoff

C = (K − S(i))+ =

K − S(i) if S(i) ≤ K,

0 if S(i) > K,

Definition 1.4. A contingent claim with payoff C is said to be attainable ifthere exists a portfolio strategy ξ such that

C = ξ · S.

When a contingent claim C is attainable, a trader will be able to:

1. at time t = 0, build a portfolio allocation ξ = (ξ(0), ξ(1), . . . , ξ(d)) ∈ Rd+1,

2. invest the amount

ξ · π =d∑i=0

ξ(i)π(i)

in this portfolio at time t = 0,

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N. Privault

3. at time t = 1, pay the claim amount C using the value ξ·S of the portfolio.

The above shows that in order to attain the claim, an initial investment ξ · πis needed at time t = 0. This amount, to be paid by the buyer to the issuerof the option (the option writer), is also called the arbitrage price of thecontingent claim C, and denoted by

π(C) := ξ · π. (1.3)

The action of allocating a portfolio ξ such that

C = ξ · S (1.4)

is called hedging, or replication, of the contingent claim C.

As a rough illustration of the principle of hedging, one may buy oil-relatedstocks in order to hedge oneself against a potential price rise of gasoline. Inthis case, any increase in the price of gasoline that would result in a highervalue of the derivative C would be correlated to an increase in the underlyingstock value, so that the equality (1.4) would be maintained.

In case the value ξ · S exceeds the amount of the claim, i.e. if

ξ · S ≥ C,

we talk about super-hedging.

In this book we focus on hedging (i.e. replication of the contingent claimC) and we will not consider super-hedging.

1.6 Market Completeness

Market completeness is a strong property saying that any contingent claimcan be perfectly hedged.

Definition 1.5. A market model is said to be complete if every contingentclaim C is attainable.

The next result is the second fundamental theorem of mathematical fi-nance.

Theorem 1.2. A market model without arbitrage is complete if and only ifit admits only one risk-neutral measure.

Proof. cf. Theorem 1.40 of [25].

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Theorem 1.2 will give us a concrete way to verify market completeness bysearching for a unique solution P∗ to Equation (1.2).

1.7 Example

In this section we work out a simple example that allows us to apply Theo-rem 1.1 and Theorem 1.2.

We take d = 1, i.e. there is only a riskless asset no 0 and a risky assetS(1). In addition we choose

Ω = ω−, ω+,

which is the simplest possible non-trivial choice of a probability space, madeof only two possible outcomes with

P(ω−) > 0 and P(ω+) > 0,

in order for the setting to be non-trivial. In other words the behavior of themarket is subject to only two possible outcomes, for example, one is expectingan important binary decision of yes/no type, which can lead to two distinctscenarios called ω− and ω+.

In this context, the asset price S(1) is given by a random variable

S(1) : Ω −→ R

whose value depends whether the scenario ω−, resp. ω+, occurs.

Precisely, we set

S(1)(ω−) = a, and S(1)(ω+) = b,

i.e. the value of S(1) becomes equal a under the scenario ω−, and equal to bunder the scenario ω+, where 0 < a ≤ b.

The first natural question we ask is:

- are there arbitrage opportunities in such a market ?

We will answer this question using Theorem 1.1, which amounts to searchingfor a risk-neutral measure P∗. In this simple framework, any measure P∗ onΩ = ω−, ω+ is characterized by the data of two numbers P∗(ω−) ∈ [0, 1]and P∗(ω+) ∈ [0, 1], such that

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N. Privault

P∗(Ω) = P∗(ω−) + P∗(ω+) = 1. (1.5)

Here, saying that P∗ is equivalent to P simply means that

P∗(ω−) > 0 and P∗(ω+) > 0.

In addition, according to Definition 1.2 a risk-neutral measure P∗ shouldsatisfy

IE∗[S(1)] = (1 + r)π(1). (1.6)

Although we should solve this equation for P∗, at this stage it is not yet clearhow P∗ appears in (1.6).

In order to make (1.6) more explicit we write the expectation as

IE∗[S(1)] = aP∗(S(1) = a) + bP∗(S(1) = b),

hence Condition (1.6) for the existence of a risk-neutral measure P∗ reads

aP∗(S(1) = a) + bP∗(S(1) = b) = (1 + r)π(1).

Using the Condition (1.5) we obtain the system of two equationsaP∗(ω−) + bP∗(ω+) = (1 + r)π(1)

P∗(ω−) + P∗(ω+) = 1,(1.7)

with solution

P∗(ω−) =b− (1 + r)π(1)

b− a and P∗(ω+) =(1 + r)π(1) − a

b− a .

In order for a solution P∗ to exist as a probability measure, the numbersP∗(ω−) and P∗(ω+) must be non-negative.

We deduce that if a, b and r satisfy the condition

a < (1 + r)π(1) < b, (1.8)

then there exists a risk-neutral measure P∗ which is unique, hence by Theo-rems 1.1 and 1.2 the market is without arbitrage and complete.

If a = b = (1 + r)π(1) then (1.2) admits an infinity of solutions, hence themarket is without arbitrage but it is not complete. More precisely, in thiscase both the riskless and risky assets yield a deterministic return rate r andthe value of the portfolio becomes

ξ · S = (1 + r)ξ · π,

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at time t = 1, hence the terminal value ξ · S is deterministic and this singlevalue can not always match the value of a random contingent claim C thatwould be allowed to take two distinct values C(ω−) and C(ω+). Therefore,market completeness does not hold when a = b = (1 + r)π(1).

Note that if a = (1 + r)π(1), resp. b = (1 + r)π(1), then P∗(ω+) = 0,resp. P∗(ω−) = 0, and P∗ is not equivalent to P.

On the other hand, under the conditions

a < b < (1 + r)π(1) or (1 + r)π(1) < a < b, (1.9)

no risk neutral measure exists and as a consequence there exist arbitrageopportunities in the market.

Let us give a financial interpretation of Conditions (1.9).

1. If (1 + r)π(1) < a < b, let ξ(1) = 1 and choose ξ(0) such that ξ(0)π(0) +ξ(1)π(1) = 0, i.e.

ξ(0) = −ξ(1)π(1)/π(0) < 0.

This means that the investor borrows the amount −ξ(0)π(0) > 0 on theriskless asset and uses it to buy one unit ξ(1) = 1 of the risky asset. Attime t = 1 she sells the risky asset S(1) at a price at least equal to a andrefunds the amount −(1 + r)ξ(0)π(0) > 0 she borrowed, with interests. Herprofit is

ξ · S = (1 + r)ξ(0)π(0) + ξ(1)S(1)

≥ (1 + r)ξ(0)π(0) + ξ(1)a

= −(1 + r)ξ(1)π(1) + ξ(1)a

= ξ(1)(−(1 + r)π(1) + a)

> 0.· ·^

2. If a 0 and choose ξ(1) such that ξ(0)π(0) +ξ(1)π(1) = 0, i.e.

ξ(1) = −ξ(0)π(0)/π(1) < 0.

This means that the investor borrows a (possibly fractional) quantity−ξ(1) > 0 of the risky asset, sells it for the amount −ξ(1)π(1), and in-vests this money on the riskless account for the amount ξ(0)π(0) > 0. Asmentioned in Section 1.2, in this case one says that the investor shortsellsthe risky asset. At time t = 1 she obtains (1 + r)ξ(0)π(0) > 0 from theriskless asset and she spends at most b to buy the risky asset and returnit to its original owner. Her profit is

ξ · S = (1 + r)ξ(0)π(0) + ξ(1)S(1)

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N. Privault

≥ (1 + r)ξ(0)π(0) + ξ(1)b

= −(1 + r)ξ(1)π(1) + ξ(1)b

= ξ(1)(−(1 + r)π(1) + b)

> 0,· ·^

since ξ(1) < 0. Note that here, a ≤ S(1) ≤ b became

ξ(1)b ≤ ξ(1)S(1) ≤ ξ(1)a

because ξ(1) < 0.

Under Condition (1.8) there is absence of arbitrage and Theorem 1.1 showsthat no portfolio strategy can yield ξ · S ≥ 0 and P(ξ · S > 0) > 0 startingfrom ξ(0)π(0) + ξ(1)π(1) ≤ 0, although this is less simple to show directly.

Finally if a = b 6= (1+r)π(1) then (1.2) admits no solution as a probabilitymeasure P∗ hence arbitrage opportunities exist and can be constructed bythe same method as above.

The second natural question is:

- is the market complete, i.e. are all contingent claims attainable ?

In the sequel we work under the condition

a < (1 + r)π(1) < b,

under which Theorems 1.1 and 1.2 show that the market is without arbitrageand complete since the risk-neutral measure P∗ exists and is unique.

Let us recover this fact by elementary calculations. For any contingentclaim C we need to show that there exists a portfolio ξ = (ξ(0), ξ(1)) suchthat C = ξ · S, i.e. ξ(0)(1 + r)π(0) + ξ(1)a = C(ω−)

ξ(0)(1 + r)π(0) + ξ(1)b = C(ω+).(1.10)

These equations can be solved as

ξ(0) =bC(ω−)− aC(ω+)

π(0)(1 + r)(b− a)and ξ(1) =

C(ω+)− C(ω−)

b− a . (1.11)

In this case we say that the portfolio (ξ(0), ξ(1)) hedges the contingent claimC. In other words, any contingent claim C is attainable and the market isindeed complete. Here, the quantity

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ξ(0)π(0) =bC(ω−)− aC(ω+)

(1 + r)(b− a)

represents the amount invested on the riskless asset.

Note that if C(ω+) ≥ C(ω−) then ξ(1) ≥ 0 and there is not short selling.This occurs in particular if C has the form C = h(S(1)) with x 7−→ h(x) anon-decreasing function, since

ξ(1) =C(ω+)− C(ω−)

b− a

=h(S(1)(ω+))− h(S(1)(ω−))

b− a=h(b)− h(a)

b− a≥ 0,

thus there is no short-selling.

The arbitrage price π(C) of the contingent claim C is defined in (1.3) asthe initial value at t = 0 of the portfolio hedging C, i.e.

π(C) = ξ · π, (1.12)

where (ξ(0), ξ(1)) are given by (1.11). Note that π(C) cannot be 0 since thiswould entail the existence of an arbitrage opportunity according to Defini-tion 1.1.

The next proposition shows that the arbitrage price π(C) of the claim canbe computed as the expected value of its payoff C under the risk-neutralmeasure, after discounting at the rate 1 + r for the time value of money.

Proposition 1.1. The arbitrage price π(C) = ξ · π of the contingent claimC is given by

π(C) =1

1 + rIE∗[C]. (1.13)

Proof. We have

π(C) = ξ · π= ξ(0)π(0) + ξ(1)π(1)

=bC(ω−)− aC(ω+)

(1 + r)(b− a)+ π(1)C(ω+)− C(ω−)

b− a

=1

1 + r

(C(ω−)

b− π(1)(1 + r)

b− a + C(ω+)(1 + r)π(1) − a

b− a

)

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N. Privault

=1

1 + r

(C(ω−)P∗(S(1) = a) + C(ω+)P∗(S(1) = b)

)=

1

1 + rIE∗[C].

In the case of a European call option with strike K ∈ [a, b] we have C =(S(1) −K)+ and

π((S(1) −K)+) = π(1) b−Kb− a −

(b−K)a

(1 + r)(b− a).

Here, (π(1) −K)+ is called the intrinsic value at time 0 of the call option.

The simple setting described in this chapter raises several questions andremarks.

Remarks

1. The fact that π(C) can be obtained by two different methods, i.e. analgebraic method via (1.11) and (1.12) and a probabilistic method from(1.13) is not a simple coincidence. It is actually a simple example of thedeep connection that exists between probability and analysis.

In a continuous time setting, (1.11) will be replaced with a partial differ-ential equation (PDE) and (1.13) will be computed via the Monte Carlomethod. In practice, the quantitative analysis departments of major fi-nancial institutions can be split into the PDE team and the Monte Carloteam, often trying to determine the same option prices by two differentmethods.

2. What if we have three possible scenarios, i.e. Ω = ω−, ωo, ω+ and therandom asset S(1) is allowed to take more than two values, e.g. S(1) ∈a, b, c according to each scenario ? In this case the system (1.7) wouldbe rewritten asaP∗(ω−) + bP∗(ωo) + cP∗(ω+) = (1 + r)π(1)

P∗(ω−) + P∗(ωo) + P∗(ω+) = 1,

and this system of two equations for three unknowns does not have aunique solution, hence the market can be without arbitrage but it cannotbe complete. Completeness can be reached by adding a second risky asset,i.e. taking d = 2, in which case we will get three equations and threeunknowns. More generally, when Ω has n ≥ 2 elements, completeness

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of the market can be reached provided we consider d risky assets withd+ 1 ≥ n. This is related to the Meta-Theorem 8.3.1 of [4] in which thenumber d of traded underlying risky assets is linked to the number ofrandom sources through arbitrage and completeness.

Exercises

Exercise 1.1 Consider a financial model with two instants t = 0 and t = 1and two assets:

- a riskless asset π with price π0 at time t = 0 and value π1 = π0(1 + r) attime t = 1,- a risky asset S with price S0 at time t = 0 and random value S1 at timet = 1.

We assume that S1 can take only the values S0(1 + a) and S0(1 + b), where−1 < a < r < b. The return of the risky asset is defined as

R =S1 − S0

S0.

1. What are the possible values of R ?2. Show that under the probability measure P∗ defined by

P∗(R = a) =b− rb− a, P∗(R = b) =

r − ab− a ,

the expected return IE∗[R] of S is equal to the return r of the risklessasset.

3. Does there exist arbitrage opportunities in this model ? Explain why.4. Is this market model complete ? Explain why.5. Consider a contingent claim with payoff C given by

C =

α if R = a,

β if R = b.

Show that the portfolio (η, ξ) defined2 by

η =α(1 + b)− β(1 + a)

π0(1 + r)(b− a)and ξ =

β − αS0(b− a)

,

hedges the contingent claim C, i.e. show that at time t = 1 we have

2 Here, η is the (possibly fractional) quantity of asset π and ξ is the quantity held ofasset S.

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N. Privault

ηπ1 + ξS1 = C.

Hint: distinguish two cases R = a and R = b.6. Compute the arbitrage price π(C) of the contingent claim C using η, π0,ξ, and S0.

7. Compute IE∗[C] in terms of a, b, r, α, β.8. Show that the arbitrage price π(C) of the contingent claim C satisfies

π(C) =1

1 + rIE∗[C]. (1.14)

9. What is the interpretation of Relation (1.14) above ?10. Let C denote the payoff at time t = 1 of a put option with strike K = $11

on the risky asset. Give the expression of C as a function of S1 and K.11. Letting π0 = S0 = 1 and a = 8, b = 11, compute the portfolio (ξ, η)

hedging the contingent claim C.12. Compute the arbitrage price π(C) of the claim C.

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Chapter 2

Discrete-Time Model

A basic limitation of the two time step model considered in Chapter 1 is that itdoes not allow for trading until the end of the time period is reached. In orderto be able to re-allocate the portfolio over time we need to consider a discrete-time financial model with N + 1 time instants t = 0, 1, . . . , N . The practicalimportance of this model lies also in its direct computer implementability.

2.1 Stochastic Processes

A stochastic process on a probability space (Ω,F ,P) is a family (Xt)t∈T ofrandom variables Xt : Ω → R indexed by a set T . Examples include:

• the two-instant model: T = 0, 1,

• the discrete-time model with finite horizon: T = 0, 1, 2, . . . , N,

• the discrete-time model with infinite horizon: T = N,

• the continuous-time model: T = R+.

For real-world examples of stochastic processes one can mention:

• the time evolution of a risky asset - in this case Xt represents the price ofthe asset at time t ∈ T .

• the time evolution of a physical parameter - for example, Xt represents atemperature observed at time t ∈ T .

In this chapter we will focus on the finite horizon discrete-time model withT = 0, 1, 2, . . . , N.

Here the vector

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N. Privault

π = (π(0), π(1), . . . , π(d))

denotes the prices at time t = 0 of d+ 1 assets numbered 0, 1, . . . , d.

The random vector

St = (S(0)t , S

(1)t , . . . , S

(d)t )

on Ω denotes the values at time t = 1, . . . , N of assets 0, 1, . . . , d, and formsa stochastic process (St)t=0,1,...,N with S0 = π.

Here we still assume that asset 0 is a riskless asset (of savings accounttype) yielding an interest rate r, i.e. we have

S(0)t = (1 + r)tπ(0), t = 0, 1, . . . , N.

2.2 Portfolio Strategies

A portfolio strategy is a stochastic process (ξt)t=1,...,N ⊂ Rd+1 where ξ(i)t

denotes the (possibly fractional) quantity of asset i held in the portfolio overthe period (t− 1, t], t = 1, . . . , N .

Note that the portfolio allocation

ξt = (ξ(0)t , ξ

(1)t , . . . , ξ

(d)t )

remains constant over the period (t− 1, t] while the stock price changes fromSt−1 to St over this period.

In other terms,

ξ(i)t S

(i)t−1

represents the amount invested in asset i at the beginning of the time period(t− 1, t], and

ξ(i)t S

(i)t

represents the value of this investment at the end of (t− 1, t], t = 1, . . . , N .

The value of the porfolio at the beginning of the time period (t− 1, t] is

ξt · St−1 =d∑i=0

ξ(i)t S

(i)t−1,

when the market “opens” at time t− 1, and becomes

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ξt · St =d∑i=0

ξ(i)t S

(i)t (2.1)

at the end of (t− 1, t], i.e. when the market “closes”, t = 1, . . . , N .

At the beginning of the next trading period (t, t + 1] the value of theportfolio becomes

ξt+1 · St =d∑i=0

ξ(i)t+1S

(i)t . (2.2)

Note that the stock price St is assumed to remain constant “overnight”, i.e.from the end of (t− 1, t] to the beginning of (t, t+ 1].

Obviously the question arises whether (2.1) should be identical to (2.2). Inthe sequel we will need such a consistency hypothesis, called self-financing,on the portfolio strategy ξt.

Definition 2.1. We say that the portfolio strategy (ξt)t=1,...,N is self-financingif

ξt · St = ξt+1 · St, t = 1, . . . , N − 1. (2.3)

The meaning of the self-financing condition (2.3) is simply that one cannottake any money in or out of the portfolio during the “overnight” transitionperiod at time t. In other words, at the beginning of the new trading period(t, t+1] one should re-invest the totality of the portfolio value obtained at theend of period (t− 1, t]. The next figure is an illustration of the self-financingcondition.

St St St+1St−1

t+ 1t− 1 t t

ξt+1ξt ξt ξt+1

ξt+1St+1ξtSt−1 ξtSt ξt+1St=Portfolio value

Asset value

Time scale

Portfolio allocation

“Morning”

“Evening”

“Morning”@

@@I

“Evening”@@

@@I

- -

Fig. 2.1: Illustration of the self-financing condition (2.3).

Note that portfolio re-allocation happens “overnight” durig which time theportfolio global value remains the same due to the self-financing condition.The portfolio allocation ξt remains the same throughout the day, howeverthe portfolio value changes from morning to evening due to a change in the

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N. Privault

stock price. Also, ξ0 is not defined and its value is actually not needed in thisframework.

Of course the chosen unit of time may not be the day, and it can be replacedby weeks, hours, minutes, or even fractions of seconds in high-frequency trad-ing.

We will denote byVt := ξt · St

the value of the portfolio at time t = 1, . . . , N , with

Vt = ξt+1 · St, t = 0, . . . , N − 1,

by the self-financing condition (2.3), and in particular

V0 = ξ1 · S0.

Let alsoXt := (X

(0)t , X

(1)t , . . . , X

(d)t )

denote the vector of discounted asset prices defined as:

X(i)t =

1

(1 + r)tS(i)t , i = 0, 1, . . . , d, t = 0, 1, . . . , N,

or

Xt :=1

(1 + r)tSt, t = 0, 1, . . . , N.

The discounted value at time 0 of the portfolio is defined by

Vt =1

(1 + r)tVt, t = 0, 1, . . . , N.

We have

Vt =1

(1 + r)tξt · St

=1

(1 + r)t

d∑i=0

ξ(i)t S

(i)t

=d∑i=0

ξ(i)t X

(i)t

= ξt · Xt, t = 1, . . . , N,

andV0 = ξ1 ·X0 = ξ1 · S0.

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The effect of discounting from time t to time 0 is to divide prices by (1 + r)t,making all prices comparable at time 0.

2.3 Arbitrage

The definition of arbitrage in discrete time follows the lines of its analog inthe two-step model.

Definition 2.2. A portfolio strategy (ξt)t=1,...,N constitutes an arbitrage op-portunity if all three following conditions are satisfied:

(i) V0 ≤ 0, [start from 0 or even with a debt]

(ii) VN ≥ 0, [finish with a non-negative amount]

(iii) P(VN > 0) > 0. [a profit is made with non-zero probability]

2.4 Contingent Claims

Recall that from Defition 1.3, a contingent claim is given by the non-negativerandom payoff C of an option contract at time t = N . For example, in the caseof the European call of Definition 0.2, the payoff C is given by X = (SN−K)+

where K is called the strike price.

In a discrete-time setting we are able to consider path-dependent optionsin addition to European type options. One can distinguish between vanillaoptions whose payoff depends on the terminal value of the underlying asset,such as simple European contracts, and exotic or path-dependent optionssuch as Asian, barrier, or lookback options, whose payoff may depend on thewhole path of the underlying asset price until expiration time.

The list provided below is actually very restricted and there exists manymore option types, with new ones appearing constantly on the markets.

European options

The payoff of a European call on the underlying asset no i with maturity Nand strike K is

C = (S(i)N −K)+.

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N. Privault

The payoff of a European put on the underlying asset no i with exercise dateN and strike K is

C = (K − S(i)N )+.

Let us mention also the existence of binary, or digital options, also calledcash-or-nothing options, whose payoffs are

C = 1[K,∞)(S(i)N ) =

$1 if S

(i)N ≥ K,

0 if S(i)N < K,

for call options, and

C = 1(−∞,K](S(i)N ) =

$1 if S

(i)N ≤ K,

0 if S(i)N > K,

for put options.

Asian options

The payoff of an Asian call option (also called average value option) on theunderlying asset no i with exercise date N and strike K is

C =

(1

N + 1

N∑t=0

S(i)t −K

)+

.

The payoff of an Asian put option on the underlying asset no i with exercisedate N and strike K is

C =

(K − 1

N + 1

N∑t=0

S(i)t

)+

.

We refer to Section 8.5 for the pricing of Asian options in continuous time.

Barrier options

The payoff of a down-an-out barrier call option on the underlying asset no iwith exercise date N , strike K and barrier B is

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C =(S(i)N −K

)+1

mint=0,1,...,N

S(i)t > B

=

S(i)N −K if min

t=0,1,...,NS(i)t > B,

0 if mint=0,1,...,N

S(i)t ≤ B.

This option is also called a Callable Bull Contract with no residual value, inwhich B denotes the call price B ≥ K. It is also called a turbo warrant withno rebate.

The payoff of an up-and-out barrier put option on the underlying asset noi with exercise date N , strike K and barrier B is

C =(K − S(i)

N

)+1

maxt=0,1,...,N

S(i)t < B

=

K − S(i)

N if maxt=0,1,...,N

S(i)t < B,

0 if maxt=0,1,...,N

S(i)t ≥ B.

This option is also called a Callable Bear Contract with no residual value, inwhich the call price B usually satisfies B ≤ K. See [23], [77] for recent resultson the pricing of CBBCs, also called turbo warrants. We refer the reader toChapter 8 for the pricing and hedging of similar exotic options in continuoustime. Barrier options in continuous time are priced in Section 8.3.

Lookback options

The payoff of a floating strike lookback call option on the underlying assetno i with exercise date N is

C = S(i)N − min

t=0,1,...,NS(i)t .

The payoff of a floating strike lookback put option on the underlying assetno i with exercise date N is

C =

(max

t=0,1,...,NS(i)t

)− S(i)

N .

We refer to Section 8.4 for the pricing of lookback options in continuous time.

2.5 Martingales and Conditional Expectation

Before proceeding to the definition of risk-neutral probability measures in dis-crete time we need to introduce more mathematical tools such as conditionalexpectations, filtrations, and martingales.

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N. Privault

Conditional expectations

Clearly, the expected value of any risky asset or random variable is dependenton the amount of available information. For example, the expected return ona real estate investment typically depends on the location of this investment.

In the probabilistic framework the available information is formalized asa collection G of events, which may be smaller than the collection F of allavailable events, i.e. G ⊂ F .1

The notation IE[F |G] represents the expected value of a random variable Fgiven (or conditionally to) the information contained in G, and it is read “theconditional expectation of F given G”. In a certain sense, IE[F |G] representsthe best possible estimate of F in mean square sense, given the informationcontained in G.

The conditional expectation satisfies the following five properties, cf. Sec-tion 16.4 for details and proofs.

(i) IE[FG | G] = G IE[F | G] if G depends only on the information con-tained in G.

(ii) IE[G | G] = G when G depends only on the information contained in G.

(iii) IE[IE[F | H] | G] = IE[F | G] if G ⊂ H, called the tower property, cf.also Relation (16.24).

(iv) IE[F | G] = IE[F ] when F “does not depend” on the informationcontained in G or, more precisely stated, when the random variable F isindependent of the σ-algebra G.

(v) If G depends only on G and F is independent of G, then

IE[h(F,G) | G] = IE[h(x, F )]x=G.

When H = ∅, Ω is the trivial σ-algebra we have IE[F | H] = IE[F ], F ∈L1(Ω). See (16.24) and (16.28) for illustrations of the tower property byconditioning with respect to discrete and continuous random variables.

1 The collection G is also called a σ-algebra, cf. Section 16.4.

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Filtrations

The total amount of “information” present in the market at time t =0, 1, . . . , N is denoted by Ft. We assume that

Ft ⊂ Ft+1, t = 0, 1, . . . , N − 1,

which means that the amount of information available on the market in-creases over time.

Usually, Ft corresponds to the knowledge of the values S(i)0 , . . . , S

(i)t , i =

1, . . . , d, of the risky assets up to time t. In mathematical notation we say

that Ft is generated by S(i)0 , . . . , S

(i)t , and we usually write

Ft = σ(S(i)0 , . . . , S

(i)t

), t = 0, 1, . . . , N.

The notation Ft is useful to represent a quantity of information available attime t. Note that different agents or traders may work with distinct filtration.For example, an insider will have access to a filtration (Gt)t=0,1,...,N largerthan the filtration (Ft)t=0,1,...,N available to an ordinary agent, in the sensethat

Ft ⊂ Gt, t = 0, 1, . . . , N.

The notation IE[F |Ft] represents the expected value of a random variable Fgiven (or conditionally to) the information contained in Ft. Again, IE[F |Ft]denotes the best possible estimate of F in mean square sense, given the in-formation known up to time t.

We will assume that no information is available at time t = 0, whichtranslates as

IE[F | F0] = IE[F ]

for any integrable random variable F .

As above, the conditional expectation with respect to Ft satisfies the fol-lowing five properties:

(i) IE[FG | Ft] = F IE[G | Ft] if F depends only on the information con-tained in Ft.

(ii) IE[F | Ft] = F when F depends only on the information known attime t and contained in Ft.

(iii) IE[IE[F | Ft+1] | Ft] = IE[F | Ft] if Ft ⊂ Ft+1 (by the tower property,cf. also Relation (6.1) below).

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N. Privault

(iv) IE[F | Ft] = IE[F ] when F does not depend on the information con-tained in Ft.

(v) If F depends only on Ft and G is independent of Ft, then

IE[h(F,G) | Ft] = IE[h(x,G)]x=F .

Note that by the tower property (iii) the process t 7→ IE[F | Ft] is a martin-gale, cf. e.g. Relation (6.1) for details.

Martingales

A martingale is a stochastic process whose value at time t+1 can be estimatedusing conditional expectation given its value at time t. Recall that a process(Mt)t=0,1,...,N is said to be Ft-adapted if the value of Mt depends only onthe information available at time t in Ft, t = 0, 1, . . . , N .

Definition 2.3. A stochastic process (Mt)t=0,1,...,N is called a discrete timemartingale with respect to the filtration (Ft)t=0,1,...,N if (Mt)t=0,1,...,N is Ft-adapted and satisfies the property

IE[Mt+1|Ft] = Mt, t = 0, 1, . . . , N − 1.

Note that the above definition implies that Mt ∈ Ft, t = 0, 1, . . . , N . Inother words, a random process (Mt)t=0,1,...,N is a martingale if the best pos-sible prediction of Mt+1 in the mean square sense given Ft is simply Mt.

As an example of the use of martingales we can mention weather forecast-ing. If Mt denotes the random temperature observed at time t, this processis a martingale when the best possible forecast of tomorrow’s temperatureMt+1 given information known up to time t is just today’s temperature Mt,t = 0, 1, . . . , N − 1.

In the sequel we will say that a stochastic process (ξk)k≥0 is predictableif ξk depends only on the information in Fk−1, k ≥ 1. In particular, ξ0 is aconstant.

An important property of martingales is that the martingale transform(2.4) of a predictable process is itself a martingale, see also Proposition 6.1for the continuous-time analog of the following proposition.

Proposition 2.1. Given (Xt)t∈N a martingale and (ξk)k∈N a square-summablepredictable process, the discrete-time process (Mt)t∈N defined by

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Mt =t∑

k=1

ξk(Xk −Xk−1), t ∈ N, (2.4)

is a martingale.

Proof. Given n ≥ 0 we have

IE [Mn | Ft] = IE

[n∑k=1

ξk(Xk −Xk−1)∣∣∣Ft]

= IE

[n∑k=1

IE [ξk(Xk −Xk−1) | Ft]]

=

t∑k=1

IE [ξk(Xk −Xk−1) | Ft] +

n∑k=t+1

IE [ξk(Xk −Xk−1) | Ft]

=t∑

k=1

ξk(Xk −Xk−1) +n∑

k=t+1

IE [ξk(Xk −Xk−1) | Ft]

= Mt +n∑

k=t+1

IE [ξk(Xk −Xk−1) | Ft] .

To conclude we need to show that

IE [ξk(Xk −Xk−1) | Ft] = 0, t+ 1 ≤ k ≤ n.

We note that when 0 ≤ t ≤ k − 1 we have Ft ⊂ Fk−1, and by the “towerproperty” of conditional expectations we get

IE [ξk(Xk −Xk−1) | Ft] = IE [IE [ξk(Xk −Xk−1) | Fk−1] | Ft] .

In addition, since the process (ξk)k∈N is predictable, ξk depends only onthe information in Fk−1, and using Property (ii) of conditional expectationswe may pull out ξk out of the expectation since it behaves as a constantparameter given Fk−1, k = 1, . . . , n, hence

IE [ξk(Xk −Xk−1) | Fk−1] = ξk IE [Xk −Xk−1 | Fk−1] = 0

because (Xt)t∈N is a martingale, and more generally,

IE [ξk(Xk −Xk−1) | Ft] = 0,

for all k = t+ 1, . . . , n. This yields

IE [Xk −Xk−1 | Fk−1] = IE [Xk | Fk−1]− IE [Xk−1 | Fk−1]

= IE [Xk | Fk−1]−Xk−1

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N. Privault

= 0, k = 1, . . . , n,

because (Xt)t∈N is a martingale.

2.6 Risk-Neutral Probability Measures

As in the two time step model, the concept of risk neutral measures will beused to price financial claims under the absence of arbitrage hypothesis.

Definition 2.4. A probability measure P∗ on Ω is called a risk-neutral mea-sure if under P∗, the expected return of each risky asset equals the return rof the riskless asset, that is

IE∗[S(i)t+1 | Ft] = (1 + r)S

(i)t , t = 0, 1, . . . , N − 1, (2.5)

i = 0, 1, . . . , d. Here, IE∗ denotes the expectation under P∗.

Since S(i)t ∈ Ft, Relation (2.5) can be rewritten in terms of asset returns

as

IE∗

[S(i)t+1 − S

(i)t

S(i)t

∣∣∣Ft] = r, t = 0, 1, . . . , N − 1.

In other words, taking risks under P∗ by buying the risky asset no i has aneutral effect, as the expected return is that of the riskless asset. The measureP∗ would be represent a risk premium if we had

IE∗[S(i)t+1 | Ft] = (1 + r)S

(i)t , t = 0, 1, . . . , N − 1,

with r > r.

The definition of risk-neutral probability measure can be reformulatedusing the notion of martingale.

Proposition 2.2. A probability measure P∗ on Ω is a risk-neutral measure

if and only if the discounted price process X(i)t is a martingale under P∗, i.e.

IE∗[X(i)t+1 | Ft] = X

(i)t , t = 0, 1, . . . , N − 1, (2.6)

i = 0, 1, . . . , d.

Proof. It suffices to check that Conditions (2.5) and (2.6) are equivalent since

IE∗[S(i)t+1 | Ft] = (1 + r)t+1 IE∗[X

(i)t+1 | Ft] and S

(i)t = (1 + r)tX

(i)t ,

t = 0, 1, . . . , N − 1, i = 1, . . . , d.

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Next we restate the first fundamental theorem of mathematical financein discrete time, which can be used to check for the existence of arbitrageopportunities.

Theorem 2.1. A market is without arbitrage opportunity if and only if itadmits at least one risk-neutral measure.



Definition 2.5. A contingent claim with payoff C is said to be attainable(at time N) if there exists a portfolio strategy (ξt)t=1,...,N such that

C = ξN · SN . (2.7)

In case (ξt)t=1,...,N is a portfolio that attains the claim C at time N , i.e.if (2.7) is satisfied, we also say that (ξt)t=1,...,N hedges the claim C. In case(2.7) is replaced by the condition

ξN · SN ≥ C,

we talk of super-hedging. When (ξt)t=1,...,N hedges C, the arbitrage priceπt(C) of the claim at time t will be given by the value

πt(C) = ξt · St

of the portfolio at time t = 0, 1, . . . , N . Note that at time t = N we have

πN (C) = ξN · SN = C,

i.e. since exercise of the claim occurs at time N , the price πN (C) of the claimequals the value C of the payoff.

Definition 2.6. A market model is said to be complete if every contingentclaim is attainable.

The next result can be viewed as the second fundamental theorem of math-ematical finance.



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N. Privault

2.8 The Cox-Ross-Rubinstein (CRR) Market Model

We consider the discrete time Cox-Ross-Rubinstein model [14] with N + 1time instants t = 0, 1, . . . , N and d = 1 risky asset, also called the binomial

model. The price S(0)t of the riskless asset evolves as

S(0)t = π(0)(1 + r)t, t = 0, 1, . . . , N.

Let the return of the risky asset S = S(1) be defined as

Rt :=St − St−1St−1

, t = 1, . . . , N.

In the CRR model the return Rt is random and allowed to take only twovalues a and b at each time step, i.e.

Rt ∈ a, b, t = 1, . . . , N,

with −1 < a < b. That means, the evolution of St−1 to St is random andgiven by

St =

(1 + b)St−1 if Rt = b

(1 + a)St−1 if Rt = a

= (1 +Rt)St−1, t = 1, . . . , N,

and

St = S0

t∏j=1

(1 +Rj), t = 0, 1, . . . , N.

Note that the price process (St)t=0,1,...,N evolves on a binary recombining (orbinomial) tree. The discounted asset price is

Xt =St

(1 + r)t, t = 0, 1, . . . , N,

with

Xt =

1 + b

1 + rXt−1 if Rt = b

1 + a

1 + rXt−1 if Rt = a

=1 +Rt1 + r

Xt−1, t = 1, . . . , N,

and

Xt =S0

(1 + r)t

t∏j=1

(1 +Rj) = X0

t∏j=1

1 +Rj1 + r

.

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In this model the discounted value at time t of the portfolio is given by

ξt · Xt = ξ(0)t π0 + ξ

(1)t Xt, t = 1, . . . , N.

The information Ft known in the market up to time t is given by the knowl-edge of S1, . . . , St, which is equivalent to the knowledge of X1, . . . , Xt orR1, . . . , Rt, i.e. we write

Ft = σ(S1, . . . , St) = σ(X1, . . . , Xt) = σ(R1, . . . , Rt), t = 0, 1, . . . , N,

where as a convention F0 = ∅, Ω contains no information.

Theorem 2.3. The CRR model is without arbitrage if and only if a < r < b.In this case the market is complete.

Proof. In order to check for arbitrage opportunities we may use Theorem 2.1and look for a risk-neutral measure P∗. According to the definition of a risk-neutral measure this probability P∗ should satisfy Condition (2.5), i.e.

IE∗[S(i)t+1 | Ft] = (1 + r)S

(i)t , t = 0, 1, . . . , N − 1.

Rewriting IE∗[St+1 | Ft] as

IE∗[St+1 | Ft] = (1 + a)StP∗(Rt+1 = a | Ft) + (1 + b)StP∗(Rt+1 = b | Ft),

it follows that any risk-neutral measure P∗ should satisfy the equations (1 + b)StP∗(Rt+1 = b | Ft) + (1 + a)StP∗(Rt+1 = a | Ft) = (1 + r)St

P∗(Rt+1 = b | Ft) + P∗(Rt+1 = a | Ft) = 1,

i.e. bP∗(Rt+1 = b | Ft) + aP∗(Rt+1 = a | Ft) = r

P∗(Rt+1 = b | Ft) + P∗(Rt+1 = a | Ft) = 1,

with solution

P∗(Rt+1 = b | Ft) =r − ab− a and P∗(Rt+1 = a | Ft) =

b− rb− a. (2.8)

Clearly, P∗ can be a non singular probability measure only if r − a > 0 andb − r > 0. In this case the solution P∗ of the problem is unique hence themarket is complete by Theorem 2.2.

Note that the values of P∗(Rt+1 = b | Ft) and P∗(Rt+1 = a | Ft) computedin (2.8) are non random, hence they are independent of the information con-tained in Ft. As a consequence, under P∗, the random variable Rt+1 is inde-pendent of the information Ft up to time t, which is generated by R1, . . . , Rt.We deduce that (R1, . . . , RN ) form a sequence of independent and identically

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N. Privault

distributed (i.i.d.) random variables.

In other words, Rt+1 is independent of R1, . . . , Rt for all t = 1, . . . , N − 1,the random variables R1, . . . , RN are independent under P∗, and by (2.8) wehave

P∗(Rt+1 = b) =r − ab− a and P∗(Rt+1 = a) =

b− rb− a.

As a consequence, letting p∗ := (r−a)/(b−a), when (k1, . . . , kn) ∈ a, bN+1

we haveP∗(R1 = k1, . . . , RN = kn) = (p∗)l(1− p∗)N−l,

where l, resp. N − l, denotes the number of times the term “a”, resp. “b”,appears in the sequence k1, . . . , kN.

Exercises

Exercise 2.1 We consider the discrete-time Cox-Ross-Rubinstein model withN + 1 time instants t = 0, 1, . . . , N , and the price πt of the riskless assetevolves as πt = π0(1 + r)t, t = 0, 1, . . . , N . The evolution of St−1 to St isgiven by

St =

(1 + b)St−1

(1 + a)St−1

with −1 < a < r < b. The return of the risky asset S is defined as


, t = 1, . . . , N,

and Ft is generated by R1, . . . , Rt, t = 1, . . . , N .

1. What are the possible values of Rt ?2. Show that under the probability measure P ∗ defined by

P ∗(Rt+1 = a | Ft) =b− rb− a, P ∗(Rt+1 = b | Ft) =

r − ab− a ,

t = 0, 1, . . . , N −1, the expected return IE∗[Rt+1 | Ft] of S is equal to thereturn r of the riskless asset.

3. Show that under P ∗ the process (St)t=0,...,N satisfies

IE∗[St+k | Ft] = (1 + r)kSt, t = 0, . . . , N − k, k = 0, . . . , N.

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Chapter 3

Pricing and hedging in discrete time

We consider the pricing and hedging of options in a discrete time financialmodel with N + 1 time instants t = 0, 1, . . . , N . Vanilla options are treatedusing backward induction, and exotic options with arbitrary payoff functionsare considered using the Clark-Ocone formula in discrete time.

3.1 Pricing of Contingent Claims

Let us consider an attainable contingent claim with payoff C ≥ 0 and matu-rity N . Recall that by the Definition 2.5 of attainability there exists a hedgingportfolio strategy (ξt)t=1,2,...,N such that

ξN · SN = C (3.1)

at time N . Clearly, if (3.1) holds, then investing the amount

V0 = ξ1 · S0 (3.2)

at time t = 0, resp.Vt = ξt · St (3.3)

at time t = 1, . . . , N , into a self-financing hedging portfolio will allow one tohedge the option and to obtain the perfect replication (3.1) at time N .

The value (3.2)-(3.3) at time t of a self-financing portfolio strategy (ξt)t=1,2,...,N

hedging an attainable claim C will be called an arbitrage price of the claimC at time t and denoted by πt(C), t = 0, 1, . . . , N .

Next we develop a second approach to the pricing of contingent claims,based on conditional expectations and martingale arguments. We will needthe following lemma.

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N. Privault

Lemma 3.1. The following statements are equivalent:

(i) The portfolio strategy (ξt)t=1,...,N is self-financing.

(ii) ξt · Xt = ξt+1 · Xt for all t = 1, . . . , N − 1.

(iii) We have

Vt = V0 +t∑

j=1

ξj · (Xj − Xj−1), t = 0, 1, . . . , N. (3.4)

Proof. First, the self-financing condition (i)

ξt−1 · St−1 = ξt · St−1, t = 1, . . . , N,

is clearly equivalent to (ii) by division of both sides by (1 + r)t−1.

Next, assuming that (ii) holds we have

Vt = V0 +t∑

j=1

(Vj − Vj−1)

= V0 +t∑

j=1

ξj · Xj − ξj−1 · Xj−1

= V0 +t∑

j=1

ξj · Xj − ξj · Xj−1

= V0 +t∑

j=1

ξj · (Xj − Xj−1), t = 1, . . . , N.

Finally, assuming that (iii) holds we get

Vt − Vt−1 = ξt · (Xt − Xt−1),

henceξt · Xt − ξt−1 · Xt−1 = ξt · (Xt − Xt−1),

andξt−1 · Xt−1 = ξt · Xt−1, t = 1, . . . , N.

In Relation (3.4), the term ξt · (Xt − Xt−1) represents the profit and loss

Vt − Vt−1 = ξt · (Xt − Xt−1),

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of the self-financing portfolio strategy (ξj)j=1,...,N over the time period[t − 1, t], computed by multiplication of the portfolio allocation ξt with thechange of price Xt − Xt−1, t = 1, . . . , N .

Relation (3.4) admits a natural interpretation by saying that when a port-

folio is self-financing the value Vt of the (discounted) portfolio at time t isgiven by summing up the (discounted) profits and losses registered over alltime periods from time 0 to time t.

The sum (3.4) is also referred to as a discrete time stochastic integralof the portfolio process (ξt)t=1,...,N with respect to the random process

(Xt)t=0,1,...,N . In particular, it can be shown from (3.4) that (Vt)t=0,1,...,N

is a martingale under P∗ by the martingale transform argument of Proposi-tion 2.1, as in the proof of Theorem 3.1 below.

As a consequence of the above Lemma 3.1, if a contingent claim C withdiscounted payoff

C :=C

(1 + r)N

is attainable by a self-financing portfolio strategy (ξt)t=1,...,N then we have

C = ξN · XN = VN = V0 +N∑t=1

ξt · (Xt − Xt−1). (3.5)

Note that in the above formula it is the use of discounted asset price Xt thatallows us to add up the profits and losses ξt · (Xt − Xt−1) since they areexpressed in units of currency “at time 0”. In general, $1 at time t = 0 and$1 at time t = 1 cannot be added without proper discounting.

Theorem 3.1. The arbitrage price πt(C) of a contingent claim C is givenby

πt(C) =1

(1 + r)N−tIE∗[C | Ft], t = 0, 1, . . . , N, (3.6)

where P∗ denotes any risk-neutral probability measure.

Proof. Let C = C/(1 + r)N denote the discounted payoff of the claim C. Wewill show that under any risk-neutral measure P∗ the discounted value of anyself-financing portfolio hedging C is given by

Vt = IE∗[C | Ft

], t = 0, 1, . . . , N, (3.7)

which shows that

Vt =1

(1 + r)N−tIE∗[C | Ft]

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N. Privault

after multiplication of both sides by (1 + r)t. To conclude we will note thatthe arbitrage price πt(C) of the claim at any time t is by definition equal tothe value Vt of the corresponding self-financing portfolio.

We now need to prove (3.7), and for this we will use the martingale trans-form argument of Proposition 2.1. Since the portfolio strategy (ξt)t=1,2,...,N

is self-financing, from Lemma 3.1 we have

IE∗[C | Ft

]= IE∗

[VN | Ft

]= IE∗

V0 +

N∑j=1

ξj · (Xj − Xj−1)∣∣∣Ft

= IE∗[V0 | Ft

]+

N∑j=1

IE∗[ξj · (Xj − Xj−1) | Ft

]= V0 +

t∑j=1


]+

N∑j=t+1


]= V0 +

t∑j=1

ξj · (Xj − Xj−1) +N∑

j=t+1


]= Vt +

N∑j=t+1


],

where we used Relation (3.4) of Lemma 3.1. In order to obtain (3.7) we needto show that

N∑j=t+1


]= 0.

Let us show thatIE∗[ξj · (Xj − Xj−1) | Ft

]= 0,

for all j = t+ 1, . . . , N . We have 0 ≤ t ≤ j − 1 hence Ft ⊂ Fj−1, and by the“tower property” of conditional expectations we get


]= IE∗

[IE∗[ξj · (Xj − Xj−1) | Fj−1

]| Ft

],

therefore it suffices to show that

IE∗[ξj · (Xj − Xj−1) | Fj−1

]= 0.

We note that the porfolio allocation ξj over the time period [j − 1, j] ispredictable, i.e. it is decided at time j − 1 and it thus depends only on theinformation Fj−1 known up to time j − 1, hence

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IE∗[ξj · (Xj − Xj−1) | Fj−1

]= ξj · IE∗

[Xj − Xj−1 | Fj−1

].

Finally we note that

IE∗[Xj − Xj−1 | Fj−1

]= IE∗

[Xj | Fj−1

]− IE∗

[Xj−1 | Fj−1

]= IE∗

[Xj | Fj−1

]− Xj−1

= 0, j = 1, . . . , N,

because (Xt)t=0,1,...,N is a martingale under the risk-neutral measure P∗, andthis concludes the proof.

Note that (3.6) admits an interpretation in an insurance framework, inwhich πt(C) represents an insurance premium and C represents the randomvalue of an insurance claim made by a subscriber. In this context, the pre-mium of the insurance contract reads as the average of the values (3.6) ofthe random claims after time discounting. In addition, the discounted priceprocess ((1 + r)−tπt(C))t=0,1,...,N is a martingale under P∗.

As a consequence of Theorem 3.1, the discounted portfolio process (Vt)t=0,1,...,N

is a martingale under P∗, since

IE∗[Vt+1 | Ft

]= IE∗

[IE∗[C | Ft+1

]| Ft

]= IE∗

[C | Ft

]= Vt, t = 0, . . . , N − 1,

from the “tower property” of conditional expectations.

In particular for t = 0 we obtain the price of the contingent claim C attime 0:

π0(C) = IE∗[C | F0

]= IE∗

[C]

=1

(1 + r)NIE∗[C].

3.2 Hedging of Contingent Claims - Backward Induction

The basic idea of hedging is to allocate assets in a portfolio in order to pro-tect oneself from a given risk. For example, a risk of increasing oil pricescan be hedged by buying oil-related stocks, whose value should be positivelycorrelated with the oil price. In this way, a loss connected to increasing oilprices could be compensated by an increase in the value of the correspondingportfolio.

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N. Privault

In the setting of this chapter, hedging an attainable contingent claim Cmeans computing a self-financing portfolio strategy (ξt)t=1,...,N such thatξN · SN = C, i.e.

ξN · XN = C, (3.8)

by first solving Equation (3.8) for ξN . The idea is then to work by backwardinduction and to compute successively ξN−1, ξN−2, . . ., ξ4, down to ξ3, ξ2,and finally ξ1.

In order to implement this algorithm we may use the self-financing condi-tion which yields N − 1 equations

ξt · Xt = ξt+1 · Xt, t = 1, . . . , N − 1, (3.9)

and allows us in principle to compute the portfolio strategy (ξt)t=1,...,N .

After solving (3.8) for ξN , we then use ξN to solve the self-financing con-dition

ξN−1 · SN−1 = ξN · SN−1for ξN−1, then

ξN−2 · SN−2 = ξN−1 · SN−2for ξN−2, and successively ξ2 down to ξ1.

Then the discounted value Vt at time t of the portfolio claim can be ob-tained from

V0 = ξ1 · X0 and Vt = ξt · Xt, t = 1, . . . , N. (3.10)

In the proof of Theorem 3.1 we actually showed that the price πt(C) of theclaim at time t coincides with the value Vt of any self-financing portfoliohedging the claim C, i.e.

πt(C) = Vt, t = 0, 1, . . . , N,

as given by (3.10). In addition, (3.6) shows that

Vt =1

(1 + r)N−tIE∗[C | Ft], t = 0, 1, . . . , N, (3.11)

hence the price of the claim can be computed either algebraically by solv-ing (3.8) and (3.9) and then using (3.10), or by a probabilistic method byevaluating the expectation (3.11).

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3.3 Pricing of Vanilla Options in the CRR Model

In this section we consider the pricing of contingent claims in the discretetime Cox-Ross-Rubinstein model, with d = 1. More precisely we are con-cerned with vanilla options whose payoffs depend on the terminal value ofthe underlying asset, as opposed to exotic options whose payoffs may dependon the whole path of the underlying asset price until expiration time.

Recall that the portfolio value process (Vt)t=0,1,...N and the discountedportfolio value process respectively satisfy

Vt = ξt · St and Vt =1

(1 + r)tVt = ξt · Xt, t = 1, 2, . . . , N.

Here we will be concerned with the pricing of vanilla options with payoffs ofthe form

C = f(SN ),

e.g. f(x) = (x −K)+ in the case of a European call. Equivalently, the dis-counted claim

C =C

(1 + r)N

satisfies C = f(SN ) with f(x) = f(x)/(1+r)N , i.e. f(x) =1

(1 + r)N(x−K)

+

in the case of a European call with strike K.

From Theorem 3.1, the discounted value of a portfolio hedging the attain-able (discounted) claim C is given by

Vt = IE∗[f(SN ) | Ft

], t = 0, 1, . . . , N,

under the risk-neutral measure P∗. Equivalently, the arbitrage price πt(C) ofthe contingent claim C = f(SN ) is given by

πt(C) =1

(1 + r)N−tIE∗[f(SN ) | Ft], t = 0, 1, . . . , N. (3.12)

In the next proposition we implement the calculation of (3.12).

Proposition 3.1. The price πt(C) of the contingent claim C = f(SN ) sat-isfies

πt(C) = v(t, St), t = 0, 1, . . . , N,

where the function v(t, x) is given by

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N. Privault

v(t, x) =1

(1 + r)N−tIE∗

fx N∏

j=t+1

(1 +Rj)

(3.13)

=1

(1 + r)N−t

N−t∑j=0

(N − tj

)(p∗)j(1− p∗)N−t−jf

(x (1 + b)

j(1 + a)

N−t−j).

Proof. From the relations

SN = St

N∏j=t+1

(1 +Rj),

and (3.12) we have, using Property (v) of the conditional expectation andthe independence of the returns R1, . . . , Rt and Rt+1, . . . , RN,

πt(C) =1

(1 + r)N−tIE∗[f(SN ) | Ft]

=1

(1 + r)N−tIE∗

fSt N∏

j=t+1

(1 +Rj)

∣∣∣Ft

=1

(1 + r)N−tIE∗

fx N∏

j=t+1

(1 +Rj)

x=St

.

Next we note that the number of times Rj is equal to b for j ∈ t+1, . . . , N,has a binomial distribution with parameter (N − t, p∗), where

p∗ =r − ab− a and 1− p∗ =

b− rb− a, (3.14)

since the set of paths from time t+ 1 to time N containing j times “(1 + b)”has cardinal

(N−tj

)and each such path has the probability (p∗)j(1−p∗)N−t−j ,

j = 0, . . . , N − t. Hence we have

πt(C) =1

(1 + r)N−tIE∗[f(SN ) | Ft]

=1

(1 + r)N−t

N−t∑j=0

(N − tj

)(p∗)j(1− p∗)N−t−jf

(St (1 + b)

j(1 + a)

N−t−j).

In the above proof we have also shown that πt(C) is given by the condi-tional expectation

πt(C) =1

(1 + r)N−tIE∗[f(SN ) | St]

48


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given the value of St at time t = 0, 1, . . . , N , i.e. the price of the claim C iswritten as the average (path integral) of the values of the contingent claimover all possible paths starting from St.

The discounted price Vt of the portfolio can also be computed via a back-ward induction procedure. Namely, by the “tower property” of conditionalexpectations we have

Vt = v(t, St)

= IE∗[f(SN ) | Ft

]= IE∗

[IE∗[f(SN ) | Ft+1

]| Ft

]= IE∗

[Vt+1 | Ft

]= IE∗ [v(t+ 1, St+1) | Ft]= v (t+ 1, (1 + a)St)P∗(Rt+1 = a) + v (t+ 1, (1 + b)St)P∗(Rt+1 = b)

= (1− p∗)v (t+ 1, (1 + a)St) + p∗v (t+ 1, (1 + b)St) ,

which shows that v(t, x) satisfies the induction relation

v(t, x) = (1− p∗)v (t+ 1, x(1 + a)) + p∗v (t+ 1, x(1 + b)) ,

while the terminal condition VN = f(SN ) implies

v(N, x) = f(x).

3.4 Hedging of Vanilla Options in the CRR model

In this section we consider the hedging of contingent claims in the discretetime Cox-Ross-Rubinstein model. Our aim is to compute a self-financingportfolio strategy hedging a vanilla option with payoff of the form

C = f(SN ).

Proposition 3.2. The replicating portfolio (ξ(0)t , ξ

(1)t )t=1,...,N hedging the

contingent claim C = f(SN ) is given by

ξ(1)t =

v (t, (1 + b)St−1)− v (t, (1 + a)St−1)

Xt−1(b− a)/(1 + r), t = 1, . . . , N,

and

ξ(0)t =

v(t− 1, St−1)− ξ(1)t Xt−1

π(0), t = 1, . . . , N,

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N. Privault

where the function v(t, x) = (1 + r)−tv(t, x) is given by (3.13).

Proof. Recall that by Lemma 3.1 the following statements are equivalent:

(i) The portfolio strategy (ξt)t=1,...,N is self-financing.

(ii) We have

Vt = V0 +

t∑j=1

ξj · (Xj − Xj−1), t = 1, . . . , N.

As a consequence, any self-financing hedging strategy (ξt)t=1,...,N should sat-isfy

v(t, St)− v(t− 1, St−1) = Vt − Vt−1 = ξt · (Xt − Xt−1).

Note that since the discounted price X(0)t of the riskless asset satisfies

X(0)t = (1 + r)−tS

(0)t = π(0),

we have

ξt · (Xt − Xt−1) = ξ(0)t (X

(0)t −X(0)

t−1) + ξ(1)t (X

(1)t −X(1)

t−1)

= ξ(0)t (π(0) − π(0)) + ξ

(1)t (X

(1)t −X(1)

t−1)

= ξ(1)t (X

(1)t −X(1)

t−1)

= ξ(1)t (Xt −Xt−1), t = 1, . . . , N.

Hence we have

v(t, St)− v(t− 1, St−1) = ξ(1)t (Xt −Xt−1), t = 1, . . . , N,

and from this we deduce the two equationsv (t, (1 + a)St−1)− v(t− 1, St−1) = ξ

(1)t

(1 + a

1 + rXt−1 −Xt−1

),

v (t, (1 + b)St−1)− v(t− 1, St−1) = ξ(1)t

(1 + b

1 + rXt−1 −Xt−1

),

t = 1, . . . , N , i.e.v (t, (1 + a)St−1)− v(t− 1, St−1) = ξ

(1)t

a− r1 + r

Xt−1,

v (t, (1 + b)St−1)− v(t− 1, St−1) = ξ(1)t

b− r1 + r

Xt−1, t = 1, . . . , N,

hence

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ξ(1)t =

v (t, (1 + a)St−1)− v (t− 1, St−1)

Xt−1(a− r)/(1 + r), t = 1, . . . , N,

and

ξ(1)t =

v (t, (1 + b)St−1)− v (t− 1, St−1)

Xt−1(b− r)/(1 + r), t = 1, . . . , N.

From the obvious relation

ξ(1)t =

b− rb− aξ

(1)t −

a− rb− a ξ

(1)t ,

we get

ξ(1)t =

v (t, (1 + b)St−1)− v (t, (1 + a)St−1)

Xt−1(b− a)/(1 + r), t = 1, . . . , N,

which only depends on St−1 as expected. This is consistent with the fact

that ξ(1)t represents the (possibly fractional) quantity of the risky asset to be

present in the portfolio over the time period [t − 1, t] in order to hedge theclaim C at time N , and is decided at time t− 1.

Concerning the quantity ξ(0)t of the riskless asset in the portfolio at time

t, recall that we have

Vt = ξt · Xt = ξ(0)t X

(0)t + ξ

(1)t X

(1)t , t = 1, . . . , N,

hence

ξ(0)t =

Vt − ξ(1)t X(1)t

X(0)t

=Vt − ξ(1)t X

(1)t

π(0)

=v(t, St)− ξ(1)t X

(1)t

π(0), t = 1, . . . , N.

Note that we have

ξ(0)t =

v(t− 1, St−1) + (v(t, St)− v(t− 1, St−1))− ξ(1)t X(1)t

π(0)

=v(t− 1, St−1) + ξ

(1)t (Xt −Xt−1)− ξ(1)t X

(1)t

π(0)

=v(t− 1, St−1)− ξ(1)t Xt−1

π(0), t = 1, . . . , N.

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N. Privault

Hence the discounted amount ξ(0)t π(0) invested on the riskless asset is

ξ(0)t π(0) = v(t− 1, St−1)− v (t, (1 + b)St−1)− v (t, (1 + a)St−1)

(b− a)/(1 + r), (3.15)

t = 1, . . . , N , and we recover the fact that ξ(0)t depends only on St−1 and not

on St.

Using the relation

v(t− 1, St−1) = (1 + r)t−1v(t− 1, St−1)

the amount (3.15) can be rewritten without discount as

ξ(0)t S

(0)t = (1 + r)tξ

(0)t π(0)

= (1 + r)tv(t− 1, St−1)− (1 + r)tv (t, (1 + b)St−1)− v (t, (1 + a)St−1)

(b− a)/(1 + r)

= (1 + r)v(t− 1, St−1)− (1 + r)v (t, (1 + b)St−1)− v (t, (1 + a)St−1)

b− a=

1 + r

b− a ((b− a)v(t− 1, St−1)− v (t, (1 + b)St−1) + v (t, (1 + a)St−1)) ,

t = 1, . . . , N . Recall that this portfolio strategy (ξ(0)t , ξ

(1)t )t=1,...,N hedges the

claim C = f(SN ), i.e. at time N we have

VN = f(SN ),

and it is self-financing by Lemma 3.1.

3.5 Hedging of Exotic Options in the CRR Model

In this section we take p = p∗ given by (3.14) and we consider the hedging ofpath dependent options. Here we choose to use the finite difference gradientand the discrete Clark-Ocone formula of stochastic analysis, see also [25],[47], [58], Chapter 1 of [59], [67], or §15-1 of [76]. See [54] and Section 8.2 of[59] for a similar approach in continuous time. Given

ω = (ω1, . . . , ωN ) ∈ Ω = −1, 1N ,

and k ∈ 1, 2, . . . , N, let

ωk+ = (ω1, . . . , ωk−1,+1, ωk+1, . . . , ωN )

and

52


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ωk− = (ω1, . . . , ωk−1,−1, ωk+1, . . . , ωN ).

We also assume that the return Rt(ω) is constructed as

Rt(ωt+) = b and Rt(ω

t−) = a, t = 1, . . . , N, ω ∈ Ω.

Definition 3.1. The operator Dt is defined on any random variable F andt ≥ 1 by

DtF (ω) = F (ωt+)− F (ωt−), t = 1, . . . , N. (3.16)

Recall the following predictable representation formula for the functionalsof the binomial process.

Definition 3.2. Let the centered and normalized return Yt be defined by

Yt :=Rt − rb− a =

b− rb− a = q, ωt = +1,

a− rb− a = −p, ωt = −1,

t = 1, . . . , N.

Note that under the risk-neutral measure P∗ we have

IE∗[Yt] = IE∗[Rt − rb− a

]=a− rb− aP

∗(Rt = a) +b− rb− aP

∗(Rt = b)

=a− rb− a

b− rb− a +

b− rb− a

r − ab− a

= 0,

andVar [Yt] = pq2 + qp2 = pq, t = 1, . . . , N.

In addition the discounted asset price increment reads

Xt −Xt−1 = Xt−11 +Rt1 + r

−Xt−1

=1

1 + rXt−1(Rt − r)

=b− a1 + r

YtXt−1, t = 1, . . . , N.

We also have

DtYt =b− rb− a +

r − ab− a = 1, t = 1, . . . , N,

and

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N. Privault

DkSN = S0(1 + b)N∏t=1t 6=k

(1 +Rt)− S0(1 + a)N∏t=1t6=k

(1 +Rt)

= S0(b− a)N∏t=1t 6=k

(1 +Rt)

= S0b− a

1 +Rk

N∏t=1

(1 +Rt)

=b− a

1 +RkSN , k = 1, . . . , N.

The next proposition is the Clark-Ocone predictable representation formulain discrete time, cf. e.g. [59], Proposition 1.7.1.

Proposition 3.3. For any square-integrable random variables F on Ω wehave

F = IE∗[F ] +

∞∑k=1

IE∗[DkF |Fk−1]Yk. (3.17)

The Clark-Ocone formula has the following consequence.

Corollary 3.1. Assume that (Mk)k∈N is a square-integrable Ft-martingale.Then we have

MN = IE∗[MN ] +

N∑k=1

YkDkMk, N ≥ 0.

Proof. We have

MN = IE∗[MN ] +∞∑k=1

IE∗[DkMN |Fk−1]Yk

= IE∗[MN ] +

∞∑k=1

Dk IE∗[MN |Fk]Yk

= IE∗[MN ] +∞∑k=1

YkDkMk

= IE∗[MN ] +N∑k=1

YkDkMk.

In addition to the Clark-Ocone formula we also state a discrete-time ana-log of Ito’s change of variable formula, which can be useful for option hedging.The next result extends Proposition 1.13.1 of [59] by removing the unneces-sary martingale requirement on (Mt)n∈N.

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Proposition 3.4. Let (Zn)n∈N be an Fn-adapted process and let f : R×N→R be a given function. We have

f(Zt, t) = f(Z0, 0) +t∑

k=1

Dkf(Zk, k)Yk

+

t∑k=1

(IE∗[f(Zk, k)|Fk−1]− f(Zk−1, k − 1)) . (3.18)

Proof. First, we note that the process

t 7−→ f(Zt, t)−t∑

k=1

(IE∗[f(Zk, k)|Fk−1]− f(Zk−1, k − 1))

is a martingale under P∗. Indeed we have

IE∗

[f(Zt, t)−

t∑k=1

(IE∗[f(Zk, k)|Fk−1]− f(Zk−1, k − 1))∣∣∣Ft−1]

= IE∗[f(Zt, t)|Ft−1]

−t∑

k=1

(IE∗[IE∗[f(Zk, k)|Fk−1]|Ft−1]− IE∗[IE∗[f(Zk−1, k − 1)|Fk−1]|Ft−1])

= IE∗[f(Zt, t)|Ft−1]−t∑

k=1

(IE∗[f(Zk, k)|Fk−1]− f(Zk−1, k − 1))

= f(Zt−1, t− 1)−t−1∑k=1

(IE∗[f(Zk, k)|Fk−1]− f(Zk−1, k − 1)) , t ≥ 1.

Note that if (Zt)t∈N is a martingale in L2(Ω) with respect to (Ft)t∈N andwritten as

Zt = Z0 +t∑

k=1

ukYk, t ∈ N,

where (ut)t∈N is a predictable process locally in L2(Ω×N), (i.e. u(·)1[0,N ](·) ∈L2(Ω × N) for all N > 0), then we have

Dtf(Zt, t) = f (Zt−1 + qut, t)− f (Zt−1 − put, t) , (3.19)

t = 1, . . . , N . On the other hand, the term

IE[f(Zt, t)− f(Zt−1, t− 1)|Ft−1]

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N. Privault

is analog to the finite variation part in the continuous time Ito formula, andcan be written as

pf (Zt−1 + qut, t) + qf (Zt−1 − put, t)− f (Zt−1, t− 1) .

Naturally, if (f(Zt, t))t∈N is a martingale we recover the decomposition

f(Zt, t) = f(Z0, 0)

+t∑

k=1

(f (Zk−1 + quk, k)− f (Zk−1 − puk, k))Yk

= f(Z0, 0) +

t∑k=1

YkDkf(Zk, k). (3.20)

This identity follows from Corollary 3.1 as well as from Proposition 3.3. Inthis case the Clark-Ocone formula (3.17) and the change of variable formula(3.20) both coincide and we have in particular

Dkf(Zk, k) = IE[Dkf(ZN , N)|Fk−1],

k = 1, . . . , N . For example this recovers the martingale representation

Xt = S0 +t∑

k=1

YkDkXk

= S0 +b− a1 + r

t∑k=1

Xk−1Yk

= S0 +t∑

k=1

Xk−1Rk − r1 + r

= S0 +t∑

k=1

(Xk −Xk−1),

of the discounted asset price.

Our goal is to hedge an arbitrary claim C on Ω, i.e. given an FN -measurable random variable C we search for a portfolio (ξt, ηt)t=1,...,N suchthat the equality

C = VN = ηNAN + ξNSN (3.21)

holds, where AN = A0(1 + r)N denotes the value of the riskless asset at timeN ∈ N.

The next proposition is the main result of this section, and provides asolution to the hedging problem under the constraint (3.21).

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Proposition 3.5. Given C a contingent claim, let

ξt = (1 + r)−(N−t)1

St−1(b− a)IE∗[DtC|Ft−1], (3.22)

t = 1 . . . , N , and

ηt =1

At

((1 + r)−(N−t) IE∗[C|Ft]− ξtSt

), (3.23)

t = 1 . . . , N . Then the portfolio (ξt, ηt)t=1...,N is self financing and satisfies

Vt = ηtAt + ξtSt = (1 + r)−(N−t) IE∗[C|Ft], t = 1 . . . , N,

in particular we have VN = C, hence (ξt, ηt)t=1...,N is a hedging strategyleading to C.

Proof. Let (ξt)t=1...,N be defined by (3.22), and consider the process (ηt)t=0,1...,N

defined by

η0 = (1 + r)−NIE∗[C]

S0and ηt+1 = ηt −

(ξt+1 − ξt)StAt

,

t = 0, . . . , N − 1. Then (ξt, ηt)t=1,...,N satisfies the self-financing condition

At(ηt+1 − ηt) + St(ξt+1 − ξt) = 0, t = 1, . . . , N − 1.

Let now

V0 = IE∗[C](1 + r)−N , and Vt = ηtAt + ξtSt, t = 1, . . . , N,

andVt = Vt(1 + r)−t t = 0, . . . , N.

Since (ξt, ηt)t=1,...,N is self-financing, by Lemma 3.1 we have

Vt = V0 + (b− a)

t∑k=1

YkξkSk−1(1 + r)−k, (3.24)

t = 1, . . . , N . On the other hand, from the Clark-Ocone formula (3.17) andthe definition of (ξt)t=1,...,N we have

(1 + r)−N IE∗[C|Ft]

= IE∗

[IE∗[C](1 + r)−N +

N∑i=0

IE∗[DiC|Fi−1](1 + r)−N∣∣∣Ft]Yi

= IE∗[C](1 + r)−N +

t∑i=0

IE∗[DiC|Fi−1](1 + r)−NYi

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N. Privault

= IE∗[C](1 + r)−N + (b− a)t∑i=0

ξiSi−1(1 + r)−iYi

= Vt

from (3.24). Hence

Vt = (1 + r)−N IE∗[C|Ft], t = 0, 1, . . . , N,

andVt = (1 + r)−(N−t) IE∗[C|Ft], t = 0, 1, . . . , N. (3.25)

In particular, (3.25) shows that we have VN = C. To conclude the proofwe note that from the relation Vt = ηtAt + ξtSt, t = 1, . . . , N , the process(ηt)t=1,...,N coincides with (ηt)t=1,...,N defined by (3.23).

From Proposition 3.1, when C = f(SN ), the price πt(C) of the contingentclaim C = f(SN ) is given by

πt(C) = v(t, St),

where the function v(t, x) is given by

v(t, St) =1

(1 + r)N−tIE∗[C|Ft] =

1

(1 + r)N−tIE∗

fx N∏

j=t+1

(1 +Rj)

x=St

.

Note that in this case we have C = v(N,SN ), IE[C] = v(0,M0), and the

discounted claim payoff C = (1 + r)−NC = v(N,SN ) satisfies

C = IE[C]

+n∑t=1

Yt IE [Dtv(N,SN )|Ft−1]

= IE[C]

+n∑t=1

YtDtv(t, St)

= IE[C]

+

n∑t=1

(1 + r)−tYtDtv(t, St)

= IE[C]

+n∑t=1

YtDt IE [v(N,SN )|Ft]

= IE[C]

+ (1 + r)−Nn∑t=1

YtDt IE[C|Ft],

hence we have

IE[Dtv(N,SN )|Ft−1] = (1 + r)N−tDtv(t, St), t = 1, . . . , N,

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and by Proposition 3.5 the hedging strategy for C = f(SN ) is given by

ξt =(1 + r)−(N−t)

St−1(b− a)IE[Dtv(N,SN )|Ft−1]

=1

St−1(b− a)Dtv(t, St)

=1

St−1(b− a)(v (t, St−1(1 + b))− v (t, St−1(1 + a)))

=1

Xt−1(b− a)/(1 + r)(v (t, St−1(1 + b))− v (t, St−1(1 + a))) ,

t = 1, . . . , N , which recovers Proposition 3.2 as a particular case. Note thatξt is non-negative (i.e. there is no short-selling) when f is a non decreasingfunction, because a < b. This is in particular true in the case of a Europeancall option for which we have f(x) = (x−K)+.

3.6 Convergence of the CRR Model

In this section we consider the convergence of the discrete time model to thecontinuous-time Black Scholes model.

Continuous compounding - riskless asset

Consider the subdivision[0,T

N,

2T

N, . . . ,

(N − 1)T

N, T

]of the time interval [0, T ] into N time steps.

Note thatlimN→∞

(1 + r)N =∞,

thus we need to renormalize r so that the interest rate on each time intervalbecomes rN , with limN→∞ rN = 0.

It turns out that the correct renormalization is

rN = rT

N,

so that

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N. Privault

limN→∞

(1 + rN )N = limN→∞

(1 + r

T

N

)N= erT , T ∈ R+. (3.26)

Hence the price of the riskless asset satifies

At = A0ert,

with the differential equation

dAtdt

= rAt,

also written asdAt = rAtdt,

ordAtAt

= rdt,

which means that the return of the riskless asset is rdt on the small timeinterval [t, t+ dt]. Equivalently, one says that r is the instantaneous interestrate per unit of time.

The same equation rewrites in integral form as

AT −A0 =w T

0dAt = r

w T

0Atdt.

Continuous compounding - risky asset

We need to apply a similar renormalization to the coefficients a and b of theCRR model. Let σ > 0 denote a positive parameter called the volatility andlet aN , bN be defined from

1 + aN1 + rN

= e−σ√

TN and

1 + bN1 + rN

= eσ√

TN ,

i.e.

aN = (1 + rN )e−σ√

TN − 1 and bN = (1 + rN )eσ

√TN − 1.

Consider the random return R(N)k ∈ aN , bN and the price process defined

as

S(N)t = S0

t∏k=1

(1 +R(t)k ), t = 1, . . . , N.

Note that the risk-neutral probabilities are given by

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P∗(Rt = aN ) =bN − rNbN − aN

=eσ√

TN − 1

2 sinh√

σ2TN

, t = 1, . . . , N,

and

P∗(Rt = bN ) =rN − aNbN − aN

=1− e−σ

√TN

2 sinh√

σ2TN

, t = 1, . . . , N,

which both converge to 1/2 as N goes to infinity.

Continuous-time limit

We have the following convergence result.

Proposition 3.6. Let f be a continuous and bounded function on R. The

price at time t = 0 of a contingent claim with payoff C = f(S(N)N

)converges

as follows:

limN→∞

1

(1 + rT/N)NIE∗[f(S

(N)N )

]= e−rT IE

[f(S0e

σ√TX+rT−σ2T/2)

](3.27)

where X ' N (0, 1) is a standard Gaussian random variable.

Proof. This result is consequence of the weak convergence of the sequence

(S(N)N )N≥1 to a lognormal distribution, cf. Theorem 5.53 of [25]. The con-

vergence of the discount factor follows directly from (3.26).

Note that the expectation (3.27) can be written as a Gaussian integral:

e−rT IE[f(S0e

σ√TX+rT−σ2T/2)

]= e−rT

w∞−∞

f(S0eσ√Tx+rT−σ2T/2)

e−x2/2

√2π

dx,

hence we have

limN→∞

1

(1 + rT/N)NIE∗[f(S

(N)N )

]= e−rT

w∞−∞

f(S0eσ√Tx+rT−σ2T/2)

e−x2/2

√2π

dx.

It is a remarkable fact that in case f(x) = (x−K)+, i.e. when C = (ST−K)+

is the payoff of a European call option with strike K, the above integral canbe computed according to the Black-Scholes formula:

e−rT IE[(S0e

σ√TX+rT−σ2T/2 −K)+

]= S0Φ(d+)−Ke−rTΦ(d−),

where

d− =(r − 1

2σ2)T + log S0

K

σ√T

, d+ = d− + σ√T ,

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N. Privault

and

Φ(x) =1√2π

w x

−∞e−y

2/2dy, x ∈ R,

is the Gaussian cumulative distribution function.

The Black-Scholes formula will be derived explicitly in the subsequentchapters using both the PDE and probabilistic method, cf. Propositions 1.8nd6.4. It can be considered as a building block for the pricing of financial deriva-tives, and its importance is not restricted to the pricing of options on stocks.Indeed, the complexity of the interest rate models makes it in general difficultto obtain closed form expressions, and in many situations one has to rely onthe Black-Scholes framework in order to find pricing formulas, for examplein the case of interest rate derivatives as in the Black caplet formula of theBGM model, cf. Proposition 12.3 in Section 12.3.

Our aim later on will be to price and hedge options directly in continuoustime using stochastic calculus, instead of applying the limit procedure de-scribed in the previous section. In addition to the construction of the risklessasset price (At)t∈R+

via

dAtAt

= rtdt, A0 = 1, t ∈ R+,

i.e.At = A0e

rt, t ∈ R+,

we now need to construct a mathematical model for the price of the rislyasset in continuous time.

The return of the risky asset St over the time period [t, d + dt] will bedefined as

dStSt

= µdt+ σdBt,

where σdBt is a “small” Gaussian random component, also called Brownianincrement, parametrized by the volatility parameter σ > 0. In the next Chap-ter 4 we will turn to the formal definition of the stochastic process (Bt)t∈R+

which will be used for the modeling of risky assets in continuous time.

Exercises

Exercise 3.1 (Exercise 2.1 continued)

1. We consider a forward contract on SN with strike K and payoff

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C := SN −K.

Find a portfolio allocation (ηN , ξN , ) with price VN = ηNπN + ξNSN attime N , such that

VN = C, (3.28)

by writing Condition (3.28) as a 2 system of equations.2. Find a portfolio allocation (ηN−1, ξN−1) with price VN−1 = ηN−1πN−1 +ξN−1SN−1 at time N − 1, and verifying the self-financing condition

VN−1 = ηNπN−1 + ξNSN−1.

Next, at all times t = 1, . . . , N−1, find a portfolio allocation (ηt, ξt) withprice Vt = ηtπt + ξtSt verifying (3.28) and the self-financing condition

Vt = ηt+1πt + ξt+1St,

where ηt, resp. ξt, represents the quantity of the riskless, resp. risky, assetin the portfolio over the time period [t− 1, t], t = 1, . . . , N .

3. forward contract C, at time t = 0, 1, . . . , N .4. Check that the arbitrage price πt(C) satisfies the relation

πt(C) =1

(1 + r)N−tIE∗[C | Ft], t = 0, 1, . . . , N.

Exercise 3.2 Consider the discrete-time Cox-Ross-Rubinstein model withN + 1 time instants t = 0, 1, . . . , N . The price S0

t of the riskless asset evolvesas S0

t = π0(1 + r)t, t = 0, 1, . . . , N . The return of the risky asset, defined as


, t = 1, . . . , N,

is random and allowed to take only two values a and b, with −1 < a < r < b.

The discounted asset price is Xt = St/(1 + r)t, t = 0, 1, . . . , N .

1. Show that this model admits a unique risk-neutral measure P∗ and ex-plicitly compute P ∗(Rt = a) and P (Rt = b) for all t = 1, . . . , N .

2. Does there exist arbitrage opportunities in this model ? Explain why.3. Is this market model complete ? Explain why.4. Consider a contingent claim with payoff1

C = (SN )2.

Compute the discounted arbitrage price Vt, t = 0, . . . , N , of a self-financing portfolio hedging the claim C, i.e. such that

1 This is the payoff of a power call option with strike K = 0.

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N. Privault

VN = C =(SN )2

(1 + r)N.

5. Compute the portfolio strategy

(ξt)t=1,...,N = (ξ0t , ξ1t )t=1,...,N

associated to Vt, i.e. such that

Vt = ξt · Xt = ξ0tX0t + ξ1tX

1t , t = 1, . . . , N.

6. Check that the above portfolio strategy is self-financing, i.e.

ξt+1 · St = ξt · St, t = 1, . . . , N − 1.

Exercise 3.3 We consider the discrete-time Cox-Ross-Rubinstein model withN + 1 time instants t = 0, 1, . . . , N .

The price πt of the riskless asset evolves as πt = π0(1+r)t, t = 0, 1, . . . , N .The evolution of St−1 to St is given by

St =

(1 + b)St−1

(1 + a)St−1

with −1 < a < r < b. The return of the risky asset is defined as


, t = 1, . . . , N.

Let ξt, resp. ηt, denote the (possibly fractional) quantities of the risky, resp.riskless, asset held over the time period [t− 1, t] in the portfolio with value

Vt = ξtSt + ηtπt, t = 0, . . . , N. (3.29)

1. Show that

Vt = (1 +Rt)ξtSt−1 + (1 + r)ηtπt−1, t = 1, . . . , N. (3.30)

2. Show that under the probability P ∗ defined by

P ∗(Rt = a | Ft−1) =b− rb− a, P ∗(Rt = b | Ft−1) =

r − ab− a ,

where Ft−1 represents the information generated by R1, . . . , Rt−1, wehave

E∗[Rt | Ft−1] = r.

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3. Under the self-financing condition

Vt−1 = ξtSt−1 + ηtπt−1 t = 1, . . . , N, (3.31)

show that

Vt−1 =1

1 + rE∗[Vt | Ft−1],

using the result of Question 1.4. Let a = 5%, b = 25% and r = 15%. Assume that the price Vt at time t

of the portfolio is $3 if Rt = a and $8 if Rt = b, and compute the priceVt−1 of the portfolio at time t− 1.

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Chapter 4

Brownian Motion and Stochastic Calculus

The modeling of random assets in finance is based on stochastic processes,which are families (Xt)t∈I of random variables indexed by a time interval I. Inthis chapter we present a description of Brownian motion and a constructionof the associated Ito stochastic integral.

4.1 Brownian Motion

We start by recalling the definition of Brownian motion, which is a funda-mental example of a stochastic process. The underlying probability space(Ω,F ,P) of Brownian motion can be constructed on the space Ω = C0(R+)of continuous real-valued functions on R+ started at 0.

Definition 4.1. The standard Brownian motion is a stochastic process(Bt)t∈R+ such that

(i) B0 = 0 almost surely,

(ii) The sample trajectories t 7→ Bt are continuous, with probability 1.

(iii) For any finite sequence of times t0 < t1 < · · · < tn, the increments

Bt1 −Bt0 , Bt2 −Bt1 , . . . , Btn −Btn−1

are independent.

(iv) For any given times 0 ≤ s < t, Bt−Bs has the Gaussian distributionN (0, t− s) with mean zero and variance t− s.

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N. Privault

We refer to Theorem 10.28 of [24] and to Chapter 1 of [65] for the proof ofthe existence of Brownian motion as a stochastic process (Bt)t∈R+

satisfyingthe above properties (i)-(iv).

In particular, Condition (iv) above implies

IE[Bt −Bs] = 0 and Var[Bt −Bs] = t− s, 0 ≤ s ≤ t.

In the sequel the filtration (Ft)t∈R+will be generated by the Brownian paths

up to time t, in other words we write

Ft = σ(Bs : 0 ≤ s ≤ t), t ≥ 0. (4.1)

A random variable F is said to be Ft-measurable if the knowledge of Fdepends only on the information known up to time t. As an example, ift =today,

• the date of the past course exam is Ft-measurable, because it belongs tothe past.

• the date of the next Chinese new year, although it refers to a future event,is also Ft-measurable because it is known at time t.

• the date of the next typhoon is not Ft-measurable since it is not knownat time t.

• the maturity date T of a European option is Ft-measurable for all t ∈ R+,because it has been determined at time 0.

• the exercise date τ of an American option after time t (see Section 9.4) isnot Ft-measurable because it refers to a future random event.

Property (iii) above shows that Bt − Bs is independent of all Brownian in-crements taken before time s, i.e.

(Bt −Bs) ⊥⊥ (Bt1 −Bt0 , Bt2 −Bt1 , . . . , Btn −Btn−1),

0 ≤ t0 ≤ t1 ≤ · · · ≤ tn ≤ s ≤ t, hence Bt − Bs is also independent of thewhole Brownian history up to time s, hence Bt − Bs is in fact independentof Fs, s ≥ 0.

For convenience we will informally regard Brownian motion as a randomwalk over infinitesimal time intervals of length ∆t, with increments

∆Bt := Bt+∆t −Bt

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over the time interval [t, t+∆t] given by

∆Bt = ±√∆t (4.2)

with equal probabilities (1/2, 1/2).

The choice of the square root in (4.2) is in fact not fortuitous. Indeed, anychoice of ±(∆t)α with a power α > 1/2 would lead to explosion of the processas dt tends to zero, whereas a power α ∈ (0, 1/2) would lead to a vanishingprocess.

Note that we have

IE[∆Bt] =1

2

√∆t− 1

2

√∆t = 0,

and

Var[∆Bt] = IE[(∆Bt)2] =

1

2∆t+

1

2∆t = ∆t.

According to this representation, the paths of Brownian motion are not dif-ferentiable, although they are continuous by Property (ii), as we have

dBtdt' ±√dt

dt= ± 1√

dt' ±∞. (4.3)

After splitting the interval [0, T ] into N intervals(k − 1

NT,

k

NT

], k = 1, . . . , N,

of length ∆t = T/N with N “large”, and letting

Xk = ±√T = ±

√N√∆t =

√N∆Bt

with probabilities (1/2, 1/2) we have V ar(Xk) = T and

∆Bt =Xk√N

= ±√∆t

is the increment of Bt over ((k − 1)∆t, k∆t], and we get

BT '∑

0<t<T

∆Bt 'X1 + · · ·+XN√

N.

Hence by the central limit theorem we recover the fact that BT has a centeredGaussian distribution with variance T , cf. point (iv) of the above definitionof Brownian motion. Indeed, the central limit theorem states that given any

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N. Privault

sequence (Xk)k≥1 of independent identically distributed centered randomvariables with variance σ2 = V ar(Xk) = T , the normalized sum

X1 + · · ·+Xn√n

converges (in distribution) to a centered Gaussian random variable N (0, σ2)with variance σ2 as n goes to infinity. As a consequence, ∆Bt could in factbe replaced by any centered random variable with variance ∆t in the abovedescription.

Note that there is no point in “computing” the value of Bt as it is arandom variable for all t > 0, however we can generate samples of Bt, whichare distributed according to the centered Gaussian distribution with variancet. Below we draw three sample paths of a standard Brownian motion obtainedby computer simulation using (4.2).

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

B t

Fig. 4.1: Sample paths of one-dimensional Brownian motion.

The n-dimensional Brownian motion can be constructed as (B1t , . . . , B

nt )t∈R+

where (B1t )t∈R+

, . . .,(Bnt )t∈R+are independent copies of (Bt)t∈R+

.

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-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

Fig. 4.2: Two sample paths of a two-dimensional Brownian motion.

Next we turn to simulations of 2 dimensional and 3 dimensional Brownianmotions. Recall that the movement of pollen particles originally observed byR. Brown in 1827 was indeed 2-dimensional.

-2-1.5

-1-0.5

0 0.5

1 1.5

2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Fig. 4.3: Sample paths of a three-dimensional Brownian motion.

4.2 Wiener Stochastic Integral

In this section we construct the Ito stochastic integral of square-integrabledeterministic function with respect to Brownian motion.

Recall that Bachelier originally modeled the price St of a risky asset bySt = σBt where σ is a volatility parameter. The stochastic integral

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N. Privault

w T

0f(t)dSt = σ

w T

0f(t)dBt

can be used to represent the value of a portfolio as a sum of profits andlosses f(t)dSt where dSt represents the stock price variation and f(t) is thequantity invested in the asset St over the short time interval [t, t+ dt].

A naive definition of the stochastic integral with respect to Brownian mo-tion would consist in writing

w∞0f(t)dBt =

w∞0f(t)

dBtdt

dt,

and evaluating the above integral with respect to dt. However this definitionfails because the paths of Brownian motion are not differentiable, cf. (4.3).Next we present Ito’s construction of the stochastic integral with respect toBrownian motion. Stochastic integrals will be first constructed as integralsof simple step functions of the form

f(t) =n∑i=1

ai1(ti−1,ti](t), t ∈ R+, (4.4)

i.e. the function f takes the value ai on the interval (ti−1, ti], i = 1, . . . , n,with 0 ≤ t0 < · · · < tn, as illustrated in Figure 4.4.

6f

-t0

a1

t1

a2

t2 t3 t4

a4

t

Fig. 4.4: Step function.

Note that the set of simple step functions f of the form (4.4) is a linear spacewhich is dense in L2(R+) for the norm

‖f‖L2(R+) :=

√w∞0|f(t)|2dt.

Recall also that the classical integral of f given in (4.4) is interpreted as thearea under the curve f and computed as

w∞0f(t)dt =

n∑i=1

ai(ti − ti−1).

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In the next definition we adapt this construction to the setting of integrationwith respect to Brownian motion.

Definition 4.2. The stochastic integral with respect to Brownian motion(Bt)t∈R+

of the simple step functions f of the form (4.4) is defined by

w∞0f(t)dBt :=

n∑i=1

ai(Bti −Bti−1). (4.5)

In the next Proposition 4.1 we determine the probability distribution ofw∞0f(t)dBt and we show that it is independent of the particular representa-

tion (4.4) chosen for f(t). In the sequel we will make a repeated use of thespace L2(R+) of functions f : R+ → R such that

‖f‖2L2(R+) :=w∞0|f(t)|2dt <∞,

called square-integrable functions.

Proposition 4.1. The definition of the stochastic integralr∞0f(t)dBt can

be extended to any measurable function f ∈ L2(R+), i.e. to f such that

w∞0|f(t)|2dt <∞. (4.6)

In this case,w∞0f(t)dBt has a centered Gaussian distribution

w∞0f(t)dBt ' N

(0,

w∞0|f(t)|2dt

)with variance

w∞0|f(t)|2dt and we have the Ito isometry

IE

[(w∞0f(t)dBt

)2]=

w∞0|f(t)|2dt. (4.7)

Proof. Recall that if X1, . . . , Xn are independent Gaussian random variableswith probability laws N (m1, σ

21), . . . ,N (mn, σ

2n) then then sum X1+· · ·+Xn

is a Gaussian random variable with probability law N (m1 + · · · + mn, σ21 +

· · ·+ σ2n).

As a consequence, when f is the simple function

f(t) =

n∑i=1

ai1(ti−1,ti](t), t ∈ R+,

the sum w∞0f(t)dBt =

n∑k=1

ak(Btk −Btk−1)

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N. Privault

has a centered Gaussian distribution with variance

n∑k=1

|ak|2(tk − tk−1),

since

Var [ak(Btk −Btk−1)] = a2kVar [Btk −Btk−1

] = a2k(tk − tk−1),

hence the stochastic integral

w∞0f(t)dBt =

n∑k=1

ak(Btk −Btk−1)

of the step function

f(t) =n∑k=1

ak1(tk−1,tk](t)

has a centered Gaussian distribution with variance

Var[w∞

0f(t)dBt

]=

n∑k=1

|ak|2(tk − tk−1)

=n∑k=1

|ak|2w tk

tk−1

dt

=w∞0

n∑k=1

|ak|21(tk−1,tk](t)dt

=w∞0|f(t)|2dt.

Finally we note that

Var[w∞

0f(t)dBt

]= IE

[(w∞0f(t)dBt

)2]−(

IE[w∞

0f(t)dBt

])2= IE

[(w∞0f(t)dBt

)2].

The extension of the stochastic integral to all functions satisfying (4.6) isobtained by density and a Cauchy sequence argument, based on the isometryrelation (4.7). Namely, given f a function satisfying (4.6) and (fn)n∈N asequence of simple functions converging to f for the norm

‖f − fn‖L2(R+) :=(w∞

0|f(t)− fn(t)|2dt

)1/2

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i.e. in L2(R+), the isometry (4.7) shows that(r∞

0fn(t)dBt

)n∈N is a Cauchy

sequence in the space L2(Ω) of square-integrable random variables F : Ω → Rsuch that

‖F‖2L2(Ω×R+) := IE[F 2] <∞.Indeed, we have∥∥∥w∞

0fk(t)dBt −

w∞0fn(t)dBt

∥∥∥L2(Ω)

=

(IE

[(w∞0fk(t)dBt −

w∞0fn(t)dBt

)2])1/2

= ‖fk − fn‖L2(R+)

≤ ‖f − fk‖L2(R+) + ‖f − fn‖L2(R+),

which tends to 0 as k, n tend to infinity, hence(r∞

0fn(t)dBt

)n∈N is it con-

verges for the L2-norm as L2(Ω) is a complete space, cf. e.g. Chapter 4 of[18]. In this case we let

w∞0f(t)dBt := lim

n→∞

w∞0fn(t)dBt

and the limit is unique from (4.7).

For example,w∞0e−tdBt has a centered Gaussian distribution with vari-

ance w∞0e−2tdt =

[−1

2e−2t

]∞0

=1

2.

Again, the Wiener stochastic integralw∞0f(s)dBs is nothing but a Gaussian

random variable and it cannot be “computed” in the way standard integralare computed via the use of primitives. However, when f ∈ L2(R+) is C1 onR+, we have the following formula

w∞0f(t)dBt = −

w∞0f ′(t)Btdt,

provided limt→∞ t|f(t)|2 = 0 and f ∈ L2(R+), cf. e.g. Remark 2.5.9 in [59].

4.3 Ito Stochastic Integral

In this section we extend the Wiener stochastic integral to square-integrableadapted processes. Recall that a process (Xt)t∈R+

is said to be Ft-adapted ifXt is Ft-measurable for all t ∈ R+, where the information flow (Ft)t∈R+ hasbeen defined in (4.1).

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N. Privault

In other words, a process (Xt)t∈R+is Ft-adapted if the value of Xt at time

t depends only on information known up to time t. Note that the value of Xt

may still depend on “known” future data, for example a fixed future date inthe calendar, such as a maturity time T > t, as long as its value is known attime t.

The extension of the stochastic integral to adapted random processes isactually necessary in order to compute a portfolio value when the portfolioprocess is no longer deterministic. This happens in particular when one needsto update the portfolio allocation based on random events occurring on themarket.

Stochastic integrals of adapted processes will be first constructed as integralsof simple predictable processes (ut)t∈R+

of the form

ut =n∑i=1

Fi1(ti−1,ti](t), t ∈ R+, (4.8)

where Fi is an Fti−1-measurable random variable for i = 1, . . . , n. The notion

of simple predictable process is natural in the context of portfolio investment,in which Fi will represent an investment allocation decided at time ti−1 andto remain unchanged over the time period (ti−1, ti].

By convention, u : Ω × R+ → R is denoted by ut(ω), t ∈ R+, ω ∈ Ω, andthe random outcome ω is often dropped for convenience of notation.

Definition 4.3. The stochastic integral with respect to Brownian motion(Bt)t∈R+

of any simple predictable process (ut)t∈R+of the form (4.8) is de-

fined by w∞0utdBt :=

n∑i=1

Fi(Bti −Bti−1). (4.9)

The next proposition gives the extension of the stochastic integral fromsimple predictable processes to square-integrable Ft-adapted processes (Xt)t∈R+

for which the value of Xt at time t only depends on information containedin the Brownian path up to time t. This also means that knowing the futureis not permitted in the definition of the Ito integral, for example a portfoliostrategy that would allow the trader to “buy at the lowest” and “sell at thehighest” is not possible as it would require knowledge of future market data.

Note that the difference between Relation (4.10) below and Relation (4.7)is the expectation on the right hand side.

Proposition 4.2. The stochastic integral with respect to Brownian motion(Bt)t∈R+

extends to all adapted processes (ut)t∈R+such that

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IE[w∞

0|ut|2dt

]<∞,

with the Ito isometry

IE

[(w∞0utdBt

)2]= IE

[w∞0|ut|2dt

]. (4.10)

In addition, the Ito integral of an adapted process (ut)t∈R+is always a cen-

tered random variable:IE[w∞

0usdBs

]= 0. (4.11)

Proof. We start by showing that the Ito isometry (4.10) holds for the simplepredictable process u of the form (4.8). We have

IE

[(w∞0utdBt

)2]= IE

( n∑i=1

Fi(Bti −Bti−1)

)2

= IE

n∑i,j=1

FiFj(Bti −Bti−1)(Btj −Btj−1

)

= IE

[n∑i=1

|Fi|2(Bti −Bti−1)2

]

+2 IE

∑1≤i<j≤n

FiFj(Bti −Bti−1)(Btj −Btj−1

)

=

n∑i=1

IE[IE[|Fi|2(Bti −Bti−1)2|Fti−1 ]]

+2∑

1≤i<j≤n

IE[IE[FiFj(Bti −Bti−1)(Btj −Btj−1

)|Ftj−1]]

=n∑i=1

IE[|Fi|2 IE[(Bti −Bti−1)2|Fti ]]

+2∑

1≤i<j≤n

IE[FiFj(Bti −Bti−1) IE[(Btj −Btj−1

)|Ftj−1]]

=n∑i=1

IE[|Fi|2 IE[(Bti −Bti−1)2]]

+2∑

1≤i<j≤n

IE[FiFj(Bti −Bti−1) IE[(Btj −Btj−1)]]

=

n∑i=1

IE[|Fi|2(ti − ti−1)]

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N. Privault

= IE

[n∑i=1

|Fi|2(ti − ti−1)

]= IE

[w∞0|ut|2dt

],

where we used the “tower property” (16.24) of conditional expectations andthe facts that Bti −Bti−1

is independent of Fti−1and

IE[Bti −Bti−1 ] = 0, IE[(Bti −Bti−1)2

]= ti − ti−1, i = 1, . . . , n.

The extension of the stochastic integral to square-integrable adapted pro-cesses (ut)t∈R+ is obtained as in Proposition 4.1 by density and a Cauchy se-quence argument using the isometry (4.10), in the same way as in the proofof Proposition 4.1. Let L2(Ω × R+) denote the space of square-integrablestochastic processes u : Ω × R+ → R such that

‖u‖2L2(Ω×R+) := IE[w∞

0|ut|2dt

]<∞.

By Lemma 1.1 of [35], p. 22 and p. 46, or Proposition 2.5.3 of [59], theset of simple predictable processes forms a linear space which is dense inthe subspace L2

ad(Ω × R+) made of square-integrable adapted processes inL2(Ω × R+). In other words, given u a square-integrable adapted processthere exists a sequence (un)n∈N of simple predictable processes convergingto u in L2(Ω × R+), and the isometry (4.10) shows that

(runt dBt

)n∈N is a

Cauchy sequence in L2(Ω), hence it converges in the complete space L2(Ω).In this case we let w∞

0utdBt := lim

n→∞

w∞0unt dBt

and the limit is unique from (4.10).

Note also that the Ito isometry (4.10) can also be written as

IE[w∞

0utdBt

w∞0vtdBt

]= IE

[w∞0utvtdt

],

for all square-integrable adapted processes u, v.

In addition, when the integrand (ut)t∈R+is not a deterministic function,

the random variablew∞0usdBs no longer has a Gaussian distribution, except

in some exceptional cases.

The stochastic integral of u over the interval [a, b] is defined as

w b

autdBt :=

w∞0

1[a,b](t)utdBt.

In particular we have

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w∞0

1[a,b](t)dBt = Bb −Ba, 0 ≤ a ≤ b,

and w b

adBt = Bb −Ba, 0 ≤ a ≤ b.

We also have the Chasles relation

w c

autdBt =

w b

autdBt +

w c

butdBt, 0 ≤ a ≤ b ≤ c,

and the stochastic integral has the following linearity property:

w∞0

(ut + vt)dBt =w∞0utdBt +

w∞0vtdBt, u, v ∈ L2(R+).

In the sequel we will define the return at time t ∈ R+ of the risky asset(St)t∈R+

asdStSt

= µdt+ σdBt,

with µ ∈ R and σ > 0. This equation can be formally rewritten in integralform as

ST = S0 + µw T

0Stdt+ σ

w T

0StdBt,

hence the need to define an integral with respect to dBt, in addition to theusual integral with respect to dt.

In Proposition 4.2 we have defined the stochastic integral of square-integrable processes with respect to Brownian motion, thus we have madesense of the equation

ST = S0 + µw T

0Stdt+ σ

w T

0StdBt,

for (St)t∈R+an Ft-adapted process, which can be rewritten in differential

notation asdSt = µStdt+ σStdBt,

ordStSt

= µdt+ σdBt. (4.12)

This model will be used to represent the random price St of a risky asset attime t. Here the return dSt/St of the asset is made of two components: a con-stant return µdt and a random return σdBt parametrized by the coefficientσ, called the volatility.

Our goal is now to solve Equation (4.12) and for this we will need tointroduce Ito’s calculus in Section 4.5 after reviewing classical deterministiccalculus in Section 4.4.

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N. Privault

4.4 Deterministic Calculus

The fundamental theorem of calculus states that for any continuously differ-entiable (deterministic) function f we have

f(x) = f(0) +w x

0f ′(y)dy.

In differential notation this relation is written as the first order expansion

df(x) = f ′(x)dx, (4.13)

where dx is “small”. Higher order expansions can be obtained from Taylor’sformula, which, letting

df(x) = f(x+ dx)− f(x),

states that

df(x) = f ′(x)dx+1

2f ′′(x)(dx)2 +

1

3!f ′′′(x)(dx)3 +

1

4!f (4)(x)(dx)4 + · · · .

Note that Relation (4.13) can be obtained by neglecting the terms of orderlarger than one in Taylor’s formula, since (dx)n << dx when n ≥ 2 and dxis “small”.

4.5 Stochastic Calculus

Let us now apply Taylor’s formula to Brownian motion, taking

dBt = Bt+dt −Bt,

and lettingdf(Bt) = f(Bt+dt)− f(Bt),

we have

df(Bt) = f ′(Bt)dBt+1

2f ′′(Bt)(dBt)

2+1

3!f ′′′(Bt)(dBt)

3+1

4!f (4)(Bt)(dBt)

4+· · · .

From the construction of Brownian motion by its small increments dBt =±√dt, it turns out that the terms in (dt)2 and dtdBt = ±(dt)3/2 can be ne-

glected in Taylor’s formula at the first order of approximation in dt. However,the term of order two

(dBt)2 = (±

√dt)2 = dt

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can no longer be neglected in front of dt.

Hence Taylor’s formula written at the second order for Brownian motionreads

df(Bt) = f ′(Bt)dBt +1

2f ′′(Bt)dt, (4.14)

for “small” dt. Note that writing this formula as

df(Bt)

dt= f ′(Bt)

dBtdt

+1

2f ′′(Bt)

does not make sense because the derivative

dBtdt' ±√dt

dt' ± 1√

dt' ±∞

does not exist.

Integrating (4.14) on both sides and using the relation

f(Bt)− f(B0) =w t

0df(Bs)

we get the integral form of Ito’s formula for Brownian motion, i.e.

f(Bt) = f(B0) +w t

0f ′(Bs)dBs +

1

2

w t

0f ′′(Bs)ds.

We now turn to the general expression of Ito’s formula which applies to Itoprocesses of the form

Xt = X0 +w t

0vsds+

w t

0usdBs, t ∈ R+, (4.15)

or in differential notation

dXt = vtdt+ utdBt,

where (ut)t∈R+and (vt)t∈R+

are square-integrable adapted processes.

Given f(t, x) a smooth function of two variables, from now on we let∂f

∂xdenote partial differentiation with respect to the second variable in f(t, x),

while∂f

∂sdenote partial differentiation with respect to the first (time) variable

in f(t, x).

Theorem 4.1. (Ito formula for Ito processes). For any Ito process (Xt)t∈R+

of the form (4.15) and any f ∈ C1,2(R+ × R) we have

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N. Privault

f(t,Xt) = f(0, X0) +w t

0vs∂f

∂x(s,Xs)ds+

w t

0us∂f

∂x(s,Xs)dBs

+w t

0

∂f

∂s(s,Xs)ds+

1

2

w t

0|us|2

∂2f

∂x2(s,Xs)ds. (4.16)

Proof. cf. [64], Theorem II-32.

Using the relation

w t

0df(s,Xs) = f(t,Xt)− f(0, X0),

we get

w t

0df(s,Xs) =

w t

0vs∂f

∂x(s,Xs)ds+

w t

0us∂f

∂x(s,Xs)dBs

+w t

0

∂f

∂s(s,Xs)ds+

1

2

w t

0|us|2

∂2f

∂x2(s,Xs)ds,

which allows us to rewrite Ito’s formula (4.16) in differential notation as

df(t,Xt) =∂f

∂t(t,Xt)dt+ut

∂f

∂x(t,Xt)dBt+vt

∂f

∂x(t,Xt)dt+

1

2|ut|2

∂2f

∂x2(t,Xt)dt,

(4.17)or

df(t,Xt) =∂f

∂t(t,Xt)dt+

∂f

∂x(t,Xt)dXt +

1

2|ut|2

∂2f

∂x2(t,Xt)dt.

Next, given two processes (Xt)t∈R+ and (Yt)t∈R+ written as

Xt = X0 +w t

0vsds+

w t

0usdBs, t ∈ R+,

and

Yt = Y0 +w t

0bsds+

w t

0asdBs, t ∈ R+,

the Ito formula also shows that

d(XtYt) = XtdYt + YtdXt + dXt · dYt

where the product dXt · dYt is computed according to the Ito rule

(dt)2 = 0, dtdBt = 0, (dBt)2 = dt, (4.18)

i.e.

dXt · dYt = (vtdt+ utdBt)(btdt+ atdBt)

= btvt(dt)2 + btutdtdBt + atvtdtdBt + atut(dBt)

2

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= utatdt.

Hence we have

(dXt)2 = (vtdt+ utdBt)

2

= (vt)2(dt)2 + (ut)

2(dBt)2 + 2utvtdt · dBt

= (ut)2dt,

according to the Ito multiplication table

· dt dBtdt 0 0

dBt 0 dt(4.19)

and (4.17) can also be rewriten as

df(t,Xt) =∂f

∂t(t,Xt)dt+

∂f

∂x(t,Xt)dXt +

1

2

∂2f

∂x2(t,Xt)(dXt)

2.

Taking ut = 1 and vt = 0 in (4.15) yields Xt = Bt, in which case the Itoformula (4.16) reads

f(t, Bt) = f(0, B0)+w t

0

∂f

∂s(s,Bs)ds+

w t

0

∂f

∂x(s,Bs)dBs+

1

2

w t

0

∂2f

∂x2(s,Bs)ds,

i.e. in differential notation:

df(t, Bt) =∂f

∂t(t, Bt)dt+

∂f

∂x(t, Bt)dBt +

1

2

∂2f

∂x2(t, Bt)dt. (4.20)

As another example, applying Ito’s formula (4.20) to B2t with

B2t = f(t, Bt) and f(t, x) = x2,

we get

dB2t = df(Bt)

=∂f

∂t(t, Bt)dt+

∂f

∂x(t, Bt)dBt +

1

2

∂2f

∂x2(t, Bt)dt

= 2BtdBt + dt,

since

∂f

∂t(t, x) = 0,

∂f

∂x(t, x) = 2x, and

1

2

∂2f

∂x2(t, x) = 1,

hence by integration we find

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N. Privault

B2T = B0 + 2

w T

0BsdBs +

w T

0dt

= 2w T

0BsdBs + T,

and w T

0BsdBs =

B2T

2− T

2.

We close this section with some comments on the practice of Ito’s calculus.In some finance textbooks, Ito’s formula for e.g. geometric Brownian motioncan be found written in the notation

f(T, ST ) = f(0, X0) + σw T

0St∂f

∂St(t, St)dBt + µ

w T

0St∂f

∂St(t, St)dt

+w T

0

∂f

∂t(t, St)dt+

1

2σ2

w T

0S2t

∂2f

∂S2t

(t, St)dt,

or

df(St) = σSt∂f

∂St(St)dBt + µSt

∂f

∂St(St)dt+

1

2σ2S2

t

∂2f

∂S2t

(St)dt.

The notation∂f

∂St(St) can in fact be easily misused in combination with the

fundamental theorem of classical calculus, and lead to the wrong identity

df(St) =∂f

∂St(St)dSt.

Similarly, writing

df(Bt) =df

dx(Bt)dBt +

1

2

d2f

dx2(Bt)dt

is consistent, while writing

df(Bt) =df(Bt)

dBtdBt +

1

2

d2f(Bt)

dB2t

dt

is potentially a source of confusion.

4.6 Geometric Brownian Motion

Our aim in this section is to solve the stochastic differential equation

dSt = µStdt+ σStdBt (4.21)

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that will defined the price St of a risky asset at time t, where µ ∈ R andσ > 0. This equation is rewritten in integral form as

St = S0 + µw t

0Ssds+ σ

w t

0SsdBs, t ∈ R+. (4.22)

It can be solved by applying Ito’s formula to f(St) = logSt with f(x) = log x,which shows that

d logSt = µStf′(St)dt+ σStf

′(St)dBt +1

2σ2S2

t f′′(St)dt

= µdt+ σdBt −1

2σ2dt,

hence

logSt − logS0 =w t

0d logSr

=w t

0

(µ− 1

2σ2

)dr +

w t

0σdBr

=

(µ− 1

2σ2

)t+ σBt, t ∈ R+,

and

St = S0 exp

((µ− 1

2σ2

)t+ σBt

), t ∈ R+.

The above provides a proof of the next proposition.

Proposition 4.3. The solution of (4.21) is given by

St = S0eµt+σBt−σ2t/2, t ∈ R+.

Proof. Let us provide an alternative proof by searching for a solution of theform

St = f(t, Bt)

where f(t, x) is a function to be determined. By Ito’s formula (4.20) we have

dSt = df(t, Bt) =∂f

∂t(t, Bt)dt+

∂f

∂x(t, Bt)dBt +

1

2

∂2f

∂x2(t, Bt)dt.

Comparing this expression to (4.21) and identifying the terms in dBt we get

∂f

∂x(t, Bt) = σSt,

and∂f

∂t(t, Bt) +

1

2

∂2f

∂x2(t, Bt) = µSt.

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N. Privault

Using the relation St = f(t, Bt) these two equations rewrite as

∂f

∂x(t, Bt) = σf(t, Bt),

and∂f

∂t(t, Bt) +

1

2

∂2f

∂x2(t, Bt) = µf(t, Bt).

Since Bt is a Gaussian random variable taking all possible values in R, theequations should hold for all x ∈ R, as follows:

∂f

∂x(t, x) = σf(t, x), (4.23)

and∂f

∂t(t, x) +

1

2

∂2f

∂x2(t, x) = µf(t, x). (4.24)

Letting g(t, x) = log f(t, x), the first equation (4.23) shows that

∂g

∂x(t, x) =

∂ log f

∂x(t, x) =

1

f(t, x)

∂f

∂x(t, x) = σ,

i.e.∂g

∂x(t, x) = σ,

which is solved asg(t, x) = g(t, 0) + σx,

hencef(t, x) = eg(t,0)eσx = f(t, 0)eσx.

Plugging back this expression into the second equation (4.24) yields

eσx∂f

∂t(t, 0) +

1

2σ2eσxf(t, 0) = µf(t, 0)eσx,

i.e. after division by eσx:

∂f

∂t(t, 0) =

(µ− σ2/2

)f(t, 0),

or∂g

∂t(t, 0) = µ− σ2/2,

i.e.g(t, 0) = g(0, 0) +

(µ− σ2/2

)t,

and

f(t, x) = eg(t,x)

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= eg(t,0)+σx

= eg(0,0)+σx+(µ−σ2/2)t

= f(0, 0)eσx+(µ−σ2/2)t,

henceSt = f(t, Bt) = f(0, 0)eσBt+(µ−σ2/2)t,

and the solution to (4.21) is given by

St = S0eσBt+(µ−σ2/2)t t ∈ R+.

Conversely, taking St = f(t, Bt) with f(t, x) = S0eσx−σ2t/2+µt we may

apply Ito’s formula to check that

dSt = df(t, Bt)

=∂f

∂t(t, Bt)dt+

∂f

∂x(t, Bt)dBt +

1

2

∂2f

∂x2(t, Bt)dt

=(µ− σ2/2

)S0e

σBt+(µ−σ2/2)tdt+ σS0eσBt+(µ−σ2/2)tdBt

+1

2σ2S0e

σBt+(µ−σ2/2)tdt

= µS0eσBt+(µ−σ2/2)tdt+ σS0e

σBt+(µ−σ2/2)tdBt

= µStdt+ σStdBt.

4.7 Stochastic Differential Equations

In addition to geometric Brownian motion there exists a large family ofstochastic differential equations that can be studied, although most of thetime they cannot be explicitly solved. Let now

σ : R+ × Rn → Rd ⊗ Rn

where Rd ⊗ Rn denotes the space of d× n matrices, and

b : R+ × Rn → R

satisfy the global Lipschitz condition

‖σ(t, x)− σ(t, y)‖2 + ‖b(t, x)− b(t, y)‖2 ≤ K2‖x− y‖2,

t ∈ R+, x, y ∈ Rn. Then there exists a unique strong solution to the stochasticdifferential equation

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N. Privault

Xt = X0 +w t

0σ(s,Xs)dBs +

w t

0b(s,Xs)ds, t ∈ R+,

where (Bt)t∈R+is a d-dimensional Brownian motion, see e.g. [64], Theorem V-

7.

Next we consider a few examples of stochastic differential equations thatcan be solved explicitly using Ito calculus.

Examples

1. Consider the stochastic differential equation

dXt = −αXtdt+ σdBt, X0 = x0,

with α > 0 and σ > 0.

Looking for a solution of the form

Xt = a(t)

(x0 +

w t

0b(s)dBs

)where a(·) and b(·) are deterministic functions, yields

Xt = x0e−αt + σ

w t

0e−α(t−s)dBs, t > 0,

after applying Theorem 4.1 to the Ito process x0 +r t0b(s)dBs of the form

(4.15) with ut = b(t) and v(t) = 0, and to the function f(t, x) = a(t)x.

Remark: the solution of this equation cannot be written as a functionf(t, Bt) of t and Bt as in the proof of Proposition 4.3.


dXt = tXtdt+ et2/2dBt, X0 = x0.

Looking for a solution of the form Xt = a(t)(X0 +

r t0b(s)dBs

), where

a(·) and b(·) are deterministic functions we get a′(t)/a(t) = t and

a(t)b(t) = et2/2, hence a(t) = et

2/2 and b(t) = 1, which yields Xt =

et2/2(X0 +Bt), t ∈ R+.


dYt = (2µYt + σ2)dt+ 2σ√YtdBt,

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where µ, σ > 0.

Letting Xt =√Yt we have dXt = µXtdt+ σdBt, hence

Yt =

(eµt√Y0 + σ

w t

0eµ(t−s)dBs

)2

.

Exercises

Exercise 4.1 Let (Bt)t∈R+denote a standard Brownian motion.

1. Let c > 0. Among the following processes, tell which is a standard Brow-nian motion and which is not. Justify your answer.

a. (Bc+t −Bc)t∈R+ .b. (cBt/c2)t∈R+

.c. (Bct2)t∈R+

.

2. Compute the stochastic integrals

w T

02dBt and

w T

0(2× 1[0,T/2](t) + 1(T/2,T ](t))dBt

and determine their probability laws (including mean and variance).3. Determine the probability law (including mean and variance) of the

stochastic integral w 2π

0sin(t) dBt.

4. Compute IE[BtBs] in terms of s, t ≥ 0.5. Let T > 0. Show that if f is a differentiable function with f(0) = f(T ) = 0

we have w T

0f(t)dBt = −

w T

0f ′(t)Btdt.

Hint: Apply Ito’s calculus to t 7→ f(t)Bt.

Exercise 4.2 Let f ∈ L2([0, T ]). Compute the conditional expectation

E[er T0f(s)dBs

∣∣∣Ft] , 0 ≤ t ≤ T,

where (Ft)t∈[0,T ] denotes the filtration generated by (Bt)t∈[0,T ].

Exercise 4.3 Compute the expectation

E

[exp

(β

w T

0BtdBt

)]" 89



N. Privault

for all β < 1/T . Hint: expand (BT )2 using Ito’s formula.

Exercise 4.4 Solve the ordinary differential equation df(t) = cf(t)dt and thestochastic differential equation dSt = rStdt+σStdBt, t ∈ R+, where r, σ ∈ Rare constants and (Bt)t∈R+ is a standard Brownian motion.

Exercise 4.5 Given T > 0, let (XTt )t∈[0,T ] denote the solution of the stochastic

differential equation

dXTt = σdBt −

XTt

T − tdt, t ∈ [0, T ], (4.25)

under the initial condition XT0 = 0 and σ > 0.

1. Show that

XTt = σ(T − t)

w t

0

1

T − sdBs, t ∈ [0, T ].

Hint: start by computing d(XTt /(T − t)) using Ito’s calculus.

2. Show that IE[XTt ] = 0 for all t ∈ [0, T ].

3. Show that Var[XTt ] = σ2t(T − t)/T for all t ∈ [0, T ].

4. Show that XTT = 0. The process (XT

t )t∈[0,T ] is called a Brownian bridge.

Exercise 4.6 Exponential Vasicek model. Consider a short term rate interestrate proces (rt)t∈R+

in the exponential Vasicek model:

drt = rt(η − a log rt)dt+ σrtdBt, (4.26)

where η, a, σ are positive parameters.

1. Find the solution (zt)t∈R+of the stochastic differential equation

dzt = −aztdt+ σdBt

as a function of the initial condition z0, where a and σ are positive pa-rameters.

2. Find the solution (yt)t∈R+of the stochastic differential equation

dyt = (θ − ayt)dt+ σdBt (4.27)

as a function of the initial condition y0. Hint: let zt = yt − θ/a.3. Let xt = eyt , t ∈ R+. Determine the stochastic differential equation

satisfied by (xt)t∈R+.

4. Find the solution (rt)t∈R+of (4.26) in terms of the initial condition r0.

5. Compute the mean1 E[rt] of rt, t ≥ 0.

1 You will need to use the generating function E[eX ] = eα2/2 for X ' N (0, α2).

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6. Compute the asymptotic mean limt→∞E[rt].

Exercise 4.7 Cox-Ingerson-Ross model. Consider the equation

drt = (α− βrt)dt+ σ√rtdBt (4.28)

modeling the variations of a short term interest rate process rt, where α, β, σand r0 are positive parameters.

1. Write down the equation (4.28) in integral form.2. Let u(t) = E[rt]. Show, using the integral form of (4.28), that u(t) satisfies

the differential equation

u′(t) = α− βu(t).

3. By an application of Ito’s formula to r2t , show that

dr2t = rt(2α+ σ2 − 2βrt)dt+ 2σr3/2t dBt. (4.29)

4. Using the integral form of (4.29), find a differential equation satisfied byv(t) = E[r2t ].

Exercise 4.8 Let (Bt)t∈R+denote a standard Brownian motion generating

the filtration (Ft)t∈R+.

1. Consider the Ito formula

f(Xt) = f(X0)+w t

0us∂f

∂x(Xs)dBs+

w t

0vs∂f

∂x(Xs)ds+

1

2

w t

0u2s∂2f

∂x2(Xs)ds,

(4.30)

where Xt = X0 +w t

0usdBs +

w t

0vsds.

Compute St := eXt by the Ito formula (4.30) applied to f(x) = ex andXt = σBt + νt, σ > 0, ν ∈ R.

2. Let r > 0. For which value of ν does (St)t∈R+satisfy the stochastic

differential equation

dSt = rStdt+ σStdBt ?

3. Let the process (St)t∈R+ be defined by St = S0eσBt+νt, t ∈ R+. Using the

result of Exercise 16.2, show that the conditional probability P (ST >K | St = x) is given by

P (ST > K | St = x) = Φ

(log(x/K) + ντ

σ√τ

),

where τ = T − t. Hint: use the decomposition ST = Steσ(BT−Bt)+ντ .

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N. Privault

4. Given 0 ≤ t ≤ T and σ > 0, let

X = σ(BT −Bt) and η2 = Var[X], η > 0.

What is η equal to ?

Exercise 4.9 Let (Bt)t∈R+be a standard Brownian motion generating the

information flow (Ft)t∈R+ .

1. Let 0 ≤ t ≤ T . What is the probability law of BT −Bt ?2. From the answer to Exercise 16.5, show that

IE[(BT )+ | Ft] =

√τ

2πe−

B2t

2τ +BtΦ

(Bt√τ

),

0 ≤ t ≤ T , where τ = T − t. Hint: write BT = BT −Bt +Bt.3. Let σ > 0, ν ∈ R, and Xt = σBt+νt. Compute eXt using the Ito formula

f(Xt) = f(X0)+w t

0us∂f

∂x(Xs)dBs+

w t

0vs∂f

∂x(Xs)ds+

1

2

w t

0u2s∂2f

∂x2(Xs)ds

stated here for a process Xt = X0 +w t

0usdBs +

w t

0vsds, t ∈ R+, and

applied to f(x) = ex.4. Let St = eXt , t ∈ R+, and r > 0. For which value of ν does (St)t∈R+

satisfy the stochastic differential equation

dSt = rStdt+ σStdBt ?

Exercise 4.10 From the answer to Exercise 16.4-(2), show that

IE[(β −BT )+ | Ft] =

√τ

2πe−

(β−Bt)22τ + (β −Bt)Φ

(β −Bt√

τ

), 0 ≤ t ≤ T,

where τ = T − t. Hint: write BT = BT −Bt +Bt.

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Chapter 5

The Black-Scholes PDE

In this section we review the notions of assets, self-financing portfolios, risk-neutral measures, and arbitrage in continuous time. We also derive the Black-Scholes PDE for self-financing portfolios, and we solve this equation using theheat kernel method.

5.1 Continuous-Time Market Model

Let (At)t∈R+be the riskless asset given by

dAtAt

= rdt, t ∈ R+, i.e. At = A0ert, t ∈ R+.

For t > 0, let (St)t∈R+be the price process defined as

dSt = µStdt+ σStdBt, t ∈ R+.

By Proposition 4.3 we have

St = S0 exp

(σBt +

(µ− 1

2σ2

)t

), t ∈ R+.

5.2 Self-Financing Portfolio Strategies

Let ξt and ηt denote the (possibly fractional) quantities invested at time t,respectively in the assets St and At, and let

ξt = (ηt, ξt), St = (At, St), t ∈ R+,

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N. Privault

denote the associated portfolio and asset price processes. The value of theportfolio Vt at time t is given by

Vt = ξt · St = ηtAt + ξtSt, t ∈ R+. (5.1)

The portfolio strategy (ηt, ξt)t∈R+ is self-financing if the portfolio value re-mains constant after updating the portfolio from (ηt, ξt) to (ηt+dt, ξt+dt), i.e.

ξt · St+dt = At+dtηt+St+dtξt = At+dtηt+dt+St+dtξt+dt = ξt+dt · St+dt, (5.2)

which is the continuous-time equivalent of the self-financing condition alreadyencountered in the discrete setting of Chapter 2, see Definition 2.1. A majordifference with the discrete-time case of Definition 2.1, however, is that thecontinuous-time differentials dSt and dξt do not make pathwise sense as thestochastic integral is defined by an L2 limit, cf. Proposition 4.2, or by con-vergence in probability.

St+dt St+dt St+2dtSt

t+ 2dtt t+ dt t+ dt

ξt+2dtξt ξt ξt+dt

ξt+dt · St+2dtξt · St ξt · St+dt ξt+dt · St+dt=Portfolio value

Asset value

Time scale

Portfolio allocation

- -

Fig. 5.1: Illustration of the self-financing condition (5.2).

Equivalently, Condition (5.2) can be rewritten as

At+dtdηt + St+dtdξt = 0, (5.3)

orAt+dt(ηt+dt − ηt) = −St+dt(ξt+dt − ξt),

i.e. when one sells a quantity −dξt > 0 of the risky asset St+dt between thetime periods [t, t+ dt] and [t+ dt, t+ 2dt] for a total amount −St+dtdξt, oneshould entirely use this income to buy a quantity dηt > 0 of the riskless assetfor an amount At+dtdηt > 0.

Similarly, if one sells a (possibly fractional) quantity −dηt > 0 of theriskless asset At+dt between the time periods [t, t + dt] and [t + dt, t + 2dt]for a total amount −At+dtdηt, one should entirely use this income to buy aquantity dξt > 0 of the risky asset for an amount St+dtdξt > 0, i.e.

St+dtdξt = −At+dtdηt,

or

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At+dt(ηt+dt − ηt) + St(ξt+dt − ξt) + (St+dt − St)(ξt+dt − ξt) = 0,

which rewrites asAtdηt + Stdξt + dSt · dξt = 0

in differential notation, since

dAt · dηt = (At+dt −At)(ηt+dt − ηt) = dAt · dηt = rAtdt · dηt ' 0

in the sense of the Ito calculus. In practice we will use the following definitionfor the self-financing portfolio property.

Definition 5.1. The portfolio Vt is said to be self-financing if

dVt = ηtdAt + ξtdSt. (5.4)

Again we check that by Ito’s calculus we have

dVt = ηtdAt + ξtdSt +Atdηt + Stdξt + dηt · dAt + dξt · dSt= ηtdAt + ξtdSt +Atdηt + Stdξt + dξt · dSt,

since dηt · dAt = rAtdt · dηt = 0, hence Condition (5.4) rewrites as

Atdηt + Stdξt + dξt · dSt = 0,

which is equivalent to (5.2) and (5.3).

LetVt = e−rtVt and Xt = e−rtSt

respectively denote the discounted portfolio value and discounted risky assetprices at time t ≥ 0. We have

dXt = d(e−rtSt)

= −re−rtStdt+ e−rtdSt

= −re−rtStdt+ µe−rtStdt+ σe−rtStdBt

= Xt((µ− r)dt+ σdBt).

In the next lemma we show that when a portfolio is self-financing, its dis-counted value is a gain process given by the sum of discounted profits andlosses (number of risky assets ξt times discounted price variation dXt) overtime.

Lemma 5.1. Let (ηt, ξt)t∈R+be a portfolio strategy with value

Vt = ηtAt + ξtSt, t ∈ R+.

The following statements are equivalent:

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N. Privault

i) the portfolio strategy (ηt, ξt)t∈R+is self-financing,

ii) we have

Vt = V0 +w t

0ξudXu, t ∈ R+. (5.5)

Proof. Assuming that (i) holds, the self-financing condition shows that

dVt = ηtdAt + ξtdSt

= rηtAtdt+ µξtStdt+ σξtStdBt

= rVtdt+ (µ− r)ξtStdt+ σξtStdBt t ∈ R+,

hence

dVt = d(e−rtVt

)= −re−rtVtdt+ e−rtdVt

= (µ− r)ξte−rtStdt+ σξte−rtStdBt

= (µ− r)ξtXtdt+ σξtXtdBt

= ξtdXt, t ∈ R+,

i.e. (5.5) holds by integrating on both sides as

Vt − V0 =w t

0dVu =

w t

0ξudXu, t ∈ R+.

Conversely, if (5.5) is satisfied we have

dVt = d(ertVt)

= rertVtdt+ ertdVt

= rertVtdt+ ertξtdXt

= rVtdt+ ertξtdXt

= rVtdt+ ertξtXt((µ− r)dt+ σdBt)

= rVtdt+ ξtSt((µ− r)dt+ σdBt)

= rηtAtdt+ µξtStdt+ σξtStdBt

= ηtdAt + ξtdSt,

hence the portfolio is self-financing according to Definition 5.1.

As a consequence of (5.5), the hedging problem of a claim C with maturiryT is reduced to that of finding the representation of the discounted claimC = e−rTC as a stochastic integral:

C = V0 +w T

0ξudXu.

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Note also that (5.5) shows that the value of a self-financing portfolio can alsobe written as

Vt = ertV0 + (µ− r)w t

0er(t−u)ξuSudu+ σ

w t

0er(t−u)ξuSudBu, t ∈ R+.

(5.6)

5.3 Arbitrage and Risk-Neutral Measures

In continuous-time, the definition of arbitrage follows the lines of its analogsin the discrete and two-step models. In the sequel we will only consider ad-missible portfolio strategies whose total value Vt remains non-negative forall times t ∈ [0, T ].

Definition 5.2. A portfolio strategy (ξt, ηt)t∈[0,T ] constitutes an arbitrageopportunity if all three following conditions are satisfied:

i) V0 ≤ 0,ii) VT ≥ 0,

iii) P(VT > 0) > 0.

Roughly speaking, (ii) means that the investor wants no loss, (iii) meansthat he wishes to sometimes make a strictly positive gain, and (i) means thathe starts with zero capital or even with a debt.

Next we turn to the definition of risk-neutral measures in continuous time.Recall that the filtration (Ft)t∈R+

is generated by Brownian motion (Bt)t∈R+,

i.e.Ft = σ(Bu : 0 ≤ u ≤ t), t ∈ R+.

Definition 5.3. A probability measure P∗ on Ω is called a risk-neutral mea-sure if it satisfies

IE∗[St|Fu] = er(t−u)Su, 0 ≤ u ≤ t, (5.7)

where IE∗ denotes the expectation under P∗.From the relation

At = er(t−u)Au, 0 ≤ u ≤ t,we interpret (5.7) by saying that the expected return of the risky asset Stunder P∗ equals the return of the riskless asset At. The discounted price Xt

of the risky asset is defined by

Xt = e−rtSt =St

At/A0, t ∈ R+,

i.e. At/A0 plays the role of a numeraire in the sense of Chapter 10.

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N. Privault

Definition 5.4. A continuous time process (Zt)t∈R+of integrable random

variables is a martingale with respect to the filtration (Ft)t∈R+if

IE[Zt|Fs] = Zs, 0 ≤ s ≤ t.

Note that when (Zt)t∈R+ is a martingale, Zt is in particular Ft-measurablefor all t ∈ R+.

As in the discrete case, the notion of martingale can be used to characterizerisk-neutral measures.

Proposition 5.1. The measure P∗ is risk-neutral if and only if the discountedprice process (Xt)t∈R+

is a martingale under P∗.

Proof. This follows from the equalities

IE∗[Xt|Fu] = IE∗[e−rtSt|Fu]

= e−rt IE∗[St|Fu]

= e−rter(t−u)Su

= e−ruSu

= Xu, 0 ≤ u ≤ t.

As in the discrete time case, P∗ would be called a risk-premium measure if itsatisfied

IE∗[St|Fu] > er(t−u)Su, 0 ≤ u ≤ t,meaning that by taking risks in buying St, one could make an expected returnhigher than that of

At = er(t−u)Au, 0 ≤ u ≤ t.

Next we note that the first fundamental theorem of mathematical financealso holds in continuous time, and can be used to check for the existence ofarbitrage opportunities.

Theorem 5.1. A market is without arbitrage opportunity if and only if itadmits at least one risk-neutral measure.

Proof. cf. Chapter VII-4a of [70].


Definition 5.5. A contingent claim with payoff C is said to be attainable ifthere exists a portfolio strategy (ηt, ξt)t∈[0,T ] such that

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C = VT .

In this case the price of the claim at time t will be equal to the value Vt ofany self-financing portfolio hedging C.

Definition 5.6. A market model is said to be complete if every contingentclaim C is attainable.

The next result is a continuous-time restatement of the second fundamentaltheorem of mathematical finance.


Proof. cf. Chapter VII-4a of [70].

In the Black-Scholes model one can show the existence of a unique risk-neutralmeasure, hence the model is without arbitrage and complete.

5.5 The Black-Scholes PDE

We start by deriving the Black-Scholes partial differential equation (PDE)for the price of a self-financing portfolio.

Proposition 5.2. Let (ηt, ξt)t∈R+be a portfolio strategy such that

(i) (ηt, ξt)t∈R+ is self-financing,

(ii) the value Vt := ηtAt + ξtSt, t ∈ R+, takes the form

Vt = g(t, St), t ∈ R+,

for some g ∈ C1,2((0,∞)× (0,∞)).

Then the function g(t, x) satisfies the Black-Scholes PDE

rg(t, x) =∂g

∂t(t, x) + rx

∂g

∂x(t, x) +

1

2x2σ2 ∂

2g

∂x2(t, x), t, x > 0,

and ξt is given by

ξt =∂g

∂x(t, St), t ∈ R+. (5.8)

Proof. First, note that the self-financing condition implies

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N. Privault


= rηtAtdt+ µξtStdt+ σξtStdBt (5.9)

= rVtdt+ (µ− r)ξtStdt+ σξtStdBt,

t ∈ R+.

We now rewrite (4.22) under the form of an Ito process

St = S0 +w t

0vsds+

w t

0usdBs, t ∈ R+,

as in (4.15), by taking

ut = σSt, and vt = µSt, t ∈ R+.

The application of Ito’s formula Theorem 4.1 to g(t, x) leads to

dg(t, St) = g(0, S0) + vt∂g

∂x(t, St)dt+ ut

∂g

∂x(t, St)dBt

+∂g

∂t(t, St)dt+

1

2|ut|2

∂2g

∂x2(t, St)dt

=∂g

∂t(t, St)dt+ µSt

∂g

∂x(t, St)dt+

1

2S2t σ

2 ∂2g

∂x2(t, St)dt+ σSt

∂g

∂x(t, St)dBt.

(5.10)

By respective identification of the terms in dBt and dt in (5.9) and (5.10) weget

rηtAtdt+ µξtStdt =∂g

∂t(t, St)dt+ µSt

∂g

∂x(t, St)dt+

1

2S2t σ

2 ∂2g

∂x2(t, St)dt,

ξtStσdBt = Stσ∂g

∂x(t, St)dBt,

hence rVt − rξtSt =

∂g

∂t(t, St) +

1

2S2t σ

2 ∂2g

∂x2(t, St),

ξt =∂g

∂x(t, St),

i.e. rg(t, St) =

∂g

∂t(t, St) + rSt

∂g

∂x(t, St) +

1

2S2t σ

2 ∂2g

∂x2(t, St),

ξt =∂g

∂x(t, St).

(5.11)

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The derivative giving ξt in (5.8) is called the Delta of the option price.

The amount invested on the riskless asset is

ηtAt = Vt − ξtSt = g(t, St)− St∂g

∂x(t, St),

and ηt is given by

ηt =Vt − ξtSt

At

=g(t, St)− St

∂g

∂x(t, St)

At

=g(t, St)− St

∂g

∂x(t, St)

A0ert.

In the next proposition we add a terminal condition g(T, x) = f(x) to theBlack-Scholes PDE in order to hedge claim C of the form C = f(ST ).

Proposition 5.3. The price of any self-financing portfolio of the form Vt =g(t, St) hedging an option with payoff C = f(ST ) satisfies the Black-ScholesPDE

rg(t, x) =

∂g

∂t(t, x) + rx

∂g

∂x(t, x) +

1

2x2σ2 ∂

2g

∂x2(t, x),

g(T, x) = f(x).

The Black-Scholes PDE admits an easy solution when C = ST − K is the(linear) payoff of a forward contract, i.e. f(x) = x−K. In this case we find

g(t, x) = x−Ke−r(T−t), t, x > 0,

and the Delta of the option price is given by

ξt =∂g

∂x(t, St) = 1, 0 ≤ t ≤ T,

cf. Exercise 5.3. The forward contract can be realized by the option issuer asfollows:

a) At time t, receive the option premium St − e−r(T−t)K from the optionbuyer.

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N. Privault

b) Borrow e−r(T−t)K from the bank, to be refunded at maturity.c) Buy the risky asset using the amount St − e−r(T−t)K + e−r(T−t)K = St.d) Hold the risky asset until maturity (do nothing, constant portfolio strat-

egy).e) At maturity T , hand in the asset to the option holder, who gives the price

K in exchange.f) Use the amount K = er(T−t)e−r(T−t)K to refund the bank of the sume−r(T−t)K borrowed at time t.

Recall that in the case of a European call option with strike K the payofffunction is given by f(x) = (x−K)+ and the Black-Scholes PDE reads

rgc(t, x) =∂gc∂t

(t, x) + rx∂gc∂x

(t, x) +1

2x2σ2 ∂

2gc∂x2

(t, x)

gc(T, x) = (x−K)+.

In the next sections we will prove that the solution of this PDE is given bythe Black-Scholes formula

gc(t, x) = BS(K,x, σ, r, T − t) = xΦ(d+)−Ke−r(T−t)Φ(d−), (5.12)

cf. Proposition 5.7 below, where

Φ(x) =1√2π

w x

−∞e−y

2/2dy, x ∈ R,

denotes the standard Gaussian distribution function and

d+ =log(x/K) + (r + σ2/2)(T − t)

σ√T − t , d− =

log(x/K) + (r − σ2/2)(T − t)σ√T − t ,

withd+ = d− + σ

√T − t.

One checks easily that when t = T ,

d+ = d− =

+∞, x > K,

−∞, x < K,

which allows one to recover the boundary condition

gc(T, x) =

xΦ(+∞)−KΦ(+∞) = x−K, x > K

xΦ(−∞)−KΦ(−∞) = 0, x < K

= (x−K)+

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at t = T .

Figure 5.2 presents the graph of the solution

(t, x) 7−→ gc(t, x) = xΦ(d+)−Ke−r(T−t)Φ(d−)

of the Black-Scholes PDE for a call option.

Black-Scholes call price

60 70 80 90 100 110 120 130 140 150

underlying HK$

0 1 2 3 4 5 6 7 8 9 10Time to maturity T-t

0

10

20

30

40

50

60

70

80

Fig. 5.2: Graph of the Black-Scholes call price function with strike K = 100.

In Figure 5.3 we consider the stock price of HSBC Holdings (0005.HK) overone year:

Fig. 5.3: Graph of the stock price of HSBC Holdings.

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N. Privault

Consider a call option issued by Societe Generale on 31 December 2008 withstrike K=$63.704, maturity T = October 05, 2009, and an entitlement ratio of100, meaning that one option contract is divided into 100 warrants, cf. page6. The next graph gives the time evolution of the Black-Scholes portfolioprice

t 7−→ gc(t, St)

driven by the market price t 7−→ St of the underlying risky asset as given inFigure 5.3, in which the number of days is counted from the origin and notfrom maturity.

40 50 60 70 80 90 100 0 50

100 150

200

0

5

10

15

20

25

30

35

40

Black-Scholes call priceOption price path

underlying HK$ Time in days

Fig. 5.4: Path of the Black-Scholes price for a call option on HSBC.

The next proposition is proved by direct differentiation of the Black-Scholesfunction, and will be recovered later using a probabilistic argument in Propo-sition 6.7 below.

Proposition 5.4. The Black-Scholes Delta of a European call option is givenby

ξt = Φ(d+).

Proof. We have

∂g

∂x(x, t) = (5.13)

∂

∂x

(xΦ

(log(x/K) + (r + σ2/2)(T − t)

σ√T − t

)−KΦ

(log(x/K) + (r − σ2/2)(T − t)

σ√T − t

))= x

∂

∂xΦ

(log(x/K) + (r + σ2/2)(T − t)

σ√T − t

)

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−K ∂

∂xΦ

(log(x/K) + (r − σ2/2)(T − t)

σ√T − t

)+Φ

(log(x/K) + (r − σ2/2)(T − t)

σ√T − t

)=

1√2πσ√T − t

exp

(−1

2

(log(x/K) + (r + σ2/2)(T − t)

σ√T − t

)2)

− K√2πσx

√T − t

exp

(−1

2

(log(x/K) + (r − σ2/2)(T − t)

σ√T − t

)2)

+Φ

(log(x/K) + (r − σ2/2)(T − t)

σ√T − t

)= Φ

(log(x/K) + (r − σ2/2)(T − t)

σ√T − t

).

As a consequence of Proposition 5.4, in the Black-Scholes model the amountinvested on the risky asset is

Stξt = StΦ(d+) = StΦ

(log(St/K) + (r + σ2/2)(T − t)

σ√T − t

)≥ 0,

which is always positive, i.e. there is no short-selling, and the amount investedon the riskless asset is

ηtAt = −KA0e−r(T−t)Φ

(log(St/K) + (r − σ2/2)(T − t)

σ√T − t

)≤ 0,

which is always negative, i.e. we are constantly borrowing money.Similarly, in the case of a European put option with strike K the payofffunction is given by f(x) = (K − x)+ and the Black-Scholes PDE reads

rgp(t, x) =∂gp∂t

(t, x) + rx∂gp∂x

(t, x) +1

2x2σ2 ∂

2gp∂x2

(t, x),

gp(T, x) = (K − x)+,

with explicit solution

gp(t, x) = Ke−r(T−t)Φ(−d−)− xΦ(−d+),

as illustrated in the next graph.Note that the call-put parity relation

g(t, St) = gc(t, St)− gp(t, St), 0 ≤ t ≤ T,

is satisfied here.

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N. Privault

-60

-40

-20

0

20

40

60

80

100

0 50 100 150 200

HK$

K

Black-Scholes priceRisky investment

Riskless investmentUnderlying

Fig. 5.5: Time evolution of the hedging portfolio for a call option on HSBC.

Black-Scholes put price

90 95 100 105 110 115 120underlying HK$

0 1 2 3 4 5 6 7 8 9 10

Time to maturity T-t

0

2

4

6

8

10

12

14

Fig. 5.6: Graph of the Black-Scholes put price function with strike K = 100.

For one more example we consider a put option issued by BNP Paribas on04 November 2008 with strike K=$77.667, maturity T = October 05, 2009,and entitlement ratio 92.593, cf. page 6. In the next Figure 5.7 the numberof days is counted from the origin and not from maturity.

In the case of a Black-Scholes put option the Delta is given by

ξt = −Φ(−d+),

and the amount invested on the risky asset is

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40 50 60 70 80 90 100

0 50 100 150 200

0 5

10 15 20 25 30 35 40 45

Black-Scholes put priceOption price path

underlying HK$Time in days

Fig. 5.7: Path of the Black-Scholes price for a put option on HSBC.

StΦ(d+) = −StΦ(− log(St/K) + (r + σ2/2)(T − t)

σ√T − t

)≤ 0,

i.e. there is always short-selling, and the amount invested on the riskless assetis

Ke−r(T−t)Φ

(− log(St/K) + (r − σ2/2)(T − t)

σ√T − t

)≥ 0,

which is always positive, i.e. we are constantly investing on the riskless asset.

-60

-40

-20

0

20

40

60

80

100

0 50 100 150 200

HK$

K

Black-Scholes priceRisky investment

Riskless investmentUnderlying

Fig. 5.8: Time evolution of the hedging portfolio for a put option on HSBC.

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N. Privault

5.6 The Heat Equation

In this section we study the heat equation which is used to model the diffusionof heat in solids. We refer the reader to [75] for a complete treatment of thistopic.

In Section 5.7 this equation will be shown to be equivalent to the Black-Scholes PDE after a change of variables. In particular this will lead to theexplicit solution of the Black-Scholes PDE.

Proposition 5.5. The heat equation∂g

∂t(t, y) =

1

2

∂2g

∂y2(t, y)

g(0, y) = ψ(y)

(5.14)

with initial condition ψ(y) has solution

g(t, y) =w∞−∞

ψ(z)e−(y−z)2

2tdz√2πt

. (5.15)

Proof. We have

∂g

∂t(t, y) =

∂

∂t

w∞−∞

ψ(z)e−(y−z)2

2tdz√2πt

=w∞−∞

ψ(z)∂

∂t

(e−

(y−z)22t√

2πt

)dz

=1

2

w∞−∞

ψ(z)

((y − z)2

t2− 1

t

)e−

(y−z)22t

dz√2πt

=1

2

w∞−∞

ψ(z)∂2

∂z2e−

(y−z)22t

dz√2πt

=1

2

w∞−∞

ψ(z)∂2

∂y2e−

(y−z)22t

dz√2πt

=1

2

∂2

∂y2

w∞−∞

ψ(z)e−(y−z)2

2tdz√2πt

=1

2

∂2g

∂y2(t, y).

On the other hand it can be checked that at time t = 0,

limt→0

w∞−∞

ψ(z)e−(y−z)2

2tdz√2πt

= limt→0

w∞−∞

ψ(y + z)e−z2

2tdz√2πt

= ψ(y),

y ∈ R.

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Let us provide a second proof of Proposition 5.5 using stochastic calculus andBrownian motion. Note that under the change of variable x = z − y we have

g(t, y) =w∞−∞

ψ(z)e−(y−z)2

2tdz√2πt

=w∞−∞

ψ(y + x)e−x2

2tdx√2πt

= IE[ψ(y +Bt)],

where (Bt)t∈R+is a standard Brownian motion. Applying Ito’s formula we

have

IE[ψ(y +Bt)] = ψ(y) + IE

[w t

0ψ′(y +Bs)dBs

]+

1

2IE

[w t

0ψ′′(y +Bs)ds

]= ψ(y) +

1

2

w t

0IE [ψ′′(y +Bs)] ds

= ψ(y) +1

2

w t

0

∂2

∂y2IE [ψ(y +Bs)] ds,

since the expectation of the stochastic integral is zero. Hence

∂g

∂t(t, y) =

∂

∂tIE[ψ(y +Bt)]

=1

2

∂2

∂y2IE [ψ(y +Bt)]

=1

2

∂2g

∂y2(t, y).

Concerning the initial condition we check that

g(0, y) = IE[ψ(y +B0)] = IE[ψ(y)] = ψ(y).

5.7 Solution of the Black-Scholes PDE

In this section we will solve the Black-Scholes PDE by the kernel method ofSection 5.6 and a change of variables.

Proposition 5.6. Assume that f(t, x) solves the Black-Scholes PDErf(t, x) =

∂f

∂t(t, x) + rx

∂f

∂x(t, x) +

1

2x2σ2 ∂

2f

∂x2(t, x),

f(T, x) = (x−K)+,

(5.16)

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N. Privault

with terminal condition h(x) = (x −K)+. Then the function g(t, y) definedby

g(t, y) = ertf(T − t, eσy+(σ2/2−r)t

)(5.17)

solves the heat equation (5.14), i.e.∂g

∂t(t, y) =

1

2

∂2g

∂y2(t, y)

g(0, y) = h(eσy),

with initial conditiong(0, y) = h(eσy). (5.18)

Proof. Letting s = T − t and x = eσy+(σ2/2−r)t we have

∂g

∂t(t, y) = rertf(T − t, eσy+(σ2/2−r)t)− ert ∂f

∂s(T − t, eσy+(σ2/2−r)t)

+

(σ2

2− r)erteσy+(σ2/2−r)t ∂f

∂x(T − t, eσy+(σ2/2−r)t)

= rertf(T − t, x)− ert ∂f∂s

(T − t, x) +

(σ2

2− r)ertx

∂f

∂x(T − t, x)

=1

2ertx2σ2 ∂

2f

∂x2(T − t, x) +

σ2

2ertx

∂f

∂x(T − t, x), (5.19)

where on the last step we used the Black-Scholes PDE.

On the other hand we have

∂g

∂y(t, y) = σerteσy+(σ2/2−r)t ∂f

∂x(T − t, eσy+(σ2/2−r)t)

and

1

2

∂g2

∂y2(t, y) =

σ2

2erteσy+(σ2/−r)t/2 ∂f

∂x(T − t, eσy+(σ2/2−r)t)

+σ2

2erte2σy+2(σ2/2−r)t ∂

2f

∂x2(T − t, eσy+(σ2/2−r)t)

=σ2

2ertx

∂f

∂x(T − t, x) +

σ2

2ertx2

∂2f

∂x2(T − t, x),

which, in view of (5.19), means that g(t, x) satisfies the heat equation (5.14)with initial condition

g(0, y) = f(T, eσy) = h(eσy).

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In the next proposition we recover the Black-Scholes formula by solving thePDE (5.16). The Black-Scholes will also be recovered by probabilistic argu-ments and the computation of an expectation in Proposition 6.4.

Proposition 5.7. When h(x) = (x−K)+, the solution of the Black-ScholesPDE (5.16) is given by

f(t, x) = xΦ(d+)−Ke−r(T−t)Φ(d−),

where

Φ(x) =1√2π

w x

−∞e−y

2/2dy, x ∈ R,

and

d+ =log(x/K) + (r + σ2/2)(T − t)

σ√T − t

, d− =log(x/K) + (r − σ2/2)(T − t)

σ√T − t

.

Proof. By inversion of (5.17) with s = T − t and x = eσy+(σ2/2−r)t we get

f(s, x) = e−r(T−s)g

(T − s, −(σ

2

2 − r)(T − s) + log x

σ

).

Hence using the solution (5.15) and Relation (5.18) we get

f(t, x) = e−r(T−t)g

(T − t, −(σ

2

2 − r)(T − t) + log x

σ

)

= e−r(T−t)w∞−∞

ψ

(−(σ

2

2 − r)(T − t) + log x

σ+ z

)e−

z2

2(T−t)dz√

2π(T − t)

= e−r(T−t)w∞−∞

h(xeσz−(σ

2/2−r)(T−t))e−

z2

2(T−t)dz√

2π(T − t)

= e−r(T−t)w∞−∞

(xeσz−(σ

2/2−r)(T−t) −K)+

e−z2

2(T−t)dz√

2π(T − t)

= e−r(T−t)w∞

(−r+σ2/2)(T−t)+log(K/x)σ

(xeσz−(σ

2/2−r)(T−t) −K)e−

z2

2(T−t)dz√

2π(T − t)

= xe−r(T−t)w∞−d−

√T−t

eσz−(σ2/2−r)(T−t)e−

z2

2(T−t)dz√

2π(T − t)

−Ke−r(T−t)w∞−d−

√T−t

e−z2

2(T−t)dz√

2π(T − t)

= xw∞−d−

√T−t

eσz−σ2(T−t)/2− z2

2(T−t)dz√

2π(T − t)


√T−t

e−z2

2(T−t)dz√

2π(T − t)

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N. Privault

= xw∞−d−

√T−t

e−1

2(T−t) (z−σ(T−t))2 dz√

2π(T − t)


√T−t

e−z2

2(T−t)dz√

2π(T − t)

= xw∞−d−

√T−t−σ(T−t)

e−z2

2(T−t)dz√

2π(T − t)


√T−t

e−z2

2(T−t)dz√

2π(T − t)

= xw∞−d−−σ

√T−t

e−z2/2 dz√

2π−Ke−r(T−t)

w∞d−e−z

2/2 dz√2π

= x (1− Φ (−d+))−Ke−r(T−t) (1− Φ (−d−))

= xΦ (d+)−Ke−r(T−t)Φ (d−) ,

where we used the relation

1− Φ(a) = Φ(−a), a ∈ R.

Exercises

Exercise 5.1

1. Solve the stochastic differential equation

dSt = αStdt+ σdBt

in terms of α, σ > 0, and the initial condition S0.2. For which values αM of α is the discounted price process St = e−rtSt,t ∈ [0, T ], a martingale under P ?

3. Compute the arbitrage price C(t, St) = e−r(T−t) IE[exp(ST ) | Ft] at timet ∈ [0, T ] of the contingent claim of exp(ST ), with α = αM .

4. Explicitly compute the strategy (ζt, ηt)t∈[0,T ] that hedges the contingentclaim exp(ST ).

Exercise 5.2 In the Black-Scholes model, the price at time t of a Europeanclaim on the underlying asset St, with strike price K, maturity T , interestrate r and volatility σ is given by the Black-Scholes formula as

f(t, St) = StΦ(d+)−Ke−r(T−t)Φ(d−),

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where

d− =(r − 1

2σ2)(T − t) + log(St/K)

σ√T − t and d+ = d− + σ

√T − t.

Recall that∂f

∂x(t, St) = Φ(d+),

cf. Proposition 5.4.

On December 18, 2007, a call warrant has been issued by Fortis Bank on thestock price S of the MTR Corporation with maturity T = 23/12/2008, StrikeK = HK$ 36.08 and Entitlement ratio=10.

1. Using the values of the Gaussian cumulative distribution function, com-pute the Black-Scholes price of the corresponding call option at timet =November 07, 2008 with St = HK$ 17.200, assuming a volatility σ =90% = 0.90 and an annual risk-free interest rate r = 4.377% = 0.04377,

2. Still using the values of the Gaussian cumulative distribution function,compute the quantity of the risky asset required in your portfolio at timet =November 07, 2008 in order to hedge one such option at maturityT = 23/12/2008.

3. Figure 1 represents the Black-Scholes price of the call option as a functionof σ ∈ [0.5, 1.5] = [50%, 150%].

0

0.1

0.2

0.3

0.4

0.5

0.6

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

HK$

sigma

Black-Scholes price

Fig. 5.9: Option price as a function of the volatility σ.

Knowing that the closing price of the warrant on November 07, 2008 wasHK$ 0.023, which value can you infer for the implied volatility σ at thisdate ?

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N. Privault

Exercise 5.3 Forward contracts. Recall that the price πt(C) of a claim C =h(ST ) of maturity T can be written as πt(C) = g(t, St), where the functiong(t, x) satisfies the Black-Scholes PDE

rg(t, x) =∂g

∂t(t, x) + rx

∂g

∂x(t, x) +

1

2x2σ2 ∂

2g

∂x2(t, x),

g(T, x) = h(x), (1)

with terminal condition g(T, x) = h(x).

1. Assume that C is a forward contract with payoff

C = ST −K,

at time T . Find the function h(x) in (1).2. Find the solution g(t, x) of the above PDE and compute the price πt(C)

at time t ∈ [0, T ].Hint: search for a solution of the form g(t, x) = x− α(t) where α(t) is afunction of t to be determined.

3. Compute the quantity

ξt =∂g

∂x(t, St)

of risky assets in a self-financing portfolio hedging C.

Exercise 5.4 Forward contracts revisited. Consider a risky asset whose priceSt is given by St = S0e

σBt+rt−σ2t/2, t ∈ R+, where (Bt)t∈R+ is a standardBrownian motion. Consider a forward contract with maturity T and payoffST − κ.

1. Compute the price Ct of this claim at any time t ∈ [0, T ].

2. Compute a hedging strategy for the option with payoff ST − κ.

Exercise 5.5 Computation of Greeks. Consider an option with payoff functionφ and price

C(x, T ) = e−rT IE[φ(ST )

∣∣∣S0 = x],

where φ(x) is a twice continuously differentiable (C2) function, on the under-lying (St)t∈R+

given by the stochastic differential equation

dSt = rStdt+ σ(St)dWt, ,

with S0 = x and Lipschitz coefficient σ(x).

Using the Ito formula, show that the sensitivity

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ThetaT =∂

∂T

(e−rT IE [φ(ST )]

)of the option price with respect to maturity T can be expressed as

ThetaT = −re−rT IE [φ(ST )]+e−rT IE [φ′(ST )σ(ST )]+1

2e−rT IE

[φ′′(ST )σ2(ST )

].

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Chapter 6

Martingale Approach to Pricing andHedging

In this chapter we present the probabilistic martingale approach method tothe pricing and hedging of options. In particular, this allows one to computeoption prices as the expectations of the discounted option payoffs, and todetermine the associated hedging portfolios.

6.1 Martingale Property of the Ito Integral

Recall (Definition 5.4) that an integrable process (Xt)t∈R+is said to be a

martingale with respect to the filtration (Ft)t∈R+if

IE[Xt | Fs] = Xs, 0 ≤ s ≤ t.

The following result shows that the indefinite Ito integral is a martingale withrespect to the Brownian filtration (Ft)t∈R+

. It is the continuous-time analogof the discrete-time Proposition 2.1.

Proposition 6.1. The indefinite stochastic integral(r t

0usdBs

)t∈R+

of a

square-integrable adapted process u ∈ L2ad(Ω × R+) is a martingale, i.e.:

IE

[w t

0uτdBτ

∣∣∣Fs] =w s

0uτdBτ , 0 ≤ s ≤ t.

Proposition 6.1 is a consequence of Proposition 6.2 below.

Proposition 6.2. For any u ∈ L2ad(Ω × R+) we have

IE[w∞

0usdBs

∣∣∣Ft] =w t

0usdBs, t ∈ R+.

In particular,r t0usdBs is Ft-measurable, t ∈ R+.

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N. Privault

Proof. The statement is first proved in case u is a simple predictable process,and then extended to the general case, cf. e.g. Proposition 2.5.7 in [59].

In particular, since F0 = ∅, Ω, this recover the fact that the Ito integral isa centered random variable:

IE[w∞

0usdBs

]= IE

[w∞0usdBs

∣∣∣F0

]=

w 0

0usdBs = 0.

Examples

1. Given any square-integrable random variable F ∈ L2(Ω), the process(Xt)t∈R+

defined by Xt := IE[F | Ft], t ∈ R+, is a martingale under P,as follows from the “tower property”

IE[Xt | Fs] = IE[IE[F | Ft] | Fs] = IE[F | Fs] = Xs, 0 ≤ s ≤ t, (6.1)

cf. (16.24) in appendix.2. Any integrable stochastic process (Xt)t∈R+

with centered and indepen-dent is a martingale:

IE[Xt|Fs] = IE[Xt −Xs +Xs|Fs]= IE[Xt −Xs|Fs] + IE[Xs|Fs]= IE[Xt −Xs] +Xs

= Xs, 0 ≤ s ≤ t. (6.2)

In particular, the standard Brownian motion (Bt)t∈R+is a martingale

because it has centered, independent increments. This fact can also berecovered from Proposition 6.1 since Bt can be written as

Bt =w t

0dBs, t ∈ R+.

3. The discounted asset price

Xt = X0e(µ−r)t+σBt−σ2t/2

is a martingale when µ = r. Indeed we have

IE[Xt|Fs] = IE[X0eσBt−σ2t/2|Fs]

= X0e−σ2t/2 IE[eσBt |Fs]

= X0e−σ2t/2 IE[eσ(Bt−Bs)+σBs |Fs]

= X0e−σ2t/2+σBs IE[eσ(Bt−Bs)|Fs]

= X0e−σ2t/2+σBs IE[eσ(Bt−Bs)]

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Nico

Inserted Text

increment


= X0e−σ2t/2+σBseσ

2(t−s)/2

= X0eσBs−σ2s/2

= Xs, 0 ≤ s ≤ t.

This fact can also be recovered from Proposition 6.1 since Xt satisfies theequation

dXt = σXtdBt,

i.e. it can be written as the Brownian stochastic integral

Xt = X0 + σw t

0XudBu, t ∈ R+.

4. The discounted valueVt = e−rtVt

of a self-financing portfolio is given by

Vt = V0 +w t

0ξudXu, t ∈ R+,

is a martingale when µ = r because

Vt = V0 + σw t

0ξuXudBu, t ∈ R+,

sincedXt = Xt((µ− r)dt+ σdBt).

Since the Black-Scholes theory is in fact valid for any value of the parameterµ we will look forward to including the case µ 6= r in the sequel.

6.2 Risk-neutral Measures

Recall that by definition, a risk-neutral measure is a probability measure P∗under which the discounted asset price (Xt)t∈R+

= (e−rtSt)t∈R+is a martin-

gale. From the analysis of Section 6.1 it appears that when µ = r, (Xt)t∈R+

is a martingale and P∗ = P is risk-neutral.

In this section we address the construction of a risk-neutral measure inthe general case µ 6= r and for this we will use the Girsanov theorem.

Note that the relation

dXt = Xt((µ− r)dt+ σdBt)

can be written as

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N. Privault

dXt = σXtdBt,

where

Bt :=µ− rσ

t+Bt, t ∈ R+.

Therefore the search for a risk-neutral measure can be replaced by the searchfor a probability measure P∗ under which (Bt)t∈R+ is a standard Brownianmotion.

Let us come back to the informal interpretation of Brownian motion viaits infinitesimal increments:

∆Bt = ±√dt,

with

P(∆Bt = +√dt) = P(∆Bt = −

√dt) =

1

2.

0

0.4

0.8

1.2

1.6

2

0 0.2 0.4 0.6 0.8 1

Drifted Brownian motionDrift

Fig. 6.1: Drifted Brownian path.

Clearly, given ν ∈ R, the drifted process νt + Bt is no longer a standardBrownian motion because it is not centered:

IE[νt+Bt] = νt+ IE[Bt] = νt 6= 0,

cf. Figure 6.1. This identity can be formulated in terms of infinitesimal in-crements as

IE[νdt+ dBt] =1

2(νdt+

√dt) +

1

2(νdt−

√dt) = νdt 6= 0.

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In order to make νt+Bt a centered process (i.e. a standard Brownian motion,since νt + Bt conserves all the other properties (i)-(iii) in the definition ofBrownian motion, one may change the probabilities of ups and downs, whichhave been fixed so far equal to 1/2.

That is, the problem is now to find two numbers p, q ∈ [0, 1] such thatp(νdt+√dt) + q(νdt−

√dt) = 0

p+ q = 1.

The solution to this problem is given by

p =1

2(1− ν

√dt) and q =

1

2(1 + ν

√dt).

Still considering Brownian motion as a discrete random walk with indepen-dent increments ±

√dt, the corresponding probability density will be obtained

by taking the product of the above probabilities divided by the referenceprobability 1/2N corresponding to the symmetric random walk, that is:

2N∏

0<t<T

(1

2∓ 1

2ν√dt

)

where 2N is a normalization factor and N = T/dt is the (infinitely large)number of discrete time steps. Using elementary calculus, this density can beinformally shown to converge as follows:

2N∏

0<t<T

(1

2∓ 1

2ν√dt

)=

∏0<t<T

(1∓ ν

√dt)

= exp

(log

∏0<t<T

(1∓ ν

√dt))

= exp

( ∑0<t<T

log(

1∓ ν√dt))

' exp

(ν∑

0<t<T

∓√dt− 1

2

∑0<t<T

(∓ν√dt)2

)

= exp

(ν∑

0<t<T

∓√dt− 1

2ν2

∑0<t<T

dt

)

= exp

(−νBT −

1

2ν2T

).

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N. Privault

6.3 Girsanov Theorem and Change of Measure

In this section we restate the Girsanov theorem in a more rigorous way, usingchanges of probability measures. Given Q a probability measure on Ω, thenotation

dQdP

= F

means that the probability measure Q has a density F with respect to P,where F is a non-negative random variable such that IE[F ] = 1. We alsowrite

dQ = FdP,

which is equivalent to stating that

wΩξ(ω)dQ(ω) =

wΩF (ω)ξ(ω)dP(ω),

or, under a different notation,

IEQ[ξ] = IE [Fξ] .

In addition we say that Q is equivalent to P when F > 0 with P-probabilityone.

Recall that here, Ω = C0([0, T ]) is the Wiener space and ω ∈ Ω is acontinuous function on [0, T ] starting at 0 in t = 0. Consider the probabilityQ defined by

dQ(ω) = exp

(−νBT −

1

2ν2T

)dP(ω).

Then the process νt + Bt is a standard (centered) Brownian motion underQ.

For example, the fact that νT +BT has a standard (centered) Gaussian lawunder Q can be recovered as follows:

IEQ[f(νT +BT )] =wΩf(νT +BT )dQ

=wΩf(νT +BT ) exp

(−νBT −

1

2ν2T

)dP

=w∞−∞

f(νT + x) exp

(−νx− 1

2ν2T

)e−

x2

2Tdx√2πT

=w∞−∞

f(y)e−y2

2Tdy√2πT

=wΩf(BT )dP

= IEP[f(BT )].

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The Girsanov theorem can actually be extended to shifts by adapted pro-cesses as follows, cf. e.g. [64], Theorem III-42. Section 14.6 will cover theextension of the Girsanov theorem to jump processes.

Theorem 6.1. Let (ψt)t∈[0,T ] be an adapted process satisfying the Novikovintegrability condition

IE

[exp

(1

2

w T

0|ψt|2dt

)]<∞, (6.3)

and let Q denote the probability measure defined by

dQdP

= exp

(−

w T

0ψsdBs −

1

2

w T

0ψ2sds

).

Then

Bt := Bt +w t

0ψsds, t ∈ [0, T ],

is a standard Brownian motion under Q.

When applied to

ψt :=µ− rσ

,

the Girsanov theorem shows that

Bt :=µ− rσ

t+Bt, t ∈ [0, T ], (6.4)

is a standard Brownian motion under the probability measure P∗ defined by

dP∗

dP= exp

(−µ− r

σBT −

(µ− r)22σ2

T

). (6.5)

Hence the discounted price process given by

dXt

Xt= (µ− r)dt+ σdBt = σdBt, t ∈ R+,

is a martingale under P∗, hence P∗ is a risk-neutral measure. We easily findthat P = P∗ when µ = r.

6.4 Pricing by the Martingale Method

In this section we give the expression of the Black-Scholes price using expec-tations of discounted payoffs.

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N. Privault

Recall that from the first fundamental theorem of mathematical finance,a continuous market is without arbitrage opportunities if there exists (atleast) a risk-neutral probability measure P∗ under which the discounted priceprocess

Xt := e−rtSt, t ∈ R+,

is a martingale under P∗. In addition, when the risk-neutral measure isunique, the market is said to be complete.

In case the price process (St)s∈[t,∞) satisfies the equation

dStSt

= µdt+ σdBt, t ∈ R+, S0 > 0

we have

St = S0eσBt−σ2t/2+µt, and Xt = S0e

(µ−r)t+σBt−σ2t/2, t ∈ R+,

hence from Section 6.2 the discounted price process is a martingale under theprobability measure P∗ defined by (6.5), and P∗ is a martingale measure.

We have

dXt = (µ− r)Xtdt+ σXtdBt = σXtdBt, t ∈ R+, (6.6)

hence the discounted value Vt of a self-financing portfolio is written as

Vt = V0 +w t

0ξudXu

= V0 + σw t


and becomes a martingale under P∗.

As in Chapter 3, the value Vt at time t of a self-financing portfolio strategy(ξt)t∈[0,T ] hedging an attainable claim C will be called an arbitrage price ofthe claim C at time t and denoted by πt(C), t ∈ [0, T ].

Proposition 6.3. Let (ξt, ηt)t∈[0,T ] be a portfolio strategy with price

Vt = ηtAt + ξtSt, t ∈ [0, T ],

and let C be a contingent claim, such that

(i) (ξt, ηt)t∈[0,T ] is a self-financing portfolio, and

(ii) (ξt, ηt)t∈[0,T ] hedges the claim C, i.e. we have VT = C.

Then the arbitrage price of the claim C is given by

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Vt = e−r(T−t) IE∗[C|Ft], 0 ≤ t ≤ T, (6.7)

where IE∗ denotes expectation under the risk-neutral measure P∗.

Proof. Since the portfolio strategy (ξt, ηt)t∈R+ is self-financing, by Lemma 5.1and (6.6) we have

Vt = V0 + σw t


which is a martingale under P∗ from Proposition 6.1, hence

Vt = IE∗[VT | Ft

]= e−rT IE∗[VT | Ft]= e−rT IE∗[C | Ft],

which impliesVt = ertVt = e−r(T−t) IE∗[C | Ft].

In case the value Vt of the portfolio at time t ∈ [0, T ] is a function C(t, St)of t and St, it can be computed from (6.7) as

Vt = C(t, St) = e−r(T−t) IE∗[φ(ST )|Ft], 0 ≤ t ≤ T,

and by Proposition 5.2 the function C(t, x) solves the Black-Scholes PDE:rC(t, x) =

∂C

∂t(t, x) +

1

2x2σ2 ∂

2C

∂x2(t, x) + rx

∂C

∂x(t, x)

C(T, x) = φ(x).

In the case of European options with payoff function φ(x) = (x−K)+ we re-cover the Black-Scholes formula (5.12), cf. Proposition 5.7, by a probabilisticargument.

Proposition 6.4. The price at time t of a European call option with strikeK and maturity T is given by

C(t, St) = StΦ(d+)−Ke−r(T−t)Φ(d−), t ∈ [0, T ].

Proof. The proof of Proposition 6.4 is a consequence of (6.7) and theLemma 6.1 below. Using the relation

ST = Ster(T−t)+σ(BT−Bt)−σ2(T−t)/2, t ∈ [0, T ].

By Proposition 6.3 the price of the portfolio hedging C is given by

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N. Privault

Vt = e−r(T−t) IE∗[C|Ft]= e−r(T−t) IE∗[(ST −K)+|Ft]= e−r(T−t) IE∗[(Ste

r(T−t)+σ(BT−Bt)−σ2(T−t)/2 −K)+|Ft]= e−r(T−t) IE∗[(xer(T−t)+σ(BT−Bt)−σ

2(T−t)/2 −K)+]x=St= e−r(T−t) IE∗[(em(x)+X −K)+]x=St , 0 ≤ t ≤ T,

wherem(x) = r(T − t)− σ2(T − t)/2 + log x

and X = σ(BT − Bt) is a centered Gaussian random variable with variance

Var [X] = Var [σ(BT − Bt)] = σ2Var [BT − Bt] = σ2(T − t)

under P∗. Hence by Lemma 6.1 below we have

Vt = e−r(T−t) IE∗[(em(x)+X −K)+]x=St

= e−r(T−t)em(St)+σ2(T−t)/2Φ(v + (m(St)− logK)/v)

−Ke−r(T−t)Φ((m(St)− logK)/v)

= StΦ(v + (m(St)− logK)/v)−Ke−r(T−t)Φ((m(St)− logK)/v)

= StΦ(d+)−Ke−r(T−t)Φ(d−),

0 ≤ t ≤ T .

Lemma 6.1. Let X be a centered Gaussian random variable with variancev2. We have

IE[(em+X −K)+] = em+ v2

2 Φ(v + (m− logK)/v)−KΦ((m− logK)/v).

Proof. We have

IE[(em+X −K)+] =1√

2πv2

w∞−∞

(em+x −K)+e−x2

2v2 dx

=1√

2πv2

w∞−m+logK

(em+x −K)e−x2

2v2 dx

=em√2πv2

w∞−m+logK

ex−x2

2v2 dx− K√2πv2

w∞−m+logK

e−x2

2v2 dx

=em+ v2

2√2πv2

w∞−m+logK

e−(v2−x)2

2v2 dx− K√2π

w∞(−m+logK)/v

e−x2/2dx

=em+ v2

2√2πv2

w∞−v2−m+logK

e−x2

2v2 dx−KΦ((m− logK)/v)

= em+ v2

2 Φ(v + (m− logK)/v)−KΦ((m− logK)/v).

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Denoting byP (t, St) = e−r(T−t) IE∗[(K − ST )+|Ft]

the price of the put option with strike K and maturity T , we check fromProposition 6.3 that

C(t, St)− P (t, St)

= e−r(T−t) IE∗[(ST −K)+|Ft]− e−r(T−t) IE∗[(K − ST )+|Ft]= e−r(T−t) IE∗[(ST −K)+ − (K − ST )+|Ft]= e−r(T−t) IE∗[ST −K|Ft]= St − e−r(T−t)K.

This relation is called the put-call parity, and it shows that

P (t, St) = C(t, St)− St + e−r(T−t)K

= StΦ(d+) + e−r(T−t)K − St − e−r(T−t)KΦ(d−)

= −St(1− Φ(d+)) + e−r(T−t)K(1− Φ(d−))

= −StΦ(−d+) + e−r(T−t)KΦ(−d−).

6.5 Hedging Strategies

In the next proposition we compute a self-financing hedging strategy leadingto an arbitrary square-integrable random variable C admitting a stochasticintegral representation formula of the form

C = IE∗[C] +w T

0ζtdBt, (6.8)

where (ζt)t∈[0,t] is a square-integrable adapted process. Consequently, themathematical problem of finding the predictable representation (6.8) of agiven random variable has important applications in finance. For example wehave

B2T = T + 2

w T

0BtdBt.

Recall that the risky asset follows the equation

dStSt

= µdt+ σdBt, t ∈ R+, S0 > 0,

and the discounted asset price satisfies

dXt = σXtdBt, t ∈ R+, X0 = S0 > 0,

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N. Privault

where (Bt)t∈R+is a standard Brownian motion under the risk-neutral prob-

ability measure P∗.

The following proposition applies to arbitrary square-integrable payofffunctions, i.e. it covers exotic and path-dependent options.

Proposition 6.5. Consider a random payoff C ∈ L2(Ω) such that (6.8)holds, and let

ξt =e−r(T−t)

σStζt, (6.9)

ηt =e−r(T−t) IE∗[C|Ft]− ξtSt

At, t ∈ [0, T ]. (6.10)

Then the portfolio (ξt, ηt)t∈[0,T ] is self-financing, and letting

Vt = ηtAt + ξtSt, t ∈ [0, T ], (6.11)

we haveVt = e−r(T−t) IE∗[C|Ft], t ∈ [0, T ]. (6.12)

In particular we haveVT = C, (6.13)

i.e. the portfolio (ξt, ηt)t∈[0,T ] yields a hedging strategy leading to C, startingfrom the initial value

V0 = e−rT IE∗[C].

Proof. Relation (6.12) follows from (6.10) and (6.11), and it implies

V0 = e−rT IE∗[C] = η0A0 + ξ0S0

at t = 0, and (6.13) at t = T . It remains to show that the portfolio strategy(ξt, ηt)t∈[0,T ] is self-financing. By (6.8) and Proposition 6.1 we have

Vt = ηtAt + ξtSt = e−r(T−t) IE∗[C|Ft]

= e−r(T−t) IE∗[IE∗[C] +

w T

0ζudBu

∣∣∣Ft]= e−r(T−t)

(IE∗[C] +

w t

0ζudBu

)= ertV0 + e−r(T−t)

w t

0ζudBu

= ertV0 + σw t

0ξuSue

r(t−u)dBu

= ertV0 + σw t

0ξuXue

rtdBu

= ertV0 + ertw t

0ξudXu, t ∈ [0, T ],

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which shows that the discounted portfolio value Vt = e−rtVt satisfies

Vt = V0 +w t

0ξudXu, t ∈ [0, T ],

and this implies that (ξt, ηt)t∈[0,T ] is self-financing by Lemma 5.1.

The above proposition shows that there always exists a hedging strategystarting from

V0 = IE∗[C]e−rT .

In addition, since there exists a hedging strategy leading to

VT = e−rTC,

then (Vt)t∈[0,T ] is necessarily a martingale with

Vt = IE∗[VT

∣∣∣Ft] = e−rT IE∗[C|Ft], t ∈ [0, T ],

and initial valueV0 = IE∗

[VT

]= e−rT IE∗[C].

In practice, the hedging problem can now be reduced to the computationof the process (ζt)t∈[0,T ] appearing in (6.8). This computation, called theDelta hedging, can be performed by application of the Ito formula and theMarkov property, see e.g. [63]. Consider the (non homogeneous) semi-group(Ps,t)0≤s≤t≤T associated to (St)t∈[0,T ] and defined by

Pu,tf(Su) = IE∗[f(St) | Fu] = IE∗[f(St) | Su], 0 ≤ u ≤ t,

which acts on functions f ∈ C2b (Rn), with

Ps,tPt,u = Ps,u, 0 ≤ s ≤ t ≤ u.

Note that (Pt,T f(St))t∈[0,T ] is an Ft-martingale, i.e.:

IE∗[Pt,T f(St) | Fu] = IE∗[IE∗[f(ST ) | Ft] | Fu]

= IE∗[f(ST ) | Fu]

= Pu,T f(Su), (6.14)

0 ≤ u ≤ t ≤ T , and we have

Pu,tf(x) = IE∗[f(St) | Su = x] = IE∗[f(xSt/Su)], 0 ≤ u ≤ t. (6.15)

The next lemma allows us to compute the process (ζt)t∈[0,T ] in case the payoffC is of the form C = φ(ST ) for some function φ. In case C ∈ L2(Ω) is thepayoff of an exotic option, the process (ζt)t∈[0,T ] can be computed using theMalliavin gradient on the Wiener space, cf. [54], [59].

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N. Privault

Lemma 6.2. Let φ ∈ C2b (Rn). The predictable representation

φ(ST ) = IE∗[φ(ST )] +w T

0ζtdBt (6.16)

is given by

ζt = σSt∂

∂x(Pt,Tφ)(St), t ∈ [0, T ]. (6.17)

Proof. Since Pt,Tφ is in C2(R), we can apply the Ito formula to the process

t 7→ Pt,Tφ(St) = IE∗[φ(ST ) | Ft],

which is a martingale from the “tower property” (6.1) of conditional expecta-tions as in (6.14), cf. also Relation (6.1). From the fact that the finite variationterm in the Ito formula vanishes when (Pt,Tφ(St))t∈[0,T ] is a martingale, (seee.g. Corollary II-1 of [64]), we obtain:

Pt,Tφ(St) = P0,Tφ(S0) + σw t

0Su

∂

∂x(Pu,Tφ)(Su)dBu, t ∈ [0, T ], (6.18)

with P0,Tφ(S0) = IE∗[φ(ST )]. Letting t = T , we obtain (6.17) by uniquenessof the predictable representation (6.16) of C = φ(ST ).

By (6.15) we also have

ζt = σSt∂

∂xIE∗[φ(ST ) | St = x]x=St

= σSt∂

∂xIE∗[φ(xST /St)]x=St , t ∈ [0, T ],

hence

ξt =1

σSte−r(T−t)ζt (6.19)

= e−r(T−t)∂

∂xIE∗[φ(xST /St)]x=St , t ∈ [0, T ],

which recovers the formula (5.8) for the Delta of a vanilla option. As a conse-quence we have ξt ≥ 0 and there is no short selling when the payoff functionφ is non-decreasing.

In the case of European options, the process ζ can be computed via the nextproposition.

Proposition 6.6. Assume that C = (ST − K)+. Then for 0 ≤ t ≤ T wehave

ζt = σSt IE∗[STSt

1[K,∞)

(xSTSt

)]x=St

.

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Proof. This result follows from Lemma 6.2 and the relation Pt,T f(x) =IE∗[f(Sxt,T )], after approximation of x 7→ φ(x) = (x−K)+ with C2 functions.

From the above proposition we recover the formula for the Delta of a Euro-pean call option in the Black-Scholes model. Proposition 6.7 shows that theBlack-Scholes self-financing hedging strategy is to hold a (possibly fractional)quantity

ξt = Φ(d+) = Φ

(log(St/K) + (r + σ2/2)(T − t)

σ√T − t

)≥ 0 (6.20)

of the risky asset, and to borrow a quantity

− ηt = Ke−rTΦ

(log(St/K) + (r − σ2

t /2)(T − t)σ√T − t

)≤ 0 (6.21)

of the riskless (savings) account, cf. also Corollary 10.2 in Chapter 10. In thenext proposition we prove another proof of the result of Proposition 5.4.

Proposition 6.7. The Delta of a European call option with payoff functionf(x) = (x−K)+ is given by

ξt = Φ(d+) = Φ

(log(St/K) + (r + σ2/2)(T − t)

σ√T − t

), 0 ≤ t ≤ T.

Proof. By Propositions 6.5 and 6.6 we have

ξt =1

σSte−r(T−t)ζt

= e−r(T−t) IE∗[STSt

1[K,∞)

(xSTSt

)]x=St

= e−r(T−t)

× IE∗[eσ(BT−Bt)−σ

2(T−t)/2+r(T−t)1[K,∞)(xeσ(BT−Bt)−σ2(T−t)/2+r(T−t))

]x=St

=1√

2π(T − t)w∞σ(T−t)/2−r(T−t)/σ+ 1

σ log KSt

eσy−σ2(T−t)/2−y2/(2(T−t))dy

=1√

2π(T − t)w∞−d−/

√T−t

e−1

2(T−t) (y−σ(T−t))2

dy

=1√2π

w∞−d−

e−12 (y−σ(T−t))

2

dy

=1√2π

w∞−d+

e−12y

2

dy

=1√2π

w d+

−∞e−

12y

2

dy

= Φ(d+).

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N. Privault

The result of Proposition 6.7 can also be recovered by (5.8) or (6.19) anddirect differentiation of the Black-Scholes function, cf. (5.13).

Exercises

Exercise 6.1 In this problem, (ηt, ξt)t∈[0,T ] denotes a portfolio strategy withvalue

Vt = ηtAt + ξtSt, 0 ≤ t ≤ T,where St, resp. At, denotes the price at time t of a risky, resp. riskless, asset.Consider the price process (St)t∈[0,T ] given by

dStSt

= rdt+ σdBt

and a riskless asset of value At = A0ert, t ∈ [0, T ], with r > 0.

1. Compute the arbitrage price

C(t, St) = e−r(T−t) IEQ[|ST |2 | Ft],

at time t ∈ [0, T ], of the contingent claim of payoff |ST |2.2. Compute the portfolio strategy (ηt, ξt)t∈[0,T ] hedging the claim |ST |2.3. Check that this strategy is self-financing.

Exercise 6.2 Again, (ηt, ξt)t∈[0,T ] denotes a portfolio strategy with value

Vt = ηtAt + ξtSt, 0 ≤ t ≤ T,

where St, resp. At, denotes the price at time t of a risky, resp. riskless, asset.

1. Recall (without proof) the solution of the stochastic differential equation

dSt = rStdt+ σdBt.

2. Show that the discounted price process St = e−rtSt, t ∈ [0, T ], is amartingale under P.

3. Compute the arbitrage price

C(t, St) = e−r(T−t) IE[exp(ST ) | Ft]

at time t ∈ [0, T ] of the contingent claim of exp(ST ).4. Explicitly compute the portfolio strategy (ηt, ξt)t∈[0,T ] that hedges the

contingent claim exp(ST ).

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5. Check that this strategy is self-financing.

Exercise 6.3 Let (Bt)t∈R+ be a standard Brownian motion generating afiltration (Ft)t∈R+

. Recall that for f ∈ C2(R+×R), Ito’s formula for Brownianmotion reads

f(t, Bt) = f(0, B0) +w t

0

∂f

∂s(s,Bs)ds

+w t

0

∂f

∂x(s,Bs)dBs +

1

2

w t

0

∂2f

∂x2(s,Bs)ds.

1. Let r ∈ R, σ > 0, f(x, t) = ert+σx−σ2t/2, and St = f(t, Bt). Compute

df(t, Bt) by Ito’s formula, and show that St solves the stochastic differ-ential equation

dSt = rStdt+ σStdBt,

where r > 0 and σ > 0.2. Show that

E[eσBT | Ft] = eσBt+σ2(T−t)/2, 0 ≤ t ≤ T.

Hint: use the independence of increments in the decomposition

BT = (BT −Bt) + (Bt −B0)

and the Laplace transform E[eαX ] = eα2η2/2 when X ' N (0, η2).

3. Show that the process (St)t∈R+satisfies

E[ST | Ft] = er(T−t)St, 0 ≤ t ≤ T.

4. Let C = ST − K denote the payoff of a forward contract with exerciseprice K and maturity T . Compute the discounted expected payoff

Vt := e−r(T−t)E[C | Ft].

5. Find a self-financing portfolio strategy (ξt, ηt)t∈R+ such that

Vt = ξtSt + ηtAt, 0 ≤ t ≤ T,

where At = A0ert is the price of a riskless asset with interest rate r > 0.

Show that it recovers the result of Exercise 5.3-(3).6. Show that the portfolio (ξt, ηt)t∈[0,T ] found in Question 5 hedges the payoffC = ST −K at time T , i.e. show that VT = C.

Exercise 6.4 Digital options. Consider a price process (St)t∈R+ given by

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N. Privault

dStSt

= rdt+ σdBt, S0 = 1,

under the risk-neutral measure P.

The digital call, resp. put, option is a contract with maturity T , strike K,and payoff

Cd :=

$1 if ST ≥ K,

0 if ST < K,resp. Pd :=

$1 if ST ≤ K,

0 if ST > K.

Recall that the prices πt(Cd) and πt(Pd) at time t of the digital call and putoptions are given by the discounted expected payoffs

πt(Cd) = e−r(T−t) IE[Cd | Ft] and πt(Pd) = e−r(T−t) IE[Pd | Ft].(6.22)

1. Show that the payoffs Cd and Pd can be rewritten as

Cd = 1[K,∞)(ST ) and Pd = 1[0,K](ST ).

2. Using Relation (6.22) and Question 1, prove the call-put parity relation

πt(Cd) + πt(Pd) = e−r(T−t), 0 ≤ t ≤ T. (6.23)

If needed, you may use the fact that P(ST = K) = 0.3. Using Relation (6.22), Question 1, and the relation

IE[1[K,∞)(ST ) | St = x] = P (ST ≥ K | St = x),

show that the price πt(Cd) is given by

πt(Cd) = Cd(t, St),

where Cd(t, x) is the function defined by

Cd(t, x) = e−r(T−t)P (ST ≥ K | St = x).

4. Using the results of Exercise 4.8-(3) and of Question 3, show that

Cd(t, x) = e−r(T−t)Φ

((r − σ2/2)τ + log(x/K)

σ√τ

),

where τ = T − t.5. Show that the price πt(Cd) of the digital call option is given by

πt(Cd) = e−r(T−t)Φ(d−),

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where

d− =(r − σ2/2)τ + log(St/K)

σ√τ

.

6. Using the results of Questions 2 and 5, show that the price πt(Pd) of thedigital put is given by

πt(Pd) = e−r(T−t)Φ(−d−).

7. Using the result of Question 5, compute the Delta

ξt :=∂Cd∂x

(t, St)

of the digital call option. Does the Black-Scholes hedging strategy of sucha call option involve short-selling ? Why ?

8. Using the result of Question 6, compute the Delta

ξt :=∂Pd∂x

(t, St)

of the digital put option. Does the Black-Scholes hedging strategy of sucha put option involve short-selling ? Why ?

Exercise 6.5

1. Consider a market model made of a risky asset with price (St)t∈R+as in

Exercise 4.9-(4) and a riskless asset with price At = $1× ert and risklessinterest rate r = σ2/2. From the answer to Exercise 4.9-(2), show thatthe arbitrage price

Vt = e−r(T−t) IE[(logST )+ | Ft]

at time t ∈ [0, T ] of a log call option with payoff (logST )+ is equal to

Vt = σe−rτ√

τ

2πe−

B2t

2τ + σe−rτBtΦ

(Bt√τ

),

where τ = T − t denote the time to maturity.2. Show that Vt can be written as

Vt = g(τ, St),

where g(τ, x) = e−rτf(τ, log x), and

f(τ, y) = σ

√τ

2πe−

y2

2σ2τ + yΦ

(y

σ√τ

).

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N. Privault

3. Figure 6.2 represents the graph of (τ, x) 7→ g(τ, x), with r = 0.05 = 5%per year and σ = 0.1. Assume that the current underlying price is $1 andthere remains 700 days to maturity. What is the price of the option ?

Price

0 0.5

1 1.5

2

St

0 100

200 300

400 500

600 700

T-t

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Fig. 6.2: Option price as a function of the underlying and of time to maturity

4. Show1 that the (possibly fractional) quantity ∆t =∂g

∂x(τ, St) of St at

time t in a portfolio hedging the payoff (logST )+ is equal to

∆t = e−rτ1

StΦ

(logStσ√τ

), 0 ≤ t ≤ T.

5. Figure 6.3 represents the graph of (τ, x) 7→ ∂g∂x (τ, x). Assuming that the

current underlying price is $1 and that there remains 700 days to ma-turity, how much of the risky asset should you hold in your portfolio inorder to hedge one log option ?

1 Recall the chain rule of derivation∂

∂xf(τ, log x) =

1

x

∂f

∂y(τ, y)|y=log x.

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Delta

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

St

0 100

200 300

400 500

600 700

T-t

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Fig. 6.3: Delta as a function of the underlying and of time to maturity

6. Based on the framework and answers of Questions 3 and 5, should youborrow or lend the riskless asset At = $1× ert, and for what amount ?

7. Show that the Gamma of the portfolio, defined as Γt =∂2g

∂x2(τ, St), equals

Γt = e−rτ1

S2t

(1

σ√

2πτe−(log St)

2

2σ2τ − Φ(

logStσ√τ

)), 0 ≤ t ≤ T.

8. Figure 6.4 represents the graph of Gamma. Assume that there remains60 days to maturity and that St, currently at $1, is expected to increase.Should you buy or (short) sell the underlying asset in order to hedge theoption ?

Gamma

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8St 60 80

100 120

140 160

180 200

T-t-0.2

0

0.2

0.4

0.6

0.8

1

Fig. 6.4: Gamma as a function of the underlying and of time to maturity

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N. Privault

9. Let now σ = 1. Show that the function f(τ, y) of Question 2 solves theheat equation

∂f

∂τ(τ, y) =

1

2

∂2f

∂y2(τ, y)

f(0, y) = (y)+.

Exercise 6.6

1. Consider a market model made of a risky asset with price (St)t∈R+as

in Exercise 4.8 and a riskless asset with price At = $1× ert and risklessinterest rate r = σ2/2. From the answer to Exercise 16.4-(2), show thatthe arbitrage price

Vt = e−r(T−t) IE[(K − logST )+ | Ft]

at time t ∈ [0, T ] of a log call option with strike K and payoff (K −logST )+ is equal to

Vt = σe−rτ√

τ

2πe−

(Bt−K/σ)22τ + e−rτ (K − σBt)Φ

(K/σ −Bt√

τ

),

where τ = T − t denote the time to maturity.2. Show that Vt can be written as

Vt = g(τ, St),

where g(τ, x) = e−rτf(τ, log x), and

f(τ, y) = σ

√τ

2πe−

(K−y)2

2σ2τ + (K − y)Φ

(K − yσ√τ

).

3. Figure 6.5 represents the graph of (τ, x) 7→ g(τ, x), with r = 0.125 peryear and σ = 0.5. Assume that the current underlying price is $3 andthere remains 700 days to maturity. What is the price of the option ?

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Price

2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8St 0

100 200

300 400

500 600

700

T-t

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Fig. 6.5: Option price as a function of the underlying and of time to maturity

4. Show2 that the quantity ∆t =∂g

∂x(τ, St) of St at time t in a portfolio

hedging the payoff (K − logST )+ is equal to

∆t = −e−rτ 1

StΦ

(K − logStσ√τ

), 0 ≤ t ≤ T.

5. Figure 6.6 represents the graph of (τ, x) 7→ ∂g∂x (τ, x). Assuming that the

current underlying price is $3 and that there remains 700 days to ma-turity, how much of the risky asset should you hold in your portfolio inorder to hedge one log option ?

Delta

2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4

St

0 100

200 300

400 500

600 700

T-t

-0.5

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

Fig. 6.6: Delta as a function of the underlying and of time to maturity

2 Recall the chain rule of derivation∂

∂xf(τ, log x) =

1

x

∂f

∂y(τ, y)|y=log x.

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N. Privault

6. Based on the framework and answers of Questions 3 and 5, should youborrow or lend the riskless asset At = $1× ert, and for what amount ?

7. Show that the Gamma of the portfolio, defined as Γt =∂2g

∂x2(τ, St), equals

Γt = e−rτ1

S2t

(1

σ√

2πτe−

(K−log St)2

2σ2τ + Φ

(K − logStσ√τ

)), 0 ≤ t ≤ T.

8. Figure 6.7 represents the graph of Gamma. Assume that there remains10 days to maturity and that St, currently at $3, is expected to increase.Should you buy or (short) sell the underlying asset in order to hedge theoption ?

Gamma

2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8St

10 20

30 40

50 60

70 80

90 100

T-t

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Fig. 6.7: Gamma as a function of the underlying and of time to maturity

9. Show that the function f(τ, y) of Question 2 solves the heat equation∂f

∂τ(τ, y) =

1

2σ2 ∂

2f

∂y2(τ, y)

f(0, y) = (K − y)+.

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Chapter 7

Estimation of Volatility

The values of the parameters r, t, St, T , and K used to price a call op-tion via the Black-Scholes formula can be easily obtained from market data.Estimating the volatility coefficient σ can be a more difficult task, and sev-eral estimation methods are considered in this section with some examplesof how the Black-Scholes formula can be fitted to market data. We cover thehistorical, implied, and local volatility models, and refer to [26] for stochasticvolatility models.

7.1 Historical Volatility

We consider the problem of estimating the parameters µ and σ from marketdata in the stock price model

dStSt

= µdt+ σdBt.

A natural estimator for the trend parameter µ can be written as

µN :=1

N

N−1∑k=0

1

tk+1 − tkStk+1

− StkStk

,

where (Stk+1−Stk)/Stk , k = 0, . . . , N − 1 is a family of log-returns observedat different times t0, . . . , tN on the market.

Similarly the parameter σ can be estimated as by the estimator σN builtas

σ2N :=

1

N − 1

N−1∑k=0

1

tk+1 − tk

(Stk+1 − Stk

Stk− µN (tk+1 − tk)

)2

.

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N. Privault

Parameter estimation based on historical data requires a lot of samples andit can only be valid on a given time interval, or as a moving average.

7.2 Implied Volatility

In contrast with the historical volatility, the computation of the impliedvolatility can be done at a fixed time and requires much less data.

Recall that when h(x) = (x−K)+, the solution of the Black-Scholes PDEis given by

f(t, x,K, σ, r, T ) = xΦ(d+)−Ke−(T−t)rΦ(d−),

where

Φ(x) =1√2π

w x

−∞e−y

2/2dy, x ∈ R,

and

d+ =log(x/K) + (r + σ2/2)(T − t)

σ√T − t

, d− =log(x/K) + (r − σ2/2)(T − t)

σ√T − t

.

Equatingf(t, St,K, σ, r, T ) = M

to the observed value M of a given market price, when t, St, r, T are known,allows one to infer a value for σ. This value is called the implied volatilityand denoted here by σimp(K,T ). The implied volatility value can be usedas an alternative way to quote the option price, based on the knowledge ofthe remaining parameters (such as underlying asset price, time to maturity,interest rate, and strike price).

Given two European call options with strikes K1, resp. K2 and maturitiesT1, resp. T2, on the same stock S, this procedure should yield two estimatesσimp(K1, T1) and σimp(K2, T2) of implied volatilites.

Clearly, there is no reason a priori for the implied volatilites σimp(K1, T1)and σimp(K2, T2) to coincide. However, in the standard Black-Scholes modelthe value of the parameter σ should be unique for a given stock S. This con-tradiction between a model and market data is a reason for the developmentof more sophisticated stochastic volatility models.

Plotting the different values of the implied volatility σ as a function of Kand T will yield a planar curve called the volatility surface.

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Figure 7.1 presents an estimation of implied volatility for Asian optionswhose underlying asset is the price of light sweet crude oil futures traded onthe New York Mercantile Exchange (NYMEX), based on contract specifica-tions and market data obtained from the Chicago Mercantile Exchange.

5 10

15 20

25 30

35

8000 8500

9000 9500

10000 10500

11000 11500

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

Vol.

Implied volatility surface

Time to maturity

Strike

Vol.

Fig. 7.1: Implied volatility of Asian options on light sweet crude oil futures.1

As observed in Figure 7.1, the volatility surface can exhibit a smile phe-nomenon, in which implied volatility is higher at a given end (or at bothends) of the range of strike values.

7.3 The Black-Scholes Formula vs Market Data

On July 28, 2009 a call warrant has been issued by Merrill Lynch on thestock price S of Cheung Kong Holdings (0001.HK) with Strike K=$109.99and Maturity T = December 13, 2010.

1 This graph is courtesy of Tan Yu Jia.

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N. Privault

Fig. 7.2: Graph of the (market) stock price of Cheung Kong Holdings.

The market price of the option (17838.HK) on September 28 was $12.30, asobtained from http://www.hkex.com.hk/dwrc/search/listsearch.asp

The next graph shows the evolution of the market price of the option overtime. One sees that the option price is much more volatile than the underlyingstock price.

Fig. 7.3: Graph of the (market) call option price on Cheung Kong Holdings.

In Figure 7.4 we have fitted the path

t 7−→ gc(t, St)

of the Black-Scholes price to the data of Figure 7.3 using the stock price dataof Figure 7.2, by varying the values of the volatility σ.

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http://www.hkex.com.hk/dwrc/search/listsearch.asp



0.1

0.12

0.14

0.16

0.18

0.2

Jul17 Aug06 Aug26 Sep15

HK$

Black-Scholes price

Fig. 7.4: Graph of the Black-Scholes call option price on Cheung Kong Holdings.

Another example

Let us consider the stock price of HSBC Holdings (0005.HK) over one year:

Fig. 7.5: Graph of the (market) stock price of HSBC Holdings.

Next we consider the graph of the price of a call option issued by SocieteGenerale on 31 December 2008 with strike K=$63.704, maturity T = October05, 2009, and entitlement ratio 100, cf. page 6.

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N. Privault

Fig. 7.6: Graph of the (market) call option price on HSBC Holdings.

As above, in Figure 7.7 we have fitted the path t 7−→ gc(t, St) of the Black-Scholes price to the data of Figure 7.6 using the stock price data of Figure 7.5.

0

0.1

0.2

0.3

Nov 08 Jan 09 Mar 09 May 09 Jul 09 Sep 09

HK$

Black-Scholes price

Fig. 7.7: Graph of the Black-Scholes call option price on HSBC Holdings.

In this case we are in the money at maturity, and we also check that theoption is worth 100 × 0.2650 = $26.650 at that time which, by absence ofarbitrage, is very close to the value $90 - $63.703 = $26.296 of its payoff.

For one more example, consider the graph of the price of a put option is-sued by BNP Paribas on 04 November 2008 with strike K=$77.667, maturityT = October 05, 2009, and entitlement ratio 92.593.

One checks easily that at maturity, the price of the put option is worth $0.01(a market price cannot be lower), which almost equals the option payoff $0,

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Fig. 7.8: Graph of the (market) put option price on HSBC Holdings.

by absence of arbitrage opportunities. Figure 7.9 is a fit of the Black-Scholesput price graph

t 7−→ gp(t, St)

to Figure 7.8 as a function of the stock price data of Figure 7.7. Note that theBlack-Scholes price at maturity is strictly equal to 0 while the correspondingmarket price cannot be lower than one cent.

0

0.1

0.2

0.3

0.4

0.5

Nov 08 Jan 09 Mar 09 May 09 Jul 09 Sep 09

HK$

Black-Scholes price

Fig. 7.9: Graph of the Black-Scholes put option price on HSBC Holdings.

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N. Privault

7.4 Local Volatility

Since the constant volatility assumption in the Black-Scholes model appearsto be not satisfying due to the existence of volatility smiles, it makes senseto consider models of the form

dStSt

= rdt+ σtdBt

where σt is a random process. Such models are called stochastic volatilitymodels.

A particular class of stochastic volatility models can be written as

dStSt

= rdt+ σ(t, St)dBt (7.1)

where σ(t, x) is a deterministic function of time and the stock price. Suchmodels are called local volatility models. The corresponding Black-ScholesPDE can be written as

rg(t, x,K) =∂g

∂t(t, x,K) + rx

∂g

∂x(t, x,K) +

1

2x2σ2(t, x)

∂2g

∂x2(t, x,K),

g(T, x,K) = (x−K)+,(7.2)

with terminal condition g(T, x,K) = (x − K)+, i.e. we consider Europeancall options.

Note that the Black-Scholes PDE would allow one to recover the value ofσ(t, x) as a function of the option price g(t, x,K), as

σ(t, x) =

√√√√√√2rg(t, x,K)− 2∂g

∂t(t, x,K)− 2rx

∂g

∂x(t, x,K)

x2∂2g

∂x2(t, x,K)

, x, t > 0,

however this formula requires the knowledge of the option price for differentvalues of the underlying x, in addition to the knowledge of the strike price K.

The Dupire formula brings a solution to the local volatility calibrationproblem by providing an estimator of σ(t, x) as a function σ(t,K) based onthe values of the strike price K.

Proposition 7.1. Assume that a family (C(T,K))T,K>0 of market call op-tion prices with maturities T and strikes K is given at time t with St = x,

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while the values of r and x are fixed.

The Dupire formula states that, defining the volatility function σ(t, y) by

σ(t, y) :=

√√√√√√√2∂C

∂t(t, y) + 2ry

∂C

∂y(t, y)

y2∂2C

∂y2(t, y)

, (7.3)

the prices g(t, x,K) computed from the Black-Scholes PDE (7.2) will matchthe option prices C(T,K) in the sense that

g(t, x,K) = C(T,K), T,K > 0. (7.4)

Proof. We use the probabilistic approach that allows us to write g(t, x,K)as

g(t, x,K) = e−r(T−t) IE[(ST −K)+ | St = x], (7.5)

where (St)t∈R+ is defined by (7.1), and use stochastic calculus. Hence thecondition (7.4) can be written at t = 0 as

C(T,K) = e−rTw∞−∞

(y −K)+ϕT (y)dy,

where ϕT (y) is the probability density of ST . After differentiating both sidestwice with respect to K one gets

∂2C

∂K2(T,K) = e−rTϕT (K). (7.6)

On the other hand, for any sufficiently smooth function f , using the Itoformula we havew∞−∞

ϕT (y)f(y)dy = IE[f(ST )]

= IE

[f(S0) +

w T

0f ′(St)dSt +

1

2

w T

0f ′′(St)σ

2(t, St)dt

]= IE

[f(S0) + r

w T

0f ′(St)Stdt+ σ

w T

0f ′(St)StdBt +

1

2

w T

0f ′′(St)σ

2(t, St)dt

]= f(S0) + IE

[rw T

0f ′(St)Stdt+

1

2

w T

0f ′′(St)S

2t σ

2(t, St)dt

]= f(S0) + r

w∞−∞

w T

0yf ′(y)ϕt(y)dtdy +

1

2

w∞−∞

w T

0y2f ′′(y)σ2(t, y)ϕt(y)dtdy,

hence after differentiating both sides of the equality with respect to T ,

w∞−∞

∂ϕT∂T

(y)f(y)dy = rw∞−∞

yf ′(y)ϕT (y)dy+1

2

w∞−∞

y2f ′′(y)σ2(T, y)ϕT (y)dy.

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Integrating by parts in the above relation yields

w∞−∞

∂ϕT∂T

(y)f(y)dy

= −rw∞−∞

f(y)∂

∂y(yϕT (y))dy +

1

2

w∞−∞

f(y)∂2

∂y2(y2σ2(T, y)ϕT (y))dy,

for all smooth functions f(y) with compact support, hence

∂ϕT∂T

(y) = −r ∂∂y

(yϕT (y)) +1

2

∂2

∂y2(y2σ2(T, y)ϕT (y)), y ∈ R.

Making use of (7.6) we get

−r ∂2C

∂y2(T, y)− ∂

∂T

∂2C

∂y2(T, y)

= r∂

∂y

(y∂2C

∂y2(T, y)

)− 1

2

∂2

∂y2

(y2σ2(T, y)

∂2C

∂y2(T, y)

), y ∈ R.

After a first integration with respect to y under the limiting conditionlimK→+∞ C(T,K) = 0, we obtain

−r ∂C∂y

(T, y)− ∂

∂T

∂C

∂y(T, y) = ry

∂2C

∂y2(T, y)− 1

2

∂

∂y

(y2σ2(T, y)

∂2C

∂y2(T, y)

),

i.e.

−r ∂C∂y

(T, y)− ∂

∂T

∂C

∂y(T, y)

= r∂

∂y

(y∂C

∂y(T, y)

)− r ∂C

∂y(T, y)− 1

2

∂

∂y

(y2σ2(T, y)

∂2C

∂y2(T, y)

),

or

− ∂

∂y

∂C

∂T(T, y) = r

∂

∂y

(y∂C

∂y(T, y)

)− 1

2

∂

∂y

(y2σ2(T, y)

∂2C

∂y2(T, y)

).

Integrating one more time with respect to y yields

−∂C∂T

(T, y) = ry∂C

∂y(T, y)− 1

2y2σ2(T, y)

∂2C

∂y2(T, y), y ∈ R,

which conducts to (7.3) and is called the Dupire [21] PDE.

From (7.3) the local volatility σ(t, y) can be estimated by computingC(T, y) by the Black-Scholes formula, based on a value of the implied volatil-ity σ. See [1] for numerical methods applied to volatility estimation in thisframework.

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Chapter 8

Exotic Options

In this chapter we work in a continuous geometric Brownian model in whichthe asset price (St)t∈[0,T ] has the dynamics

dSt = rStdt+ σStdBt, t ∈ R+,


ability measure P∗. In particular the value Vt of a self-financing portfoliosatisfies

VT e−rT = V0 + σ

w T

0ξtSte

−rtdBt, t ∈ [0, T ].

8.1 Generalities

An exotic option is an option whose payoff may depend on the whole pathSt : t ∈ [0, T ] of the price process via a “complex” operation such asaveraging or computing a maximum. They are opposed to vanilla optionswhose payoff

C = φ(ST ),

where φ is called a payoff function, depends only on the terminal value ST ofthe price process.

An option with payoff C = φ(ST ) can be priced as

e−rT IE[φ(ST )] = e−rTw∞−∞

φ(y)fST (y)dy

where fST (y) is the (one parameter) probability density function of ST , whichsatisfies

P(ST ≤ y) =w y

−∞fST (v)dv, y ∈ R.

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N. Privault

Recall that typically we have

φ(x) = (x−K)+ =

x−K if x ≥ K,

0 if x < K,

for a European call option with strike K, and

φ(x) = 1[K,∞)(x) =

$1 if x ≥ K,

0 if x < K,

for a binary call option with strike K.

Exotic Options

Exotic options, also called path-dependent options, are options whose payoffC may depend on the whole path

St : 0 ≤ t ≤ T

of the underlying price process instead of its terminal value ST . Next wereview some examples of exotic options.

Options on Extrema

We takeC := φ(MT ),

whereMT = max

t∈[0,T ]St

is the maximum of (St)t∈R+over the time interval [0, T ].

Figure 8.1 represents the running maximum process (Mt)t∈R+of Brownian

motion (Bt)t∈R+ .

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Fig. 8.1: Brownian motion Bt and its supremum Xt.

Barrier Options

The payoff of an up-and-out barrier put option on the underlying asset Stwith exercise date T , strike K and barrier B is

C = (K − ST )+

1max0≤t≤T

St < B =

(K − ST )+ if max

0≤t≤TSt < B,

0 if max0≤t≤T

St ≥ B.

This option is also called a Callable Bear Contract with no residual value, inwhich the call price B usually satisfies B ≤ K.

The payoff of a down-and-out barrier call option on the underlying assetSt with exercise date T , strike K and barrier B is

C = (ST −K)+

1min

0≤t≤TSt > B

=

(ST −K)+ if min

0≤t≤TSt > B,

0 if min0≤t≤T

St ≤ B.

This option is also called a Callable Bull Contract with no residual value, inwhich B denotes the call price B ≥ K. It is also called a turbo warrant withno rebate.

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N. Privault

Lookback Options

The payoff of a floating strike lookback call option on the underlying assetSt with exercise date T is

C = ST − min0≤t≤T

St.

The payoff of a floating strike lookback put option on the underlying assetSt with exercise date T is

C =

(max0≤t≤T

St

)− ST .

Options on Average

In this case we can take

C = φ

(1

T

w T

0Stdt

)where

1

T

w T

0Stdt

represents the average of (St)t∈R+ over the time interval [0, T ] and φ : R −→ Ris a payoff function.

Fig. 8.2: Brownian motion Bt and its moving average.

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Figure 8.2 shows a graph of Brownian motion and its moving average processXt.

Asian Options

Asian options are particular cases of options on average, and they were firsttraded in Tokyo in 1987. The payoff of the Asian call option on the underlyingasset St with exercise date T and strike K is given by

C =

(1

T

w T

0Stdt−K

)+

.

Similarly, the payoff of the Asian put option on the underlying asset St withexercise date T and strike K is

C =

(K − 1

T

w T

0Stdt

)+

.

Due to the fact that their dependence on averaged asset prices, Asian op-tions are less volatile than plain vanilla options whose payoffs depend onlyon the terminal value of the underlying asset. Asian options have becomeparticularly popular in commodities trading.

8.2 The Reflexion Principle

In order to price barrier options we will have to derive the probability densityof the maximum

MT = maxt∈[0,T ]

St

of geometric Brownian motion (St)t∈R+over a given time interval [0, T ].

In such situations the option price at time t = 0 can be expressed as

e−rT IE[φ(MT , ST )] = e−rTw∞−∞

w∞−∞

φ(x, y)f(MT ,ST )(x, y)dxdy

where f(MT ,ST ) is the joint probability density function of (MT , ST ), whichsatisfies

P(MT ≤ x, ST ≤ y) =w x

−∞

w y

−∞f(MT ,ST )(u, v)dudv, x, y ∈ R.

In order to price such options by the above probabilistic method, we willcompute f(MT ,ST )(u, v) by the reflection principle.

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N. Privault

Maximum of Standard Brownian Motion

Let (Bt)t∈R+denote a standard Brownian motion started at B0 = 0. While

it is well-known that BT ' N (0, T ), computing the law of the maximum

XT = maxt∈[0,T ]

Bt

might seem a difficult problem. However this is not the case, due to thereflection principle. Note that since B0 = 0 we have

XT ≥ 0,

almost surely.

Given a > B0 = 0, let

τa = inft ∈ R+ : Bt = a

denote the first time (Bt)t∈R+hits the level a > 0.

Due to the space symmetry of Brownian motion we have the identity

P(BT > a | τa < T ) =1

2= P(BT < a | τa < T ).

This identity is clearly equivalent to

2P(BT > a & τa < T ) = P(τa < T ) = 2P(BT < a & τa < T ),

and to

2P(BT > a & XT ≥ a) = P(τa < T ) = 2P(BT < a & XT ≥ a),

due to the equivalence

XT ≥ a = τa < T. (8.1)

In other words, we have

P(XT ≥ a) = P(BT > a & XT ≥ a) + P(BT < a & XT ≥ a)

= 2P(BT > a & XT ≥ a)

= 2P(BT > a)

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"



= P(BT > a) + P(BT < −a)

= P(|BT | > a),

where we used the fact that

BT > a ⊂ BT > a & XT ≥ a ⊂ BT > a.

Figure 8.3 shows a graph of Brownian motion and its reflected path.

Fig. 8.3: Reflected Brownian motion with a = 1.

Consequently, the maximum XT of Brownian motion has same distribution asthe absolute value |BT | of BT . In other words, XT is a non-negative randomvariable with distribution function

P(XT ≤ a) = P(|BT | ≤ a)

=1√2πT

w a

−ae−x

2/(2T )dx

=2√2πT

w a

0e−x

2/(2T )dx, a ∈ R+,

and probability density

fXT (a) =dP(XT ≤ a)

da=

√2

πTe−a

2/(2T )1[0,∞)(a), a ∈ R. (8.2)

" 157



N. Privault

0

0.2

0.4

0.6

0.8

1

-4 -3 -2 -1 0 1 2 3 4

dens

ity

x

Density function

Fig. 8.4: Probability density of the maximum of Brownian motion over [0, 1].

Using the density of XT we can price an option with payoff φ(XT ), as

e−rT IE [φ(XT )] = e−rTw∞−∞

φ(x)dP(XT = x)

= e−rT√

2

πT

w∞0φ(x)e−|x|

2/(2T )dx.

Next we consider

MT = maxt∈[0,T ]

St

= S0 maxt∈[0,T ]

eσBt

= S0eσmaxt∈[0,T ] Bt

= S0eσXT ,

since σ > 0. When the payoff takes the form

C = φ(MT ),

whereST = S0e

σBT ,

we haveC = φ(MT ) = φ(S0e

σXT ),

hence

e−rT IE [C] = e−rT IE[φ(S0e

σXT )]

= e−rTw∞−∞

φ(S0eσx)dP(XT = x)

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"



=

√2

πTe−rT

w∞0φ(S0e

σx)e−x2/(2T )dx.

This however is not sufficient since this imposes the condition r = σ2/2. Inorder to do away with this condition we need to consider the maximum ofdrifted Brownian motion, and for this we have to compute the joint densityof XT and BT .

Joint Density

The reflection principle also allows us to compute the joint density of Brow-nian motion BT and its maximum XT . Indeed, for b ∈ [0, a] we also have

P(BT > a+ (a− b) | τa < T ) = P(BT 2a− b & τa < T ) = P(BT 2a− b & XT ≥ a) = P(BT 2a− b & XT ≥ a) = P(BT 2a− b & XT ≥ 2a− b⊂ BT > 2a− b & XT ≥ a ⊂ BT > a,

which shows that BT ≥ 2a− b = BT > 2a− b & XT ≥ a.

Hence by (8.3) we have

P(BT < b & XT ≥ a) = P(BT ≥ 2a− b) =1√2πT

w∞2a−b

e−x2/(2T )dx,

0 ≤ b ≤ a, which yields the joint probability density

fXT ,BT (a, b) = −dP(XT ≥ a & BT ≤ b)dadb

=dP(XT ≤ a & BT ≤ b)

dadb,

" 159



N. Privault

a, b ∈ R, by (16.15), i.e., letting a ∨ b := max(a, b),

fXT ,BT (a, b) =

√2

πT

(2a− b)T

e−(2a−b)2/(2T )1a≥b∨0 (8.4)

=

√

2

πT

(2a− b)T

e−(2a−b)2/(2T ), a > b ∨ 0,

0, a < b ∨ 0.

-1-0.5 0 0.5 1 1.5 2 2.5 3

-1-0.5

0 0.5

1 1.5

2 2.5

3

0 0.05

0.1 0.15

0.2 0.25

0.3 0.35

0.4 0.45

0.5

Density function

ab

Fig. 8.5: Joint probability density of B1 and its maximum over [0,1].

Maximum of Drifted Brownian Motion

Using the Girsanov theorem, it is even possible to compute the probabilitydensity function of the maximum

XT = maxt∈[0,T ]

Bt = maxt∈[0,T ]

(Bt + µt)

of the drifted Brownian motion Bt = Bt + µt, µ ∈ R. The arguments previ-ously applied to Bt cannot be directly applied to Bt because drifted Brownianmotion is no longer symmetric in space when µ 6= 0.

On the other hand, Bt is a standard Brownian motion under the proba-bility measure P defined from

160


"



dPdP

= e−µBT−µ2T/2, (8.5)

hence the density of XT under P is given by (8.4).

Now, using the density (8.5) we get

P(XT ≤ a & BT ≤ b) = IE[1XT≤a & BT≤b

]= IE

[eµBT+µ

2T/21XT≤a & BT≤b

]= IE

[eµBT−µ

2T/21XT≤a & BT≤b

]=

√2

πT

w a

0

w b

−∞1(−∞,x](y)eµy−µ

2T/2 (2x− y)

Te−(2x−y)

2/(2T )dxdy,

0 ≤ b ≤ a, which yields the joint probability density

fXT ,BT (a, b) =dP(XT ≤ a & BT ≤ b)

dadb,

i.e.

fXT ,BT (a, b) = 1a≥b∨01

T

√2

πT(2a− b)eµb−(2a−b)2/(2T )−µ2T/2 (8.6)

=

1

T

√2

πT(2a− b)e−µ2T/2+µb−(2a−b)2/(2T ), a > b ∨ 0,

0, a < b ∨ 0.

We also find

P(XT ≤ a) =

√2

πT

w a

0

w∞−∞

1(−∞,x](y)eµy−µ2T/2 (2x− y)

Te−(2x−y)

2/(2T )dydx

=

√2

πTe−µ

2T/2w a

−∞eµy

w a

y∨0

(2x− y)

Te−(2x−y)

2/(2T )dxdy

=

√1

2πTe−µ

2T/2w a

−∞

(eµy−(2(y∨0)−y)

2/(2T ) − eµy−(2a−y)2/(2T ))dy

=

√1

2πT

w a

−∞

(eµy−y

2/(2T )−µ2T/2 − eµy−2a2/T+2ay/T−y2/(2T )−µ2T/2)dy

" 161



N. Privault

=

√1

2πT

w a

−∞

(e−(y−µT )2/(2T ) − e−(y−(µT+2a))2/(2T )+2aµ

)dy

=

√1

2πT

w a

−∞e−(y−µT )2/(2T )dy − e2aµ

√1

2πT

w a

−∞e−(y−(µT+2a))2/(2T )dy

=

√1

2πT

w a−µT

−∞e−y

2/(2T )dy − e2aµ√

1

2πT

w −a−µT−∞

e−y2/(2T )dy

= Φ

(a− µT√

T

)− e2µaΦ

(−a− µT√T

), (8.7)

cf. Corollary 7.2.2 and pages 297-299 of [71] for another derivation. Thisyields the density

dP(XT ≤ a)

da=

√2

πTe−(a−µT )2/(2T ) − 2µe2µaΦ

(−a− µT√T

),

of the supremum of drifted Brownian motion, and recovers (8.2) for µ = 0.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

-1 0 1 2 3 4

dens

ity

x

µ=0µ=-0.5µ=0.5

Fig. 8.6: Probability density of the maximum of drifted Brownian motion.

Note from Figure 8.2 that small values of the maximum are more likely tooccur when µ takes large negative values.

Based on the relation mint∈[0,T ]

Bt = − maxt∈[0,T ]

(−Bt), the joint density fRT ,BT

of the minimumRT = min

t∈[0,T ]Bt = min

t∈[0,T ](Bt + µt)

of the drifted Brownian motion Bt := Bt + µt and its value BT at time Tcan similarly be computed as follows, letting a ∧ b := min(a, b):

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"



fRT ,BT (a, b) = 1a≤b∧01

T

√2

πT(b− 2a)eµb−(2a−b)

2/(2T )−µ2T/2 (8.8)

=

1

T

√2

πT(b− 2a)e−µ

2T/2+µb−(2a−b)2/(2T ), a b ∧ 0.

8.3 Barrier Options

General Case

Using the joint density of BT and XT we are able to price any exotic optionwith payoff φ(BT , XT ), as

e−r(T−t) IE

[φ(XT , BT )

∣∣∣∣∣ Ft],

with in particular

e−rT IE[φ(XT , BT )

]= e−rT

w∞−∞

w∞y∨0

φ(x, y)dP(XT = x, BT = y).

When the payoff takes the form

C = φ(MT , ST ),

whereST = S0e

σBT−σ2T/2+rT = S0eσBT ,

with µ = −σ/2 + r/σ and BT = BT + µT , and

MT = maxt∈[0,T ]

St

= S0 maxt∈[0,T ]

eσBt−σ2t/2+rt

= S0 maxt∈[0,T ]

eσBt

= S0eσmaxt∈[0,T ] Bt

= S0eσXT ,

" 163



N. Privault

we have

C = φ(ST ,MT )

= φ(S0eσBT−σ2T/2+rT ,MT )

= φ(S0eσBT , S0e

σXT ),

hence

e−rT IE[C] = e−rT IE[φ(S0e

σBT , S0eσXT )

]= e−rT

w∞−∞

w∞y∨0

φ(S0eσy, S0e

σx)dP(XT = x, BT = y)

=1

T

√2

πTe−rT

w∞−∞

w∞y∨0

φ(S0eσy, S0e

σx)(2x− y)e−µ2T/2+µy−(2x−y)2/(2T )dxdy

=1

Te−rT

√2

πT

w∞0

w∞yφ(S0e

σy, S0eσx)(2x− y)e−µ

2T/2+µy−(2x−y)2/(2T )dxdy

+1

Te−rT

√2

πT

w 0

−∞

w∞0φ(S0e

σy, S0eσx)(2x− y)e−µ

2T/2+µy−(2x−y)2/(2T )dxdy.

We can distinguish 8 different versions of barrier options according to thefollowing table.

164


"



option type behavior payoff

down-and-out (ST −K)+

1min

0≤t≤TSt > B

down-and-in (ST −K)+

1min

0≤t≤TSt B

down-and-out (K − ST )+

1min

0≤t≤TSt > B

down-and-in (K − ST )+

1min

0≤t≤TSt B

We have the following obvious relations between the prices of barrier andvanilla call and put options:

Cup−in(t) + Cup−out(t) = C(t) = e−r(T−t) IE∗[(ST −K)+],

Cdown−in(t) + Cdown−out(t) = C(t) = e−r(T−t) IE∗[(ST −K)+],

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N. Privault

Pup−in(t) + Pup−out(t) = P (t) = e−r(T−t) IE∗[(K − ST )+],

Pdown−in(t) + Pdown−out(t) = P (t) = e−r(T−t) IE∗[(K − ST )+],

where C(t), resp. P (t) denotes the price of a European call, resp. put optionwith strike K as obtained from the Black-Scholes formula. Consequently, inthe sequel we will only compute the prices of the up-and-out call and put,and down-and-out barrier call and put options.

Up-and-Out Barrier Call Option

Let us consider an up-and-out call option with maturity T , strike K, barrier(or call price) B, and payoff

C = (ST −K)+

1max0≤t≤T

St ≤ B =

ST −K if max

0≤t≤TSt ≤ B,

0 if max0≤t≤T

St > B,

with B > K. Our goal is to prove the following result.

Proposition 8.1. When K < B, the price

e−r(T−t)1Mt ≤ B IE∗

(xST−tS0−K

)+

1x max

0≤r≤T−tSr/S0 ≤ B

x=St

of the up-and-out call option with maturity T , strike K and barrier B is givenby

e−r(T−t) IE∗[C | Ft] (8.9)

= St1Mt ≤ B(Φ

(δT−t+

(StK

))− Φ

(δT−t+

(StB

))−(B

St

)1+2r/σ2 (Φ

(δT−t+

(B2

KSt

))− Φ

(δT−t+

(B

St

))))

−e−r(T−t)K1Mt ≤ B(Φ

(δT−t−

(StK

))− Φ

(δT−t−

(StB

))−(StB

)1−2r/σ2 (Φ

(δT−t−

(B2

KSt

))− Φ

(δT−t−

(B

St

)))),

where

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"



δτ±(s) =1

σ√τ

(log s+

(r ± 1

2σ2

)τ

), s > 0. (8.10)

Note that taking B = +∞ in the above identity (8.9) recovers the Black-Scholes formula for the price of a European call option, and that the price ofthe up-and-out barrier call option is 0 when B < K.

The following graph represents the up-and-out call option price given thevalue St of the underlying and the time t ∈ [0, T ] with T = 220 days.

50 55 60 65 70 75 80 85 90 100 120

140 160

180 200

220 0

2

4

6

8

10

12

14

16

up and out call priceOption price path

underlying

Time in days

Fig. 8.7: Graph of the up-and-out call option price.

Proof of Proposition 8.1. We have C = φ(ST ,MT ) with

φ(x, y) = (x−K)+

1y≤B =

x−K if y ≤ B,

0 if y > B,

hence

e−r(T−t) IE∗[C | Ft] = e−r(T−t) IE∗

[(ST −K)

+1MT≤B

∣∣∣∣∣ Ft]

= e−r(T−t) IE∗

[(ST −K)

+1MT ≤ B

∣∣∣∣∣ Ft]


(ST −K)+

1Mt ≤ B1

maxt≤r≤T

Sr ≤ B∣∣∣∣∣ Ft

" 167



N. Privault

= e−r(T−t)1Mt ≤ B IE∗

(ST −K)+

1maxt≤r≤T

Sr ≤ B∣∣∣∣∣ Ft


(xSTSt−K

)+

1x maxt≤r≤T

Sr/St ≤ Bx=St


(xST−tS0−K

)+

1x max


x=St

.

It suffices to compute

e−rτ IE∗[C] = e−rτ IE∗[(Sτ −K)

+1Mτ≤B

]= e−rτ IE∗

[(S0e

σBτ −K)+

1S0eσXτ≤B

]= e−rτ

w∞−∞

w∞y∨0

(S0eσy −K)

+1S0eσx≤BdP(Xτ = x, Bτ = y)

=1

τ

√2

πτe−rτ

w σ−1 log(B/S0)

−∞w∞y∨0

(S0eσy −K)

+1S0eσx≤B(2x− y)e−µ

2τ/2+µy−(2x−y)2/(2τ)dxdy

=1

τe−rτ

√2

πτ

w σ−1 log(B/S0)

0w∞y

(S0eσy −K)



+1

τe−rτ

√2

πτ

w 0

−∞w∞0

(S0eσy −K)



=1

τe−rτ

√2

πτ

w σ−1 log(B/S0)

0w∞y

(S0eσy −K)

+1x≤σ−1 log(B/S0)(2x− y)e−µ


+1

τe−rτ

√2

πτ

w 0

−∞w∞0

(S0eσy −K)

+1x≤σ−1 log(B/S0)(2x− y)e−µ


=1

τe−rτ

√2

πτ

w σ−1 log(B/S0)

0

168


"



w σ−1 log(B/S0)

y(S0e

σy −K)+

(2x− y)e−µ2τ/2+µy−(2x−y)2/(2τ)dxdy

+1

τe−rτ

√2

πτ

w 0

−∞w σ−1 log(B/S0)

0(S0e

σy −K)+

(2x− y)e−µ2τ/2+µy−(2x−y)2/(2τ)dxdy

=1

τe−rτ

√2

πτ

w σ−1 log(B/S0)

σ−1 log(K/S0)w σ−1 log(B/S0)

y∨0(S0e

σy −K) (2x− y)e−µ2τ/2+µy−(2x−y)2/(2τ)dxdy

=1

τe−rτ−µ

2τ/2

√2

πτ

w σ−1 log(B/S0)

σ−1 log(K/S0)(S0e

σy −K) eµy−y2/(2τ)

w σ−1 log(B/S0)

y∨0(2x− y)e2x(y−x)/τdxdy,

if B ≥ S0 (otherwise the option price is 0), with µ = r/σ − σ/2 andy ∨ 0 = max(y, 0).

Letting a = y ∨ 0 and b = σ−1 log(B/S0), we have

w b

a(2x− y)e2x(y−x)/τdx =

w b

a(2x− y)e2x(y−x)/τdx

= −τ2

[e2x(y−x)/τ

]x=bx=a

=τ

2(e2a(y−a)/τ − e2b(y−b)/τ )

=τ

2(e2(y∨0)(y−y∨0)/τ − e2b(y−b)/τ )

=τ

2(1− e2b(y−b)/τ ),

hence, letting c = σ−1 log(K/S0), we have

e−rτ IE∗[C] = e−τ(r+µ2/2) 1√

2πτ

w b

c(S0e

σy −K) eµy−y2/(2τ)(1− e2b(y−b)/τ )dy

= S0e−τ(r+µ2/2) 1√

2πτ

w b

cey(σ+µ)−y

2/(2τ)(1− e2b(y−b)/τ )dy

−Ke−τ(r+µ2/2) 1√2πτ

w b

ceµy−y

2/(2τ)(1− e2b(y−b)/τ )dy

= S0e−τ(r+µ2/2) 1√

2πτ

w b

cey(σ+µ)−y

2/(2τ)dy

−S0e−τ(r+µ2/2)−2b2/τ 1√

2πτ

w b

cey(σ+µ+2b/τ)−y2/(2τ)dy

" 169



N. Privault

−Ke−τ(r+µ2/2) 1√2πτ

w b

ceµy−y

2/(2τ)dy

+Ke−τ(r+µ2/2)−2b2/τ 1√

2πτ

w b

cey(µ+2b/τ)−y2/(2τ)dy.

Using the relation

1√2πτ

w b

ceγy−y

2/(2τ)dy = eγ2τ/2

(Φ

(−c+ γτ√τ

)− Φ

(−b+ γτ√τ

)),

we find

e−rτ IE∗[C] = e−rT IE∗[(ST −K)

+1MT≤B

]= S0e

−τ(r+µ2/2)+(σ+µ)2τ/2

(Φ

(−c+ (σ + µ)τ√τ

)− Φ

(−b+ (σ + µ)τ√τ

))−S0e

−τ(r+µ2/2)−2b2/τ+(σ+µ+2b/τ)2τ/2

×(Φ

(−c+ (σ + µ+ 2b/τ)τ√τ

)− Φ

(−b+ (σ + µ+ 2b/τ)τ√τ

))−Ke−rτ

(Φ

(−c+ µτ√τ

)− Φ

(−b+ µτ√τ

))+Ke−τ(r+µ

2/2)−2b2/τ+(µ+2b/τ)2τ/2

×(Φ

(−c+ (µ+ 2b/τ)τ√τ

)− Φ

(−b+ (µ+ 2b/τ)τ√τ

))= S0

(Φ

(δτ+

(S0

K

))− Φ

(δτ+

(S0

B

)))−S0e

−τ(r+µ2/2)−2b2/τ+(σ+µ+2b/τ)2τ/2

(Φ

(δτ+

(B2

KS0

))− Φ

(δτ+

(B

S0

)))−Ke−rτ

(Φ

(δτ−

(S0

K

))− Φ

(δτ−

(S0

B

)))+Ke−τ(r+µ

2/2)−2b2/τ+(µ+2b/τ)2τ/2

(Φ

(δ−

(B2

KS0

))− Φ

(δ−

(B

S0

))),

0 ≤ x ≤ B, where δτ±(s) is defined in (8.10). Given the relations

−τ(r+µ2/2)−2b2/τ+(σ+µ+2b/τ)2τ/2 = 2b(r/σ+σ/2) = (1+2r/σ2) log(B/S0),

and

−τ(r+µ2/2)−2b2/τ+(µ+2b/τ)2τ/2 = −rτ+2µb = −rτ+(−1+2r/σ2) log(B/S0),

this yields

e−rτ IE∗[C] = e−rτ IE∗[(Sτ −K)

+1Mτ≤B

](8.11)

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"



= S0

(Φ

(δτ+

(S0

K

))− Φ

(δτ+

(S0

B

)))−e−rτK

(Φ

(δτ−

(S0

K

))− Φ

(δτ−

(S0

B

)))−B

(B

S0

)2r/σ2 (Φ

(δτ+

(B2

KS0

))− Φ

(δτ+

(B

S0

)))+e−rτK

(S0

B

)1−2r/σ2 (Φ

(δτ−

(B2

KS0

))− Φ

(δτ−

(B

S0

)))= S0

(Φ

(δτ+

(S0

K

))− Φ

(δτ+

(S0

B

)))−S0

(B

S0

)1+2r/σ2 (Φ

(δτ+

(B2

KS0

))− Φ

(δτ+

(B

S0

)))−e−rτK

(Φ

(δτ−

(S0

K

))− Φ

(δτ−

(S0

B

)))−e−rτK

(S0

B

)1−2r/σ2 (Φ

(δτ−

(B2

KS0

))− Φ

(δτ−

(B

S0

))),

and this yields the result of Proposition 8.1, cf. § 7.3.3 pages 304-307 of [71]for a different calculation. This concludes the proof of Proposition 8.1.

Up-and-Out Barrier Put Option

The price

e−r(T−t)1Mt ≤ B IE∗

(K − xST−tS0

)+

1x max


x=St

of the up-and-out put option with maturity T , strike K and barrier B isgiven by

e−r(T−t) IE∗[P | Ft]

= St1Mt ≤ B(Φ

(δT−t+

(StK

))− 1

−(B

St

)1+2r/σ2 (Φ

(δT−t+

(B2

KSt

))− 1

))


(δT−t−

(StK

))− 1

" 171



N. Privault

−(StB

)1−2r/σ2 (Φ

(δT−t−

(B2

KSt

))− 1

))

= St1Mt ≤ B

(−Φ

(−δT−t+

(StK

))+

(B

St

)1+2r/σ2

Φ

(−δT−t+

(B2

KSt

)))−Ke−r(T−t)

×1Mt ≤ B

(−Φ

(−δT−t−

(StK

))+

(StB

)1−2r/σ2

Φ

(−δT−t−

(B2

KSt

))),

if B > K, and

e−r(T−t) IE∗[P | Ft] = St1Mt ≤ B(Φ

(δT−t+

(StB

))− 1

−(B

St

)1+2r/σ2 (Φ

(δT−t+

(B

St

))− 1

))


(δT−t−

(StB

))− 1

−(StB

)1−2r/σ2 (Φ

(δT−t−

(B

St

))− 1

))

= St1Mt ≤ B

(−Φ

(−δT−t+

(StB

))+

(B

St

)1+2r/σ2

Φ

(−δT−t+

(B

St

)))−Ke−r(T−t)

×1Mt ≤ B

(−Φ

(−δT−t−

(StB

))+

(StB

)1−2r/σ2

Φ

(−δT−t−

(B

St

))),

(8.12)

if B < K.

172


"



50 55 60 65 70 75 80 85 90 100 120

140 160

180 200

220

-2

0

2

4

6

8

10

12

up and out put price

underlying

Time in days

Fig. 8.8: Graph of the up-and-out put option price with B > K.

50 55 60 65 70 75 80 85 90 100 120

140 160

180 200

220

0

5

10

15

20

25

30

35

40

45

50

up and out put price

underlying

Time in days

Fig. 8.9: Graph of the up-and-out put option price with K > B.

Down-and-Out Barrier Call Option

Let us now consider a down-and-out barrier call option on the underlyingasset St with exercise date T , strike K, barrier B, and payoff

C = (ST −K)+

1min

0≤t≤TSt > B

=

ST −K if min

0≤t≤TSt > B,

0 if min0≤t≤T

St ≤ B,

with 0 ≤ B ≤ K. This option is also called a Callable Bull Contract with noresidual value, in which B denotes the call price, or a turbo warrant with norebate.

" 173



N. Privault

We have

e−r(T−t) IE∗[C | Ft]

= g(t, St) = StΦ

(δT−t+

(StK

))− e−r(T−t)KΦ

(δT−t−

(StK

))(8.13)

−B(B

St

)2r/σ2

Φ

(δT−t+

(B2

Kx

))+e−r(T−t)K

(StB

)1−2r/σ2

Φ

(δT−t−

(B2

KSt

))(8.14)

= BSc(St, r, T − t,K)

−B(B

St

)2r/σ2

Φ

(δT−t+

(B2

KSt

))+e−r(T−t)K

(StB

)1−2r/σ2

Φ

(δT−t−

(B2

KSt

))= BSc(St, r, T − t,K)− 1

B

(StB

)1−2r/σ2

BSc(B/St, r, T − t,K/B),

St > B, 0 ≤ t ≤ T , and

IE∗

(ST −K)+

1min

0≤t≤TSt > B

∣∣∣∣∣ Ft

= 1mint∈[0,T ] St>Bg(t, St),

t ∈ [0, T ]. When B > K we find

e−r(T−t) IE∗[C | Ft] = g(t, St)

= StΦ

(δT−t+

(StB

))− e−r(T−t)KΦ

(δT−t−

(StB

))−B

(B

St

)2r/σ2

Φ

(δT−t+

(B

St

))+e−r(T−t)K

(StB

)1−2r/σ2

Φ

(δT−t−

(B

St

)), (8.15)

St > B, 0 ≤ t ≤ T , cf. Exercise 8.3 below.

174


"



50 55 60 65 70 75 80 85 90 100 105

110 115

120 125

130 135

140

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

down and out call price

underlying

Time in days

Fig. 8.10: Graph of the down-and-out call option price with B < K.

50 55 60 65 70 75 80 85 90 100 120

140 160

180 200

220

0

10

20

30

40

50

60

down and out call price

underlying

Time in days

Fig. 8.11: Graph of the down-and-out call option price with K > B.

Down-and-Out Barrier Put Option

When B > K, the price

e−r(T−t)1mt ≥ B IE∗

(K − xST−tS0

)+

1x min

0≤r≤T−tSr/S0 ≥ B

x=St

of the down-and-out put option with maturity T , strike K and barrier B isgiven by

" 175



N. Privault

e−r(T−t) IE∗[P | Ft] (8.16)

= St1mt ≥ B(Φ

(δT−t+

(StK

))− Φ

(δT−t+

(StB

))−(B

St

)1+2r/σ2 (Φ

(δT−t+

(B2

KSt

))− Φ

(δT−t+

(B

St

))))

−e−r(T−t)K1mt ≥ B(Φ

(δT−t−

(StK

))− Φ

(δT−t−

(StB

))+

(StB

)1−2r/σ2 (Φ

(δT−t−

(B2

KSt

))− Φ

(δT−t−

(B

St

)))),

while the corresponding price vanishes when B < K.

50 55 60 65 70 75 80 85 90 100 120

140 160

180 200

220

-2

0

2

4

6

8

10

12

14

down and out put price

underlying

Time in days

Fig. 8.12: Graph of the down-and-out put option price with K > B.

Note that although Figures 8.8 and 8.10, resp. 8.9 and 8.11, appear to sharesome symmetry property, the functions themselves are not exactly symmetric.Concerning 8.7 and 8.12 the pricing function is actually the same, but theconditions B < K and B > K play opposite roles.

PDE Method

Having computed the up-and-out call option price by probabilistic arguments,we are now interested in deriving a PDE for this price.

The option price can be written as

e−r(T−t) IE∗

[(ST −K)

+1MT≤B

∣∣∣∣∣ Ft]

176


"



= e−r(T−t)1max0≤r≤t

Sr ≤ B IE∗

(ST −K)+

1maxt≤r≤T

Sr ≤ B∣∣∣∣∣ Ft

= g(t, St,Mt),

where the function g(t, x) of t and St is given by

g(t, x, y) = 1y≤Be−r(T−t) IE∗

(ST −K)+

1maxt≤r≤T

Sr ≤ B∣∣∣∣∣ St = x

.(8.17)

Next, by the same argument as in the proof of Proposition 5.2 we derive theBlack-Scholes partial differential equation (PDE) satisfied by g(t, x), for theprice of a self-financing portfolio.


(i) (ηt, ξt)t∈R+is self-financing,


Vt = g(t, St,Mt), t ∈ R+.

Then the function g(t, x, y) satisfies the Black-Scholes PDE

rg(t, x, y) =∂g

∂t(t, x, y) + ry

∂g

∂x(t, x, y) +

1

2x2σ2 ∂

2g

∂x2(t, x, y), (8.18)

t > 0, x > 0, 0 < y < B, and ξt is given by

ξt =∂g

∂x(t, St,Mt), t ∈ [0, T ], (8.19)

provided Mt < B.

Proof. By (8.17) the price at time t of the down-and-out call barrier optiondiscounted to time 0 is given by

e−rtg(t, St,Mt) = 1Mt≤Be−rT IE∗

(ST −K)+

1maxt≤r≤T

Sr ≤ B∣∣∣∣∣ St = x

= e−rT IE∗

(ST −K)+

1Mt≤B1

maxt≤r≤T

Sr ≤ B∣∣∣∣∣ St = x

" 177



N. Privault

= e−rT IE∗

(ST −K)+

1max

0≤r≤TSr ≤ B

∣∣∣∣∣ St = x

,which is a martingale. We conclude by applying the Ito formula to t 7→e−rtg(t, St,Mt) “on Mt ≤ y, 0 ≤ t ≤ T” and noting that the sum ofcomponents in factor of dt vanishes.

In the sequel we will drop the variable y in g(t, x, y) and simply write g(t, x)since

∂g

∂y(t, x, y) = 0, 0 < y < B,

and the function g(t, x, y) is constant in y ∈ (0, B).

In the next proposition we add a boundary condition to the Black-ScholesPDE (8.18) in order to hedge the up-and-out call option with maturity T ,strike K, barrier (or call price) B, and payoff

C = (ST −K)+

1max0≤t≤T

St ≤ B =

ST −K if max

0≤t≤TSt ≤ B,

0 if max0≤t≤T

St > B,

with B > K.

Proposition 8.3. The price of any self-financing portfolio of the form Vt =g(t, St) hedging the up-and-out barrier call option satisfies the Black-ScholesPDE

rg(t, x) =∂g

∂t(t, x) + rx

∂g

∂x(t, x) +

1

2x2σ2 ∂

2g

∂x2(t, x),

g(t, x) = 0, x ≥ B, t ∈ [0, T ],

g(T, x) = (x−K)+1x<B,

on the time-space domain [0, T ]× [0, B] with terminal condition

g(T, x) = (x−K)+1x<B

and additional boundary condition

g(t, B) = 0. (8.20)

Condition (8.20) holds since the price of the claim at time t is 0 wheneverSt = B, cf. e.g. [23].

178


"



The closed-form solution for this PDE is given by (8.11), as

g(t, x) = x(Φ(δT−t+

( xK

))− Φ

(δT−t+

( xB

)))(8.21)

−x( xB

)1−2r/σ2 (Φ

(δT−t+

(B2

Kx

))− Φ

(δT−t+

(B

x

)))−Ke−r(T−t)

(Φ(δT−t−

( xK

))− Φ

(δT−t−

( xB

)))+Ke−r(T−t)

( xB

)1−2r/σ2 (Φ

(δT−t−

(B2

Kx

))− Φ

(δT−t−

(B

x

))),

0 < x ≤ B, 0 ≤ t ≤ T .

We note that the expression (8.21) can be rewritten using the standardBlack-Scholes formula

BSc(S,K, r, σ, τ) = SΦ

(δτ+

(S

K

))−Ke−rτΦ

(δτ−

(S

K

))for the price of a European call option, as

g(t, x) = BSc(x,K, r, σ, T − t)− xΦ(δT−t+

( xB

))+ e−r(T−t)KΦ

(δT−t−

( xB

))−B

(B

x

)2r/σ2 (Φ

(δT−t+

(B2

Kx

))− Φ

(δT−t+

(B

x

)))+e−r(T−t)K

( xB

)1−2r/σ2 (Φ

(δT−t−

(B2

Kx

))− Φ

(δT−t−

(B

x

))),

0 < x ≤ B, 0 ≤ t ≤ T .

Figure 8.13 represents the value of Delta obtained from (8.19) for theup-and-out call option, cf. Exercise 8.3-(1).

" 179



N. Privault

Fig. 8.13: Delta for the up-and-out option.

Checking the Boundary Conditions

For x = B we check that

g(t, B) = B

(Φ

(δT−t+

(B

K

))− Φ

(δT−t+ (1)

))−e−r(T−t)K

(Φ

(δT−t−

(B

K

))− Φ

(δT−t− (1)

))−B

(Φ

(δT−t+

(B

K

))− Φ

(δT−t+ (1)

))+e−r(T−t)K

(Φ

(δT−t−

(B

K

))− Φ

(δT−t− (1)

))= 0,

and the function g(t, x) is extended to x > B by letting

g(t, x) = 0, x > B.

For x = K and t = T we find

180


"



δ0±(s) = −∞× 1s<1 +∞× 1s>1 =

+∞ if s > 1,

0 if s = 1,

−∞ if s < 1,

hence when x < K < B we have

g(T,K) = x (Φ (−∞)− Φ (−∞))

−K (Φ (−∞)− Φ (−∞))

−B(B

x

)2r/σ2

(Φ (+∞)− Φ (+∞))

+K

(B

K

)2r/σ2

(Φ (+∞)− Φ (+∞))

= 0,

when K < x B we obtain

g(T,K) = x (Φ (+∞)− Φ (+∞))

−K (Φ (+∞)− Φ (+∞))

−B(B

x

)2r/σ2

(Φ (−∞)− Φ (−∞))

+K

(B

K

)2r/σ2

(Φ (−∞)− Φ (−∞))

= 0.

Down-and-Out Barrier Call Option

Similarly the price g(t, St) at time t of the down-and-out barrier call optionsatisfies the Black-Scholes PDE

" 181



N. Privault

rg(t, x) =

∂g

∂t(t, x) + rx

∂g

∂x(t, x) +

1

2x2σ2 ∂

2g

∂x2(t, x),

g(t, B) = 0, t ∈ [0, T ],

g(T, x) = (x−K)+1x≥B,

on the time-space domain [0, T ] × [0, B] with terminal condition g(T, x) =(x−K)+1x≥B and the additional boundary condition g(t, B) = 0 since theprice of the claim at time t is 0 whenever St = B.

8.4 Lookback Options

Letmts = inf

u∈[s,t]Su

andM ts = sup

u∈[s,t]Su,

0 ≤ s ≤ t ≤ T , and let Mts be either mt

s or M ts . In the lookback option case

the payoff φ(ST ,MT0 ) depends not only on the price of the underlying asset

at maturity but it also depends on all price values of the underlying assetover the period which starts from the initial time and ends at maturity.

The payoff of such of an option is of the form φ(ST ,MT0 ) with φ(x, y) =

x− y in the case of lookback call options, and φ(x, y) = y − x in the case oflookback put options. We let

e−r(T−t) IE∗[φ(ST ,MT0 )|Ft]

denote the price at time t ∈ [0, T ] of such an option.

The Lookback Put Option

The standard lookback put option gives its holder the right to sell the un-derlying asset at its historically highest price. In this case the strike is MT

0

and the payoff isC = MT

0 − ST .Our goal is to prove the following pricing formula for lookback put options.

Proposition 8.4. The price at time t ∈ [0, T ] of the lookback put option withpayoff MT

0 − ST is given by

182


"



e−r(T−t) IE∗[MT0 − ST | Ft]

= M t0e−r(T−t)Φ

(−δT−t−

(StM t

0

))+ St

(1 +

σ2

2r

)Φ

(δT−t+

(StM t

0

))−Ste−r(T−t)

σ2

2r

(M t

0

St

)2r/σ2

Φ

(−δT−t−

(M t

0

St

))− St.

Figure 8.14 represents the lookback put price as a function of St and M t0, for

different values of the time to maturity T − t.

Fig. 8.14: Graph of the lookback put option price.

Proof of Proposition 8.4. We have

IE∗[MT0 − ST | Ft] = IE∗[MT

0 | Ft]− IE∗[ST | Ft]= IE∗[MT

0 | Ft]− er(T−t)St,

and

IE∗[MT0 | Ft] = IE∗[M t

0 ∨MTt | Ft]

= IE∗[M t01Mt

0>MTt | Ft] + IE∗[MT

t 1MTt >M

t0 | Ft]

= M t0 IE∗[1Mt

0>MTt | Ft] + IE∗[MT

t 1MTt >M

t0 | Ft]

= M t0P(M t

0 > MTt | Ft) + IE∗[MT

t 1MTt >M

t0 | Ft].

Next, we have

" 183



N. Privault

P(M t0 > MT

t | Ft) = P

(M t

0

St>MTt

St

∣∣∣∣∣ Ft)

= P

(x >

MTt

St

∣∣∣∣∣ Ft)x=Mt

0/St

= P

(MT−t

0

S0< x

)x=Mt

0/St

.

On the other hand, letting µ := r/σ − σ/2, from (8.7) we have

P(Mτ

0

S0< x

)= P(Xτ < σ−1 log x)

= Φ

(−µτ + σ−1 log x√τ

)− e2µσ−1 log xΦ

(−µτ − σ−1 log x√τ

)= Φ

(−δτ−(1/x)

)− x−1+2r/σ2

Φ(−δτ−(x)

).

Hence

P(M t0 > MT

t ) = P

(MT−t

0

S0< x

)x=Mt

0/St

= Φ

(−δT−

(StM t

0

))−(M t

0

St

)−1+2r/σ2

Φ

(−δT−

(M t

0

St

)).

Next, we have

IE∗[MTt 1MT

t >Mt0 | Ft] = St IE∗

[MTt

St1MT

t /St>Mt0/St

∣∣∣∣∣ Ft]

= St IE∗

[maxr∈[t,T ]

SrSt

1maxr∈[t,T ] Sr/St>x

∣∣∣∣∣ Ft]x=Mt

0/St

= St IE∗[

maxr∈[0,T−t]

SrS0

1maxr∈[0,T−t] Sr/S0>x

]x=Mt

0/St

,

and

IE∗[

maxr∈[0,τ ]

SrS0

1maxr∈[0,τ] Sr/S0>x

]= IE∗

[maxr∈[0,τ ]

eσBr1maxr∈[0,τ] eσBr>x

]= IE∗

[eσmaxr∈[0,τ] Br1maxr∈[0,τ] Br>σ−1 log x

]

184


"



= IE∗[eσXτ1Xτ>σ−1 log x

]=

w∞σ−1 log x

eσxfXτ (z)dz

=w∞σ−1 log x

eσz

(√2

πτe−(z−µτ)

2/(2τ) − 2µe2µzΦ

(−z − µτ√τ

))dz

=

√2

πτ

w∞σ−1 log x

eσz−(z−µτ)2/(2τ)dz − 2µ

w∞σ−1 log x

ez(σ+2µ)Φ

(−z − µτ√τ

)dz.

By standard arguments we have

1√2πτ

w∞σ−1 log x

eσz−(z−µτ)2/(2τ)dz

=1√2πτ

w∞σ−1 log x

e−(z2+µ2τ2−2(µ+σ)τz)/(2τ)dz

=1√2πτ

eσ2τ/2+µστ

w∞σ−1 log x

e−(z−(µ+σ)τ)2/(2τ)dz

=1√2πτ

erτw∞−(µ+σ)τ+σ−1 log x

e−z2/(2τ)dz

= erτΦ

(δτ+

(1

x

)),

since µσ + σ2/2 = r. The second integral

w∞σ−1 log x

ez(σ+2µ)Φ

(−z − µτ√τ

)dz

can be computed by integration by parts using the identity

w∞av′(z)u(z)dz = u(+∞)v(+∞)− u(a)v(a)−

w∞av(z)u′(z)dz,

with a = σ−1 log x. We let

u(z) = Φ

(−z − µτ√τ

)and v′(z) = ez(σ+2µ)

which satisfy

u′(z) = − 1√2πτ

e−(z+µτ)2/(2τ) and v(z) =

1

σ + 2µez(σ+2µ),

and

w∞aez(σ+2µ)Φ

(−z − µτ√τ

)dz =

w∞av′(z)u(z)dz

" 185



N. Privault

= u(+∞)v(+∞)− u(a)v(a)−w∞av(z)u′(z)dz

= − 1

σ + 2µea(σ+2µ)Φ

(−a− µτ√τ

)+

1

(σ + 2µ)√

2πτ

w∞aez(σ+2µ)e−(z+µτ)

2/(2τ)dz

= − 1


(−a− µτ√τ

)+

1

(σ + µ)√

2πτe(τ(σ+µ)

2−µ2τ)/2w∞ae−(z−τ(σ+µ))

2/(2τ)dz

= − 1


(−a− µτ√τ

)+

1

(σ + 2µ)√

2πe(τ(σ+µ)

2−µ2τ)/2w∞(a−τ(σ+µ))/

√τe−z

2/2dz

= − 1


(−a− µτ√τ

)+

1

σ + 2µe(τ(σ+µ)

2−µ2τ)/2Φ

(−a+ τ(σ + µ)√τ

)= −2r

σ(x)

2r/σ2

Φ

(−(r/σ − σ/2)τ − σ−1 log x√τ

)+

2r

σeστ(σ+2µ)/2Φ

(+τ(r/σ + σ/2)− σ−1 log x√

τ

)=

σ

2rerτΦ

(δτ+

(1

x

))− σ

2rx2r/σ

2

Φ(−δτ− (x)

),

cf. pages 317-319 of [71] for a different derivation using double integrals.

Hence we have

IE∗

[MTt 1MT

t >Mt0

∣∣∣∣∣ Ft]

= St IE∗[

maxr∈[0,T−t]

SrS0

1maxr∈[0,T−t] Sr/S0>x

]x=Mt

0/St

= 2Ster(T−t)Φ

(δT−t+

(StM t

0

))− St

µσ

rer(T−t)Φ

(δT−t+

(StM t

0

))+St

µσ

r

(M t

0

St

)2r/σ2

Φ

(−δT−t−

(M t

0

St

)),

and consequently this yields, since µσ/r = 1− σ2/(2r),

IE∗[MT0 | Ft] = IE∗[MT

0 |M t0]

= M t0P(M t

0 > MTt |M t

0) + IE∗[MTt 1MT

t >Mt0 |M

t0]

186


"



= M t0Φ

(−δT−t−

(StM t

0

))− St

(M t

0

St

)2r/σ2

Φ

(−δT−t−

(M t

0

St

))+2Ste

r(T−t)Φ

(δT−t+

(StM t

0

))−St

(1− σ2

2r

)er(T−t)Φ

(δT−t+

(StM t

0

))+St

(1− σ2

2r

)(M t

0

St

)2r/σ2

Φ

(−δT−t−

(M t

0

St

))= M t

0Φ

(−δT−t−

(StM t

0

))+ Ste

r(T−t)(

1 +σ2

2r

)Φ

(δT−t+

(StM t

0

))−St

σ2

2r

(M t

0

St

)2r/σ2

Φ

(−δT−t−

(M t

0

St

)),

hence

e−r(T−t) IE∗[MT0 − ST | Ft] = e−r(T−t) IE∗[MT

0 | Ft]− e−r(T−t) IE∗[ST | Ft]= e−r(T−t) IE∗[MT

0 |M t0]− St


(−δT−t−

(StM t

0

))− StΦ

(−δT−t+

(StM t

0

))+St

σ2

2rΦ

(δT−t+

(StM t

0

))− St

σ2

2re−r(T−t)

(M t

0

St

)2r/σ2

Φ

(−δT−t−

(M t

0

St

)).

This concludes the proof of Proposition 8.4.

PDE Method

If the couple (St,Mt) is Markov, the price can be written as a function

f(t, St,Mt) = e−rT IE∗[φ(ST ,MT ) | Ft],

and in this case the function f(t, x, y) can solve a PDE.

Next we derive the Black-Scholes partial differential equation (PDE) forthe price of a self-financing portfolio.

Black-Scholes PDE for Lookback Put Options



" 187



N. Privault

(ii) the portfolio value Vt := ηtAt + ξtSt, t ∈ R+, takes the form

Vt = f(t, St,Mt0), t ∈ R+,

for some f ∈ C2((0,∞)× (0,∞)2).

Then the function f(t, x, y) satisfies the Black-Scholes PDE

rf(t, x, y) =∂f

∂t(t, x, y) + rx

∂f

∂x(t, x, y) +

1

2x2σ2 ∂

2f

∂x2(t, x, y), t, x, y > 0,

(8.22)under the boundary conditions

f(t, 0, y) = e−r(T−t)y, 0 ≤ t ≤ T, y ∈ R+,

∂f

∂y(t, x, y)x=y = 0, 0 ≤ t ≤ T, y > 0,

f(T, x, y) = y − x, 0 ≤ x ≤ y.

(8.23a)

(8.23b)

(8.23c)

The replicating portfolio of the lookback put option is given by

ξt =∂f

∂x(t, St,M

t0), t ∈ [0, T ], (8.24)

where f(t, x, y) is given by

f(t, St,Mt0) = e−r(T−t) IE∗[φ(ST ,M

T0 ) | Ft], 0 ≤ t ≤ T. (8.25)

Proof. The existence of f(t, x, y) follows from the Markov property, moreprecisely the function f(t, x, y) satisfies

f(t, x, y) = e−r(T−t) IE∗[φ(ST ,MT0 ) | St = x, M t

0 = y]

= e−r(T−t) IE∗[φ

(xSTSt,y

x∧ M

Tt

St

)]= e−r(T−t) IE∗

[φ

(xST−tS0

,y

x∧ M

T−t0

x

)], t ∈ [0, T ],

from the time homogeneity of the asset price process (St)t∈R+. Applying the

change of variable formula to the discounted portfolio value

f(t, x, y) = e−rtf(t, x, y) = e−rT IE∗[φ(ST ,MT0 ) | St = x, M t

0 = y]

which is a martingale for t ∈ [0, T ], we have

df(t, St,Mt0) = −re−rtf(t, St,M

t0)dt+ e−rtdf(t, St,M

t0)

188


"



= −re−rtf(t, St,Mt0)dt+ e−rt

∂f

∂t(t, St,M

t0)dt+ re−rtSt

∂f

∂x(t, St,M

t0)dt

+1

2e−rtσ2S2

t

∂2f

∂x2(t, St,M

t0)dt

+e−rt∂f

∂y(t, St,M

t0)dM t

0 + e−rtσSt∂f

∂x(t, St,M

t0)dBt.

Since(IE∗[φ(ST ,M

T0 ) | Ft]

)t∈[0,T ]

is a P-martingale and (M t0)t∈[0,T ] has finite

variation (it is in fact a non-decreasing process), we have:

df(t, St,Mt0) = σSt

∂f

∂x(t, St,M

t0)dBt, t ∈ [0, T ], (8.26)

and the function f(t, x, y) satisfies the equation

∂f

∂t(t, St,M

t0)dt+ rSt

∂f

∂xf(t, St,M

t0)dt

+1

2σ2S2

t

∂2f

∂x2(t, St,M

t0)dt+

∂f

∂y(t, St,M

t0)dM t

0 = rf(t, St,Mt0)dt,

which implies

∂f

∂t(t, St,M

t0) + rSt

∂f

∂x(t, St,M

t0) +

1

2σ2S2

t

∂2f

∂x2(t, St,M

t0) = rf(t, St,M

t0),

which is (8.22), and∂f

∂y(t, St,M

t0)dM t

0 = 0,

because M t0 increases only on a set of zero measure (which has no isolated

points). This implies∂f

∂y(t, St, St) = 0,

which shows the boundary condition (8.23b), since M t0 hits St when M t

0

increases. On the other hand, (8.26) shows that

φ(ST ,MT0 ) = IE∗[φ(ST ,M

T0 )] + σ

w T

0St∂f

∂x(t, x,M t

0)|x=StdBt,

0 ≤ t ≤ T , which implies (8.24) as in the proof of Proposition 5.2.

In other words, the price of the lookback put option takes the form

f(t, St,Mt) = e−r(T−t) IE∗

[MT

0 − ST∣∣∣∣∣ Ft

],

where the function f(t, x, y) is given by

" 189



N. Privault

f(t, x, y) = ye−r(T−t)Φ(−δT−t− (x/y)

)+ x

(1 +

σ2

2r

)Φ(δT−t+ (x/y)

)−xσ

2

2re−r(T−t) (y/x)

2r/σ2

Φ(−δT−t− (y/x)

)− x.


The boundary condition (8.23a) is explained by the fact that

f(t, 0, y) = e−r(T−t) IE∗[MT0 − ST | St = 0, M t

0 = x]

= e−r(T−t) IE∗[M t0 − ST | St = 0, M t

0 = x]

= e−r(T−t) IE∗[M t0 |M t

0 = x]− e−r(T−t) IE∗[ST | St = 0]

= xe−r(T−t),

since IE∗[ST | St = 0] = 0 as St = 0 implies ST = 0. On the other hand,(8.23c) follows from the fact that

f(T, x, y) = IE∗[MT0 − ST | ST = x, MT

0 = y] = y − x.

Note that we havef(t, x, x) = xC(T − t),

with

C(τ) = e−rτΦ(−δτ−(1)

)+

(1 +

σ2

2r

)Φ(δτ+ (1)

)− σ2

2re−rτΦ

(−δτ− (1)

)− 1,

τ > 0, hence∂f

∂x(t, x, x) = C(T − t), t ∈ [0, T ],

while we also have

∂f

∂y(t, x, y)y=x = 0, 0 ≤ x ≤ y.

Scaling Property of Lookback Put Prices

We note the scaling property

190


"



f(t, x, y) = e−r(T−t) IE∗

[MT

0 − ST∣∣∣∣∣ St = x, Mt = y

]


[M t

0 ∨MTt − ST

∣∣∣∣∣ St = x, Mt = y

]

= e−r(T−t)x IE∗

[M t

0

St∨ M

Tt

St− 1

∣∣∣∣∣ St = x, Mt = y

]


[y

x∨ M

Tt

St− 1

∣∣∣∣∣ St = x, Mt = y

]


[M t

0 ∨MTt − 1

∣∣∣∣∣ St = 1, Mt =y

x

]= xf(t, 1, y/x)

= xg(T − t, x/y),

where we let

g(τ, z) :=1

ze−rτΦ

(−δτ−(z)

)+

(1 +

σ2

2r

)Φ(δτ+ (z)

)−σ

2

2re−rτ

(1

z

)2r/σ2

Φ(−δτ−(1/z))− 1,

with the boundary condition∂g

∂z(τ, 1) = 0, τ > 0,

g(0, z) =1

z− 1, z ∈ (0, 1].

(8.27a)

(8.27b)

The next Figure 8.15 shows a graph of the function g(τ, z).

" 191



N. Privault

0.6 0.7

0.8 0.9

1 0 0.2

0.4 0.6

0.8 1 0

0.2

0.4

0.6

0.8

1

1.2

normalized lookback put price

z τ

Fig. 8.15: Graph of the normalized lookback put option price.

Black-Scholes Approximation of Lookback Put Prices

Letting

BSp(S,K, r, σ, τ) = Ke−rτΦ

(−δτ−

(S

K

))− SΦ

(−δτ+

(S

K

))denote the standard Black-Scholes formula for the price of a European putoption, we observe that the lookback put option price satisfies

e−r(T−t) IE∗[MT0 − ST | Ft] = BSp(St,M

t0, r, σ, T − t)

+Stσ2

2r

(Φ

(δT−t+

(StM t

0

))− e−r(T−t)

(M t

0

St

)2r/σ2

Φ

(−δT−t−

(M t

0

St

))),

i.e.

e−r(T−t) IE∗

[MT

0 − ST∣∣∣∣∣ Ft

]= BSp(St,M

t0, r, σ, T − t) + Sthp

(T − t, St

M t0

)where the function

hp(τ, z) =σ2

2r

(Φ(δτ+ (z)

)− e−rτz−2r/σ2

Φ(−δτ− (1/z)

)), (8.28)

depends only on time τ and z = St/Mt0. In other words, due to the relation

BSp(x, y, r, σ, τ) = ye−rτΦ

(−δτ−

(x

y

))− xΦ

(−δτ+

(x

y

))= xBSp(1, y/x, r, σ, τ)

192


"



for the standard Black-Scholes put formula, we observe that f(t, x, y) satisfies

f(t, x, y) = xBSp(1, y/x, r, σ, T − t) + xh(T − t, x/y),

i.e.f(t, x, y) = xg(T − t, x/y),

withg(τ, z) = BSp(1, 1/z, r, σ, τ) + hp(τ, z), (8.29)

where hp(τ, z) is the function given by (8.28), and (x, y) 7→ xhp(T − t, x/y)also satisfies the Black-Scholes PDE (8.22), i.e. (τ, z) 7→ BSp(1, 1/z, r, σ, τ)and hp(τ, z) both satisfy the PDE

∂hp∂τ

(τ, z) = z(r + σ2

) ∂hp∂z

(τ, z) +1

2σ2z2

∂2hp∂z2

(τ, z), (8.30)

τ ∈ R+, z ∈ [0, 1], under the boundary condition

hp(0, z) = 0, 0 ≤ z ≤ 1.

The next Figures 8.16 and 8.17 show the decompositions (8.29) of the normal-ized lookback put option price g(τ, z) in Figure 8.15 into the Black-Scholesput function BSp(1, 1/z, r, σ, τ) and hp(τ, z).

0.6 0.7

0.8 0.9

1 0 0.2

0.4 0.6

0.8 1 0

0.2

0.4

0.6

0.8

1

1.2

normalized Black-Scholes put price BSp(1,1/z,r,σ,τ)

z τ

Fig. 8.16: Black-Scholes put price in the decomposition (8.29).

" 193



N. Privault

0.6 0.7

0.8 0.9

1 0 0.2

0.4 0.6

0.8 1 0

0.2

0.4

0.6

0.8

1

1.2

h(τ,x)

z τ

Fig. 8.17: Function hp(τ, z) in the decomposition (8.29).

Note that in Figures 8.16-8.17 the condition hp(0, z) = 0 is not fully respectedas z → 1 due to numerical error in the approximation of the function Φ.

The Lookback Call Option

The standard Lookback call option gives the right to buy the underlying assetat its historically lowest price. In this case the strike is mT

0 and the payoff is

C = ST −mT0 .

The following result gives the price of the lookback call option, cf. e.g. Propo-sition 9.5.1, page 270 of [15].

Proposition 8.6. The price at time t ∈ [0, T ] of the lookback call optionwith payoff ST −mT

0 is given by

e−r(T−t) IE∗[ST −mT0 | Ft]

= StΦ

(δT−t+

(Stmt

0

))−mt

0e−r(T−t)Φ

(δT−t−

(Stmt

0

))+e−r(T−t)St

σ2

2r

(mt

0

St

)2r/σ2

Φ

(δT−t−

(mt

0

St

))− St

σ2

2rΦ

(−δT−t+

(Stmt

0

)).

Figure 8.18 represents the price of the lookback call option as a function ofmt

0 and St for different values of the time to maturity T − t.

194


"



Fig. 8.18: Graph of the lookback call option price.

Proof of Proposition 8.6. We have

e−r(T−t) IE∗[ST −mT0 | Ft] = e−r(T−t) IE∗[ST | Ft]− e−r(T−t) IE∗[mT

0 | Ft],

and

IE∗[mT0 | Ft] = IE∗[mt

0 ∧mTt | Ft]

= IE∗[mt01mt0<mTt | Ft] + IE∗[mT

t 1mt0>mTt | Ft]= mt

0 IE∗[1mt0<mTt | Ft] + IE∗[mTt 1mt0>mTt | Ft]

= mt0P(mt

0 < mTt | Ft) + IE∗[mT

t 1mt0>mTt | Ft].

By computations similar to those of the lookback put option case we find

P(mt0 < mT

t | Ft) = P

(mt

0

St<mTt

St

∣∣∣∣∣ Ft)

= P

(x <

mTt

St

∣∣∣∣∣ Ft)x=mt0/St

= P

(mT−t

0

S0> x

)x=mt0/St

= Φ

(δT−t−

(Stmt

0

))−(mt

0

St

)−1+2r/σ2

Φ

(δT−t−

(mt

0

St

)),

and

" 195



N. Privault

IE∗[mTt 1mt0>mTt | Ft] = St IE∗

[minr∈[t,T ]

SrSt

1mt0/St>mTt /St

]x=mt0,y=St

= St IE∗[

minr∈[t,T ]

SrSt

1minr∈[t,T ] Sr/St<x

]x=mt0/St

= St IE∗[

minr∈[0,T−t]

SrS0

1minr∈[0,T−t] Sr/S0<x

]x=mt0/St

= 2Ster(T−t)Φ

(−δT−t+

(Stmt

0

))− St

µσ

rer(T−t)Φ

(−δT−t+

(Stmt

0

))+St

µσ

r

(mt

0

St

)2r/σ2

Φ

(δT−t−

(mt

0

St

)).

Given the relation µσ/r = 1− σ2/(2r), this yields

e−r(T−t) IE∗[ST −mT0 | Ft] = St −mt

0e−r(T−t)P

(mT−t

0

S0> x

)x=mt0/St

−Ste−r(T−t) IE∗[

minr∈[0,T−t]

SrS0

1minr∈[0,T−t] Sr/S0<x

]x=mt0/St

= St −mt0e−r(T−t)Φ

(δT−t−

(Stmt

0

))+mt

0e−r(T−t)

(mt

0

St

)−1+2r/σ2

Φ

(δT−t−

(mt

0

St

))−2StΦ

(−δT−t+

(Stmt

0

))+ St

µσ

rΦ

(−δT−t+

(Stmt

0

))−Ste−r(T−t)

µσ

r

(mt

0

St

)2r/σ2

Φ

(δT−t−

(mt

0

St

))= St −mt

0e−r(T−t)Φ

(δT−t−

(Stmt

0

))− St

(1 +

σ2

2r

)Φ

(−δT−t+

(Stmt

0

))+Ste

−r(T−t)σ2

2r

(mt

0

St

)2r/σ2

Φ

(δT−t−

(mt

0

St

))= StΦ

(δT−t+

(Stmt

0

))− e−r(T−t)mt

0Φ

(δT−t−

(Stmt

0

))+e−r(T−t)

Stσ2

2r

((mt

0

St

)2r/σ2

Φ

(δT−t−

(mt

0

St

))− er(T−t)Φ

(−δT−t+

(Stmt

0

))).

Black-Scholes Approximation of Lookback Call Prices

Letting

196


"



BSc(S,K, r, σ, τ) = SΦ

(δτ+

(S

K

))−Ke−rτΦ

(δτ−

(S

K

))denote the standard Black-Scholes formula for the price of a European calloption, we observe that the lookback call option price satisfies

e−r(T−t) IE∗[ST −mT0 | Ft] = BSc(St,m

t0, r, σ, T − t)

−Stσ2

2r

(Φ

(−δT−t+

(Stmt

0

))− e−r(T−t)

(mt

0

St

)2r/σ2

Φ

(δT−t−

(mt

0

St

))),

i.e.

e−r(T−t) IE∗[ST −mT0 | Ft] = BSc(St,m

t0, r, σ, T − t) + Sthc

(T − t, St

mt0

)where the function

hc(τ, z) = −σ2

2r

(Φ(−δτ+ (z)


Φ(δτ− (1/z)

)), (8.31)

depends only on z = St/mt0 and satisfies

hc(τ, z) = hp(τ, z)−σ2

2r

(1− e−rτz−2r/σ2

), τ ∈ R+, z ∈ R+,

where (z, τ) 7→ e−rτz−2r/σ2

also solves the PDE (8.30).

Black-Scholes PDE for Lookback Call Options

By the same argument as in the proof of Proposition 8.5, the functionf(t, x, y) satisfies the Black-Scholes PDE

rf(t, x, y) =∂f

∂t(t, x, y) + rx

∂f

∂x(t, x, y) +

1

2x2σ2 ∂

2f

∂x2(t, x, y), t, x > 0,

under the boundary conditions

limy0

f(t, x, y) = x, 0 ≤ t ≤ T, x > 0,

∂f

∂y(t, x, y)x=y = 0, 0 ≤ t ≤ T, y > 0,

f(T, x, y) = x− y, 0 ≤ y ≤ x,

(8.32a)

(8.32b)

(8.32c)

" 197



N. Privault

and the corresponding self-financing hedging strategy is given by

ξt =∂f

∂x(t, St,m

t0), t ∈ [0, T ], (8.33)

which represents the quantity of the risky asset St to be held at time t in thehedging portfolio.

In other words, the price of the lookback call option takes the form

f(t, St,mt) = e−r(T−t) IE∗[ST −mT0 | Ft],


f(t, x, y) = xΦ

(δT−t+

(x

y

))− e−r(T−t)yΦ

(δT−t−

(x

y

))(8.34)

+e−r(T−t)xσ2

2r

((yx

)2r/σ2

Φ(δT−t−

(yx

))− er(T−t)Φ

(−δT−t+

(x

y

)))= x− ye−r(T−t)Φ

(δT−t−

(x

y

))− x

(1 +

σ2

2r

)Φ

(−δT−t+

(x

y

))+xe−r(T−t)

σ2

2r

(yx

)2r/σ2

Φ(δT−t−

(yx

)).


The boundary condition (8.32a) is explained by the fact that

f(t, x, 0) = e−r(T−t) IE∗[ST −mT0 | St = x, mt

0 = 0]

= e−r(T−t) IE∗[ST | St = x, mt0 = 0]

= e−r(T−t) IE∗[ST | St = x]

= e−r(T−t)x.

On the other hand, (8.32b) follows from the fact that

f(T, x, y) = IE∗[ST −mT0 | ST = x, mT

0 = y] = x− y.

We havef(t, x, x) = xC(T − t),

with

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"



C(τ) = 1− e−rτΦ(δτ− (1)

)−(

1 +σ2

2r

)Φ(−δτ+ (1)

)+ e−rτ

σ2

2rΦ(δτ− (1)

),

τ > 0, hence∂f

∂x(t, x, x) = C(T − t), t ∈ [0, T ],

while we also have

∂f

∂y(t, x, y)y=x = 0, 0 ≤ x ≤ y.

Scaling Property of Lookback Call Prices

We note the scaling property


[ST −mT

0

∣∣∣∣∣ St = x, mt = y

]


[mt

0 ∧mTt − ST

∣∣∣∣∣ St = x, mt = y

]


[mt

0

St∨ m

Tt

St− 1

∣∣∣∣∣ St = x, mt = y

]


[y

x∨ m

Tt

St− 1

∣∣∣∣∣ St = x, mt = y

]


[mt

0 ∨mTt − 1

∣∣∣∣∣ St = 1, mt = y/x

]= xf(t, 1, y/x),

hence letting

g(τ, z) = 1− 1

ze−rτΦ

(δτ− (z)

)−(

1 +σ2

2r

)Φ(−δτ+ (z)

)+σ2

2re−rτz−2r/σ

2

Φ(δτ−(1/z)),

we have g(τ, 1) = C(T − t), and

f(t, x, y) = xg(T − t, x/y)

and the boundary condition

" 199



N. Privault

∂g

∂z(τ, 1) = 0, τ > 0,

g(0, z) = 1− 1

z, z ≥ 1.

(8.35a)

(8.35b)

The next Figure 8.19 shows a graph of the function g(τ, z).

1 1.5

2 2.5

3

0 50

100 150

200

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

normalized lookback call priceoption price path

zt

Fig. 8.19: Normalized lookback call option price.

The next Figure 8.20 represents the path of the underlying asset price usedin Figure 8.19.

Fig. 8.20: Graph of the underlying asset price.

200


"



Next we represent the option price as a function of time.

0

10

20

30

40

50

60

0 50 100 150 200

t

option price pathSt-mt

Fig. 8.21: Graph of the lookback call option price.

The next Figure 8.22 represents the corresponding underlying asset price andits running minimum.

20

30

40

50

60

70

80

90

100

0 50 100 150 200

t

Stmt

Fig. 8.22: Running minimum of the underlying asset price.

Due to the relation

BSc(x, y, r, σ, τ) = xΦ

(δτ+

(x

y

))− ye−rτΦ

(δτ−

(x

y

))= xBSc(1, y/x, r, σ, τ)

for the standard Black-Scholes call formula, we observe that f(t, x, y) satisfies

" 201



N. Privault

f(t, x, y) = xBSc(1, y/x, r, σ, T − t) + xhc(T − t, x/y),

i.e.f(t, x, y) = xg(T − t, x/y),

withg(τ, z) = BSc(1, 1/z, r, σ, τ) + hc(τ, z), (8.36)

where hc(τ, z) is the function given by (8.31), and (x, y) 7→ xhc(T − t, x/y)also satisfies the Black-Scholes PDE (8.22), i.e. (τ, z) 7→ BSc(1, 1/z, r, σ, τ)and hc(τ, z) both satisfy the PDE (8.30) under the boundary condition

hc(0, z) = 0, z ≥ 1.

The next Figures 8.23 and 8.24 show the decomposition of g(t, z) in (8.36) andFigures 8.19-8.20 into the sum of the Black-Scholes call function BSc(1, 1/z, r, σ, τ)and h(t, z).

1 1.5

2 2.5

3

0 50

100 150

200

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

normalized Black-Scholes put price BSc(1,1/z,r,σ,T-t)

zt

Fig. 8.23: Black-Scholes call price in the decomposition (8.36) of the normalizedlookback call option price g(τ, z).

202


"



1 1.5

2 2.5

3

0 50

100 150

200

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

h(T-t,x)

zt

Fig. 8.24: Function hc(τ, z) in the decomposition (8.36) of the normalized lookbackcall option price g(τ, z).

We also note that

IE∗[MT0 −mT

0 | S0 = x] = x− xe−r(T−t)Φ(δT−t− (1)

)−x(

1 +σ2

2r

)Φ(−δT−t+ (1)

)+ xe−r(T−t)

σ2

2rΦ(δT−t− (1)

)+xe−r(T−t)Φ

(−δT−t− (1)

)+ x

(1 +

σ2

2r

)Φ(δT−t+ (1)

)−xσ

2

2re−r(T−t)Φ

(−δT−t− (1)

)− x

= x

(1 +

σ2

2r

)(Φ(δT−t+ (1)

)− Φ

(−δT−t+ (1)

))+xe−r(T−t)

(σ2

2r− 1

)(Φ(δT−t− (1)

)− Φ

(−δT−t− (1)

)).

Hedging of Lookback Options

In this section we compute hedging strategies for lookback options by ap-plication of the Delta hedging formula (8.33). See [3], § 2.6.1, page 29, foranother approach to the following result using the Clark-Ocone formula.Here we use (8.33) instead, cf. Proposition 4.6 of [45].

Proposition 8.7. The hedging strategy of the lookback call option is givenby

ξt = Φ

(δT−t+

(Stmt

0

))− σ2

2rΦ

(−δT−t+

(Stmt

0

))(8.37)

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N. Privault

+e−r(T−t)(mt

0

St

)2r/σ2 (σ2

2r− 1

)Φ

(δT−t−

(mt

0

St

)), t ∈ [0, T ].

Proof. We need to differentiate

f(t, x, y) = BSc(x, y, r, σ, T − t) + xhc

(T − t, x

y

)with respect to the variable x, where

hc(τ, z) = −σ2

2r

(Φ(−δτ+ (z)


Φ(δτ− (1/z)

))is given by (8.31) First we note that the relation

∂

∂xBSc(x, y, r, σ, τ) = Φ

(δτ+

(x

y

))is known, cf. Propositions 5.4 and 6.7. Next, we have

∂

∂x

(xhc

(τ,x

y

))= hc

(τ,x

y

)+x

y

∂hc∂z

(τ,x

y

),

and

∂hc∂z

(τ, z) = −σ2

2r

(∂

∂x

(Φ(−δτ+(z)

))− e−rτz−2r/σ2 ∂

∂z

(Φ(δτ−(1/z))

))−σ

2

2r

(2r

σ2e−rτz−1−2r/σ

2

Φ(δτ−(1/z))

)=

σ

2rz√

2πτexp

(−1

2

(δτ+(z)

)2)−e−rτz−2r/σ2 σ

2rz√

2πτexp

(−1

2(δτ−(1/z))

2

+2r

σ2e−rτz−1−2r/σ

2

Φ(δτ−(1/z))

).

Next we note that

e−(δτ−(1/z))

2/2 = exp

(−1

2

(δτ+(z)

)2 − 1

2

(4r2

σ2τ − 4r

σδτ+(z)

√τ

))= e−

12 (δτ+(z))

2

exp

(−1

2

(4r2

σ2τ − 4r

σ2

(log z + (r +

1

2σ2)τ

)))= e−

12 (δτ+(z))

2

exp

(−2r2

σ2τ +

2r

σ2log z +

2r2

σ2τ + rτ

)= erτz2r/σ

2

e−(δτ+(z))2/2 (8.38)

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"



as in the proof of Proposition 5.4, hence

∂hc∂z

(τ,x

y

)= −e−rτz−1−2r/σ2

Φ(δτ−(1/z)),

and

∂

∂x

(xhc

(τ,x

y

))= hc

(τ,x

y

)− e−rτ

(yx

)2r/σ2

Φ(δτ−

(yx

)),

which concludes the proof.

Similar calculations using (8.24) can be carried out for other types of look-back options, such as options on extrema and partial lookback options, cf.[44].

As a consequence of (8.37) we have

e−r(T−t) IE∗[ST −mT0 | Ft]

= StΦ

(δT−t+

(Stmt

0

))−mt

0e−r(T−t)Φ

(δT−t−

(Stmt

0

))+e−r(T−t)St

σ2

2r

(Stmt

0

)−2r/σ2

Φ

(δT−t−

(mt

0

St

))− St

σ2

2rΦ

(−δT−t+

(Stmt

0

))= ξtSt −mt

0e−r(T−t)

(Φ

(δT−t−

(Stmt

0

))+

(Stmt

0

)1−2r/σ2

Φ

(δT−t−

(mt

0

St

))),

and the quantity of the riskless asset ert in the portfolio is given by

ηt = −mt0e−rT

(Φ

(δT−t−

(Stmt

0

))+

(Stmt

0

)1−2r/σ2

Φ

(δT−t−

(mt

0

St

)))≤ 0,

so that the portfolio value Vt at time t satisfies

Vt = ξtSt + ηtert, t ∈ R+,

and one has to constantly borrow from the riskless account in order to hedgethe lookback option.

8.5 Asian Options

As we will see below there exists no easily tractable closed form solution forthe price of an arithmetically averaged Asian option.

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N. Privault

General Results

An option on average is an option whose payoff has the form

C = φ(YT , ST ),

where

YT = S0

w T

0eσBu+ru−σ

2u/2du =w T

0Sudu, T ∈ R+.

• For example when φ(y, x) = (y/T −K)+ this yields the Asian call optionwith payoff (

1

T

w T

0Sudu−K

)+

=

(YTT−K

)+

, (8.39)

which is a path-dependent option whose price at time t ∈ [0, T ] is givenby

e−r(T−t) IE∗

[(1

T

w T

0Sudu−K

)+∣∣∣∣∣ Ft

]. (8.40)

• As another example, when φ(y, x) = e−y this yields the price

P (0, T ) = IE∗[e−

r T0Sudu

]= IE∗

[e−YT

]at time 0 of a bond with underlying short term rate process St.

The option with payoff C = φ(YT , ST ) can be priced as

e−r(T−t) IE∗[φ(YT , ST ) | Ft] = e−r(T−t) IE∗

[φ

(Yt +

w T

tSudu, ST

) ∣∣∣∣∣ Ft]


[φ

(y + x

w T

t

SuStdu, x

STSt

) ∣∣∣∣∣ Ft]y=Yt,x=St

= e−r(T−t) IE∗[φ

(y + x

w T−t

0

SuS0du, x

ST−tS0

)]y=Yt,x=St

. (8.41)

Hence the option can be priced as

f(t, St, Yt) = e−r(T−t) IE∗[φ(YT , ST ) | Ft],


f(t, x, y) = e−r(T−t) IE∗[φ

(y + x

w T−t

0

SuS0du, x

ST−tS0

)].

First we note that the numerical computation of Asian option prices canbe done using the joint probability density ψYT−t,BT−t of (YT−t, BT−t), as

206


"



follows:

f(t, x, y) =

e−r(T−t)w∞0

w∞−∞

φ(y + xz, xeσu+r(T−t)−σ

2(T−t)/2)ψYT−t,BT−t(z, u)dzdu.

In [79], Proposition 2, the joint probability density of

(Yt, Bt) =

(w t

0S0e

σBs−pσ2s/2ds,Bt − pσt/2), t > 0,

has been computed in the case σ = 2, cf. also [52]. In the next propositionwe restate this result for an arbitrary variance parameter σ after rescaling.Let θ(v, τ) denote the function defined as

θ(v, τ) =veπ

2/(2τ)

√2π3τ

w∞0e−ξ

2/(2τ)e−v cosh ξ sinh(ξ) sin (πξ/τ) dξ, v, τ > 0.

(8.42)

Proposition 8.8. For all t > 0 we have

P(w t

0eσBs−pσ

2s/2ds ∈ dz, Bt − pσt/2 ∈ du)

=σ

2e−pσu/2−p

2σ2t/8 exp

(−2

1 + eσu

σ2z

)θ

(4eσu/2

σ2z,σ2t

4

)dz

zdu,

u ∈ R, z > 0.

The expression of this probability density can then been used for the pricingof options on average such as (8.41), as

f(t, x, y) = e−r(T−t) IE∗[φ

(y + x

w T−t

0

SvS0dv, x

ST−tS0

)]= e−r(T−t)

×w∞0φ(y + xz, xeσu+r(T−t)−σ

2(T−t)/2)P(w T−t

0

SvS0dv ∈ dz,BT−t ∈ du

)=σ

2e−r(T−t)+p

2σ2(T−t)/8w∞0

w∞−∞

φ(y + xz, xeσu+r(T−t)−σ

2(1+p)(T−t)/2)

× exp

(−2

1 + eσu−pσ2(T−t)/2

σ2z− p

2σu

)θ

(4eσu/2−pσ

2(T−t)/4

σ2z,σ2(T − t)

4

)dudz

z

= e−r(T−t)−p2σ2(T−t)/8

w∞0

w∞0φ(y + x/z, xv2er(T−t)−σ

2(T−t)/2)

×v−1−p exp

(−2z

1 + v2

σ2

)θ

(4vz

σ2,σ2(T − t)

4

)dvdz

z,

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N. Privault

which actually stands as a triple integral due to the definition (8.42) of θ(v, τ).Note that here the order of integration between du and dz cannot be ex-changed without particular precautions, at the risk of wrong computations.

The Asian Call Option

We have

e−r(T−t) IE∗

[(1

T

w T

0Sudu−K

)+∣∣∣∣∣ Ft

]


[(1

T

(Yt +

w T

tSudu

)−K

)+∣∣∣∣∣ Ft

]


[(1

T

(y + x

w T

t

SuStdu

)−K

)+∣∣∣∣∣ Ft

]x=St, y=Yt


[(1

T

(y + x

w T−t

0

SuS0du

)−K

)+]x=St, y=Yt

.

Hence the option can be priced as

f(t, St, Yt) = e−r(T−t) IE∗

[(1

T

w T

0Sudu−K

)+∣∣∣∣∣ Ft

],

where the function f(t, x, y) is defined by


[(1

T

(y + x

w T−t

0

SuS0du

)−K

)+]


[(1

T

(y +

x

S0YT−t

)−K

)+].

Probabilistic Approach

First we note that the numerical computation of Asian option prices can bedone using the probability density of

YT =w T

0Stdt.

From Proposition 8.8 we deduce the marginal density of

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"



Yt =w t

0eσBs−pσ

2s/2ds,

as follows:

P(w t

0eσBs−pσ

2s/2ds ∈ du)

=σ

2uep

2σ2t/8w∞−∞

exp

(−2

1 + eσv−pσ2t/2

σ2u− p

2σv

)θ

(4eσv/2−pσ

2t/4

σ2u,σ2t

4

)dvdu

= e−p2σ2t/8

w∞0v−1−p exp

(−2

1 + v2

σ2u

)θ

(4v

σ2u,σ2t

4

)dvdu

u,

u > 0. From this we get

P(Yt/S0 ∈ du) = P(w t

0Stdt ∈ du

)(8.43)

= e−p2σ2t/8

w∞0v−1−p exp

(−2

1 + v2

σ2u

)θ

(4v

σ2u,σ2t

4

)dvdu

u,

where St = S0eσBt−pσ2t/2 and p = 1 − 2r/σ2. This probability density can

then be used for the pricing of Asian options, as


[(1

T

(y +

x

S0YT−t

)−K

)+]

(8.44)

= e−r(T−t)w∞0

(y + xz

T−K

)+

P(YT−t/S0 ∈ dz)

= e−r(T−t)σ

2e−p

2σ2(T−t)/8w∞0

w∞0

(y + xz

T−K

)+

×v−1−p exp

(−2

1 + v2

σ2z

)θ

(4v

σ2z,σ2(T − t)

4

)dvdz

z

=1

Te−r(T−t)−p

2σ2(T−t)/8w∞0∨(KT−y)/x

w∞0

(xz + y −KT )

× exp

(−2

1 + v2

σ2z

)θ

(4v

σ2z,σ2(T − t)

4

)dvdz

z

=4x

σ2Te−r(T−t)−p

2σ2(T−t)/8w∞0

w∞0

(1

z− σ2(KT − y)

4x

)+

×v−1−p exp

(−z 1 + v2

2

)θ

(vz,

σ2(T − t)4

)dvdz

z,

cf. the Theorem in § 5 of [9], which is actually a triple integral due to thedefinition (8.42) of θ(v, t). Note that since the integrals are not absolutelyconvergent, here the order of integration between dv and dz cannot be ex-

" 209



N. Privault

changed without particular precautions, at the risk of wrong computations.

The time Laplace transform of Asian option prices has been computed in[29], and this expression can be used for pricing by numerical inversion of theLaplace transform. The following Figure 8.25 represents Asian option pricescomputed by the Geman-Yor [29] method.

0 50 100 150 200 250 300 350

80

85

90

95

100

5

10

15

20

25

30

Asian option price

Time in days

underlying

Fig. 8.25: Graph of the Asian option price with σ = 0.3, r = 0.1 and K = 90.

We refer to e.g. [2], [9], [20], and references therein for more on Asian op-tion pricing using the probability density of the averaged geometric Brownianmotion.

Figure 7.1 presents a graph of implied volatility surface for Asian options onlight sweet crude oil futures.

Lognormal approximation

Other numerical approaches to the pricing of Asian options include [49], [73]which relies on approximations of the average price probability based onthe Lognormal distribution. The lognormal distribution with mean µ andvariance η2 has the probability density function

g(x) =1

η√

2πe−(µ−log x)

2/(2η2) dx

x,

where x > 0, µ ∈ R, η > 0, and moments

E[X] = eµ+η2/2 and E[X2] = e2µ+2η2 . (8.45)

Under the lognormal approximation, asian options on the time integral

ΛT :=w T

0Stdt

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of geometric Brownian motion

St = eσBt+(r−σ2/2)t, t ∈ [0, T ],

are computed by approximating ΛT by a lognormal random variable, as

e−rTE

[(1

T

w T

0Stdt−K

)+]' 1

Teµ+σ

2/2Φ(d1)−KΦ(d2), (8.46)

where

d1 =log(E[ΛT ]/(KT ))

σ√T

+ σ

√T

2=µT + σ2T − log(KT )

σ√T

and

d2 = d1 − σ√T =

log(E[ΛT ]/(KT ))

σ√T

− σ√T

2,

and µ, σ are estimated as

σ2 =1

Tlog

(E[Λ2

T ]

(E[ΛT ])2

)and

µ =1

TlogE[ΛT ]− 1

2σ2,

based on the first two moments of the lognormal distribution, cf. (8.45) below.The next Figure 8.26 compares the lognormal approximation to a MonteCarlo estimate of Asian option prices with σ = 0.5, r = 0.05 and K/St = 1.1

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0 1 2 3 4 5 6 7 8 9 10

asia

n op

tion

pric

e

time t

lognormal approximationMonte Carlo estimate

Fig. 8.26: Lognormal approximation to the Asian option price.

" 211



N. Privault

For reference, in the next proposition we compute the unconditional meanand variance of ΛT , which have been used in (8.46), cf. also (7) and (8) page480 of [49].

Proposition 8.9. We have

E[ΛT ] = S0erT − 1

r,

and

E[(ΛT )2] = 2S20

re(2r+σ2)T − (2r + σ2)erT + (r + σ2)

r(r + σ2)(2r + σ2).

Proof. The computation of the first moment is straightforward. For thesecond moment we have, letting p = 1− 2r/σ2,

E[(ΛT )2] = S20

w T

0

w T

0e−pσ

2a/2−pσ2b/2E[eσBaeσBb ]dbda

= 2S20

w T

0

w a

0e−pσ

2a/2−pσ2b/2eσ2(a+b)/2ebσ

2

dbda

= 2S20

w T

0e−(p−1)σ

2a/2w a

0e−(p−3)σ

2b/2dbda

=4S2

0

(p− 3)σ2

w T

0e−(p−1)σ

2a/2(1− e−(p−3)σ2a/2)da

=4S2

0

(p− 3)σ2

w T

0e−(p−1)σ

2a/2da− 4S20

(p− 3)σ2

w T

0e−(p−1)σ

2a/2e−(p−3)σ2a/2da

=8S2

0

(p− 3)(p− 1)σ4(1− e−(p−1)σ2T/2)− 4S2

0

(p− 3)σ2

w T

0e−(2p−4)σ

2a/2da

=8S2

0

(p− 3)(p− 1)σ4(1− e−(p−1)σ2T/2)− 4S2

0

(p− 3)(p− 2)σ4(1− e−(p−2)σ2T )

= 2S20

re(2r+σ2)T − (2r + σ2)erT + (r + σ2)

r(r + σ2)(2r + σ2),

since r − σ2/2 = −pσ2/2.

PDE Method - Two Variables

The price at time t of the Asian option with payoff (8.39) can be written as

f(t, St, Yt) = e−r(T−t) IE∗

[(1

T

w T

0Sudu−K

)+∣∣∣∣∣ Ft

], t ∈ [0, T ].

(8.47)Next, we derive the Black-Scholes partial differential equation (PDE) for theprice of a self-financing portfolio. Until the end of this chapter we model the

212


"



asset price (St)t∈[0,T ] as

dSt = µStdt+ σStdWt, t ∈ R+,

where (Wt)t∈R+ is a standard Brownian motion under the historical proba-bility measure P.

Proposition 8.10. Let (ηt, ξt)t∈R+ be a portfolio strategy such that

(i) (ηt, ξt)t∈R+is self-financing,


Vt = f(t, Yt, St), t ∈ R+,

for some f ∈ C2((0,∞)× (0,∞)2).

Then the function f(t, x, y) in (8.47) satisfies the PDE

rf(t, x, y) =∂f

∂t(t, x, y) + x

∂f

∂y(t, x, y) + rx

∂f

∂x(t, x, y) +

1

2x2σ2 ∂

2f

∂x2(t, x, y),

t, x > 0, under the boundary conditions

f(t, 0, y) = e−r(T−t)( yT−K

)+, 0 ≤ t ≤ T, y ∈ R+,

limy→−∞

f(t, x, y) = 0, 0 ≤ t ≤ T, x ∈ R+,

f(T, x, y) =( yT−K

)+, x, y ∈ R+,

(8.48a)

(8.48b)

(8.48c)

and ξt is given by

ξt =∂f

∂x(t, St, Yt), t ∈ R+.

Proof. We note that the self-financing condition implies


= rηtAtdt+ µξtStdt+ σξtStdWt (8.49)

= rVtdt+ (µ− r)ξtStdt+ σξtStdWt

= rηtAtdt+ µξtStdt+ σξtStdWt, (8.50)

t ∈ R+. Noting that dYt = Stdt, the application of Ito’s formula to f(t, x, y)leads to

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N. Privault

df(t, St, Yt) =∂f

∂t(t, St, Yt)dt+ St

∂f

∂y(t, St, Yt)dt

+µSt∂f

∂x(t, St, Yt)dt+

1

2S2t σ

2 ∂2f

∂x2(t, St, Yt)dt+ σSt

∂f

∂x(t, St, Yt)dWt.

(8.51)

By respective identification of the terms in dWt and dt in (8.49) and (8.51)we get

rηtAtdt+ µξtStdt =∂f

∂t(t, St, Yt)dt+ St

∂f

∂y(t, St, Yt)dt+ µSt

∂f

∂x(t, St, Yt)dt

+1

2S2t σ

2 ∂2f

∂x2(t, St, Yt)dt,

ξtStσdWt = Stσ∂f

∂x(t, St, Yt)dWt,

hencerVt − rξtSt =

∂f

∂t(t, St, Yt) + St

∂f

∂y(t, St, Yt)dt+

1

2S2t σ

2 ∂2f

∂x2(t, St, Yt),

ξt =∂f

∂x(t, St, Yt),

i.e.

rf(t, St, Yt) =∂f

∂t(t, St, Yt) + St

∂f

∂y(t, St, Yt) + rSt

∂f

∂x(t, St, Yt)

+1

2S2t σ

2 ∂2f

∂x2(t, St, Yt),

ξt =∂f

∂x(t, St, Yt).

Next we examine two methods which allow one to reduce the Asian optionpricing PDE from two variables to one variable.

PDE Method - One Variable (1) - Time Independent Coefficients

Following [47], page 91, we define the auxiliary process

Zt =1

St

(1

T

w t

0Sudu−K

)=

1

St

(YtT−K

), t ∈ [0, T ].

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With this notation, the price of the Asian option at time t becomes

e−r(T−t) IE∗

[(1

T

w T

0Sudu−K

)+∣∣∣∣∣ Ft

]= e−r(T−t) IE∗

[ST (ZT )+

∣∣∣∣∣ Ft].

Lemma 8.1. The price (8.40) at time t of the Asian option with payoff(8.39) can be written as

Stg(t, Zt), t ∈ [0, T ],

where

g(t, z) = e−r(T−t) IE∗

[(z +

1

T

w T−t

0

SuS0du

)+]

(8.52)


[(z +

YT−tS0T

)+].

Proof. For 0 ≤ s ≤ t ≤ T , we have

d (StZt) =1

Td

(w t

0Sudu−K

)=StTdt,

henceStZtSs

= Zs +1

T

w t

s

SuSsdu, t ≥ s.

Since for any t ∈ [0, T ], St is positive and Ft-measurable, and Su/St is inde-pendent of Ft, u ≥ t, we have:

e−r(T−t) IE∗

[ST (ZT )+

∣∣∣∣∣ Ft]

= e−r(T−t)St IE∗

[(STStZT

)+∣∣∣∣∣ Ft

]


[(Zt +

1

T

w T

t

SuStdu

)+∣∣∣∣∣ Ft

]


[(z +

1

T

w T

t

SuStdu

)+∣∣∣∣∣ Ft

]z=Zt


[(z +

1

T

w T−t

0

SuS0du

)+]z=Zt


[(z +

YT−tS0T

)+]z=Zt

= Stg(t, Zt),

which proves (8.52).

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N. Privault

Note that as in (8.44), g(t, z) can be computed from the density (8.43) ofYT−t, as

g(t, z) = IE∗

[(z +

YT−tS0T

)+]

=w∞0

(z +

u

T

)+P(Yt/S0 ∈ du)

= e−p2σ2t/8

×w∞0

(z +

u

T

)+ w∞0v−1−p exp

(−2

1 + v2

σ2

)θ

(4v

σ2u,σ2(T − t)

4

)dvdu

u

= e−p2σ2t/8

×w∞(−zT )∨0

(z +

u

T

)w∞0v−1−p exp

(−2

1 + v2

σ2

)θ

(4v

σ2u,σ2(T − t)

4

)dvdu

u

= ze−p2σ2t/8

w∞(−zT )∨0

w∞0v−1−p exp

(−2

1 + v2

σ2

)θ

(4v

σ2u,σ2(T − t)

4

)dvdu

u

+1

Te−p

2σ2t/8w∞(−zT )∨0

w∞0v−1−p exp

(−2

1 + v2

σ2

)θ

(4v

σ2u,σ2(T − t)

4

)dvdu.

The next proposition gives a replicating hedging strategy for Asian options.




Vt = Stg(t, Zt), t ∈ R+,

for some f ∈ C2((0,∞)× (0,∞)2).

Then the function g(t, x) satisfies the PDE

∂g

∂t(t, z) +

(1

T− rz

)∂g

∂z(t, z) +

1

2σ2z2

∂2g

∂z2(t, z) = 0, (8.53)

under the terminal condition

g(T, z) = z+,

and the corresponding replicating portfolio is given by

ξt = g(t, Zt)− Zt∂g

∂zg(t, Zt), t ∈ [0, T ].

Proof. We proceed as in [66]. From the expression of 1/St we have

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"



d

(1

St

)=

1

St

((−µ+ σ2

)dt− σdWt

),

hence

dZt = d

(1

St

(YtT−K

))= d

(YtTSt

− K

St

)=

1

Td

(YtSt

)−Kd

(1

St

)=

1

T

dYtSt

+

(YtT−K

)d

(1

St

)=dt

T+ StZtd

(1

St

)=dt

T+ Zt

(−µ+ σ2

)dt− ZtσdWt.

By self-financing we have


= rηtAtdt+ µξtStdt+ σξtStdWt, (8.54)

t ∈ R+. The application of Ito’s formula to f(t, x, y) leads to

d(Stg(t, Zt)) = g(t, Zt)dSt + Stdg(t, Zt) + dSt · dg(t, Zt)

=∂g

∂t(t, Zt)dt+

∂g

∂z(t, Zt)dZt

+1

2

∂2g

∂z2(t, Zt)(dZt)

2 + dSt · dg(t, Zt)

= St∂g

∂t(t, Zt)dt+ µStg(t, Zt)dt+ σStg(t, Zt)dWt

+StZt(−µ+ σ2

) ∂g∂z

(t, Zt)dt+1

TSt∂g

∂z(t, Zt)dt− σStZt

∂g

∂z(t, Zt)dWt

+1

2σ2Z2

t St∂2g

∂z2(t, Zt)dt− σ2StZt

∂g

∂z(t, Zt)dt

= µStg(t, Zt)dt+ St∂g

∂t(t, Zt)dt+ StZt

(−µ+ σ2

) ∂g∂z

(t, Zt)dt+1

TSt∂g

∂z(t, Zt)dt

+1

2σ2Z2

t St∂2g

∂z2(t, Zt)dt− σ2StZt

∂g

∂z(t, Zt)dt

+σStg(t, Zt)dWt − σStZt∂g

∂z(t, Zt)dWt.

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N. Privault


rηtAt + µξtSt = µStg(t, Zt) + St∂g

∂t(t, Zt)− µStZt

∂g

∂z(t, Zt)

+1

TSt∂g

∂z(t, Zt) +

1

2σ2Z2

t St∂2g

∂z2(t, Zt),

ξtStσ = σStg(t, Zt)− σStZt∂g

∂z(t, Zt),

hencerVt − rξtSt = St

∂g

∂t(t, Zt) +

1

TSt∂g

∂z(t, Zt) +

1

2σ2Z2

t St∂2g

∂z2(t, Zt),


∂z(t, Zt),

i.e. ∂g

∂t(t, z) +

(1

T− rz

)∂g

∂z(t, z) +

1

2σ2z2

∂2g

∂z2(t, z) = 0,


∂z(t, Zt),


g(T, z) = z+.

We check that

ξt = e−r(T−t)σSt∂f

∂xf(t, St, Zt)− σZt

∂f

∂zf(t, St, Zt)

= e−r(T−t)(−Zt

∂g

∂z(t, Zt) + g(t, Zt)

)= e−r(T−t)

(St∂g

∂x

(t,

1

x

(1

T

w t

0Sudu−K

))|x=St

+ g(t, Zt)

)

=∂

∂x

(xe−r(T−t)g

(t,

1

x

(1

T

w t

0Sudu−K

)))|x=St

, t ∈ [0, T ].

We also find that the amount invested on the riskless asset is given by

ηtAt = ZtSt∂g

∂z(t, Zt).

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Next we note that a PDE with no first order derivative term can be obtainedusing time-dependent coefficients.

PDE Method - One Variable (2) - Time Dependent Coefficients

Define now the auxiliary process

Ut :=1

rT(1− e−r(T−t)) + e−r(T−t)

1

St

(1

T

w t

0Sudu−K

)=

1

rT(1− e−r(T−t)) + e−r(T−t)Zt, t ∈ [0, T ],

i.e.

Zt = er(T−t)Ut +er(T−t) − 1

rT, t ∈ [0, T ].

We have

dUt = − 1

Te−r(T−t)dt+ re−r(T−t)Ztdt+ e−r(T−t)dZt

= e−r(T−t)σ2Ztdt− e−r(T−t)σZtdWt − (µ− r)e−r(T−t)Ztdt= −e−r(T−t)σZtdWt, t ∈ R+,

where

dWt = dWt − σdt+µ− rσ

dt = dWt − σdt

is a standard Brownian motion under

dP = eσWT−σ2t/2dP∗ = e−rTSTS0dP∗.

Lemma 8.2. The Asian option price can be written as

Sth(t, Ut) = e−r(T−t) IE∗

[(1

T

w T

0Sudu−K

)+∣∣∣∣∣ Ft

],

where the function h(t, y) is given by

h(t, y) = IE

[(UT )+

∣∣∣∣∣ Ut = y

], 0 ≤ t ≤ T.

Proof. We have

UT =1

ST

(1

T

w T

0Sudu−K

)= ZT ,

and

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N. Privault

dP|FtdP∗|Ft

= eσ(WT−Wt)−σ2(T−t)/2 =e−rTSTe−rtSt

,

hence the price of the Asian option is

e−r(T−t) IE∗[ST (ZT )+ | Ft] = e−r(T−t) IE∗[ST (UT )+ | Ft]

= St IE∗

[e−rTSTe−rtSt

(UT )+

∣∣∣∣∣ Ft]

= St IE∗

[dP|FtdP∗|Ft

(UT )+

∣∣∣∣∣ Ft]

= StIE[(UT )+ | Ft].

The next proposition gives a replicating hedging strategy for Asian options.See § 7.5.3 of [71] and references therein for a different derivation of thePDE (8.55).




Vt = Sth(t, Ut) t ∈ R+,

for some f ∈ C2((0,∞)× (0,∞)2).

Then the function h(t, z) satisfies the PDE

∂h

∂t(t, y) +

1

2σ2

(1− e−r(T−t)

rT− y)2

∂2h

∂y2(t, y) = 0, (8.55)


h(T, z) = z+,

and the corresponding replicating portfolio is given by

ξt = h(t, Ut)− Zt∂h

∂y(t, Ut), t ∈ [0, T ].

Proof. By the self-financing condition (8.50) we have

dVt = rVtdt+ (µ− r)ξtStdt+ σξtStdWt, (8.56)

t ∈ R+. By Ito’s formula we get

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"



d(Sth(t, Ut)) = h(t, Ut)dSt + Stdh(t, Ut) + dSt · dh(t, Ut)

= µSth(t, Ut)dt+ σSth(t, Ut)dWt

+St

(∂h

∂t(t, Ut)dt+

∂h

∂y(t, Ut)dUt +

1

2

∂2h

∂y2(t, Ut)(dUt)

2

)+∂h

∂y(t, Ut)dSt · dUt

= µSth(t, Ut)dt+ σSth(t, Ut)dWt − St(µ− r)∂h

∂y(t, Ut)Ztdt

+St

(∂h

∂t(t, Ut)dt− σ

∂h

∂y(t, Ut)ZtdWt +

1

2σ2 ∂

2h

∂y2(t, Ut)Z

2t dt

)−σ2St

∂h

∂y(t, Ut)Ztdt

= µSth(t, Ut)dt+ σSth(t, Ut)dWt − St(µ− r)∂h

∂y(t, Ut)Ztdt

+St

(∂h

∂t(t, Ut)dt− σ

∂h

∂y(t, Ut)Zt(dWt − σdt) +

1

2σ2 ∂

2h

∂y2(t, Ut)Z

2t dt

)−σ2St

∂h

∂y(t, Ut)Ztdt.


rηtAt + µξtSt = µSth(t, Ut)− (µ− r)StZt∂h

∂y(t, Ut)dt+ St

∂h

∂t(t, Ut)

+1

2Stσ

2Z2t

∂2h

∂y2(t, Ut),


∂y(t, Ut),

hencerηtAt = −rSt(ξt − h(t, Ut)) + St

∂h

∂t(t, Ut) +

1

2Stσ

2 ∂2h

∂y2(t, Ut)Z

2t ,


∂y(t, Ut),

and

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N. Privault

∂h

∂t(t, y) +

1

2σ2

(1− e−r(T−t)

rT− y)2

∂2h

∂y2(t, y) = 0,

ξt = h(t, Ut) +

(1− e−r(T−t)

rT− Ut

)∂h

∂y(t, Ut),


h(T, z) = z+.

We also find

ηtAt = er(T−t)St

(Ut −

1− e−r(T−t)rT

)∂h

∂y(t, Ut) = StZt

∂h

∂y(t, Ut).

Exercises

Exercise 8.1 Consider a risky asset whose price St is given by

dSt = σStdBt + σ2Stdt/2, (8.57)

where (Bt)t∈R+is a standard Brownian motion.

1. Solve the stochastic differential equation (8.57).2. Compute the expected stock price value E[ST ] at time T .3. What is the probability distribution of the supremum sup

t∈[0,T ]

Bt over the

interval [0, T ] ?4. Compute the expected value E[ST ] of the maximum

ST := supt∈[0,T ]

St = S0 supt∈[0,T ]

eσBt = S0 exp

(σ supt∈[0,T ]

Bt

).

of the stock price over the interval [0, T ].

Exercise 8.2 Recall that the maximum Xt := sups∈[0,t]Bs over [0, t] of stan-dard Brownian motion (Bs)s∈[0,t] has the probability density

ϕXt(x) =

√2

πte−x

2/(2t)1[0,∞)(x), x ∈ R.

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1. Let τa = infs ∈ R+ : Bs = a denote the first hitting time of a > 0by (Bs)s∈R+

. Using the relation between τa ≤ t and Xt ≥ a, writedown the probability P (τa ≤ t) as an integral from a to ∞.

2. Using integration by parts on [a,∞), compute the probability density ofτa.

Hint: the derivative of e−x2/(2t) with respect to x is −xe−x2/(2t)/t.

3. Compute the mean value E[(τa)−2] of 1/τ2a .

Exercise 8.3 Barrier options.

1. Compute the hedging strategy of the up-and-out barrier call option onthe underlying asset St with exercise date T , strike K and barrier B,with B > K.

2. Compute the joint probability density

fYT ,BT (a, b) =dP(YT ≤ a & BT ≤ b)

dadb, a, b ∈ R,

of standard Brownian motion BT and its minimum

YT = mint∈[0,T ]

Bt.

3. Compute the joint probability density


dadb, a, b ∈ R,

of drifted Brownian motion BT = BT + µT and its minimum

YT = mint∈[0,T ]

Bt = mint∈[0,T ]

(Bt + µt).

4. Compute the price at time t ∈ [0, T ] of the down-and-out barrier calloption on the underlying asset St with exercise date T , strike K, barrierB, and payoff

C = (ST −K)+

1min

0≤t≤TSt > B

=

ST −K if min

0≤t≤TSt > B,

0 if min0≤t≤T

St ≤ B,

in cases 0 K.

Exercise 8.4 Barrier forward contracts. Compute the price at time t of thefollowing barrier forward contracts on the underlying asset St with exercise

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N. Privault

date T , strike K, barrier B, and the following payoffs. In addition, computethe corresponding hedging strategies.

1. Up-and-in barrier long forward contract. Take

C = (ST −K) 1max0≤t≤T

St > B =

ST −K if max

0≤t≤TSt > B,

0 if max0≤t≤T

St ≤ B.

2. Up-and-out barrier long forward contract. Take

C = (ST −K) 1max0≤t≤T

St < B =

ST −K if max

0≤t≤TSt < B,

0 if max0≤t≤T

St ≥ B.

3. Down-and-in barrier long forward contract. Take

C = (ST −K) 1min

0≤t≤TSt B

=

ST −K if min

0≤t≤TSt > B,

0 if min0≤t≤T

St ≤ B.

5. Up-and-in barrier short forward contract. Take

C = (K − ST ) 1max0≤t≤T

St > B =

K − ST if max

0≤t≤TSt > B,

0 if max0≤t≤T

St ≤ B.

6. Up-and-out barrier short forward contract. Take

C = (K − ST ) 1max0≤t≤T

St < B =

K − ST if max

0≤t≤TSt < B,

0 if max0≤t≤T

St ≥ B.

7. Down-and-in barrier short forward contract. Take

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"



C = (K − ST ) 1min

0≤t≤TSt B

=

K − ST if min

0≤t≤TSt > B,

0 if min0≤t≤T

St ≤ B.

Exercise 8.5 Consider a risky asset whose price St is given by

dSt = σStdBt + σ2Stdt/2,

where (Bt)t∈R+is a standard Brownian motion.

1. What is the probability distribution (distribution function and probabil-ity density function) of the minimum min

t∈[0,T ]Bt over the interval [0, T ]

?2. Compute the price value

e−σ2T/2E

[ST − min

t∈[0,T ]St

]of a lookback call option on ST with maturity T .

Exercise 8.6 Lookback options. Compute the hedging strategy of the look-back put option priced in Proposition 8.4.

Exercise 8.7 Consider the short rate process rt = σBt, where (Bt)t∈R+is a

standard Brownian motion.

1. Find the probability distribution of the time integralr T0rsds.

2. Compute the price

e−rT IE

[(w T

0rudu− κ

)+]

of a caplet on the forward rater T0rsds.

Exercise 8.8 Asian call options with negative strike. Consider the asset priceprocess

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N. Privault

St = S0ert+σBt−σ2t/2, t ∈ R+,

where (Bt)t∈R+is a standard Brownian motion. Assuming that κ ≤ 0, com-

pute the price

e−r(T−t) IE∗

[(1

T

w T

0Sudu− κ

)+∣∣∣∣∣ Ft

]

of the Asian option at time t ∈ [0, T ].

Exercise 8.9 Pricing of Asian options by PDEs. Show that the functionsg(t, z) and h(t, y) are linked by the relation

g(t, z) = h

(t,

1− e−r(T−t)rT

+ e−r(T−t)z

), t ∈ [0, T ], z > 0,

and that the PDE (1.35) for h(t, y) can be derived from the PDE (1.33) forg(t, z) and the above relation.

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Chapter 9

American Options

In contrast with European option which have fixed maturities, the holder ofan American option is allowed to exercise at any given (random) time. Thistransforms the valuation problem into an optimization problem in which onehas to find the optimal time to exercise in order to maximize the payoff ofthe option. As will be seen in the first section below, not all random timescan be considered in this process, and we restrict ourselves to stopping timeswhose value at time t be can decided based on the historical data available.

9.1 Filtrations and Information Flow

Let (Ft)t∈R+ denote the filtration generated by a stochastic process (Xt)t∈R+ .In other words, Ft denotes the collection of all events possibly generated byXs : 0 ≤ s ≤ t up to time t. Examples of such events include the event

Xt0 ≤ a0, Xt1 ≤ a1, . . . , Xtn ≤ an

for a0, a1, . . . , an a given fixed sequence of real numbers and 0 ≤ t1 < · · · <tn < t, and Ft is said to represent the information generated by (Xs)s∈[0,t]up to time t.

By construction, (Ft)t∈R+is an increasing family of σ-algebras in the sense

that we have Fs ⊂ Ft (information known at time s is contained in the in-formation known at time t) when 0 < s < t.

One refers sometimes to (Ft)t∈R+as the increasing flow of information

generated by (Xt)t∈R+.

"



N. Privault

9.2 Martingales, Submartingales, and Supermartingales

Let us recall the definition of martingale (cf. Definition 5.4) and introduce inaddition the definitions of supermartingale and submartingale.

Definition 9.1. An integrable stochastic process (Zt)t∈R+is a martingale

(resp. a supermartingale, resp. a submartingale) with respect to (Ft)t∈R+if

it satisfies the property

Zs = IE[Zt | Fs], 0 ≤ s ≤ t,

resp.Zs ≥ IE[Zt | Fs], 0 ≤ s ≤ t,

resp.Zs ≤ IE[Zt | Fs], 0 ≤ s ≤ t.

Clearly, a process (Zt)t∈R+is a martingale if and only if it is both a su-

permartingale and a submartingale.

A particular property of martingales is that their expectation is constant.

Proposition 9.1. Let (Zt)t∈R+be a martingale. We have

IE[Zt] = IE[Zs], 0 ≤ s ≤ t.

The above proposition follows from the “tower property” (16.24) of con-ditional expectations, which shows that

IE[Zt] = IE[IE[Zt | Fs]] = IE[Zs], 0 ≤ s ≤ t. (9.1)

Similarly, a supermartingale has a decreasing expectation, while a submartingalehas a increasing expectation.

Proposition 9.2. Let (Zt)t∈R+ be a supermartingale, resp. a submartingale.Then we have

IE[Zt] ≤ IE[Zs], 0 ≤ s ≤ t,resp.

IE[Zt] ≥ IE[Zs], 0 ≤ s ≤ t.Proof. As in (9.1) above we have

IE[Zt] = IE[IE[Zt | Fs]] ≤ IE[Zs], 0 ≤ s ≤ t.

The proof is similar in the submartingale case.

Independent increments processes whose increments have negative expecta-tion give examples of supermartingales. For example, if (Xt)t∈R+

is such aprocess then we have

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IE[Xt | Fs] = IE[Xs | Fs] + IE [Xt −Xs | Fs]= IE[Xs | Fs] + IE[Xt −Xs]

≤ IE[Xs | Fs]= Xs, 0 ≤ s ≤ t.

Similarly, a process with independent increments which have positive expec-tation will be a submartingale. Brownian motion Bt + µt with positive driftµ > 0 is such an example, as in Figure 9.1 below.

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 2 4 6 8 10 12 14 16 18 20

drifted Brownian motion

drift

Fig. 9.1: Drifted Brownian path.

The following example comes from gambling.

Fig. 9.2: Evolution of the fortune of a poker player vs number of games played.

A natural way to construct submartingales is to take convex functions ofmartingales. Indeed, if (Mt)t∈R+ is a martingale and φ is a convex function,Jensen’s inequality states that

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N. Privault

φ(IE[Mt | Fs]) ≤ IE[φ(Mt) | Fs], 0 ≤ s ≤ t, (9.2)

which shows that

φ(Ms) = φ(IE[Mt | Fs]) ≤ IE[φ(Mt) | Fs], 0 ≤ s ≤ t,

i.e. (φ(Mt))t∈R+is a submartingale. More generally, the above shows that

φ(Mt)t∈R+ remains a submartingale when φ is convex non-decreasing and(Mt)∈R+ is a submartingale. Similarly, (φ(Mt))t∈R+ will be supermartingalewhen (Mt)∈R+

is a martingale and the function φ is concave.

Other examples of (super, sub)-martingales include geometric Brownianmotion


which is a martingale for r = 0, a supermartingale for r ≤ 0, and asubmartingale for r ≥ 0.

9.3 Stopping Times

Next we turn to the definition of stopping time.

Definition 9.2. A stopping time is a random variable τ : Ω −→ R+∪+∞such that

τ > t ∈ Ft, t ∈ R+. (9.3)

The meaning of Relation (9.3) is that the knowledge of the event τ > tdepends only on the information present in Ft up to time t, i.e. on the knowl-edge of (Xs)0≤s≤t.

In other words, an event occurs at a stopping time τ if at any time t itcan be decided whether the event has already occured (τ ≤ t) or not (τ > t)based on the information generated by (Xs)s∈R+

up to time t.

For example, the day you bought your first car is a stopping time (onecan always answer the question “did I ever buy a car”), whereas the day youwill buy your last car may not be a stopping time (one may not be able toanswer the question “will I ever buy another car”).

Note that a constant time is always a stopping time, and if τ and θ arestopping times then the smallest τ ∧ θ of τ and θ is also a stopping time,since

τ ∧ θ > t = τ > t and θ > t = τ > t ∩ θ > t ∈ Ft, t ∈ R+.

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Hitting times provide natural examples of stopping times. The hitting timeof level x by the process (Xt)t∈R+

, defined as

τx = inft ∈ R+ : Xt = x,

is a stopping time,1 as we have (here in discrete time)

τx > t = Xs 6= x, 0 ≤ s ≤ t= X0 6= x ∩ X1 6= x ∩ · · · ∩ Xt 6= x ∈ Ft, t ∈ N.

In gambling, a hitting time can be used as an exit strategy from the game.For example, letting

τx,y := inft ∈ R+ : Xt = x or Xt = y (9.4)

defines a hitting time (hence a stopping time) which allows a gambler to exitthe game as soon as losses become equal to x = −10, or gains become equalto y = +100, whichever comes first.

However, not every R+-valued random variable is a stopping time. Forexample the random time

τ = inf

t ∈ [0, T ] : Xt = sup

s∈[0,T ]

Xs

,

which represents the first time the process (Xt)t∈[0,T ] reaches its maximumover [0, T ], is not a stopping time with respect to the filtration generated by(Xt)t∈[0,T ]. Indeed, the information known at time t ∈ (0, T ) is not sufficientto determine whether τ > t.

Given (Zt)t∈R+a stochastic process and τ : Ω −→ R+ ∪+∞ a stopping

time, the stopped process (Zt∧τ )t∈R+is defined as

Zt∧τ =

Zτ if t ≥ τ,

Zt if t < τ,

Using indicator functions we may also write

Zt∧τ = Zτ1τ≤t + Zt1τ>t, t ∈ R+.

The following figure is an illustration of the path of a stopped process.

1 As a convention we let τ = +∞ in case there exists no t ∈ R+ such that Xt = x.

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N. Privault

0.025

0.03

0.035

0.04

0.045

0.05

0.055

0.06

0.065

0 5 10 15 20

t

Fig. 9.3: Stopped process

Theorem 9.1 below is called the stopping time (or optional sampling, or op-tional stopping) theorem, it is due to the mathematician J.L. Doob (1910-2004). It is also used in Exercise 9.2 below.

Theorem 9.1. Assume that (Mt)t∈R+ is a martingale with respect to (Ft)t∈R+ .Then the stopped process (Mt∧τ )t∈R+ is also a martingale with respect to(Ft)t∈R+

.

Proof. We only give the proof in discrete time by applying the martingaletransform argument of Proposition 2.1. Writing

Mn = M0 +

n∑l=1

(Ml −Ml−1) = M0 +

∞∑l=1

1l≤n(Ml −Ml−1),

we have

Mτ∧n = M0 +τ∧n∑l=1

(Ml −Ml−1) = M0 +∞∑l=1

1l≤τ∧n(Ml −Ml−1),

and for k ≤ n,

IE[Mτ∧n | Fk] = M0 +∞∑l=1

IE[1l≤τ∧n(Ml −Ml−1) | Fk]

= M0 +k∑l=1


+∞∑

l=k+1


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http://en.wikipedia.org/wiki/Joseph_Leo_Doob



= M0 +

k∑l=1

(Ml −Ml−1) IE[1l≤τ∧n | Fk]

+∞∑

l=k+1

IE[IE[(Ml −Ml−1)1l≤τ∧n | Fl−1] | Fk]

= M0 +k∑l=1

(Ml −Ml−1)1l≤τ∧n

+

∞∑l=k+1

IE[1l≤τ∧n IE[(Ml −Ml−1) | Fl−1] | Fk]

= M0 +

τ∧n∧k∑l=1


= M0 +τ∧k∑l=1


= Mτ∧k, k = 0, 1, . . . , n.

Since the stopped process (Mτ∧t)t∈R+ is a martingale by Theorem 9.1 wefind that its expectation is constant by Proposition 9.1. More generally, if(Mt)t∈R+

is a supermartingale with respect to (Ft)t∈R+, then the stopped

process (Mt∧τ )t∈R+remains a supermartingale with respect to (Ft)t∈R+

.

As a consequence, if τ is a stopping time bounded by T > 0, i.e. τ ≤ Talmost surely, we have

IE[Mτ ] = IE[Mτ∧T ] = IE[Mτ∧0] = IE[M0]. (9.5)

In case τ is finite with probability one but not bounded we may also write

IE[Mτ ] = IE[

limt→∞

Mτ∧t

]= limt→∞

IE[Mτ∧t] = IE[M0], (9.6)

provided|Mτ∧t| ≤ C, a.s., t ∈ R+. (9.7)

More generally, (9.6) will hold provided the limit and expectation signs canbe exchanged, and this can be done using e.g. the Dominated ConvergenceTheorem.

In case P(τ = +∞) > 0, (9.6) will hold under the above conditions,provided

M∞ := limt→∞

Mt (9.8)

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N. Privault

exists with probability one.

In addition, if τ and ν are two bounded stopping times such that τ ≤ ν,a.s., we have

IE[Mτ ] ≥ IE[Mν ] (9.9)

if (Mt)t∈R+is a supermartingale, and

IE[Mτ ] ≤ IE[Mν ] (9.10)

if (Mt)t∈R+is a submartingale, cf. Exercise 9.2 below for a proof in discrete

time. As a consequence of (9.9) and (9.10) (or directly from (9.5)), if τ andν are two bounded stopping times such that τ ≤ ν, a.s., we have

IE[Mτ ] = IE[Mν ] (9.11)

if (Mt)t∈R+is a martingale.

Relations (9.9), (9.10) and (9.11) can be extended to unbounded stoppingtimes along the same lines and conditions as (9.6), such as (9.7) applied toboth τ and ν. Dealing with unbounded stopping times can be necessary inthe case of hitting times.

In general, for all stopping times τ (bounded or unbounded) it remainstrue that

IE[Mτ ] = IE[ limt→∞

Mτ∧t] ≤ limt→∞

IE[Mτ∧t] ≤ limt→∞

IE[M0] = IE[M0], (9.12)

provided (Mt)t∈R+is a nonnegative supermartingale, where we used Fatou’s

lemma.2 As in (9.6), the limit (9.8) is required to exist with probability oneif P(τ = +∞) > 0.

In the case of the exit strategy τx,y of (9.4) the stopping time theoremshows that IE[Mτx,y ] = 0 if M0 = 0, which shows that on average this exitstrategy does not increase the average gain of the player. More precisely wehave

0 = M0 = IE[Mτx,y ] = xP(Mτx,y = x) + yP(Mτx,y = y)

= −10P(Mτx,y = −10) + 100P(Mτx,y = 100),

which shows that

P(Mτx,y = −10) =10

11and P(Mτx,y = 100) =

1

11,

2 IE[limn→∞ Fn] ≤ limn→∞ IE[Fn] for any sequence (Fn)n∈N of nonnegative randomvariables, provided the limits exist.

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provided the relation P(Mτx,y = x) +P(Mτx,y = y) = 1 is satisfied, see belowfor further applications to Brownian motion.

As a counterexample to (9.11), the random time

τ := inf

t ∈ [0, T ] : Mt = sup

s∈[0,T ]

Ms

,

which is not a stopping time, will satisfy

IE[Mτ ] > IE[MT ],

although τ ≤ T almost surely.

Similarly,

τ := inf

t ∈ [0, T ] : Mt = inf

s∈[0,T ]Ms

,

is not a stopping time and satisfies

IE[Mτ ] < IE[MT ].

When Xt is a martingale, e.g. a centered random walk with independent in-crements, the message of the stopping time Theorem 9.1 is that the expectedgain of the strategy (9.4) remains zero on average since

IE[Xτx,y ] = IE[X0] = 0.

Similar arguments are used in the examples below.

In the table below we summarize some of the results of this section forbounded stopping times.

bounded stopping times τ ≤ ν

Mt

supermartingale IE[Mτ ] ≥ IE[Mν ]

martingale IE[Mτ ] = IE[Mν ]

submartingale IE[Mτ ] ≤ IE[Mν ]

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N. Privault

Examples of application

In this section we note that, as an application of the stopping time theorem,a number of expectations can be computed in a simple and elegant way.

Brownian motion hitting a barrier

Given a, b ∈ R, a < b, let the hitting3 time τa,b : Ω −→ R+ be defined by

τa,b = inft ≥ 0 : Bt = a or Bt = b,

which is the hitting time of the boundary a, b of Brownian motion (Bt)t∈R+ ,a, b ∈ R, a < b.

Recall that Brownian motion (Bt)t∈R+is a martingale since it has inde-

pendent increments, and those increments are centered:

IE[Bt −Bs] = 0, 0 ≤ s ≤ t.

Consequently, (Bτa,b∧t)t∈R+is still a martingale and by (9.6) we have

IE[Bτa,b | B0 = x] = IE[B0 | B0 = x] = x,

as the exchange between limit and expectation in (9.6) can be justified since

|Bt∧τa,b | ≤ max(|a|, |b|), t ∈ R+.

Hence we have

x = IE[Bτa,b | B0 = x] = a×P(Bτa,b = a | B0 = x)+b×P(Bτa,b = b | B0 = x),

under the additional condition

P(Xτa,b = a | X0 = x

)+ P(Xτa,b = b | X0 = x) = 1.

which yields

P(Bτa,b = b | B0 = x) =x− ab− a , a ≤ x ≤ b,

which also shows that

P(Bτa,b = a | B0 = x) =b− xb− a , a ≤ x ≤ b.

3 A hitting time is a stopping time

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Note that the above result and its proof actually apply to any continuousmartingale, and not only to Brownian motion.

Drifted Brownian motion hitting a barrier

Next, let us turn to the case of drifted Brownian motion

Xt = x+Bt + µt, t ∈ R+.

In this case the process (Xt)t∈R+ is no longer a martingale and in order touse Theorem 9.1 we need to construct a martingale of a different type. Herewe note that the process

Mt := eσBt−σ2t/2, t ∈ R+,

is a martingale with respect to (Ft)t∈R+. Indeed, we have

IE[Mt | Fs] = IE[eσBt−σ

2t/2∣∣∣Fs] = eσBs−σ

2s/2, 0 ≤ s ≤ t.

By Theorem 9.1 we know that the stopped process (Mτa,b∧t)t∈R+ is a mar-tingale, hence its expectation is constant by Proposition 9.1, and (9.6) gives

1 = IE[M0] = IE[Mτa,b ],

as the exchange between limit and expectation in (9.6) can be justified since

|Mt∧τa,b | ≤ max(eσ|a|, eσ|b|), t ∈ R+.

Next we note that letting σ = −2µ we have

eσXt = eσx+σBt+σµt = eσx+σBt−σ2t/2 = eσxMt,

hence

1 = IE[Mτa,b ]

= e−σx IE[eσXτa,b ]

= eσ(a−x)P(Xτa,b = a | X0 = x

)+ eσ(b−x)P(Xτa,b = b | X0 = x),

under the additional condition

P(Xτa,b = a | X0 = x

)+ P(Xτa,b = b | X0 = x) = 1.

Finally this gives

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N. Privault

P(Xτa,b = a | X0 = x) =eσx − eσbeσa − eσb =

e−2µx − e−2µbe−2µa − e−2µb , (9.13)

a ≤ x ≤ b, and

P(Xτa,b = b | X0 = x) =e−2µa − e−2µxe−2µa − e−2µb , a ≤ x ≤ b.

Letting b tend to infinity in the above equalities shows that the probabilityof escape to infinity of Brownian motion started from x ∈ [a,∞) is equal to

1− P(Xτa,∞ = a | X0 = x) = 1− e−2µ(x−a), x > a,

when µ > 0, and equal to 0 when µ ≤ 0.

Mean hitting time for Brownian motion

The martingale method also allows us to compute the expectation IE[Bτa,b ],after checking that (B2

t − t)t∈R+ is also a martingale. Indeed we have

IE[B2t − t | Fs] = IE[(Bs + (Bt −Bs))2 − t | Fs]

= IE[B2s + (Bt −Bs)2 + 2Bs(Bt −Bs)− t | Fs]

= IE[B2s − s | Fs]− (t− s) + IE[(Bt −Bs)2 | Fs] + 2 IE[Bs(Bt −Bs) | Fs]

= B2s − s− (t− s) + IE[(Bt −Bs)2 | Fs] + 2Bs IE[Bt −Bs | Fs]

= B2s − s− (t− s) + IE[(Bt −Bs)2] + 2Bs IE[Bt −Bs]

= B2s − s, 0 ≤ s ≤ t.

Consequently the stopped process (B2τa,b∧t− τa,b∧ t)t∈R+

is still a martingale

by Theorem 9.1 hence the expectation IE[B2τa,b∧t − τa,b ∧ t] is constant in

t ∈ R+, hence by (9.6) we get4

x2 = IE[B20 − 0 | B0 = x]

= IE[B2τa,b− τa,b | B0 = x]

= IE[B2τa,b| B0 = x]− IE[τa,b | B0 = x]

= b2P(Bτa,b = b | B0 = x) + a2P(Bτa,b = a | B0 = x)− IE[τa,b | B0 = x],

i.e.

IE[τa,b | B0 = x] = b2P(Bτa,b = b | B0 = x) + a2P(Bτa,b = a | B0 = x)− x2

= b2x− ab− a + a2

b− xb− a − x

2

4 Here we note that it can be showed that IE[τa,b] <∞ in order to apply (9.6).

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= (x− a)(b− x), a ≤ x ≤ b.

Mean hitting time for drifted Brownian motion

Finally we show how to recover the value of the mean hitting time of driftedBrownian motion Xt = Bt+µt. As above, the process Xt−µt is a martingalethe stopped process (Xτa,b∧t − µ(τa,b ∧ t))t∈R+

is still a martingale by The-orem 9.1. Hence the expectation IE[Xτa,b∧t−µ(τa,b∧t)] is constant in t ∈ R+.

Since the stopped process (Xτa,b∧t − µt)t∈R+ is a martingale, we have

x = IE[Xτa,b − µτa,b | X0 = x],

which gives

x = IE[Xτa,b − µτa,b | X0 = x]

= IE[Xτa,b | X0 = x]− µ IE[τa,b | X0 = x]

= bP(Xτa,b = b | X0 = x) + aP(Xτa,b = a | X0 = x)− µ IE[τa,b | X0 = x],

i.e. by (9.13),

µ IE[τa,b | X0 = x] = bP(Xτa,b = b | X0 = x) + aP(Xτa,b = a | X0 = x)− x

= be−2µa − e−2µxe−2µa − e−2µb + a

e−2µx − e−2µbe−2µa − e−2µb − x

=b(e−2µa − e−2µx) + a(e−2µx − e−2µb)− x(e−2µa − e−2µb)

e−2µa − e−2µb ,

hence

IE[τa,b | X0 = x] =b(e−2µa − e−2µx) + a(e−2µx − e−2µb)− x(e−2µa − e−2µb)

µ(e−2µa − e−2µb) ,

a ≤ x ≤ b.

9.4 Perpetual American Options

The price of an American put option expiring at time T > 0 with strike Kcan be expressed as the expected value of its discounted payoff:

f(t, St) = supt≤τ≤T

τ stopping time

IE∗[e−r(τ−t)(K − Sτ )+

∣∣∣St] ,

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N. Privault

under the risk-neutral probability measure P∗, where the supremum is takenover stopping times between t and a fixed maturity T . Similarly, the price ofa finite expiration American call option with strike K is expressed as


τ stopping time

IE∗[e−r(τ−t)(Sτ −K)+

∣∣∣St] .In this section we take T = +∞, in which case we refer to these options asperpetual options, and the corresponding put and call are respectively pricedas

f(t, St) = supτ≥t

τ stopping time


∣∣∣St] ,and


τ stopping time


∣∣∣St] .

Two-choice optimal stopping at a fixed price level for perpetualput options

In this section we consider the pricing of perpetual put options. Given L ∈(0,K) a fixed price, consider the following choices for the exercise of a putoption with strike K:

1. If St ≤ L, then exercise at time t.

2. Otherwise if St > L, wait until the first hitting time

τL := infu ≥ t : Su ≤ L (9.14)

and exercise the option at time τL.

Note that by definition of (9.14) we have τL = t if St ≤ L.

In case St ≤ L, the payoff will be

(K − St)+ = K − St

since K > L ≥ St, however in this case one would buy the option at priceK − St only to exercise it immediately for the same amount.

In case St > L, the price of the option will be

fL(t, St) = IE∗[e−r(τL−t)(K − SτL)+

∣∣∣St]= IE∗

[e−r(τL−t)(K − L)+

∣∣∣St]240


"



= (K − L) IE∗[e−r(τL−t)

∣∣∣St] . (9.15)

We note that the starting date t does not matter when pricing perpetualoptions, hence fL(t, x) is actually independent of t ∈ R+, and the pricing ofthe perpetual put option can be performed by taking t = 0 and in the sequelwe will work under

fL(t, x) = fL(x), x > 0.

Recall that the underlying asset price is written as

St = S0eσBt−σ2t/2+rt, t ∈ R+, (9.16)


ability measure P∗, r is the risk-free interest rate, and σ > 0 is the volatilitycoefficient.

Proposition 9.3. Assume that r ≥ 0. We have

fL(x) = IE∗[e−r(τL−t)(K − SτL)+

∣∣∣St = x]

=

K − x, 0 < x ≤ L,

(K − L)( xL

)−2r/σ2

, x ≥ L.

Proof. The result is obvious for St = x ≤ L since in this case we have τL = tand SτL = St = x, so that we only focus on the case x ≥ L. In addition wetake t = 0 without loss of generality. By the relation

IE∗[e−r(τL−t)(K − SτL)+

∣∣∣St = x]

= IE∗[e−r(τL−t)(K − L)

∣∣∣St = x],

(9.17)

we check that it suffices to compute IE∗[e−r(τL−t)

∣∣∣St = x]. For this we note

that from (9.16), for all λ ∈ R the process (Zt)t∈R+defined as

Zt :=

(StS0

)λe−rλt+λσ

2t/2−λ2σ2t/2 = eλσBt−λ2σ2t/2, t ∈ R+,

is a martingale under the risk-neutral probability measure P∗, hence we have5

IE∗[ZτL ] = IE∗[Z0] = 1,

with

5 Here the exchange of limit and expectation can be justified by monotone conver-gence, cf. p. 347 of [71].

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N. Privault

ZτL =

(SτLS0

)λe−(rλ−λσ

2/2+λ2σ2/2)τL

=

(L

S0

)λe−(rλ−λσ

2/2+λ2σ2/2)τL ,

which yields

IE∗

[(L

S0

)λe−(rλ−λσ

2/2+λ2σ2/2)τL

]= 1,

i.e.

IE∗[e−(rλ−λσ

2/2+λ2σ2/2)τL]

=

(S0

L

)λ,

or

IE∗[e−rτL

]=

(S0

L

)λ, (9.18)

provided we choose λ such that

−(rλ− λσ2/2 + λ2σ2/2) = −r,

i.e.

0 = λ2σ2/2 + λ(r − σ2/2)− r =σ2

2(λ+ 2r/σ2)(λ− 1). (9.19)

This equation admits two solutions and we choose the negative solution λ =−2r/σ2 since S0 = x > L and the expectation IE∗[e−rτL ] < 1 in (9.18) isstrictly smaller than 1 as r ≥ 0. Consequently we have

IE∗[e−r(τL−t)

∣∣∣St = x]

=( xL

)−2r/σ2

x ≥ L,(9.20)

and we conclude by (9.17), which shows that

IE∗[e−r(τL−t)(K − SτL)+

∣∣∣St = x]

= (K − L) IE∗[e−r(τL−t)

∣∣∣St = x]

= (K − L)( xL

)−2r/σ2

,

when St = x > L.

We note that taking L = K would yield a payoff always equal to 0 for theoption holder, hence the value of L should be strictly lower than K. On theother hand, if L = 0 the value of τL will be infinite almost surely, hencethe option price will be 0 when r ≥ 0 from (9.15). Therefore there should

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be an optimal value L∗, which should be strictly comprised between 0 and K.

Figure 9.4 shows for K = 100 that there exists an optimal value L∗ = 85.71which maximizes the option price for all values of the underlying.

0

5

10

15

20

25

30

35

70 80 90 100 110 120

Opt

ion

pric

e

Underlying

L=75L=L*=85.71

L=98(K-x)+

Fig. 9.4: Graphs of the option price by exercise at τL for several values of L.

In order to compute L∗ we observe that, geometrically, the slope of fL(x) atx = L∗ is equal to −1, i.e.

f ′L∗(L∗) = − 2r

σ2(K − L∗) (L∗)−2r/σ

2−1

(L∗)−2r/σ2 = −1,

i.e.2r

σ2(K − L∗) = L∗,

or

L∗ =2r

2r + σ2K < K,

cf. [71] page 351 for another derivation.

The next figure is a 2-dimensional animation that also shows the optimalvalue L∗ of L.

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N. Privault

Fig. 9.5: Animated graph of the option price depending on the values of L.

The next figure gives another view of the put option prices according todifferent values of L, with the optimal value L∗ = 85.71.

(K-x)+fL(x)

fL*(x)K-L

70 75 80 85 90 95 100 105 110Underlying x 60

65 70

75 80

85 90

95 100

L

0

5

10

15

20

25

30

Fig. 9.6: Graph of the option price as a function of L and of the underlying assetprice.

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In Figure 9.7 which is based on the stock price of HSBC Holdings (0005.HK)over year 2009, the optimal exercise strategy for an American put optionwith strike K=$77.67 would have been to exercise whenever the underlyingprice goes above L∗ = $62, i.e. at approximately 54 days, for a payoff of$38. Note that waiting a longer time, e.g. until 85 days, would have yielded ahigher payoff of at least $65. This is due to the fact that, here, optimizationis done based on the past information only and makes sense in expectation(or average) over all possible future paths.

Payoff (K-x)+

American put priceOption price path

L*

30 40

50 60

70 80

90 100

underlying HK$

0 50

100 150

200Time in days

0

10

20

30

40

50

60

70

80

Fig. 9.7: Path of the American put option price on the HSBC stock.

PDE approach

We can check by hand that

fL∗(x) =

K − x, 0 < x ≤ L∗ =

2r

2r + σ2K,

Kσ2

2r + σ2

(2r + σ2

2r

x

K

)−2r/σ2

, x ≥ L∗ =2r

2r + σ2K,

satisfies the PDE

− rfL∗(x) + rxf ′L∗(x) +1

2σ2x2f ′′L∗(x) =

−rK < 0, 0 < x ≤ L∗ < K,

0, x > L∗.(9.21)

in addition to the condition fL∗(x) = K − x, 0 < x < L∗ < K,

fL∗(x) > (K − x)+, x ≥ L∗." 245



N. Privault

This can be summarized in the following differential inequalities, or varia-tional differential equation:

fL∗(x) ≥ (K − x)+,

rxf ′L∗(x) +1

2σ2x2f ′′L∗(x) ≤ rfL∗(x),

(rfL∗(x)− rxf ′L∗(x)− 1

2σ2x2f ′′L∗(x)

)(fL∗(x)− (K − x)+) = 0,

(9.22a)

(9.22b)

(9.22c)

which admits an interpretation in terms of absence of arbitrage, as shownbelow.

By Ito’s formula the discounted portfolio price fL∗(St) = e−rtfL∗(St)satisfies

d(fL∗(St)) =

(−rfL∗(St) + rStf

′L∗(St) +

1

2σ2S2

t f′′L∗(St)

)e−rtdt

+e−rtσStf′L∗(St)dBt, (9.23)

hence from Equation (9.22c), fL∗(St) is a martingale when

fL∗(St) > (K − St)+, i .e. St > L∗,

and the expected rate of return of the option price fL∗(St) then equals therate r of the risk-free asset as

d(fL∗(St)) = d(ertfL∗(St)) = rfL∗(St)dt+ ertdfL∗(St),

and the investor prefers to wait.

On the other hand if fL∗(St) = (K − St)+, i.e. 0 < St < L∗, it is notworth waiting as (9.22b) and (9.22c) show that the return of the option islower than that of the risk-free asset, i.e.:

−rfL∗(St) + rStf′L∗(St) +

1

2σ2S2

t f′′L∗(St) < 0,

and exercise becomes immediate since the process fL∗(St) becomes a (strict)supermartingale. In this case, (9.22c) implies fL∗(x) = (K − x)+.

In view of the above derivation it should make sense to assert that fL∗(St)is the price at time t of the perpetual American put option. The next propo-

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sition shows that this is indeed the case, and that the optimal exercise timeis τ∗ = τL∗ .

Proposition 9.4. The price of the perpetual American put option is givenfor all t ≥ 0 by

fL∗(St) = supτ≥t

τ stopping time


∣∣∣St]= IE∗

[e−r(τL∗−t)(K − SτL∗ )+

∣∣∣St]

=

K − St, 0 < St ≤ L∗,

Kσ2

2r + σ2

(2r + σ2

2r

StK

)−2r/σ2

, St ≥ L∗.

Proof. By Ito’s formula (9.23) and the inequality (9.22b) one checks that thediscounted portfolio price

u 7→ e−rufL∗(Su), u ≥ t,

is a supermartingale. As a consequence, for all stopping times τ we have, by(9.12),

e−rtfL∗(St) ≥ IE∗[e−rτfL∗(Sτ )

∣∣∣St]≥ IE∗

[e−rτ (K − Sτ )+

∣∣∣St] ,from (9.22a), which implies

e−rtfL∗(St) ≥ supτ≥t

τ stopping time

IE∗[e−rτ (K − Sτ )+

∣∣∣St] . (9.24)

The converse is obvious by Proposition 9.3 as

fL∗(St) = IE∗[e−r(τL∗−t)(K − SτL∗ )+

∣∣∣St]≤ sup

τ≥tτ stopping time


∣∣∣St] ,since τL∗ is a stopping time larger than t ≥ 0.

Note that that converse statement can also be obtained from the varia-tional PDE (9.22a)-(9.22c) itself, without relying on Proposition 9.3. For this,taking τ = τL∗ we note that the process

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N. Privault

u 7→ e−ru∧τL∗ fL∗(Su∧τL∗ ), u ≥ t,

is not only a supermartingale, it is also a martingale until exercise at timeτL∗ by (9.21) since Su∧τL∗ ≥ L∗, hence we have

e−rtfL∗(St) = IE∗[e−r(u∧τL∗ )fL∗(Su∧τL∗ )

∣∣∣St] , u ≥ t,

hence after letting u tend to infinity we obtain

e−rtfL∗(St) = IE∗[e−rτL∗ fL∗(SτL∗ )

∣∣∣St]= IE∗

[e−rτL∗ fL∗(L

∗)∣∣∣St]

= IE∗[e−rτL∗ (K − SτL∗ )+

∣∣∣St]≤ sup


IE∗[e−rτL∗ (K − SτL∗ )+

∣∣∣St] ,which shows that

e−rtfL∗(St) ≤ supτ≥t

τ stopping time


∣∣∣St] , t ≥ 0.

Two-choice optimal stopping at a fixed price level for perpetualcall options

In this section we consider the pricing of perpetual call options. Given L > Ka fixed price, consider the following choices for the exercise of a call optionwith strike K:

1. If St ≥ L, then exercise at time t.

2. Otherwise, wait until the first hitting time

τL = infu ≥ t : Su = L

and exercise the option and time τL.

In case St ≥ L, the payoff will be

(St −K)+ = St −K

since K < L ≤ St.

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In case St < L, the price of the option will be

fL(St) = IE∗[e−r(τL−t)(SτL −K)+

∣∣∣St]= IE∗

[e−r(τL−t)(L−K)+

∣∣∣St]= (L−K) IE∗

[e−r(τL−t)

∣∣∣St] .Proposition 9.5. We have

fL(x) =

x−K, x ≥ L > K,

(L−K)x

L, 0 < x ≤ L.

(9.25)

Proof. We only need to consider the case x < L. Note that for all λ ∈ R,

Zt :=

(StS0

)λe−rλt+λσ

2t/2−λ2σ2t/2 = eλσBt−λ2σ2t/2, t ∈ R+,

is a martingale under the risk-neutral probability measure P. Hence thestopped process (Zt∧τL)t∈R+ is a martingale and it has constant expecta-tion. Hence we have

IE∗[Zt∧τL ] = IE∗[Z0] = 1,

and by letting t go to infinity we get

IE∗

[(SτLS0

)λe−(rλ−λσ

2/2+λ2σ2/2)τL

]= 1,

which yields

IE∗[e−(rλ−λσ

2/2+λ2σ2/2)τL]

=

(S0

L

)λ,

i.e.

IE∗[e−rτL

]=

(S0

L

)λ, (9.26)

provided that λ is chosen such that

−(rλ− λσ2/2 + λ2σ2/2) = −r,

i.e.

0 = λ2σ2/2 + λ(r − σ2/2)− r =σ2

2(λ+ 2r/σ2)(λ− 1).

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N. Privault

Here we choose the positive solution λ = 1 since S0 = x < L and theexpectation (9.26) is lower than 1.

One sees from Figure 9.8 that the situation completely differs from the per-petual put option case, as there does not exist an optimal value L∗ that wouldmaximize the option price for all values of the underlying.

0

50

100

150

200

250

300

350

400

450

0 50 100 150 200 250 300 350 400 450

Opt

ion

pric

e

Underlying

L=150L=250L=400(x-K)

+

x

Fig. 9.8: Graphs of the option price by exercising at τL for several values of L.

The intuition behind this picture is that there is no upper limit above whichone should exercise the option, and in order to price the American perpetualcall option we have to let L go to infinity, i.e. the “optimal” exercise strategyis to wait indefinitely.

(x-K)+

fL(x)K-L

100

150

200

250

Underlying x

100 150 200 250 300

L

0

20

40

60

80

100

120

140

160

180

Fig. 9.9: Graphs of the option prices parametrized by different values of L.

We check from (9.25) that

limL→∞

fL(x) = x− limL→∞

Kx

L= x, x > 0. (9.27)

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As a consequence we have the following proposition.

Proposition 9.6. The price of the perpetual American call option is givenby

supτ≥t

τ stopping time


∣∣∣St] = St, t ∈ R+. (9.28)

Proof. For all L > K we have

fL(St) = IE∗[e−r(τL−t)(SτL −K)+

∣∣∣St]≤ sup



∣∣∣St] , t ≥ 0,

hence taking the limit as L→∞ yields

St ≤ supτ≥t

τ stopping time


∣∣∣St] (9.29)

from (9.27). On the other hand, for all stopping times τ ≥ t we have, by(9.12),


∣∣∣St] ≤ IE∗[e−r(τ−t)Sτ

∣∣∣St] ≤ St, t ≥ 0,

since u 7→ e−r(u−t)Su is a martingale, hence

supτ≥t

τ stopping time


∣∣∣St] ≤ St, t ≥ 0,

which shows (9.28) by (9.29).

We may also check that since (e−rtSt)t∈R+is a martingale, the process t 7→

(e−rtSt−K)+ is a submartingale since the function x 7→ (x−K)+ is convex,hence for all bounded stopping times τ such that t ≤ τ we have

(St −K)+ ≤ IE∗[(e−r(τ−t)Sτ −K)+

∣∣∣St] ≤ IE∗[e−r(τ−t)(Sτ −K)+

∣∣∣St] ,t ≥ 0, showing that it is always better to wait than to exercise at time t,and the optimal exercise time is τ∗ = +∞. This argument does not apply toAmerican put options.

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N. Privault

9.5 Finite Expiration American Options

In this section we consider finite expirations American put and call optionswith strike K, whose prices can be expressed as


τ stopping time


∣∣∣St] ,and


τ stopping time


∣∣∣St] .

Two-choice optimal stopping at fixed times with finite expiration

We start by considering the optimal stopping problem in a simplified settingwhere τ ∈ t, T is allowed at time t to take only two values t (which corre-sponds to immediate exercise) and T (wait until maturity).

Call options

Since x 7→ (x − K)+ is a non-decreasing convex function and the process(e−rtSt)t∈R+

is a martingale under P∗, we know that t 7→ (e−rtSt−e−rTK)+

becomes a submartingale by the Jensen inequality (9.2), hence by (9.12), forany stopping time τ ≤ t we have

(St −K)+ = ert(e−rtSt − e−rtK)+

≤ ert(e−rtSt − e−rτK)+

≤ ert IE∗[(e−rτSτ − e−rτK)+ | Ft]≤ IE∗[e−r(τ−t)(Sτ −K)+ | Ft],

i.e.(x−K)+ ≤ IE∗[e−r(τ−t)(Sτ −K)+|St = x], x, t > 0.

In particular for τ = T ≥ t a deterministic time we get

(x−K)+ ≤ e−r(T−t) IE∗[(ST −K)+|St = x], x, t > 0.

as illustrated in Figure 9.10 using the Black-Scholes formula for Europeancall options.

In other words, taking x = St, the payoff (St−K)+ of immediate exercise attime t is always lower than the expected payoff e−r(T−t) IE∗[(ST−K)+|St = x]given by exercise at maturity T . As a consequence, the optimal strategy for

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the investor is to wait until time T to exercise an American call option, ratherthan exercising earlier at time t.

Black-Scholes European call pricePayoff (x-K)+

60 80 100 120 140underlying HK$


0 10 20 30 40 50 60 70 80

Fig. 9.10: Expected Black-Scholes European call price vs (x, t) 7→ (x−K)+.

More generally it can be in fact shown that the price of an American call op-tion equals the price of the corresponding European call option with maturityT , i.e.

f(t, St) = e−r(T−t) IE∗[(ST −K)+

∣∣∣St] ,i.e. T is the optimal exercise date, cf. e.g. §14.4 of [72] for a proof.

Put options

For put options the situation is entirely different. The Black-Scholes formulafor European put options shows that the inequality

(K − x)+ ≤ e−r(T−t) IE∗[(K − ST )+|St = x],

does not always hold, as illustrated in Figure 9.11.

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N. Privault

Black-Scholes European put pricePayoff (K-x)+

90 100

110 120

underlying HK$


0 2 4 6 8

10 12 14 16

Fig. 9.11: Black-Scholes put price function vs (x, t) 7→ (K − x)+.

As a consequence, the optimal exercise decision for a put option depends onwhether (K − St)+ ≤ e−r(T−t) IE∗[(K − ST )+|St] (in which case one choosesto exercise at time T ) or (K − St)+ > e−r(T−t) IE∗[(K − ST )+|St] (in whichcase one chooses to exercise at time t).

A view from above of the graph of Figure 9.11 shows the existence ofan optimal frontier depending on time to maturity and on the value of theunderlying asset instead of being given by a constant level L∗ as in Section 9.4,cf. Figure 9.12:

underlying HK$

T-t

90 95 100 105 110 115 120

0

1

2

3

4

5

6

7

8

9

10

Fig. 9.12: Optimal frontier for the exercise of a put option.

At a given time t, one will choose to exercise immediately if (St, T−t) belongsto the blue area on the right, and to wait until maturity if (St, T − t) belongsto the red area on the left.

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PDE characterization of the finite expiration American put price

Let us describe the PDE associated to American put options. After discretiza-tion 0 = t0, t1, . . . , tN = T of the time interval [0, T ], the optimal exercisestrategy for an American put option can be described as follow at each timestep:

If f(t, St) > (K − St)+, wait.

If f(t, St) = (K − St)+, exercise the option at time t.

Note that we cannot have f(t, St) < (K − St)+.

If f(t, St) > (K − St)+ the expected return of the option equals that ofthe risk-free asset. This means that f(t, St) follows the Black-Scholes PDE

rf(t, St) =∂f

∂t(t, St) + rSt

∂f

∂x(t, St) +

1

2σ2S2

t

∂2f

∂x2(t, St),

whereas if f(t, St) = (K − St)+ it is not worth waiting as the return of theoption is lower than that of the risk-free asset:

rf(t, St) ≥∂f

∂t(t, St) + rSt

∂f

∂x(t, St) +

1

2σ2S2

t

∂2f

∂x2(t, St).

As a consequence, f(t, x) should solve the following variational PDE:

f(t, x) ≥ (K − x)+,

∂f

∂t(t, x) + rx

∂f

∂x(t, x) +

1

2σ2x2

∂2f

∂x2(t, x) ≤ rf(t, x),

(∂f

∂t(t, x) + rx

∂f

∂x(t, x) +

1

2σ2x2

∂2f

∂x2(t, x)− rf(t, x)

)× (f(t, x)− (K − x)+) = 0,

(9.30a)

(9.30b)

(9.30c)

subject to the terminal condition f(T, x) = (K −x)+. In other words, equal-ity holds either in (9.30a) or in (9.30b) due to the presence of the term(f(t, x)− (K − x)+) in (9.30c).

The optimal exercise strategy consists in exercising the put option as soonas the equality f(u, Su) = (K − Su)+ holds, i.e. at the time

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N. Privault

τ∗ = infu ≥ t : f(u, Su) = (K − Su)+,

after which the process fL∗(St) ceases to be a martingale and becomes a(strict) supermartingale.

A simple procedure to compute numerically the price of an American putoption is to use a finite difference scheme while simply enforcing the conditionf(t, x) ≥ (K − x)+ at every iteration by adding the condition

f(ti, xj) := max(f(ti, xj), (K − xj)+)

right after the computation of f(ti, xj).

The next figure shows a numerical resolution of the variational PDE(9.30a)-(9.30c) using the above simplified (implicit) finite difference scheme.In comparison with Figure 9.7, one can check that the PDE solution becomestime-dependent in the finite expiration case.

Finite expiration American put priceImmediate payoff (K-x)

+

L*=2r/(2r+sigma

2)

90 100

110 120

underlying

0 2

4 6

8 10


0

2

4

6

8

10

12

14

16

Fig. 9.13: Numerical values of the finite expiration American put price.

In general, one will choose to exercise the put option when

f(t, St) = (K − St)+,

i.e. within the blue area in Figure (9.13). We check that the optimal thresholdL∗ = 90.64 of the corresponding perpetual put option is within the exerciseregion, which is consistent since the perpetual optimal strategy should allowone to wait longer than in the finite expiration case.

The numerical computation of the put price


τ stopping time


∣∣∣St]

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"



can also be done by dynamic programming and backward optimization usingthe Longstaff-Schwartz (or Least Square Monte Carlo, LSM) algorithm [50],as in Figure 9.14.

Longstaff-Schwartz algorithmImmediate payoff (K-x)

+

L*=2r/(2r+sigma

2)

90 100

110 120

underlying

0 2

4 6

8 10


0

2

4

6

8

10

12

14

16

Fig. 9.14: Longstaff-Schwartz algorithm for the finite expiration American put price.

In Figure 9.14 above and Figure 9.15 below the optimal threshold of thecorresponding perpetual put option is again L∗ = 90.64 and falls within theexercise region. Also, the optimal threshold is closer to L∗ for large time tomaturities, which shows that the perpetual option approximates the finiteexpiration option in that situation. In the next Figure 9.15 we compare thenumerical computation of the American put price by the finite difference andLongstaff-Schwartz methods.

0

2

4

6

8

10

90 100 110 120

Tim

e to

mat

urity

T-t

underlying

Longstaff-Schwartz algorithmImplicit finite differencesImmediate payoff (K-x)

+

Fig. 9.15: Comparison between Longstaff-Schwartz and finite differences.

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N. Privault

It turns out that, although both results are very close, the Longstaff-Schwartzmethod performs better in the critical area close to exercise at it yields theexpected continuously differentiable solution, and the simple numerical PDEsolution tends to underestimate the optimal threshold. Also, a small errorin the values of the solution translates into a large error on the value of theoptimal exercise threshold.

The finite expiration American call option

In the next proposition we compute the price of a finite expiration Americancall option with an arbitrary convex payoff function φ.

Proposition 9.7. Let φ : R −→ R be a nonnegative convex function suchthat φ(0) = 0. The price of the finite expiration American call option withpayoff function φ on the underlying asset (St)t∈R+

is given by


τ stopping time

IE∗[e−r(τ−t)φ(Sτ )

∣∣∣St] = e−r(T−t) IE∗[φ(ST )

∣∣∣St] ,i.e. the optimal strategy is to wait until the maturity time T to exercise theoption, and τ∗ = T .

Proof. Since the function φ is convex and φ(0) = 0 we have

φ(px) = φ((1− p)× 0 + px) ≤ (1− p)× φ(0) + pφ(x) ≤ pφ(x),

for all p ∈ [0, 1] and x ≥ 0. Hence the process s 7→ e−rsφ(St+s) is asubmartingale since taking p = e−r(τ−t) we have

e−rs IE∗ [φ (St+s) | Ft] ≥ e−rsφ (IE∗ [St+s | Ft])≥ φ

(e−rs IE∗ [St+s | Ft]

)= φ(St),

where we used Jensen’s inequality (9.2) applied to the convex function φ andthe fact that

φ(px) = φ((1− p)× 0 + px) ≤ (1− p)φ(0) + pφ(x) = pφ(x), x > 0,

by the convexity of φ and the fact that φ(0) = 0. Hence by the optionalstopping theorem for submartingales, cf (9.10), for all (bounded) stoppingtimes τ comprised between t and T we have,

IE∗[e−r(τ−t)φ(Sτ ) | Ft] ≤ e−r(T−t) IE∗[φ(ST ) | Ft],

i.e. it is always better to wait until time T than to exercise at time τ ∈ [t, T ],and this yields

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supt≤τ≤T

τ stopping time


∣∣∣St] ≤ e−r(T−t) IE∗[φ(ST )

∣∣∣St] .The converse inequality

e−r(T−t) IE∗[φ(ST )

∣∣∣St] ≤ supt≤τ≤T

τ stopping time


∣∣∣St] ,being obvious because T is a stopping time.

As a consequence of Proposition 9.7 applied to the convex function φ(x) =(x−K)+, the price of the finite expiration American call option is given by


τ stopping time


∣∣∣St]= e−r(T−t) IE∗

[(ST −K)+

∣∣∣St] ,i.e. the optimal strategy is to wait until the maturity time T to exercise theoption.

In the following table we summarize the optimal exercise strategies for thepricing of American options.

option perpetual finite expirationtype

putoption

K − St, 0 < St ≤ L∗,

(K − L∗)(St

L∗

)−2r/σ2

, St ≥ L∗.

Solve the PDE (9.30a)-(9.30c) for f(t, x)or use Longstaff-Schwartz [50]

τ∗ = T ∧ infu ≥ t : f(u, Su) = (K − Su)+τ∗ = τL∗

calloption

St τ∗ = +∞ e−r(T−t) IE∗[(ST −K)+ | St], τ∗ = T

Exercises

Exercise 9.1 Stopping times. Let (Bt)t∈R+be a standard Brownian motion

started at 0.

1. Consider the random time ν defined by

ν := inft ∈ R+ : Bt = B1,

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N. Privault

which represents the first time Brownian motion Bt hits the level B1. Isν a stopping time ?

2. Consider the random time τ defined by

τ := inft ∈ R+ : eBt = αe−t/2,

which represents the first time the exponential of Brownian motion Btcrosses the path of t 7→ αe−t/2, where α > 1.

Is τ a stopping time ? If τ is a stopping time, compute E[e−τ ] by thestopping time theorem.

3. Consider the random time τ defined by

τ := inft ∈ R+ : B2t = 1 + αt,

which represents the first time Brownian motion Bt crosses the straightline t 7→ 1 + αt, with α ∈ (−∞, 1).

Is τ a stopping time ? If τ is a stopping time, compute E[τ ] by the Doobstopping time theorem.

Exercise 9.2 (Doob-Meyer decomposition in discrete time). Let (Mn)n∈Nbe a discrete-time submartingale with respect to a filtration (Fn)n∈N, withF−1 = ∅, Ω.1. Show that there exists two processes (Nn)n∈N and (An)n∈N such that

(i) (Nn)n∈N is a martingale with respect to (Fn)n∈N,

(ii) (An)n∈N is non-decreasing, i.e. An ≤ An+1 a.s., n ∈ N,

(iii) (An)n∈N is predictable in the sense that An is Fn−1-measurable,n ∈ N, and

(iv) Mn = Nn +An, n ∈ N.

Hint: Let A0 = 0,

An+1 = An + IE[Mn+1 −Mn | Fn], n ≥ 0,

and define (Nn)n∈N in such a way that it satisfies the four required prop-erties.

2. Show that for all bounded stopping times σ and τ such that σ ≤ τ a.s.,we have

IE[Mσ] ≤ IE[Mτ ].

Hint: Use the stopping time Theorem 9.1 for martingales and (9.11).

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Exercise 9.3 American digital options. An American digital call (resp. put)option with maturity T > 0 can be exercised at any time t ∈ [0, T ], at thechoice of the option holder.

The call (resp. put) option exercised at time t yields the payoff 1[K,∞)(St)(resp. 1[0,K](St)), and the option holder wants to find an exercise strategythat will maximize his payoff.

1. Consider the following possible situations at time t:

(i) St ≥ K,(ii) St < K.

In each case (i) and (ii), tell whether you would choose to exercise thecall option immediately or to wait.

2. Consider the following possible situations at time t:

(i) St > K,(ii) St ≤ K.

In each case (i) and (ii), tell whether you would choose to exercise theput option immediately or to wait.

3. The price CAmd (t, St) of an American digital call option is known to satisfy

the Black-Scholes PDE

rCAmd (t, x) =

∂

∂tCAmd (t, x) + rx

∂

∂xCAmd (t, x) +

1

2σ2x2

∂2

∂x2CAmd (t, x).

Based on your answers to Question 1, how would you set the boundaryconditions CAm

d (t,K), 0 ≤ t < T , and CAmd (T, x), 0 ≤ x < K ?

4. The price PAmd (t, St) of an American digital put option is known to satisfy

the same Black-Scholes PDE

rPAmd (t, x) =

∂

∂tPAmd (t, x) + rx

∂

∂xPAmd (t, x) +

1

2σ2x2

∂2

∂x2PAmd (t, x).

(9.31)Based on your answers to Question 2, how would you set the boundaryconditions PAm

d (t,K), 0 ≤ t < T , and PAmd (T, x), x > K ?

5. Show that the optimal exercise strategy for the American digital calloption with strike K is to exercise as soon as the underlying reaches thelevel K, at the time

τK = infu ≥ t : Su = K,

starting from any level St ≤ K, and that the price CAmd (t, St) of the

American digital call option is given by

CAmd (t, x) = IE[e−r(τK−t)1τK<T | St = x].

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N. Privault

6. Show that the price CAmd (t, St) of the American digital call option is equal

to

CAmd (t, x) =

x

KΦ

((r + σ2/2)τ + log(x/K)

σ√τ

)+( xK

)−2r/σ2

Φ

(−(r + σ2/2)τ + log(x/K)

σ√τ

), 0 ≤ x ≤ K,

where τ = T − t. Show that this formula is consistent with your answerto Question 3.

7. Show that the optimal exercise strategy for the American digital putoption with strike K is to exercise as soon as the underlying reaches thelevel K, at the time

τK = infu ≥ t : Su = K,

starting from any level St ≥ K, and that the price PAmd (t, St) of the

American digital put option is

PAmd (t, x) = IE[e−r(τK−t)1τK<T | St = x], x ≥ K.

8. Show that the price PAmd (t, St) of the American digital put option is equal

to

PAmd (t, x) =

x

KΦ

(−(r + σ2/2)τ − log(x/K)

σ√τ

)+( xK

)−2r/σ2

Φ

((r + σ2/2)τ − log(x/K)

σ√τ

), x ≥ K,

and that this formula is consistent with your answer to Question 4.9. Does the call-put parity hold for American digital options ?

Exercise 9.4 American forward contracts. Consider (St)t∈R+ an asset priceprocess given by

dStSt

= rdt+ σdBt,

where (Bt)t∈R+ is a standard Brownian motion.



τ stopping time

IE∗[e−r(τ−t)(K − Sτ )

∣∣∣St] ,and optimal exercise strategy of a finite expiration American type shortforward contract with strike K on the underlying asset (St)t∈R+ , withpayoff K − ST .

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τ stopping time

IE∗[e−r(τ−t)(Sτ −K)

∣∣∣St] ,and optimal exercise strategy of a finite expiration American type longforward contract with strike K on the underlying asset (St)t∈R+

, withpayoff ST −K.

3. How are the answers to Questions 1 and 2 modified in the case of per-petual options ?

Exercise 9.5 Consider an underlying asset price process written as


where (Bt)t∈R+ is a standard Brownian motion under the risk-neutral prob-ability measure P∗, with σ, r > 0.

1. Show that the processes (Yt)t∈R+ and (Zt)t∈R+ defined as

Yt := e−rtS−2r/σ2

t and Zt := e−rtSt, t ∈ R+,

are both martingales under P∗.2. Let τL denote the hitting time

τL = infu ∈ R+ : Su = L.

By application of the stopping time theorem to the martingales (Yt)t∈R+

and (Zt)t∈R+, show that

IE∗[e−rτL | S0 = x

]=

x/L, 0 < x ≤ L,

(x/L)−2r/σ2

, x ≥ L.

3. Compute the price IE∗[e−rτL(K−SτL)] of a short forward contract underthe exercise strategy τL.

4. Show that for every value of S0 = x there is an optimal value L∗x of Lthat maximizes L 7→ E[e−rτL(K − SτL)].

5. Would you use the strategy

τL∗x = infu ∈ R+ : Su = L∗x

as an optimal exercise strategy for the short forward contract with payoffK − Sτ ?

Exercise 9.6 Let p ≥ 1 and consider a power put option with payoff

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N. Privault

((κ− Sτ )+)p =

(κ− Sτ )p if Sτ ≤ κ,

0 if Sτ > κ,

when exercised at time τ , on an underlying asset whose price St is written as


where (Bt)t∈R+ is a standard Brownian motion under the risk-neutral prob-ability measure P∗, r ≥ 0 is the risk-free interest rate, and σ > 0 is thevolatility coefficient.

Given L ∈ (0, κ) a fixed price, consider the following choices for the exerciseof a put option with strike κ:

(a) If St ≤ L, then exercise at time t.

(b) Otherwise, wait until the first hitting time τL := infu ≥ t : Su = L,and exercise the option at time τL.

1. Under the above strategy, what is the option payoff equal to if St ≤ L ?2. Show that in case St > L, the price of the option is equal to

fL(St) = (κ− L)p IE∗[e−r(τL−t)

∣∣∣St] .3. Compute the price fL(St) of the option at time t.

Hint. Recall that by (9.20) we have IE∗[e−r(τL−t) | St = x] = (x/L)−2r/σ2

,x ≥ L.

4. Compute the optimal value L∗ that maximizes L 7→ fL(x) for all fixedx > 0.

Hint. Observe that, geometrically, the slope of x 7→ fL(x) at x = L∗ isequal to −p(κ− L∗)p−1.

5. How would you compute the American option price


τ stopping time

IE∗[e−r(τ−t)((κ− Sτ )+)p

∣∣∣St] ?

Exercise 9.7 Same questions as in Exercise 9.6 for the option with payoffκ− (Sτ )p when exercised at time τ , with p > 0.

Exercise 9.8 (cf. Exercise 8.5 page 372 of [71]). Consider an underlying assetprice process written as

St = S0e(r−a)t+σBt−σ2t/2, t ∈ R+,

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ability measure P∗, r > 0 is the risk-free interest rate, σ > 0 is the volatilitycoefficient, and a > 0 is a constant dividend rate.

1. Show that for all x ≥ L and λ ∈ R the process (Zt)t∈R+defined as

Zt :=

(StS0

)λe−(r−a)λt+λσ

2t/2−λ2σ2t/2, t ∈ R+,

is a martingale under P∗.2. Let τL denote the hitting time

τL = infu ∈ R+ : Su ≤ L.

By application of the stopping time theorem to the martingale (Zt)t∈R+ ,show that

IE∗[e−rτL

]=

(S0

L

)λ, (9.32)

with

λ =−(r − a− σ2/2)−

√(r − a− σ2/2)2 + 4rσ2/2

σ2. (9.33)

3. Show that for all L ∈ (0,K) we have

IE∗[e−rτL(K − SτL)+

∣∣∣S0 = x]

=

K − x, 0 < x ≤ L,

(K − L)( xL

)−(r−a−σ2/2)−√

(r−a−σ2/2)2+4rσ2/2

σ2

, x ≥ L.

4. Show that the value L∗ of L that maximizes

fL(x) := IE∗[e−rτL(K − SτL)+

∣∣∣S0 = x]

for all x is given by

L∗ =λ

λ− 1K.

5. Show that

fL∗(x) =

K − x, 0 < x ≤ L∗ =

λ

λ− 1K,

(1− λK

)λ−1(x

−λ

)λ, x ≥ L∗ =

λ

λ− 1K,

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N. Privault

6. Show by hand computation that fL∗(x) satisfies the variational differen-tial equation

fL∗(x) ≥ (K − x)+,

(r − a)xf ′L∗(x) +1

2σ2x2f ′′L∗(x) ≤ rfL∗(x),

(rfL∗(x)− (r − a)xf ′L∗(x)− 1

2σ2x2f ′′L∗(x)

)× (fL∗(x)− (K − x)+) = 0.

(9.34a)

(9.34b)

(9.34c)

7. By Ito’s formula, check that the discounted portfolio price

t 7→ e−rtfL∗(St)

is a supermartingale.8. Show that we have

fL∗(S0) ≥ supτ stopping time


∣∣∣S0

].

9. Show that the stopped process

s 7→ e−r(s∧τL∗ )fL∗(Ss∧τL∗ ), s ∈ R+,

is a martingale, and that

fL∗(S0) ≤ supτ stopping time


].

10. Fix t ∈ R+ and let τL∗ denote the hitting time

τL∗ = infu ≥ t : Su = L∗.

Conclude that the price of the perpetual American put option with div-idend is given for all t ≥ 0 by


∣∣∣St]

=

K − St, 0 < St ≤

λ

λ− 1K,

(1− λK

)λ−1(St−λ

)λ, St ≥

λ

λ− 1K,

where λ is given by (9.33), and

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τL∗ = infu ≥ t : Su ≤ L.

Exercise 9.9 This exercise is a simplified adaptation of the paper [30].

We consider two risky assets S1 and S2 modeled by

S1(t) = S1(0)eσ1Wt+rt−σ22t/2 and S2(t) = S2(0)eσ2Wt+rt−σ2

2t/2,(9.35)

t ∈ R+, with σ2 > σ1 ≥ 0, and the perpetual optimal stopping problem

supτ stopping time

IE[e−rτ (S1(τ)− S2(τ))+],

where (Wt)t∈R+is a standard Brownian motion under P.

1. Find α > 1 such that the process

Zt := e−rtS1(t)αS2(t)1−α, t ∈ R+, (9.36)

is a martingale.2. For some fixed L ≥ 1, consider the hitting time

τL = inft ∈ R+ : S1(t) ≥ LS2(t),

and show that

IE[e−rτL(S1(τL)− S2(τL))+] = (L− 1) IE[e−rτLS2(τL)].

3. By an application of the stopping time theorem to the martingale (9.36),show that we have

IE[e−rτL(S1(τL)− S2(τL))+] =L− 1

LαS1(0)αS2(0)1−α.

4. Show that the price of the perpetual exchange option is given by

supτ stopping time

IE[e−rτ (S1(τ)− S2(τ))+] =L∗ − 1

(L∗)αS1(0)αS2(0)1−α,

whereL∗ =

α

α− 1.

5. As an application of Question 4, compute the perpetual American putoption price

supτ stopping time

IE[e−rτ (κ− S2(τ))+]

when r = σ22/2.

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Chapter 10

Change of Numeraire and ForwardMeasures

In this chapter we introduce the notion of numeraire. This allows us to con-sider pricing under random discount rates using forward measures, with thepricing of exchange options (Margrabe formula) and foreign exchange op-tions (Garman-Kohlagen formula) as main applications. A short introduc-tion to the computation of self-financing hedging strategies under change ofnumeraire is also given in Section 10.5. The change of numeraire techniqueand associated forward measures will also be applied to the pricing of bondsand interest rate derivatives such as bond options in Chapter 12.

10.1 Notion of Numeraire

A numeraire is any strictly positive Ft-adapted stochastic process (Nt)t∈R+

that can be taken as a unit of reference when pricing an asset or a claim.

In general the price St of an asset in the numeraire Nt is given by

St =StNt, t ∈ R+.

Deterministic numeraires transformations are easy to handle as a change ofnumeraire by a deterministic factor is a formal algebraic transformation thatdoes not involve any risk. This can be the case for example when a currencyis pegged to another currency, e.g. the exchange rate 6.55957 from Euro toFrench Franc has been fixed on January 1st, 1999.

On the other hand, a random numeraire may involve risk and allow forarbitrage opportunities.

Examples of numeraire include:

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N. Privault

- the money market account

Nt = exp

(w t

0rsds

),

where (rt)t∈R+is a possibly random and time-dependent risk-free interest

rate.

In this case,

St =StNt

= e−r t0rsdsSt, t ∈ R+,

represents the discounted price of the asset at time 0.

- an exchange rate Nt = Rt with respect to a foreign currency.

In this case,

St =StRt, t ∈ R+,

represents the price of the asset in units of the foreign currency. For ex-ample, if Rt = 1.7 is the exchange rate from Euro to Singapore dollar andSt = S$1, then St = St/Rt =e 0.59.

- forward numeraire: the price

P (t, T ) = IE∗[e−

r Ttrsds

∣∣∣ Ft] , 0 ≤ t ≤ T,

of a bond paying P (T, T ) = $1 at maturity T , in this case Rt = P (t, T ),0 ≤ t ≤ T . We check that

t 7−→ e−r t0rsdsP (t, T ) = IE∗

[e−

r T0rsds

∣∣∣ Ft] , 0 ≤ t ≤ T,

is a martingale.

- annuity numeraire of the form

Rt =n∑k=1

(Tk − Tk−1)P (t, Tk)

where P (t, T1), . . . , P (t, Tn) are bond prices with maturities T1 < · · · < Tnarranged according to a tenor structure.

- combinations of the above, for example a foreign money market account

er t0rfsdsRt, expressed in local (or domestic) currency, where (rft)t∈R+

rep-

270


"



resents a short term interest rate on the foreign market.

When the numeraire is a random process, the pricing of a claim whose valuehas been transformed under change of numeraire, e.g. under a change of cur-rency, has to take into account the risks existing on the foreign market.

In particular, in order to perform a fair pricing, one has to determine aprobability measure (for example on the foreign market), under which thetransformed process St = St/Nt will be a martingale.

For example in case Nt = er t0rsds, the risk-neutral measure P∗ is a measure

under which the discounted price process

St =StNt

= e−r t0rsdsSt, t ∈ R+,

is a martingale.

In the next section we will see that this property can be extended to anykind of numeraire.

10.2 Change of Numeraire

In this section we review the pricing of options by a change of measure asso-ciated to a numeraire Nt, cf. e.g. [28] and references therein.

Most of the results of this chapter rely on the following assumption, whichexpresses absence of arbitrage. In the sequel, (rt)t∈R+ denotes an Ft-adaptedshort term interest rate process.

Assumption (A) Under the risk-neutral measure P∗, the discountednumeraire

t 7−→ e−r t0rsdsNt

is an Ft-martingale.

Taking the process (Nt)t∈[0,T ] as a numeraire, we define the forward measure

P bydPdP∗

= e−r T0rsds

NTN0

. (10.1)

Recall that from Section 6.3 the above relation rewrites as

dP = e−r T0rsds

NTN0

dP∗,

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N. Privault

which is equivalent to stating that

wΩξ(ω)dP(ω) =

wΩe−

r T0rsds

NTN0

ξdP∗,

or, under a different notation,

IE[ξ] = IE∗[e−

r T0rsds

NTN0

ξ

],

for all integrable FT -measurable random variable ξ.

More generally, (10.1) and the fact that

t 7−→ e−r t0rsdsNt

is a martingale under P∗ under Assumption (A), imply that

IE∗

[dPdP∗

∣∣∣ Ft] = IE∗[NTN0

e−r T0rsds

∣∣∣ Ft] =NtN0

e−r t0rsds, (10.2)

for all integrable Ft-measurable random variables ξ, 0 ≤ t ≤ T . The nextlemma shows that for any integrable random claim F we have

IE[F | Ft] = IE∗[Fe−

r Ttrsds

NTNt

∣∣∣ Ft] , 0 ≤ t ≤ T.

Note that (10.2) should not be confused with (10.3).

Lemma 10.1. We have

dP|FtdP∗|Ft

= e−r Ttrsds

NTNt

, 0 ≤ t ≤ T. (10.3)

Proof. The proof of (10.3) relies on the abstract version of the Bayes formula.we start by noting that for all integrable Ft-measurable random variable Gwe have

IE[GIE[ξ|Ft]] = IE[IE[Gξ|Ft]]= IE[Gξ]

= IE∗[Gξe−

r T0rsds

NTN0

]= IE∗

[GNtN0

e−r t0rsds IE∗

[ξe−

r Ttrsds

NTNt

∣∣∣ Ft]]= IE

[G IE∗

[ξe−

r Ttrsds

NTNt

∣∣∣ Ft]] ,272


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which shows that

IE[ξ|Ft] = IE∗[ξe−

r Ttrsds

NTNt

∣∣∣ Ft] ,i.e. (10.3) holds.

We note that in case the numeraire Nt = er t0rsds is equal to the money market

account we simply have P = P∗.

Pricing using Change of Numeraire

The change of numeraire technique is specially useful for pricing under ran-dom interest rates.

The next proposition is the basic result of this section, it provides a wayto price an option with arbitrary payoff ξ under a random discount factor

e−r Ttrsds by use of the forward measure. It will be applied in Chapter 12 to

the pricing of bond options and caplets, cf. Propositions 12.1, 12.2 and 12.3below.

Proposition 10.1. An option with payoff ξ is priced at time t as

IE∗[e−

r Ttrsdsξ

∣∣∣ Ft] = NtIE

[ξ

NT

∣∣∣ Ft] , 0 ≤ t ≤ T, (10.4)

provided ξ/NT ∈ L1(P,FT ).

Each application of the formula (10.4) will require to

a) identify a suitable numeraire (Nt)t∈R+, and to

b) make sure that the ratio ξ/NT takes a sufficiently simple form,

in order to allow for the computation of the expectation in the right-handside of (10.4).

Proof. Proposition 10.1 relies on Relation (10.3), which shows that

NtIE

[ξ

NT

∣∣∣ Ft] = NtIE∗

[ξ

NT

dP|FtdP∗|Ft

∣∣∣ Ft]= IE∗

[e−

r Ttrsdsξ

∣∣∣ Ft] , 0 ≤ t ≤ T.

Next we consider further examples of numeraires and associated examples ofoption prices.

Examples:

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N. Privault

a) Money market account: Nt = er t0rsds, where (rt)t∈R+

is a possibly randomand time-dependent risk-free interest rate.

In this case we have P = P∗ and (10.4) simply reads

IE∗[e−

r Ttrsdsξ

∣∣∣ Ft] = er t0rsds IE∗

[e−

r T0rsdsξ

∣∣∣ Ft] , 0 ≤ t ≤ T,

which yields no particular information.

b) Forward numeraire: Nt = P (t, T ) is the price P (t, T ) of a bond ma-turing at time T , 0 ≤ t ≤ T . Here, the discounted bond price pro-

cess(e−

r t0rsdsP (t, T )

)t∈[0,T ]

is an Ft-martingale under P∗, i.e. Assump-

tion (A) is satisfied and Nt = P (t, T ) can be taken as numeraire. In thiscase, (10.4) shows that a random claim ξ can be priced as

IE∗[e−

r Ttrsdsξ

∣∣∣ Ft] = P (t, T )IE[ξ∣∣∣ Ft] , 0 ≤ t ≤ T, (10.5)

since P (T, T ) = 1, where the forward measure P satisfies

dPdP∗

= e−r T0rsds

P (T, T )

P (0, T )=e−

r T0rsds

P (0, T )(10.6)

by (10.1).c) Annuity numeraire of the form

Nt =

n∑k=1

(Tk − Tk−1)P (t, Tk)

where P (t, T1), . . . , P (t, Tn) are bond prices with maturities T1 < · · · <Tn. Here, (10.4) shows that

IE∗[e−

r Ttrsds(P (T, Tn)− P (T, T1)− κNT )+

∣∣∣ Ft]= NtIE

[(P (T, Tn)− P (T, T1)

NT− κ)+ ∣∣∣ Ft] ,

0 ≤ t ≤ T , where (P (T, Tn)− P (T, T1))/NT is a swap rate.

In the sequel, given (Xt)t∈R+an asset price process, we define the process of

forward prices

Xt :=Xt

Nt, 0 ≤ t ≤ T, (10.7)

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which represents the values at times t of Xt, expressed in units of thenumeraire Nt. It will be useful to determine the dynamics of (Xt)t∈R+

under

the forward measure P.

Proposition 10.2. Let (Xt)t∈R+denote a continuous Ft-adapted asset price

process such that

t 7−→ e−r t0rsdsXt

is a martingale under P∗. Then under change of numeraire, the process

(Xt)t∈[0,T ] of forward prices is a martingale under P, provided it is integrable

under P.

Proof. We need to show that

IE

[Xt

Nt

∣∣∣ Fs] =Xs

Ns, 0 ≤ s ≤ t, (10.8)

and we achieve this using a standard characterization of conditional expec-tation. Namely, for all bounded Fs-measurable random variables G we notethat under Assumption (A) we have

IE

[GXt

Nt

]= IE∗

[GXt

Nt

dPdP∗

]

= IE∗

[GXt

NtIE∗

[dPdP∗

∣∣∣ Ft]]

= IE∗[Ge−

r t0rudu

Xt

N0

]= IE∗

[Ge−

r s0rudu

Xs

N0

]= IE

[GXs

Ns

], 0 ≤ s ≤ t,

becauset 7−→ e−

r t0rsdsXt

is a martingale. Finally, the identity

IE

[GXt

Nt

]= IE

[GXs

Ns

], 0 ≤ s ≤ t,

for all bounded Fs-measurable G, implies (10.8).

Next we will rephrase Proposition 10.2 in Proposition 10.3 using the Girsanovtheorem, which briefly recalled below.

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N. Privault

Girsanov theorem

Recall that letting

Φt = IE

[dPdP∗

∣∣∣ Ft] , t ∈ [0, T ],

the Girsanov theorem,1 shows that the process (Wt)t∈R+defined by

dWt = dWt −1

ΦtdΦt · dWt, t ∈ R+, (10.9)

is a standard Brownian motion under P.

Next, Relation (10.2) shows that

Φt = IE

[dPdP∗

∣∣∣ Ft]

= IE

[NTN0

e−r T0rsds

∣∣∣ Ft]=NtN0

e−r t0rsds, 0 ≤ t ≤ T,

hence

dΦt = −Φtrtdt+ΦtNtdNt,

and Relation (10.9) becomes

dWt = dWt −1

NtdNt · dWt, t ∈ R+. (10.10)

The next proposition confirms the statement of Proposition 10.2, and inaddition it determines the precise dynamics of (Xt)t∈R+

under P. See Exer-cise 10.1 for another calculation based on geometric Brownian motion, andExercise 10.4 for an extension to correlated Brownian motions.

Proposition 10.3. Assume that (Xt)t∈R+and (Nt)t∈R+

satisfy the stochas-tic differential equations

dXt = rtXtdt+ σXt XtdWt, and dNt = rtNtdt+ σNt NtdWt, (10.11)

where (σXt )t∈R+and (σNt )t∈R+

are Ft-adapted volatility processes. Then wehave

dXt = (σXt − σNt )XtdWt. (10.12)

1 See e.g. Theorem III-35 in [64].

276


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Proof. First we note that by (10.10) and (10.11),

dWt = dWt −1

NtdNt · dWt = dWt − σNt dt, t ∈ R+,

is a standard Brownian motion under P. Next, by Ito’s calculus we have

dXt = d

(Xt

Nt

)=dXt

Nt− Xt

N2t

dNt −1

N2t

dNt · dXt +Xt(dNt)

2

N3t

=1

Nt(rtXtdt+ σXt XtdWt)−

Xt

N2t

(rtNtdt+ σNt NtdWt)

−XtNtN2t

σXt σNt dt+Xt

|σNt |2N2t

N3t

dt

=Xt

NtσXt dWt −

Xt

NtσNt dWt −

Xt

NtσXt σ

Nt dt+Xt

|σNt |2Nt

dt

=Xt

Nt

(σXt dWt − σNt dWt − σXt σNt dt+ |σNt |2dt

)= Xt(σ

Xt − σNt )dWt − Xt(σ

Xt − σNt )σNt dt

= Xt(σXt − σNt )dWt,

since dWt = dWt − σNt dt, t ∈ R+.

We end this section with a comment on inverse changes of measure.

Inverse Change of Measure

In the next proposition we compute conditional inverse density dP/dP.


IE

[dPdP

∣∣∣ Ft] =N0

Ntexp

(w t

0rsds

)0 ≤ t ≤ T, (10.13)

and the process

t 7−→ N0

Ntexp

(w t

0rsds

), 0 ≤ t ≤ T,

is an Ft-martingale under P.

Proof. For all bounded and Ft-measurable random variables F we have,

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N. Privault

IE

[FdPdP

]= IE∗ [F ]

= IE∗[FNtNt

]= IE∗

[FNTNt

exp

(−

w T

trsds

)]= IE

[FN0

Ntexp

(w t

0rsds

)].

By Ito’s calculus and (10.11) we also have

d

(1

Nt

)= − 1

N2t

dNt +1

N3t

(dNt)2

= − 1

N2t

(rtNtdt+ σNt NtdWt) +|σNt |2Nt

dt

= − 1

N2t

(rtNtdt+ σNt Nt(dWt + σNt dt)) +|σNt |2Nt

dt

= − 1

Nt(rtdt+ σNt dWt),

and

d

(1

Ntexp

(w t

0rsds

))= − 1

Ntexp

(w t

0rsds

)σNt dWt,

which recovers the second part of Proposition 10.4, i.e. the martingale prop-erty of

t 7−→ 1

Ntexp

(w t

0rsds

)under P.

10.3 Foreign Exchange

Currency exchange is a typical application of change of numeraire that illus-trate the principle of absence of arbitrage.

Let Rt denote the foreign exchange rate, i.e. Rt is the (possibly fractional)quantity of local currency that correspond to one unit of foreign currency.

Consider an investor that intends to exploit an “overseas investment op-portunity” by

278


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a) at time 0, changing one unit of local currency into 1/R0 units of foreigncurrency,

b) investing 1/R0 on the foreign market at the rate rf to make the amount

etrf

/R0 until time t,

c) changing back etrf

/R0 into a quantity etrf

Rt/R0 of his local currency.

In other words, the foreign money market account etrf

is valued etrf

Rt onthe local (or domestic) market, and its discounted value on the local marketis

e−tr+trf

Rt, t ∈ R+.

The outcome of this investment will be obtained by comparing etrf

Rt/R0

to the amount ert that could have been obtained by investing on the localmarket.

Taking

Nt := etrf

Rt, t ∈ R+, (10.14)

as numeraire, absence of arbitrage is expressed by stating that the discountednumeraire process

t 7−→ e−rtNt = e−t(r−rf )Rt

is a martingale under P∗, which is Assumption (A).

Next we find a characterization of this arbitrage condition under the pa-rameters of the model.

Proposition 10.5. Assume that the foreign exchange rate Rt satisfies astochastic differential equation of the form

dRt = µRtdt+ σRtdWt, (10.15)

where (Wt)t∈R+ is a standard Brownian motion under P∗. Under the absenceof arbitrage Assumption (A) for the numeraire (10.14), we have

µ = r − rf , (10.16)

hence the exchange rate process satisfies

dRt = (r − rf)Rtdt+ σRtdWt. (10.17)

under P∗.

Proof. The equation (10.15) has solution

Rt = R0eµt+σWt−σ2t/2, t ∈ R+,

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N. Privault

hence the discounted value of the foreign money market account etrf

on thelocal market is

e−tr+trf

Rt = R0et(rf−r+µ)+σWt−σ2t/2, t ∈ R+.

Under absence of arbitrage, e−t(r−rf )Rt = e−trNt should be a martingale

under P∗ and this holds provided rf − r + µ = 0, which yields (10.16) and(10.17).

As a consequence of Proposition 10.5, under absence of arbitrage a localinvestor who buys a unit of foreign currency in the hope of a higher returnrf >> r will have to face a lower (or even more negative) drift

µ = r − rf << 0

in his exchange rate Rt,

The local money market account Xt := ert is valued ert/Rt on the foreignmarket, and its discounted value on the foreign market is

t 7−→ et(r−rf )

Rt=Xt

Nt(10.18)

=1

R0et(r−r

f )−µt−σWt+σ2t/2

=1

R0et(r−r

f )−µt−σWt−σ2t/2,

where

dWt = dWt −1

NtdNt · dWt

= dWt −1

RtdRt · dWt

= dWt − σdt, t ∈ R+,

is a standard Brownian motion under P by (10.10). Under absence of arbitrage

e−t(r−rf )Rt is a martingale under P∗ and (10.18) is a martingale under P by

Proposition 10.2, which recovers (10.16).

Proposition 10.6. Under the absence of arbitrage condition (10.16), theexchange rate 1/Rt satisfies

d

(1

Rt

)= (rf − r) 1

Rtdt− σ

RtdWt, (10.19)

under P , where (Rt)t∈R+is given by (10.17).

Proof. By (10.16), the exchange rate 1/Rt is written by Ito’s calculus as

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d

(1

Rt

)= − 1

R2t

(µRtdt+ σRtdWt) +1

R3t

σ2R2tdt

= −(µ− σ2)1

Rtdt− σ

RtdWt

= − µ

Rtdt− σ

RtdWt

= (rf − r) 1

Rtdt− σ

RtdWt,

where Wt is a standard Brownian motion under P.

Consequently, under absence of arbitrage, a foreign investor who buys a unitof the local currency in the hope of a higher return r >> rf will have to facea lower (or even more negative) drift −µ = rf − r in his exchange rate 1/Rtas written in (10.19) under P.

Foreign exchange options

We now price a foreign exchange option with payoff (RT − κ)+ under P∗by the Black-Scholes formula as in the next proposition, also known as theGarman-Kohlagen [27] formula.

Proposition 10.7. (Garman-Kohlagen formula). Assume that (Rt)t∈R+ isgiven by (10.17). The price of the foreign exchange call option on RT withmaturity T and strike κ is given by

e−(T−t)r IE∗[(RT −κ)+ | Rt] = e−(T−t)rf

RtΦ+(t, Rt)−κe−(T−t)rΦ−(t, Rt),

0 ≤ t ≤ T , where

Φ+(t, x) = Φ

(log(x/κ) + (T − t)(r − rf + σ2/2)

σ√T − t

),

and

Φ−(t, x) = Φ

(log(x/κ) + (T − t)(r − rf − σ2/2)

σ√T − t

).

Proof. As a consequence of (10.17) we find the numeraire dynamics

dNt = d(etrf

Rt)

= rfetrf

Rtdt+ etrf

dRt

= retrf

Rtdt+ σetrf

RtdWt

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N. Privault

= rNtdt+ σNtdWt.

Hence a standard application of the Black-Scholes formula yields

e−(T−t)r IE∗[(RT − κ)+ | Ft] = e−(T−t)r IE∗[(e−T rf

NT − κ)+ | Ft]= e−(T−t)re−T r

f

IE∗[(NT − κeT rf

)+ | Ft]

= e−T rf

(NtΦ

(log(Nte

−Trf/κ) + (r + σ2/2)(T − t)σ√T − t

)

−κeT rf−(T−t)rΦ(

log(Nte−Trf/κ) + (r − σ2/2)(T − t)

σ√T − t

))

= e−T rf

(NtΦ

(log(Rt/κ) + (T − t)(r − rf + σ2/2)

σ√T − t

)−κeT rf−(T−t)rΦ

(log(Rt/κ) + (T − t)(r − rf − σ2/2)

σ√T − t

))= e−(T−t)r

f

RtΦ+(t, Rt)− κe−(T−t)rΦ−(t, Rt).

Similarly, from (10.19) rewritten as

d

(ert

Rt

)= rf

ert

Rtdt− σ e

rt

RtdWt,

a foreign exchange option with payoff (1/RT −κ)+ can be priced under P as aput option in a Black-Scholes model by taking ert/Rt as underlying price, rf

as risk-free interest rate, and −σ as volatility parameter. In this frameworkthe Black-Scholes formula (5.12) yields

e−(T−t)rf

IE

[(1

RT− κ)+ ∣∣∣ Rt] (10.20)

= e−(T−t)rf

e−rT IE

[(erT

RT− κerT

)+ ∣∣∣ Rt] (10.21)

=e−r(T−t)

RtΦ+

(t,

1

Rt

)− κe−(T−t)rfΦ−

(t,

1

Rt

),

where

Φ+(t, x) = Φ

(log(x/κ) + (T − t)(rf − r + σ2/2)

σ√T − t

),

and

Φ−(t, x) = Φ

(log(x/κ) + (T − t)(rf − r − σ2/2)

σ√T − t

).

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Call/put duality for foreign exchange options

Let Nt = etrf

Rt, where Rt is an exchange rate with respect to a foreign cur-rency and rf is the foreign market interest rate.

From Proposition 10.1 and (10.4) we have

e−(T−t)r IE∗

[(1

κ−RT

)+ ∣∣∣ Rt] = NtIE

[1

eT rfRT

(1

κ−RT

)+ ∣∣∣ Rt] ,and this yields the call/put duality

e−(T−t)rf

IE

[(1

RT− κ)+ ∣∣∣ Rt] = e−(T−t)r

f

IE

[κ

RT

(1

κ−RT

)+ ∣∣∣ Rt]

= etrf

IE

[κ

eT rfRT

(1

κ−RT

)+ ∣∣∣ Rt]

=κ

Ntetr

f−(T−t)r IE∗

[(1

κ−RT

)+ ∣∣∣ Rt]

=κ

Rte−(T−t)r IE∗

[(1

κ−RT

)+ ∣∣∣ Rt] , (10.22)

between a call option with strike κ and a (possibly fractional) quantity κ/Rtof put option(s) with strike 1/κ.

In the Black-Scholes case the duality (10.22) can be directly checked byverifying that (10.20) coincides with

κ

Rte−(T−t)r IE∗

[(1

κ−RT

)+ ∣∣∣ Rt]

=κ

Rte−(T−t)re−T r

f

IE∗

(eT rfκ− eT rfRT

)+ ∣∣∣ Rt

=κ

Rte−(T−t)re−T r

f

IE∗

(eT rfκ−NT

)+ ∣∣∣ Rt

=κ

Rt

(e−(T−t)r

κΦp− (t, Rt)− e−(T−t)r

f

RtΦp+ (t, Rt)

)

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N. Privault

=e−(T−t)r

RtΦp− (t, Rt)− κe−(T−t)r

f

Φp+ (t, Rt)

=e−r(T−t)

RtΦ+

(t,

1

Rt

)− κe−(T−t)rfΦ−

(t,

1

Rt

),

where

Φp−(t, x) = Φ

(− log(xκ) + (T − t)(r − rf − σ2

t /2)

σ√T − t

),

and

Φp+(t, x) = Φ

(− log(xκ) + (T − t)(r − rf + σ2/2)

σ√T − t

).

10.4 Pricing of Exchange Options

Based on Proposition 10.2 we model the process Xt of forward prices as acontinuous martingale under P, written as

dXt = σtdWt, t ∈ R+, (10.23)

where (Wt)t∈R+ is a standard Brownian motion under P and (σt)t∈R+ isan Ft-adapted process. The following lemma is a consequence of the Markovproperty of the process (Xt)t∈R+

and leads to the Margrabe formula of Propo-sition 10.8 below.

Lemma 10.2. Assume that (Xt)t∈R+has the dynamics

dXt = σt(Xt)dWt, (10.24)

where x 7−→ σt(x) is a Lipschitz function, uniformly in t ∈ R+. Then theoption with payoff ξ = NT g(XT ) is priced at time t as

NtC(t, Xt) = NtIE[g(XT )

∣∣∣ Ft] = IE∗[e−

r TtrsdsNT g(XT )

∣∣∣ Ft] , (10.25)

for some (measurable) function C(t, x) on R+ × R+.

The next proposition states the Margrabe [51] formula for the pricing ofexchange options by the zero interest rate Black-Scholes formula. It will beapplied in particular in Proposition 12.2 below for the pricing of bond options.

Proposition 10.8. (Margrabe formula). Assume that σt(Xt) = σ(t)Xt, i.e.

the martingale (Xt)t∈[0,T ] is a geometric Brownian motion under P with de-terministic volatility (σ(t))t∈[0,T ]. Then we have

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IE∗[e−

r Ttrsds (XT − κNT )

+∣∣∣ Ft] = XtΦ

0+(t, Xt)− κNtΦ0

−(t, Xt),

(10.26)

t ∈ [0, T ], where

Φ0+(t, x) = Φ

(log(x/κ)

v(t, T )+v(t, T )

2

), Φ0

−(t, x) = Φ

(log(x/κ)

v(t, T )− v(t, T )

2

),

and

v2(t, T ) =w T

tσ2(s)ds.

Proof. Taking g(x) = (x− κ)+ in (10.25), the call option with payoff

(XT − κNT )+ = NT (XT − κ)+,

and floating strike κNT is priced by (10.25) as

IE∗[e−

r Ttrsds(XT − κNT )+

∣∣∣ Ft] = IE∗[e−

r TtrsdsNT (XT − κ)+

∣∣∣ Ft]= NtIE

[(XT − κ)+

∣∣∣ Ft]= NtC(t, Xt),

where the function C(t, Xt) is given by the Black-Scholes formula

C(t, x) = xΦ0+(t, x)− κΦ0

−(t, x),

with zero interest rate, since (Xt)t∈[0,T ] is a geometric Brownian motion which

is a martingale under P. Hence we have

IE∗[e−


+∣∣∣ Ft] = NtC(t, Xt)

= NtXtΦ0+(t, Xt)− κNtΦ0

−(t, Xt),

t ∈ R+.

In particular, from Proposition 10.3 and (10.12), we can take σ(t) = σXt −σNtwhen (σXt )t∈R+

and (σNt )t∈R+are deterministic.

Examples:

a) When the short rate process (r(t))t∈[0,T ] is a deterministic function and

Nt = e−r Ttr(s)ds, 0 ≤ t ≤ T , Proposition 10.8 yields Merton’s [53] “zero

interest rate” version of the Black-Scholes formula

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N. Privault

e−r Ttr(s)ds IE∗

[(XT − κ)

+∣∣∣ Ft]

= XtΦ0+

(t, e

r Ttr(s)dsXt

)− κe−

r Ttr(s)dsΦ0

−

(t, e

r Ttr(s)dsXt

),

where (Xt)t∈R+satisfies the equation

dXt

Xt= r(t)dt+ σ(t)dWt, 0 ≤ t ≤ T.

b) In the case of pricing under a forward numeraire, i.e. Nt = P (t, T ), 0 ≤t ≤ T , we get

IE∗[e−

r Ttrsds (XT − κ)

+∣∣∣ Ft] = XtΦ+(t, Xt)− κP (t, T )Φ−(t, Xt),

t ∈ R+, since P (T, T ) = 1. In particular, when Xt = P (t, S) the aboveformula allows us to price a bond call option on P (T, S) as

IE∗[e−

r Ttrsds (P (T, S)− κ)

+∣∣∣ Ft] = P (t, S)Φ+(t, Xt)−κP (t, T )Φ−(t, Xt),

0 ≤ t ≤ T , provided the martingale Xt = P (t, S)/P (t, T ) under P is givenby a geometric Brownian motion, cf. Section 12.2.

10.5 Self-Financing Hedging by Change of Numeraire

In this section we reconsider and extend the Black-Scholes self-financing hedg-ing strategies found in (6.20)-(6.21) and Proposition 6.7 of Chapter 6, anduse the stochastic integral representation of the forward payoff ξ/NT andchange of numeraire to compute self-financing portfolio strategies. Our hedg-ing portfolios will be built on the assets (Xt, Nt), not on Xt and the money

market account Bt = er t0rsds, extending the classical hedging portfolios that

are available in from the Black-Scholes formula, using a technique from [39],cf. also [61].

Assume that the forward claim ξ/NT ∈ L2(Ω) has the stochastic integralrepresentation

ξ

NT= IE

[ξ

NT

]+

w T

0φtdXt, (10.27)

where (Xt)t∈[0,T ] is given by (10.23) and (φt)t∈[0,T ] is a square-integrable

adapted process under P, from which it follows that the forward claim price

Vt := IE

[ξ

NT

∣∣∣ Ft] , 0 ≤ t ≤ T,

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is a martingale that can be decomposed as

Vt = IE

[ξ

NT

]+

w t

0φsdXs, 0 ≤ t ≤ T. (10.28)

The next proposition extends the argument of [39] to the general frame-work of pricing using change of numeraire. Note that this result differs fromthe standard formula that uses the money market account Bt = e

r t0rsds for

hedging instead of Nt, cf. e.g. [28] pages 453-454. The notion of self-financingportfolio is similar to that of Definition 5.1.

Proposition 10.9. Letting ηt = Vt − Xtφt, 0 ≤ t ≤ T , the portfolio(φt, ηt)t∈[0,T ] with value

Vt = φtXt + ηtNt, 0 ≤ t ≤ T,

is self-financing in the sense that

dVt = φtdXt + ηtdNt,

and it hedges the claim ξ, i.e.

Vt = φtXt + ηtNt = IE∗[e−

r Ttrsdsξ

∣∣∣ Ft] , 0 ≤ t ≤ T. (10.29)

Proof. In order to check that the portfolio (φt, ηt)t∈[0,T ] hedges the claimξ it suffices to check that (10.29) holds since by (10.4) the price Vt at timet ∈ [0, T ] of the hedging portfolio satisfies

Vt = NtVt = IE∗[e−

r Ttrsdsξ

∣∣∣ Ft] , 0 ≤ t ≤ T.

Next, we show that the portfolio (φt, ηt)t∈[0,T ] is self-financing. By numeraireinvariance, cf. e.g. page 184 of [63], we have

dVt = d(NtVt)

= VtdNt +NtdVt + dNt · dVt= VtdNt +NtφtdXt + φtdNt · dXt

= φtXtdNt +NtφtdXt + φtdNt · dXt + (Vt − φtXt)dNt

= φtd(NtXt) + (Vt − φtXt)dNt

= φtdXt + ηtdNt.

We now consider an application to the forward Delta hedging of Europeantype options with payoff ξ = g(XT ) where g : R −→ R and (Xt)t∈R+

has theMarkov property as in (10.24), where σ : R+×R. Assuming that the functionC(t, x) defined by

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N. Privault

Vt := IE[g(XT )

∣∣∣ Ft] = C(t, Xt)

is C2 on R+, we have the following corollary of Proposition 10.9.

Corollary 10.1. Letting ηt = C(t, Xt) − Xt∂C

∂x(t, Xt), 0 ≤ t ≤ T , the

portfolio

(∂C

∂x(t, Xt), ηt

)t∈[0,T ]

with value

Vt = ηtNt +Xt∂C

∂x(t, Xt), t ∈ R+,

is self-financing and hedges the claim ξ = NT g(XT ).

Proof. This result follows directly from Proposition 10.9 by noting thatby Ito’s formula, and the martingale property of Vt under P the stochasticintegral representation (10.28) is given by

φt =∂C

∂x(t, Xt), 0 ≤ t ≤ T.

In the case of a call option with payoff function ξ = (XT − κNT )+

=

NT (XT − κ)+ on the geometric Brownian motion (Xt)t∈[0,T ] under P with

σt(Xt) = σ(t)Xt,

where (σ(t))t∈[0,T ] is a deterministic function, we have the following corollaryon the hedging of exchange options based on the Margrabe formula (10.26).

Corollary 10.2. The decomposition

IE∗[e−


+∣∣∣ Ft] = XtΦ

0+(t, Xt)− κNtΦ0

−(t, Xt)

yields a self-financing portfolio (Φ0+(t, Xt),−κΦ0

−(t, Xt))t∈[0,T ] in (Xt, Nt)

that hedges the claim ξ = (XT − κNT )+

.

Proof. We apply Corollary 10.1 and the classical relation

∂C

∂x(t, x) = Φ0

+(t, x), x ∈ R,

for the function C(t, x) = xΦ0+(t, x)− κΦ0

−(t, x).

Note that the Delta hedging method requires the computation of the func-tion C(t, x) and that of the associated finite differences, and may not apply

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to path-dependent claims.

Examples:

a) When the short rate process (r(t))t∈[0,T ] is a deterministic function and

Nt = er Ttr(s)ds, Corollary 10.2 yields the usual Black-Scholes hedging

strategy(Φ+(t, Xt),−κe

r T0r(s)dsΦ−(t,Xt)

)t∈[0,T ]

=(Φ0+(t, e

r Ttr(s)dsXt),−κe

r T0r(s)dsΦ0

−(t, er Ttr(s)dsXt)

)t∈[0,T ]

in (Xt, er t0r(s)ds), that hedges the claim ξ = (XT − κ)

+.

b) In case Nt = P (t, T ) and Xt = P (t, S), 0 ≤ t ≤ T < S, Corollary 10.2shows that a bond call option with payoff (P (T, S)− κ)+ can be hedgedas

IE∗[e−


+∣∣∣ Ft] = P (t, S)Φ+(t, Xt)−κP (t, T )Φ−(t, Xt)

by the self-financing portfolio

(Φ+(t, Xt),−κΦ−(t, Xt))t∈[0,T ]

in (P (t, S), P (t, T )), i.e. one needs to hold the quantity Φ+(t, Xt) of thebond maturing at time S, and to short a quantity κΦ−(t, Xt) of the bondmaturing at time T .

Exercises

Exercise 10.1 Let (Bt)t∈R+be a standard Brownian motion started at 0

under the risk-neutral measure P∗. Consider a numeraire (Nt)t∈R+given by

Nt := N0eηBt−η2t/2, t ∈ R+,

and a risky asset (Xt)t∈R+ given by

Xt := X0eσBt−σ2t/2, t ∈ R+.

Let P denote the forward measure relative to the numeraire (Nt)t∈R+ , under

which the process Xt := Xt/Nt of forward prices is known to be a martingale.

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N. Privault

1. Using the Ito formula, compute

dXt = d(Xt/Nt) = (X0/N0)d(e(σ−η)Bt−(σ

2−η2)t/2).

2. Explain why the exchange option price IE[(XT − λNT )+] at time 0 hasthe Black-Scholes form

IE[(XT − λNT )+] (10.30)

= X0Φ

(log(X0/λ)

σ√T

+σ√T

2

)− λN0Φ

(log(X0/λ)

σ√T

− σ√T

2

).

Hints:

(i) Use the change of numeraire identity

IE[(XT − λNT )+] = N0IE[(XT − λ)+].

(ii) The forward price Xt is a martingale under the forward measure Prelative to the numeraire (Nt)t∈R+

.

3. Give the value of σ in terms of σ and η.

Exercise 10.2 Bond options. Consider two bonds with maturities T and S,with prices P (t, T ) and P (t, S) given by

dP (t, T )

P (t, T )= rtdt+ ζTt dWt,

anddP (t, S)

P (t, S)= rtdt+ ζSt dWt,

where (ζT (s))s∈[0,T ] and (ζS(s))s∈[0,S] are deterministic functions.

1. Show, using Ito’s formula, that

d

(P (t, S)

P (t, T )

)=P (t, S)

P (t, T )(ζS(t)− ζT (t))dWt,

where (Wt)t∈R+ is a standard Brownian motion under P.2. Show that

P (T, S) =P (t, S)

P (t, T )exp

(w T

t(ζS(s)− ζT (s))dWs −

1

2

w T

t|ζS(s)− ζT (s)|2ds

).

Let P denote the forward measure associated to the numeraire

Nt := P (t, T ), 0 ≤ t ≤ T.

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3. Show that for all S, T > 0 the price at time t

IE[e−

r Ttrsds(P (T, S)− κ)+

∣∣∣ Ft]of a bond call option on P (T, S) with payoff (P (T, S)− κ)+ is equal to

IE∗[e−

r Ttrsds(P (T, S)− κ)+

∣∣∣ Ft] (10.31)

= P (t, S)Φ

(v

2+

1

vlog

P (t, S)

κP (t, T )

)− κP (t, T )Φ

(−v

2+

1

vlog

P (t, S)

κP (t, T )

),

where

v2 =w T

t|ζS(s)− ζT (s)|2ds.

4. Compute the self-financing hedging strategy that hedges the bond optionusing a portfolio based on the assets P (t, T ) and P (t, S).

Exercise 10.3 Consider two risky assets S1 and S2 modeled by the geometricBrownian motions

S1(t) = eσ1Wt+µt and S2(t) = eσ2Wt+µt, t ∈ R+, (10.32)

where (Wt)t∈R+ is a standard Brownian motion under P.

1. Find a condition on r, µ and σ2 so that the discounted price processe−rtS2(t) is a martingale under P.

2. Assume that r − µ = σ22/2, and let

Xt = e(σ22−σ

21)t/2S1(t), t ∈ R+.

Show that the discounted process e−rtXt is a martingale under P.3. Taking Nt = S2(t) as numeraire, show that the forward process X(t) =

Xt/Nt is a martingale under the forward measure P defined by

dPdP

= e−rTNTN0

.

Recall thatWt := Wt − σ2t

is a standard Brownian motion under P.4. Using the relation

e−rT IE[(S1(T )− S2(T ))+] = N0IE[(S1(T )− S2(T ))+/NT ],

compute the pricee−rT IE[(S1(T )− S2(T ))+].

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N. Privault

of the exchange option on the assets S1 and S2.

Exercise 10.4 Extension of Proposition 10.3 to correlated Brownian motions.Assume that (St)t∈R+

and (Nt)t∈R+satisfy the stochastic differential equa-

tions

dSt = rtStdt+ σSt StdWSt , and dNt = ηtNtdt+ σNt NtdW

Nt ,

where (WSt )t∈R+ and (WN

t )t∈R+ have the correlation

dWSt · dWN

t = ρdt,

where ρ ∈ [−1, 1].

1. Show that (WNt )t∈R+

can be written as

WNt = ρWS

t +√

1− ρ2Wt, t ∈ R+,

where (Wt)t∈R+is a standard Brownian motion under P∗, independent

of (WSt )t∈R+

.2. Letting Xt = St/Nt, show that dXt can be written as

dXt = (rt − ηt + (σNt )2 − ρσNt σSt )Xtdt+ σtXtdWXt ,

where WXt is a standard Brownian motion under P∗ and σt is to be

computed.

Exercise 10.5 Quanto options (Exercise 9.5 in [71]). Consider an asset pricedSt at time t, with

dSt = rStdt+ σSStdWSt ,

and an exchange rate (Rt)t∈R+given by

dRt = (r − rf)Rtdt+ σRRtdWRt ,

from (10.16) in Proposition 10.5, where (WRt )t∈R+

is written as

WRt = ρWS

t +√

1− ρ2Wt, t ∈ R+,

where (Wt)t∈R+is a standard Brownian motion under P∗, independent of

(WSt )t∈R+

, i.e. we have

dWRt · dWS

t = ρdt,

where ρ is a correlation coefficient.

1. Leta = r − rf + ρσRσS − (σR)2

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and Xt = eatSt/Rt, t ∈ R+, and show by Exercise 10.4 that dXt can bewritten as

dXt = rXtdt+ σXtdWXt ,

where (WXt )t∈R+

is a standard Brownian motion under P∗ and σ is to becomputed.


e−r(T−t) IE∗

[(STRT− κ)+ ∣∣∣ Ft]

at time t of a quanto option.

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Chapter 11

Forward Rate Modeling

This chapter is concerned with interest rate modeling, in which the meanreversion property plays an important role. We consider the main short ratemodels (Vasicek, CIR, CEV, affine models) and the computation of bondprices in such models. Next we consider the modeling of forward rates in theHJM and BGM models, as well as in two-factor models.

11.1 Short Term Models

Mean Reversion

The first model to capture the mean reversion property of interest rates, aproperty not possessed by geometric Brownian motion, is the Vasicek [74]model, which is based on the Ornstein-Uhlenbeck process. Here, the shortterm interest rate process (rt)t∈R+

solves the equation

drt = (a− brt)dt+ σdBt, (11.1)

where (Bt)t∈R+ is a standard Brownian motion, with solution

rt = r0e−bt +

a

b(1− e−bt) + σ

w t

0e−b(t−s)dBs, t ∈ R+. (11.2)

The law of rt is Gaussian at all times t, with mean r0e−bt + a

b (1− e−bt) andvariance

σ2w t

0(e−b(t−s))2ds = σ2

w t

0e−2bsds =

σ2

2b(1− e−2bt), t ∈ R+.

This model has the interesting properties of being statistically stationary intime in the long run, and to admit a Gaussian N (a/b, σ2/(2b)) invariant dis-

"



N. Privault

tribution when b > 0, however its drawback is to allow for negative values ofrt.

Figure 11.1 presents a random simulation of t 7−→ rt in the Vasicek modelwith r0 = a/b = 5%, i.e. the reverting property of the process is with respectto its initial value r0 = 5%. Note that the interest rate in Figure 11.1 becomesnegative for a short period of time, which is unusual for interest rates butmay nevertheless happen.

0.025

0.03

0.035

0.04

0.045

0.05

0.055

0.06

0.065

0.07

0 5 10 15 20

Fig. 11.1: Graph of t 7−→ rt in the Vasicek model.

The Cox-Ingersoll-Ross (CIR) [13] model brings a solution to the positiv-ity problem encountered with the Vasicek model, by the use the nonlinearequation

drt = β(α− rt)dt+ r1/2t σdBt, α, β ≥ 0.

Other classical mean reverting models include the Courtadon (1982) model

drt = β(α− rt)dt+ σrtdBt,

where α, β, σ are nonnegative, and the exponential Vasicek model

drt = rt(η − a log rt)dt+ σrtdBt,

where a, η, σ are nonnegative.

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Constant Elasticity of Variance (CEV)

Constant Elasticity of Variance models are designed to take into account non-constant volatilities that can vary as a power of the underlying asset. TheMarsh-Rosenfeld (1983) model

drt = (βr−(1−γ)t + αrt)dt+ σr

γ/2t dBt

where α, β, σ, γ are nonnegative constants, covers most of the CEV models.For γ = 1 this is the CIR model, and for β = 0 we get the standard CEVmodel

drt = αrtdt+ σrγ/2t dBt.

If γ = 2 this yields the Dothan [17] model

drt = αrtdt+ σrtdBt.

Affine Models

The class of short rate interest rate models admits a number of generalizationsthat can be found in the references quoted in the introduction of this chapter,among which is the class of affine models of the form

drt = (η(t) + λ(t)rt)dt+√δ(t) + γ(t)rtdBt. (11.3)

Such models are called affine because the associated zero-coupon bonds canbe priced using an affine PDE of the type (11.10) below, as will be seen afterProposition 11.2.

They also include the Ho-Lee model

drt = θ(t)dt+ σdBt,

where θ(t) is a deterministic function of time, and the Hull-White model

drt = (θ(t)− α(t)rt)dt+ σ(t)dBt

which is a time-dependent extension of the Vasicek model.

11.2 Zero-Coupon Bonds

A zero-coupon bond is a contract priced P0(t, T ) at time t < T to deliverP0(T, T ) = 1$ at time T . The computation of the arbitrage price P0(t, T ) ofa zero-coupon bond based on an underlying short term interest rate process

" 297



N. Privault

(rt)t∈R+is a basic and important issue in interest rate modeling.

In case the short term interest rate process (rt)t∈R+is a deterministic

function of time, a standard arbitrage argument shows that the price P (t, T )of the bond is given by

P (t, T ) = e−r Ttrsds, 0 ≤ t ≤ T. (11.4)

In case (rt)t∈R+ is an Ft-adapted random process the formula (11.4) is nolonger valid as it relies on future information, and we replace it with

P (t, T ) = IE∗[e−

r Ttrsds

∣∣∣ Ft] , 0 ≤ t ≤ T, (11.5)

under a risk-neutral measure P∗. It is natural to write P (t, T ) as a conditionalexpectation under a martingale measure, as the use of conditional expectation

helps to “filter out” the future information past time t contained inw T

trsds.

Expression (11.5) makes sense as the “best possible estimate” of the future

quantity e−r Ttrsds in mean square sense, given information known up to time

t.

Pricing bonds with non-zero coupon is not difficult in the case of a deter-ministic continuous-time coupon yield at rate c > 0. In this case the pricePc(t, T ) of the coupon bound is given by

Pc(t, T ) = ec(T−t)P0(t, T ), 0 ≤ t ≤ T,

In the sequel we will only consider zero-coupon bonds, and let P (t, T ) =P0(t, T ), 0 ≤ t ≤ T .

The following proposition shows that Assumption (A) of Chapter 10 issatisfied, i.e. the bond price process t 7−→ P (t, T ) can be taken as a numeraire.

Proposition 11.1. The discounted bond price process

t 7−→ e−r t0rsdsP (t, T )

is a martingale under P∗.

Proof. We have

e−r t0rsdsP (t, T ) = e−

r t0rsds IE∗

[e−

r Ttrsds

∣∣∣ Ft]= IE∗

[e−

r t0rsdse−

r Ttrsds

∣∣∣ Ft]= IE∗

[e−

r T0rsds

∣∣∣ Ft]

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"



and this suffices to conclude since by the “tower property” (16.24) of condi-tional expectations, any process of the form t 7−→ IE∗[F | Ft], F ∈ L1(Ω), isa martingale, cf. Relation (6.1).

Bond pricing PDE

We assume from now on that the underlying short rate process is solutionto the stochastic differential equation

drt = µ(t, rt)dt+ σ(t, rt)dBt (11.6)

where (Bt)t∈R+is a standard Brownian motion under P∗.

Since all solutions of stochastic differential equations such as (11.6) have theMarkov property, cf e.g. Theorem V-32 of [64], the arbitrage price P (t, T )can be rewritten as a function F (t, rt) of rt, i.e.

P (t, T ) = IE∗[e−

r Ttrsds

∣∣∣ Ft] = IE∗[e−

r Ttrsds

∣∣∣ rt] = F (t, rt),

and depends on rt only instead of depending on all information available inFt up to time t, meaning that the pricing problem can now be formulated asa search for the function F (t, x).

Proposition 11.2. (Bond pricing PDE) The bond pricing PDE for P (t, T ) =F (t, rt) is written as

xF (t, x) =∂F

∂t(t, x) + µ(t, x)

∂F

∂x(t, x) +

1

2σ2(t, x)

∂2F

∂x2(t, x), (11.7)

t ∈ R+, x ∈ R, subject to the terminal condition

F (T, x) = 1, x ∈ R. (11.8)

Proof. From Ito’s formula we have

d(e−

r t0rsdsP (t, T )

)= −rte−

r t0rsdsP (t, T )dt+ e−

r t0rsdsdP (t, T )

= −rte−r t0rsdsF (t, rt)dt+ e−

r t0rsdsdF (t, rt)

= −rte−r t0rsdsF (t, rt)dt+ e−

r t0rsds

∂F

∂x(t, rt)(µ(t, rt)dt+ σ(t, rt)dBt)

+e−r t0rsds

(1

2σ2(t, rt)

∂2F

∂x2(t, rt)dt+

∂F

∂t(t, rt)dt

)

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N. Privault

= e−r t0rsdsσ(t, rt)

∂F

∂x(t, rt)dBt

+e−r t0rsds

(−rtF (t, rt) + µ(t, rt)

∂F

∂x(t, rt)

+1

2σ2(t, rt)

∂2F

∂x2(t, rt) +

∂F

∂t(t, rt)

)dt. (11.9)

Given that t 7−→ e−r t0rsdsP (t, T ) is a martingale, the above expression (11.9)

should only contain terms in dBt (cf. Corollary II-1 of [64]), and all terms indt should vanish inside (11.9). This leads to the identity

−rtF (t, rt) + µ(t, rt)∂F

∂x(t, rt) +

1

2σ2(t, rt)

∂2F

∂x2(t, rt) +

∂F

∂t(t, rt) = 0.

Condition (11.8) is due to the fact that P (T, T ) = $1.

In the case of an interest rate process modeled by (11.3) we have

µ(t, x) = η(t) + λ(t)x and σ(t, x) =√δ(t) + γ(t)x,

hence (11.7) yields the affine PDE

xF (t, x) =∂F

∂t(t, x) + (η(t) + λ(t)x)

∂F

∂x(t, x) +

1

2(δ(t) + γ(t)x)

∂2F

∂x2(t, x),

(11.10)

t ∈ R+, x ∈ R.

The above proposition also shows that t 7−→ e−r t0rsdsP (t, T ) satisfies the

stochastic differential equations

d(e−

r t0rsdsP (t, T )

)= e−

r t0rsdsσ(t, rt)

∂F

∂x(t, rt)dBt.

Consequently we have

dP (t, T )

P (t, T )=

1

P (t, T )d(er t0rsdse−

r t0rsdsP (t, T )

)=

1

P (t, T )

(rtP (t, T )dt+ e

r t0rsdsd

(e−

r t0rsdsP (t, T )

))= rtdt+

1

P (t, T )er t0rsdsd

(e−

r t0rsdsP (t, T )

)= rtdt+

σ(t, rt)

F (t, rt)

∂F

∂x(t, rt)dBt

= rtdt+ σ(t, rt)∂ logF

∂x(t, rt)dBt. (11.11)

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In the Vasicek casedrt = (a− brt)dt+ σdWt,

the bond price takes the form P (t, T ) = eA(T−t)+C(T−t)rt , cf. (11.14) below,and by (11.11) we find

dP (t, T )

P (t, T )= rtdt−

σ

b(1− e−b(T−t))dWt. (11.12)

Note that more generally, all affine short rate models as defined in Rela-tion (11.3), including the Vasicek model, will yield a bond pricing formula ofthe form

P (t, T ) = eA(T−t)+C(T−t)rt ,

cf. e.g. § 3.2.4. of [8].

Probabilistic PDE Solution

Next we solve the PDE (11.7) by a direct computation of the conditionalexpectation

P (t, T ) = IE∗[e−

r Ttrsds

∣∣∣ Ft] , (11.13)

in the Vasicek [74] model

drt = (a− brt)dt+ σdBt,

i.e. when the short rate (rt)t∈R+has the expression

rt = g(t) +w t

0h(t, s)dBs = r0e

−bt +a

b(1− e−bt) + σ

w t

0e−b(t−s)dBs,

where g(t) and h(t, s) are deterministic functions.

Letting u∨t = max(u, t), using the fact that Wiener integrals are Gaussianrandom variables and the Gaussian moment generating function, we have

P (t, T ) = IE∗[e−

r Ttrsds

∣∣∣ Ft]= IE∗

[e−

r Tt

(g(s)+r s0h(s,u)dBu)ds

∣∣∣ Ft]= e−

r Ttg(s)ds IE∗

[e−

r Tt

r s0h(s,u)dBuds

∣∣∣ Ft]= e−

r Ttg(s)ds IE∗

[e−

r T0

r Tu∨t h(s,u)dsdBu

∣∣∣ Ft]= e−

r Ttg(s)dse−

r t0

r Tu∨t h(s,u)dsdBu IE∗

[e−

r Tt

r Tu∨t h(s,u)dsdBu

∣∣∣ Ft]= e−

r Ttg(s)dse−

r t0

r Tth(s,u)dsdBu IE∗

[e−

r Tt

r Tuh(s,u)dsdBu

∣∣∣ Ft]" 301



N. Privault

= e−r Ttg(s)dse−

r t0

r Tth(s,u)dsdBu IE∗

[e−

r Tt

r Tuh(s,u)dsdBu

]= e−

r Ttg(s)dse−

r t0

r Tth(s,u)dsdBue

12

r Tt (

r Tuh(s,u)ds)

2du

= e−r Tt

(r0e−bs+ a

b (1−e−bs))dse−σ

r t0

r Tte−b(s−u)dsdBu

×eσ2

2

r Tt (

r Tue−b(s−u)ds)

2du

= e−r Tt

(r0e−bs+ a

b (1−e−bs))dse−

σb (1−e

−b(T−t))r t0e−b(t−u)dBu

×eσ2

2

r Tte2bu

(e−bu−e−bT

b

)2du

= e−rtb (1−e−b(T−t))+ 1

b (1−e−b(T−t))(r0e

−bt+ ab (1−e

−bt))

×e−r Tt

(r0e−bs+ a

b (1−e−bs))ds+σ2

2

r Tte2bu

(e−bu−e−bT

b

)2du

= eC(T−t)rt+A(T−t), (11.14)

where

C(T − t) = −1

b(1− e−b(T−t)),

and

A(T − t) =4ab− 3σ2

4b3+σ2 − 2ab

2b2(T − t) +

σ2 − abb3

e−b(T−t) − σ2

4b3e−2b(T−t).

Analytical PDE Solution

In order to solve the PDE (11.7) analytically we may look for a solution ofthe form

F (t, x) = eA(T−t)+xC(T−t), (11.15)

where A and C are functions to be determined under the conditions A(0) = 0and C(0) = 0. Plugging (11.15) into the PDE (11.7) yields the system ofRiccati and linear differential equations

−A′(s) = −aC(s)− σ2

2C2(s)

−C ′(s) = bC(s) + 1,

which can be solved to recover the above value of P (t, T ).

Some Bond Price Simulations

In this section we consider again the Vasicek model, in which the short rate(rt)t∈R+

is solution to (11.1). Figure 11.2 presents a random simulation oft 7−→ P (t, T ) in the same Vasicek model. The graph of the corresponding

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deterministic bond price obtained for a = b = σ = 0 is also shown on theFigure 11.2.

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0 5 10 15 20

Fig. 11.2: Graphs of t 7−→ P (t, T ) and t 7−→ e−r0(T−t).

Figure 11.3 presents a random simulation of t 7−→ P (t, T ) for a non-zerocoupon bond with price Pc(t, T ) = ec(T−t)P (t, T ), and coupon rate c > 0,0 ≤ t ≤ T .

100.00

102.00

104.00

106.00

108.00

0 5 10 15 20

Fig. 11.3: Graph of t 7−→ P (t, T ) for a bond with a 2.3% coupon.

The above simulation can be compared to the actual market data of acoupon bond in Figure 11.4 below.

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N. Privault

Fig. 11.4: Bond price graph with maturity 01/18/08 and coupon rate 6.25%.

Bond pricing in the Dothan model

In the Dothan [17] model, the short term interest rate process (rt)t∈R+is

modeled according to a geometric Brownian motion

drt = λrtdt+ σrtdBt, (11.16)

where the volatility σ > 0 and the drift λ ∈ R are constant parameters and(Bt)t∈R+

is a standard Brownian motion. In this model the short term inter-est rate rt remains always positive, while the proportional volatility term σrtaccounts for the sensitivity of the volatility of interest rate changes to thelevel of the rate rt.

On the other hand, the Dothan model is the only lognormal short ratemodel that allows for an analytical formula for the zero coupon bond price

P (t, T ) = IE[e−

r Ttrsds

∣∣∣ Ft] , 0 ≤ t ≤ T.

For convenience of notation we let p = 1− 2λ/σ2 and rewrite (11.16) as

drt = (1− p)1

2σ2rtdt+ σrtdBt,

with solution

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rt = r0 exp(σBt − pσ2t/2

), t ∈ R+,

where pσ/2 identifies to the market price of risk. By the Markov property of(rt)t∈R+

, the bond price P (t, T ) is a function F (τ, rt) of rt and of the timeto maturity τ = T − t:

P (t, T ) = F (τ, rt) = IE[e−

r Ttrsds

∣∣∣ rt] , 0 ≤ t ≤ T. (11.17)

By computation of the conditional expectation (11.17) using (8.43) we easilyobtain the following result, cf. [56], where the function θ(v, t) is defined in(8.42).

Proposition 11.3. The zero-coupon bond price P (t, T ) = F (T − t, rt) isgiven for all p ∈ R by

F (τ, x) = e−σ2p2τ/8

w∞0

w∞0e−ux exp

(−2

(1 + z2

)σ2u

)θ

(4z

σ2u,σ2τ

4

)du

u

dz

zp+1,

(11.18)x > 0.

Proof. By Proposition 8.8 and (8.43) we have

F (T − t, rt) = P (t, T )

= IE

[exp

(−

w T

trsds

) ∣∣∣ Ft]= IE

[exp

(−rt

w T

teσ(Bs−Bt)−σ

2p(s−t)/2ds

) ∣∣∣ Ft] (11.19)

=w∞0

w∞−∞

e−rtuP(w T−t

0eσBs−pσ

2s/2ds ∈ du,BT−t ∈ dy)

=w∞0e−rtuP

(w T−t

0eσBs−pσ

2s/2ds ∈ du)

= e−p2σ2(T−t)/8

w∞0e−rtu

w∞0

exp

(−2

1 + z2

σ2u

)θ

(4z

σ2u,σ2(T − t)

4

)dz

zp+1

du

u

where the exchange of integrals is justified by the Fubini theorem and thenon-negativity of integrands.

See [56] and [55] for more results on bond pricing in the Dothan model, and[62] for numerical computations.

11.3 Forward Rates

A forward interest rate contract gives its holder a loan decided at presenttime t and to be delivered over a future period of time [T, S] at a rate de-

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N. Privault

noted by f(t, T, S), t ≤ T ≤ S, and called a forward rate.

Let us determine the arbitrage or “fair” value of this rate using the in-struments available in a bond market, which are bonds priced at P (t, T ) forvarious maturity dates T > t.

The loan can be realized using the bonds available on the market by proceed-ing in two steps:

1) at time t, borrow the amount P (t, S) by shortselling one unit of bondwith maturity S, which will mean refunding $1 at time S.

2) since one only needs the money at time T , it makes sense to investthe amount P (t, S) over the period [t, T ] by buying a (possibly fractional)quantity P (t, S)/P (t, T ) of a bond with maturity T priced P (t, T ) at timet. This will yield the amount

$1× P (t, S)

P (t, T )

at time T .

As a consequence the investor will receive P (t, S)/P (t, T ) at time T , and willrefund $1 at time S.

The corresponding forward rate f(t, T, S) is then given by the relation

P (t, S)

P (t, T )exp ((S − T )f(t, T, S)) = $1, 0 ≤ t ≤ T ≤ S, (11.20)

where we used exponential compounding, which leads to the following defi-nition (11.21).

Definition 11.1. The forward rate f(t, T, S) at time t for a loan on [T, S]is given by

f(t, T, S) =logP (t, T )− logP (t, S)

S − T . (11.21)

The spot forward rate f(t, t, T ) is given by

f(t, t, T ) = − logP (t, T )

T − t , or P (t, T ) = e−(T−t)f(t,t,T ), 0 ≤ t ≤ T,(11.22)

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and is also called the yield.

Figure 11.5 presents a typical forward rate curve on the LIBOR (LondonInterbank Offered Rate) market with t =07 may 2003, δ = six months.

2

2.5

3

3.5

4

4.5

5

0 5 10 15 20 25 30

years

Forward interest rate TimeSerieNb 505AsOfDate 7mai032D 2,551W 2,531M 2,562M 2,523M 2,481Y 2,342Y 2,493Y 2,794Y 3,075Y 3,316Y 3,527Y 3,718Y 3,889Y 4,0210Y 4,1411Y 4,2312Y 4,3313Y 4,414Y 4,4715Y 4,5420Y 4,7425Y 4,8330Y 4,86

Fig. 11.5: Graph of T 7−→ f(t, T, T + δ).

The instantaneous forward rate f(t, T ) is defined by taking the limit off(t, T, S) as S T , i.e.

f(t, T ) : = limST

f(t, T, S)

= − limST

logP (t, S)− logP (t, T )

S − T

= − limε0

logP (t, T + ε)− logP (t, T )

ε

= −∂ logP (t, T )

∂T

= − 1

P (t, T )

∂P (t, T )

∂T. (11.23)

The above equation can be viewed as a differential equation to be solved forlogP (t, T ) under the initial condition P (T, T ) = 1, which yields the followingproposition.


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N. Privault

P (t, T ) = exp

(−

w T

tf(t, s)ds

), 0 ≤ t ≤ T. (11.24)

Proof. We check that

logP (t, T ) = logP (t, T )− logP (t, t) =w T

t

∂ logP (t, s)

∂sds = −

w T

tf(t, s)ds.

As a consequence of (11.20) and (11.24) the forward rate f(t, T, S) can berecovered from the instantaneous forward rate f(t, s), as:

f(t, T, S) =1

S − Tw S

Tf(t, s)ds, 0 ≤ t ≤ T < S. (11.25)

Forward Swap Rates

An interest rate swap makes it possible to exchange a variable forward ratef(t, T, S) against a fixed rate κ over a time period [T, S]. Over a succession oftime intervals [T1, T2], . . . , [Tn−1, Tn], the sum of such exchanges will generatea cumulative discounted cash flow(n−1∑k=1

(Tk+1 − Tk)e−r Tk+1t rsdsf(t, Tk, Tk+1)

)−(n−1∑k=1

(Tk+1 − Tk)κe−r Tk+1t rsds

)

=n−1∑k=1

(Tk+1 − Tk)e−r Tk+1t rsds(f(t, Tk, Tk+1)− κ),

at time t, in which we use linear interest rate compounding, and priced attime t as

IE

[n−1∑k=1

(Tk+1 − Tk)e−r Tk+1t rsds(f(t, Tk, Tk+1)− κ)

∣∣∣ Ft]

=

n−1∑k=1

(Tk+1 − Tk)(f(t, Tk, Tk+1)− κ) IE

[e−

r Tk+1t rsds

∣∣∣ Ft]

=

n−1∑k=1

(Tk+1 − Tk)P (t, Tk+1)(f(t, Tk, Tk+1)− κ).

The swap rate S(t, T1, Tn) is by definition the fair value of κ that cancels thiscash flow and achieves equilibrium, i.e. S(t, T1, Tn) satisfies

n−1∑k=1

(Tk+1 − Tk)P (t, Tk+1)(f(t, Tk, Tk+1)− S(t, T1, Tn)) = 0, (11.26)

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and is given by

S(t, T1, Tn) =1

P (t, T1, Tn)

n−1∑k=1

(Tk+1 − Tk)P (t, Tk+1)f(t, Tk, Tk+1), (11.27)

where

P (t, T1, Tn) =n−1∑k=1

(Tk+1 − Tk)P (t, Tk+1), 0 ≤ t ≤ T1,

is the annuity numeraire.

LIBOR Rates

Recall that the forward rate f(t, T, S), 0 ≤ t ≤ T ≤ S, is defined usingexponential compounding, from the relation

f(t, T, S) = − logP (t, S)− logP (t, T )

S − T . (11.28)

In order to compute swaption prices one prefers to use forward rates as de-fined on the London InterBank Offered Rates (LIBOR) market instead of thestandard forward rates given by (11.28).

The forward LIBOR L(t, T, S) for a loan on [T, S] is defined using linearcompounding, i.e. by replacing (11.28) with the relation

1 + (S − T )L(t, T, S) =P (t, T )

P (t, S),

which yields the following definition.

Definition 11.2. The forward LIBOR rate L(t, T, S) at time t for a loan on[T, S] is given by

L(t, T, S) =1

S − T

(P (t, T )

P (t, S)− 1

), 0 ≤ t ≤ T < S. (11.29)

Note that (11.29) above yields the same formula for the instantaneousforward rate

f(t, T ) : = limST

L(t, T, S)

= limST

P (t, S)− P (t, T )

(S − T )P (t, S)

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N. Privault

= − 1

P (t, T )

∂P (t, T )

∂T

= −∂ logP (t, T )

∂T

as (11.23).

In addition, Relation (11.29) shows that the LIBOR rate can be viewedas a forward price Xt = Xt/Nt with numeraire Nt = (S − T )P (t, S) andXt = P (t, T ) − P (t, S), according to Relation (10.7) of Chapter 10. As aconsequence, from Proposition 10.2, the LIBOR rate (L(t, T, S))t∈[T,S] is a

martingale under the forward measure P defined by

dPdP∗

=1

P (0, S)e−

r S0rtdt.

LIBOR Swap Rates

The LIBOR swap rate S(t, T1, Tn) satisfies the same relation as (11.26) withthe forward rate f(t, Tk, Tk+1) replaced with the LIBOR rate L(t, Tk, Tk+1),i.e.

n−1∑k=1

(Tk+1 − Tk)P (t, Tk+1)(L(t, Tk, Tk+1)− S(t, T1, Tn)) = 0.


S(t, T1, Tn) =P (t, T1)− P (t, Tn)

P (t, T1, Tn), 0 ≤ t ≤ T1. (11.30)

Proof. By (11.27) and (11.29) we have

S(t, T1, Tn) =1

P (t, T1, Tn)

n−1∑k=1

(Tk+1 − Tk)P (t, Tk+1)L(t, Tk, Tk+1)

=1

P (t, T1, Tn)

n−1∑k=1

P (t, Tk+1)

(P (t, Tk)

P (t, Tk+1)− 1

)

=1

P (t, T1, Tn)

n−1∑k=1

(P (t, Tk)− P (t, Tk+1))

=P (t, T1)− P (t, Tn)

P (t, T1, Tn)(11.31)

by a telescoping summation.

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Clearly, a simple expression for the swap rate such as that of Proposition 11.5cannot be obtained using the standard (i.e. non-LIBOR) rates defined in(11.28).

When n = 2, the swap rate S(t, T1, T2) coincides with the forward rateL(t, T1, T2):

S(t, T1, T2) = L(t, T1, T2), (11.32)

and the bond prices P (t, T1) can be recovered from the forward swap ratesS(t, T1, Tn).

Similarly to the case of LIBOR rates, Relation (11.30) shows that theLIBOR swap rate can be viewed as a forward price with (annuity) numeraireNt = P (t, T1, Tn) and Xt = P (t, T1) − P (t, Tn). Consequently the LIBOR

swap rate (S(t, T1, Tn)t∈[T,S] is a martingale under the forward measure Pdefined from (10.1) by

dPdP∗

=P (T1, T1, Tn)

P (0, T1, Tn)e−

r T10 rtdt.

11.4 The HJM Model

In the previous chapter we have focused on the modeling of the short rate(rt)t∈R+ and on its consequences on the pricing of bonds P (t, T ), from whichthe forward rates f(t, T, S) and L(t, T, S) have been defined.

In this section we choose a different starting point and consider the prob-lem of directly modeling the instantaneous forward rate f(t, T ). The graphgiven in Figure 11.4 presents a possible random evolution of a forward interestrate curve using the Musiela convention, i.e. we will write

g(x) = f(t, t+ x) = f(t, T ),

under the substitution x = T − t, x ≥ 0, and represent a sample of theinstantaneous forward curve x 7−→ f(t, t+ x) for each t ∈ R+.

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0 5

10 15

20 0

5

10

15

20

0.5 1

1.5 2

2.5 3

3.5 4

4.5 5

Forward rate

x

t

Fig. 11.6: Stochastic process of forward curves.

In the HJM model, the instantaneous forward rate f(t, T ) is modeled underP by a stochastic differential equation of the form

dtf(t, T ) = α(t, T )dt+ σ(t, T )dBt, (11.33)

where t 7−→ α(t, T ) and t 7−→ σ(t, T ), 0 ≤ t ≤ T , are allowed to be ran-dom (adapted) processes. In the above equation, the date T is fixed and thedifferential dt is with respect to t.

Under basic Markovianity assumptions, a HJM model with deterministiccoefficients α(t, T ) and σ(t, T ) will yield a short rate process (rt)t∈R+ of theform

drt = (a(t)− b(t)rt)dt+ σ(t)dBt,

cf. § 6.6 of [60], which is the [34] Hull-White model, cf. Section 11.1, withexplicit solution

rt = rse−

r tsb(τ)dτ +

w t

se−

r tub(τ)dτa(u)du+

w t

sσ(u)e−

r tub(τ)dτdBu,

0 ≤ s ≤ t.

The HJM Condition

How to “encode” absence of arbitrage in the defining equation (11.33) is animportant question. Recall that under absence of arbitrage, the bond priceP (t, T ) has been defined as

P (t, T ) = IE∗[exp

(−

w T

trsds

) ∣∣∣ Ft] , (11.34)

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and the discounted bond price process

t 7−→ exp

(−

w t

0rsds

)P (t, T ) = exp

(−

w t

0rsds−

w T

tf(t, s)ds

)= exp

(−

w t

0rsds−Xt

)(11.35)

is a martingale by Proposition 11.1 and Relation (11.24). This latter propertywill be used to characterize absence of arbitrage in the HJM model.

Proposition 11.6. (HJM Condition [33]). Under the condition

α(t, T ) = σ(t, T )w T

tσ(t, s)ds, t ∈ [0, T ], (11.36)

which is known as the HJM absence of arbitrage condition, the process (11.35)becomes a martingale.

Proof. Consider the spot forward rate

f(t, t, T ) =1

T − tw T

tf(t, s)ds,

and let

Xt =w T

tf(t, s)ds = − logP (t, T ), 0 ≤ t ≤ T,

with the relation

f(t, t, T ) =1

T − tw T

tf(t, s)ds =

Xt

T − t , 0 ≤ t ≤ T, (11.37)

where the dynamics of t 7−→ f(t, s) is given by (11.33). We note that whenf(t, s) = g(t)h(s) is a smooth function which satisfies the separation of vari-ables property we have the relation

dtw T

tg(t)h(s)ds = −g(t)h(t)dt+ g′(t)

w T

tg(t)h(s)dsdt,

which extends to f(t, s) as

dtw T

tf(t, s)ds = −f(t, t)dt+

w T

tdtf(t, s)dsdt,

which can be seen as a form of the Leibniz integral rule. Therefore we have

dtXt = −f(t, t)dt+w T

tdtf(t, s)ds

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N. Privault

= −f(t, t)dt+w T

tα(t, s)dsdt+

w T

tσ(t, s)dsdBt

= −rtdt+

(w T

tα(t, s)ds

)dt+

(w T

tσ(t, s)ds

)dBt,

hence

|dtXt|2 =

(w T

tσ(t, s)ds

)2

dt.

Hence by Ito’s calculus we have

dtP (t, T ) = dte−Xt

= −e−XtdtXt +1

2e−Xt(dtXt)

2

= −e−XtdtXt +1

2e−Xt

(w T

tσ(t, s)ds

)2

dt

= −e−Xt(−rtdt+

w T

tα(t, s)dsdt+

w T

tσ(t, s)dsdBt

)+

1

2e−Xt

(w T

tσ(t, s)ds

)2

dt,

and the discounted bond price satisfies

dt

(exp

(−

w t

0rsds

)P (t, T )

)= −rt exp

(−

w t

0rsds−Xt

)dt+ exp

(−

w t

0rsds

)dtP (t, T )

= −rt exp

(−

w t

0rsds−Xt

)dt− exp

(−

w t

0rsds−Xt

)dtXt

+1

2exp

(−

w t

0rsds−Xt

)(w T

tσ(t, s)ds

)2

dt

= −rt exp

(−

w t

0rsds−Xt

)dt

− exp

(−

w t

0rsds−Xt

)(−rtdt+

w T

tα(t, s)dsdt+

w T

tσ(t, s)dsdBt

)+

1

2exp

(−

w t

0rsds−Xt

)(w T

tσ(t, s)ds

)2

dt

= − exp

(−

w t

0rsds−Xt

)w T

tσ(t, s)dsdBt

− exp

(−

w t

0rsds−Xt

)(w T

tα(t, s)dsdt− 1

2

(w T

tσ(t, s)ds

)2)dt.

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Thus the process P (t, T ) will be a martingale provided that

w T

tα(t, s)ds− 1

2

(w T

tσ(t, s)ds

)2

= 0, 0 ≤ t ≤ T. (11.38)

Differentiating the above relation with respect to T , we get

α(t, T ) = σ(t, T )w T

tσ(t, s)ds,

which is in fact equivalent to (11.38).

11.5 Forward Vasicek Rates

In this section we consider the Vasicek model, in which the short rate processis the solution (11.2) of (11.1) as illustrated in Figure 11.1.

In this model the forward rate is given by

f(t, T, S) = − logP (t, S)− logP (t, T )

S − T= −rt(C(S − t)− C(T − t)) +A(S − t)−A(T − t))

S − T

= −σ2 − 2ab

2b2

− 1

S − T

((rtb

+σ2 − abb3

)(e−b(S−t) − e−b(T−t)) − σ

2

4b3(e−2b(S−t) − e−2b(T−t))

).

In this model the forward rate t 7−→ f(t, T, S) can be represented as inFigure 11.7, with here b/a > r0.

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N. Privault

0.005

0.0055

0.006

0.0065

0.007

0.0075

0.008

0.0085

0.009

0.0095

0.01

0 2 4 6 8 10

t

f(t,T,S)

Fig. 11.7: Forward rate process t 7−→ f(t, T, S).

Note that the forward rate cure t 7−→ f(t, T, S) is flat for small values of t.

The instantaneous short rate is given by

f(t, T ) : = −∂ logP (t, T )

∂T(11.39)

= rte−b(T−t) +

a

b(1− e−b(T−t))− σ2

2b2(1− e−b(T−t))2,

and the relation limTt f(t, T ) = rt is easily recovered from this formula.

The instantaneous forward rate t 7−→ f(t, T ) can be represented as inFigure 11.8, with here t = 0 and b/a > r0:

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0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 2 4 6 8 10 12 14 16 18 20

t

f(t,T)

Fig. 11.8: Instantaneous forward rate process t 7−→ f(t, T ).

The HJM coefficients in the Vasicek model are in fact deterministic andtaking a = 0 we have

dtf(t, T ) = σ2e−b(T−t)w T

teb(t−s)dsdt+ σe−b(T−t)dBt,

i.e.

α(t, T ) = σ2e−b(T−t)w T

teb(t−s)ds, and σ(t, T ) = σe−b(T−t),

and the HJM condition reads

α(t, T ) = σ2e−b(T−t)w T

teb(t−s)ds = σ(t, T )

w T

tσ(t, s)ds. (11.40)

Random simulations of the Vasicek instantaneous forward rates are providedin Figures 11.9 and 11.10.

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N. Privault

Fig. 11.9: Forward instantaneous curve (t, x) 7−→ f(t, t+ x) in the Vasicek model.

Fig. 11.10: Forward instantaneous curve x 7−→ f(0, x) in the Vasicek model.

For x = 0 the first “slice” of this surface is actually the short rate Vasicekprocess rt = f(t, t) = f(t, t + 0) which is represented in Figure 11.11 usinganother discretization.

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0.03

0.035

0.04

0.045

0.05

0.055

0.06

0.065

0.07

0 5 10 15 20

Fig. 11.11: Short term interest rate curve t 7−→ rt in the Vasicek model.

Another example of market data is given in the next Figure 11.12, in whichthe red and blue curves refer respectively to July 21 and 22 of year 2011.

Fig. 11.12: Market example of yield curves (11.22).

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N. Privault

11.6 Modeling Issues

Parametrization of Forward Rates

In the Nelson-Siegel parametrization the forward interest rate curves areparametrized by 4 coefficients z1, z2, z3, z4, as

g(x) = z1 + (z2 + z3x)e−xz4 , x ≥ 0.

An example of a graph obtained by the Nelson-Siegel parametrization is givenin Figure 11.13, for z1 = 1, z2 = −10, z3 = 100, z4 = 10.

-10

-8

-6

-4

-2

0

2

4

0 0.2 0.4 0.6 0.8 1

z1+(z2+xz3)exp(-xz4)

Fig. 11.13: Graph of x 7−→ g(x) in the Nelson-Siegel model.

The Svensson parametrization has the advantage to reproduce two humpsinstead of one, the location and height of which can be chosen via 6 param-eters z1, z2, z3, z4, z5, z6 as

g(x) = z1 + (z2 + z3x)e−xz4 + z5xe−xz6 , x ≥ 0.

A typical graph of a Svensson parametrization is given in Figure 11.14, forz1 = 7, z2 = −5, z3 = −100, z4 = 10, z5 = −1/2, z6 = −1.

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2

2.5

3

3.5

4

4.5

5

0 5 10 15 20 25 30

lambda

x->z1+(z2+z3*x)*exp(-x*z4)+z5*x*exp(-z6*x)

Fig. 11.14: Graph of x 7−→ g(x) in the Svensson model.

Figure 11.15 presents a fit of the market data of Figure 11.5 using a Svens-son curve.

2

2.5

3

3.5

4

4.5

5

0 5 10 15 20 25 30

years

Market dataSvensson curve

Fig. 11.15: Comparison of market data vs a Svensson curve.

One may think of constructing an instantaneous rate process taking values inthe Svensson space, however this type of modelization is not consistent withabsence of arbitrage, and it can be proved that the HJM curves cannot livein the Nelson-Siegel or Svensson spaces, cf. §3.5 of [5].

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N. Privault

For example it can be easily shown that the forward curves of the Vasicekmodel are included neither in the Nelson-Siegel space, nor in the Svenssonspace, cf. [60] and §3.5 of [5]. In addition, such curves do not appear to cor-rectly model the market forward curves considered above, cf. e.g. Figure 11.5.In the Vasicek model we have

∂f

∂T(t, T ) =

(−brt + a− σ2

b+σ2

be−b(T−t)

)e−b(T−t),

and one can check that the sign of the derivatives of f can only change onceat most. As a consequence, the possible forward curves in the Vasicek modelare limited to one change of “regime” per curve, as illustrated in Figure 11.16for various values of rt, and in Figure 11.17.

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0 5 10 15 20

Fig. 11.16: Graphs of forward rates.

0 2

4 6

8 10

x

0 5

10 15

20

t

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Fig. 11.17: Forward instantaneous curve (t, x) 7−→ f(t, t+ x) in the Vasicek model.

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Another way to deal with the curve fitting problem is to use deterministicshifts for the fitting of one forward curve, such as the initial curve at t = 0,cf. e.g. § 8.2. of [60].

The Correlation Problem and a Two-Factor Model

The correlation problem is another issue of concern when using the affinemodels considered so far. Let us compare three bond price simulations withmaturity T1 = 10, T2 = 20, and T3 = 30 based on the same Brownian path,as given in Figure 11.18. Clearly, the bond prices P (t, T1) and P (t, T2) withmaturities T1 and T2 are linked by the relation

P (t, T2) = P (t, T1) exp(A(t, T2)−A(t, T1) + rt(C(t, T2)−C(t, T1))), (11.41)

meaning that bond prices with different maturities could be deduced fromeach other, which is unrealistic.

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 25 30

t

P(t,T1)P(t,T2)P(t,T3)

Fig. 11.18: Graph of t 7−→ P (t, T1).

For affine models of short rates we have the perfect correlation

Cor(logP (t, T1), logP (t, T2)) = 1,

cf. § 8.3 of [60], since by (11.41), logP (t, T1) and logP (t, T2) are linked bythe linear relation

logP (t, T2) = logP (t, T1) +A(t, T2)−A(t, T1) + rt(C(t, T2)− C(t, T1)),

involving a same random variable Z = rt. A solution to the correlation prob-lem is to consider two control processes (Xt)t∈R+

, (Yt)t∈R+which are solution

of

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N. Privault

dXt = µ1(t,Xt)dt+ σ1(t,Xt)dB

(1)t ,

dYt = µ2(t, Yt)dt+ σ2(t, Yt)dB(2)t ,

(11.42)

where (B(1)t )t∈R+ , (B

(2)t )t∈R+ have correlated Brownian motion with

Cov(B(1)s , B

(2)t ) = ρmin(s, t), s, t ∈ R+, (11.43)

anddB

(1)t dB

(2)t = ρdt, (11.44)

for some ρ ∈ [−1, 1]. In practice, (B(1))t∈R+ and (B(2))t∈R+ can be con-

structed from two independent Brownian motions (W (1))t∈R+ and (W (2))t∈R+ ,by letting

B(1)t = W

(1)t ,

B(2)t = ρW

(1)t +

√1− ρ2W (2)

t , t ∈ R+,

and Relations (11.43) and (11.44) are easily satisfied from this construction.

In two-factor models one chooses to build the short term interest rate rt via

rt = Xt + Yt, t ∈ R+.

By the previous standard arbitrage arguments we define the price of a bondwith maturity T as

P (t, T ) : = IE

[exp

(−

w T

trsds

) ∣∣∣ Ft]= IE

[exp

(−

w T

trsds

) ∣∣∣ Xt, Yt

]= F (t,Xt, Yt), (11.45)

since the couple (Xt, Yt)t∈R+is Markovian. Using the Ito formula with two

variables and the fact that

t 7−→ e−r t0rsdsP (t, T ) = e−

r t0rsds IE

[exp

(−

w T

trsds

) ∣∣∣ Ft]is an Ft-martingale under P∗ we can derive a PDE

−(x+ y)F (t, x, y) + µ1(t, x)∂F

∂x(t, x, y) + µ2(t, y)

∂F

∂y(t, x, y)

+1

2σ21(t, x)

∂2F

∂x2(t, x, y) +

1

2σ22(t, y)

∂2F

∂y2(t, x, y)

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+ρσ1(t, x)σ2(t, y)∂2F

∂x∂y(t, x, y) +

∂F

∂t(t,Xt, Yt) = 0, (11.46)

on R2 for the bond price P (t, T ). In the Vasicek modeldXt = −aXtdt+ σdB

(1)t ,

dYt = −bYtdt+ ηdB(2)t ,

this yieldsP (t, T ) = F1(t,Xt)F2(t, Yt) exp (U(t, T )) , (11.47)

where F1(t,Xt) and F2(t, Yt) are the bond prices associated to Xt and Yt inthe Vasicek model, and

U(t, T ) = ρση

ab

(T − t+

e−a(T−t) − 1

a+e−b(T−t) − 1

b− e−(a+b)(T−t) − 1

a+ b

)is a correlation term which vanishes when (B

(1)t )t∈R+

and (B(2)t )t∈R+

are in-dependent, i.e. when ρ = 0, cf [8], Chapter 4, Appendix A, and [60]. [8].

Partial differentiation of logP (t, T ) with respect to T leads to the instanta-neous forward rate

f(t, T ) = f1(t, T ) + f2(t, T )− ρσηab

(1− e−a(T−t))(1− e−b(T−t)), (11.48)

where f1(t, T ), f2(t, T ) are the instantaneous forward rates corresponding toXt and Yt respectively, cf. § 8.4 of [60].

An example of a forward rate curve obtained in this way is given in Fig-ure 11.19.

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N. Privault

0.18

0.19

0.2

0.21

0.22

0.23

0.24

0 5 10 15 20 25 30 35 40

T

Fig. 11.19: Graph of forward rates in a two-factor model.

Next in Figure 11.20 we present a graph of the evolution of forward curvein a two factor model.

0 1

2 3

4 5

6 7

8

x 0

0.2 0.4

0.6 0.8

1 1.2

1.4

t

0.215

0.22

0.225

0.23

0.235

0.24

Fig. 11.20: Random evolution of forward rates in a two-factor model.

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11.7 The BGM Model

The models (HJM, affine, etc.) considered in the previous chapter suffer fromvarious drawbacks such as non-positivity of interest rates in Vasicek model,and lack of closed form solutions in more complex models. The BGM [7]model has the advantage of yielding positive interest rates, and to permit toderive explicit formulas for the computation of prices for interest rate deriva-tives such as caps and swaptions on the LIBOR market.

In the BGM model we work with a tenor structure T1, . . . , Tn (see Sec-tion 12.1 for details) and consider the family (Pi)i=1,...,n of forward measuresdefined by taking the bond prices (P (t, T1))t∈[0,T1], i = 1, . . . , n, as respectivenumeraires, i.e.

dPidP∗i

=e−

r T10 rsds

P (0, T1),

cf. (10.6).

The forward LIBOR rate L(t, T1, T2) is modeled as a geometric Brownianmotion under P2, i.e.

dL(t, T1, T2)

L(t, T1, T2)= γ1(t)dB

(2)t , (11.49)

0 ≤ t ≤ T1, i = 1, . . . , n − 1, for some deterministic function γ1(t), withsolution

L(u, T1, T2) = L(t, T1, T2) exp

(w u

tγ1(s)dB(2)

s −1

2

w u

t|γ1|2(s)ds

),

i.e. for u = T1,

L(T1, T1, T2) = L(t, T1, T2) exp

(w T1

tγ1(s)dB(2)

s −1

2

w T1

t|γ1|2(s)ds

).

Since L(t, T1, T2) is a geometric Brownian motion under P2, standard capletscan be priced at time t ∈ [0, T1] from the Black-Scholes formula.

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N. Privault

The following graph summarizes the notions introduced in this chapter.

Short rate1 rt

Short ratert = f(t, t) = f(t, t, t)

Bond price2

P (t, T ) = E[e−r Tt rsds | Ft]

LIBOR rate3

L(t, T, S) = P (t,T )−P (t,S)(S−T )P (t,S)

Forward rate3

f(t, T, S) = logP (t,T )−logP (t,S)S−T

Instantaneous forward rate4

f(t, T ) = L(t, T ) = limST f(t, T, S)= limST L(t, T, S)

Bond price

P (t, T ) = e−r Tt f(t,s)ds

Bond priceP (t, T ) = e−(T−t)f(t,t,T )

Instantaneous forward rate4

f(t, T ) = L(t, T ) = −∂ logP (t,T )∂T

Spot forward rate (yield)

f(t, t, T ) =r Tt f(t, s)ds/(T − t)

1Can be modeled by Vasicek and other short rate models2Can be modeled from dP (t, T )/P (t, T ).3Can be modeled in the BGM model4Can be modeled in the HJM model

Fig. 11.21: Graph of stochastic interest rate modeling.

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Exercises

Exercise 11.1 Solve the stochastic differential equation

dXt = −bXtdt+ σe−btdBt, t ∈ R+,

where (Bt)t∈R+is a standard Brownian motion and σ, b > 0.

Exercise 11.2 Let (Bt)t∈R+ denote a standard Brownian motion started at0 under the risk-neutral measure P∗. We consider a short term interest rateprocess (rt)t∈R+

in a Ho-Lee model with constant deterministic volatility,defined by

drt = adt+ σdBt,

where a > 0 and σ > 0. Let P (t, T ) will denote the arbitrage price of azero-coupon bond in this model:

P (t, T ) = IE∗[exp

(−

w T

trsds

) ∣∣∣ Ft] , 0 ≤ t ≤ T. (11.50)

1. State the bond pricing PDE satisfied by the function F (t, x) defined via

F (t, x) := IE∗[exp

(−

w T

trsds

) ∣∣∣ rt = x

], 0 ≤ t ≤ T.

2. Compute the arbitrage price F (t, rt) = P (t, T ) from its expression (11.50)as a conditional expectation.

Hint. One may use the integration by parts relation

w T

tBsds = TBT − tBt −

w T

tsdBs

= (T − t)Bt + T (BT −Bt)−w T

tsdBs

= (T − t)Bt +w T

t(T − s)dBs,

and the Laplace transform identity IE[eλX ] = eλ2η2/2 for X ' N (0, η2).

3. Check that the function F (t, x) computed in question 2 does satisfy thePDE derived in question 1.

4. Compute the forward rate f(t, T, S) in this model.

From now on we let a = 0.5. Compute the instantaneous forward rate f(t, T ) in this model.

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N. Privault

6. Derive the stochastic equation satisfied by the instantaneous forward ratef(t, T ).

7. Check that the HJM absence of arbitrage condition is satisfied in thisequation.

Exercise 11.3 Given (Bt)t∈R+ a standard Brownian motion, consider a HJMmodel given by

dtf(t, T ) =σ2

2T (T 2 − t2)dt+ σTdBt. (11.51)

1. Show that the HJM condition is satisfied by (11.51).2. Compute f(t, T ) by solving (11.51).

Hint: We have f(t, T ) = f(0, T ) +r t0dsf(s, T ) = · · ·

3. Compute the short rate rt = f(t, t) from the result of Question 2.4. Show that the short rate rt satisfies a stochastic differential equation of

the formdrt = η(t)dt+ (rt − f(0, t))ψ(t)dt+ ξ(t)dBt,

where η(t), ψ(t), ξ(t) are deterministic functions to be determined.

Exercise 11.4 Let (rt)t∈R+denote a short term interest rate process. For any

T > 0, let P (t, T ) denote the price at time t ∈ [0, T ] of a zero coupon bonddefined by the stochastic differential equation

dP (t, T )

P (t, T )= rtdt+ σTt dBt, 0 ≤ t ≤ T, (11.52)

under the terminal condition P (T, T ) = 1, where (σTt )t∈[0,T ] is an adaptedprocess. Let the forward measure PT be defined by

IE

[dPTdP

∣∣∣ Ft] =P (t, T )

P (0, T )e−

r t0rsds, 0 ≤ t ≤ T.

Recall that

BTt := Bt −w t

0σTs ds, 0 ≤ t ≤ T,

is a standard Brownian motion under PT .

1. Solve the stochastic differential equation (11.52).2. Derive the stochastic differential equation satisfied by the discounted

bond price process

t 7−→ e−r t0rsdsP (t, T ), 0 ≤ t ≤ T,

and show that it is a martingale.

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3. Show that

IE[e−

r T0rsds

∣∣∣ Ft] = e−r t0rsdsP (t, T ), 0 ≤ t ≤ T.

4. Show that

P (t, T ) = IE[e−

r Ttrsds

∣∣∣ Ft] , 0 ≤ t ≤ T.

5. Compute P (t, S)/P (t, T ), 0 ≤ t ≤ T , show that it is a martingale underPT and that

P (T, S) =P (t, S)

P (t, T )exp

(w T

t(σSs − σTs )dBTs −

1

2

w T

t(σSs − σTs )2ds

).

6. Assuming that (σTt )t∈[0,T ] and (σSt )t∈[0,S] are deterministic functions,compute the price

IE[e−


+∣∣∣ Ft] = P (t, T ) IET

[(P (T, S)− κ)

+∣∣∣ Ft]

of a bond option with strike κ.

Recall that if X is a centered Gaussian random variable with mean mt

and variance v2t given Ft, we have

IE[(eX −K)+ | Ft] = emt+v2t /2Φ

(vt2

+1

vt(mt + v2t /2− logK)

)−KΦ

(−vt

2+

1


)where Φ(x), x ∈ R, denotes the Gaussian distribution function.

Exercise 11.5 (Exercise 4.5 continued). Assume that the price P (t, T ) of azero coupon bond is modeled as

P (t, T ) = e−µ(T−t)+XTt , t ∈ [0, T ],

where µ > 0.

1. Show that the terminal condition P (T, T ) = 1 is satisfied.2. Compute the forward rate

f(t, T, S) = − 1

S − T (logP (t, S)− logP (t, T )).

3. Compute the instantaneous forward rate

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N. Privault

f(t, T ) = − limST

1


4. Show that the limit limTt

f(t, T ) does not exist in L2(Ω).

5. Show that P (t, T ) satisfies the stochastic differential equation

dP (t, T )

P (t, T )= σdBt +

1

2σ2dt− logP (t, T )

T − t dt, t ∈ [0, T ].

6. Show, using the results of Exercise 11.4-(4), that

P (t, T ) = IE[e−

r TtrTs ds

∣∣∣ Ft] ,where (rTt )t∈[0,T ] is a process to be determined.

7. Compute the conditional density

IE

[dPTdP

∣∣∣∣∣ Ft]

=P (t, T )

P (0, T )e−

r t0rTs ds

of the forward measure PT with respect to P.8. Show that the process

Bt := Bt − σt, 0 ≤ t ≤ T,

is a standard Brownian motion under PT .9. Compute the dynamics of XS

t and P (t, S) under PT .Hint: Show that

−µ(S − T ) + σ(S − T )w t

0

1

S − sdBs =S − TS − t logP (t, S).

10. Compute the bond option price

IE[e−

r TtrTs ds(P (T, S)−K)+

∣∣∣ Ft] = P (t, T ) IET

[(P (T, S)−K)+

∣∣∣ Ft] ,0 ≤ t < T < S.

Exercise 11.6 (Exercise 4.7 continued). Write down the bond pricing PDEfor the function

F (t, x) = E[e−

r Ttrsds

∣∣∣ rt = x]

and show that in case α = 0 the corresponding bond price P (t, T ) equals

P (t, T ) = e−B(T−t)rt , 0 ≤ t ≤ T,

where

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B(x) =2(eγx − 1)

2γ + (β + γ)(eγx − 1),

with γ =√β2 + 2σ2.

Exercise 11.7 Let (rt)t∈R+ denote a short term interest rate process. For anyT > 0, let P (t, T ) denote the price at time t ∈ [0, T ] of a zero coupon bonddefined by the stochastic differential equation

dP (t, T )

P (t, T )= rtdt+ σTt dBt, 0 ≤ t ≤ T,

under the terminal condition P (T, T ) = 1, where (σTt )t∈[0,T ] is an adaptedprocess. Let the forward measure PT be defined by

IE

[dPTdP

∣∣∣Ft] =P (t, T )

P (0, T )e−


Recall that

BTt := Bt −w t

0σTs ds, 0 ≤ t ≤ T,


1. Solve the stochastic differential equation (11.52).2. Derive the stochastic differential equation satisfied by the discounted

bond price process

t 7→ e−r t0rsdsP (t, T ), 0 ≤ t ≤ T,

and show that it is a martingale.3. Show that

IE[e−

r T0rsds

∣∣∣Ft] = e−r t0rsdsP (t, T ), 0 ≤ t ≤ T.

4. Show that

P (t, T ) = IE[e−

r Ttrsds

∣∣∣Ft] , 0 ≤ t ≤ T.

5. Compute P (t, S)/P (t, T ), 0 ≤ t ≤ T , show that it is a martingale underPT and that

P (T, S) =P (t, S)

P (t, T )exp

(w T


1

2

w T


).

6. Assuming that (σTt )t∈[0,T ] and (σSt )t∈[0,S] are deterministic functions,compute the price

IE[e−


+∣∣∣Ft] = P (t, T ) IET

[(P (T, S)− κ)

+∣∣∣Ft]

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N. Privault

of a bond option with strike κ.

Recall that if X is a centered Gaussian random variable with mean mt

and variance v2t given Ft, we have

IE[(eX −K)+ | Ft] = emt+v2t /2Φ

(vt2

+1


)−KΦ

(−vt

2+

1


)where Φ(x), x ∈ R, denotes the Gaussian distribution function.

Exercise 11.8 (Exercise 11.5 continued).

1. Compute the forward rate

f(t, T, S) = − 1


2. Compute the instantaneous forward rate

f(t, T ) = − limST

1


3. Show that the limit limTt

f(t, T ) does not exist in L2(Ω).

4. Show that P (t, T ) satisfies the stochastic differential equation

dP (t, T )

P (t, T )= σdBt +

1


T − t dt, t ∈ [0, T ].

5. Show, using the results of Exercise 11.7-(4), that

P (t, T ) = IE[e−

r TtrTs ds

∣∣∣Ft] ,where (rTt )t∈[0,T ] is a process to be determined.

6. Compute the conditional density

IE

[dPTdP

∣∣∣Ft] =P (t, T )

P (0, T )e−

r t0rTs ds

of the forward measure PT with respect to P.7. Show that the process

Bt := Bt − σt, 0 ≤ t ≤ T,


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8. Compute the dynamics of XSt and P (t, S) under PT .

Hint: Show that

−µ(S − T ) + σ(S − T )w t

0

1

S − sdBs =S − TS − t logP (t, S).

9. Compute the bond option price

IE[e−

r TtrTs ds(P (T, S)−K)+

∣∣∣Ft] = P (t, T ) IET

[(P (T, S)−K)+

∣∣∣Ft] ,0 ≤ t < T < S.

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Chapter 12

Pricing of Interest Rate Derivatives

In this chapter we consider the pricing of caplets, caps, and swaptions, usingchange of numeraire and forward swap measures.

12.1 Forward Measures and Tenor Structure

The maturity dates are arranged according to a discrete tenor structure

0 = T0 < T1 < T2 < · · · < Tn.

An example of forward interest rate curve data is given in the table of Fig-ure 12.1, which contains the values of (T1, T2, . . . , T23) and of f(t, t+Ti, t+Ti + δ)i=1,...,23, with t = 07/05/2003 and δ = six months.

2D 1W 1M 2M 3M 1Y 2Y 3Y 4Y 5Y 6Y 7Y2.55 2.53 2.56 2.52 2.48 2.34 2.49 2.79 3.07 3.31 3.52 3.71

8Y 9Y 10Y 11Y 12Y 13Y 14Y 15Y 20Y 25Y 30Y3.88 4.02 4.14 4.23 4.33 4.40 4.47 4.54 4.74 4.83 4.86

Fig. 12.1: Forward rates arranged according to a tenor structure.

Recall that by definition of P (t, Ti) and absence of arbitrage the process

t 7−→ e−r t0rsdsP (t, Ti), 0 ≤ t ≤ Ti, i = 1, . . . , n,

is an Ft-martingale under P, and as a consequence (P (t, Ti))t∈[0,Ti] can betaken as numeraire in the definition

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N. Privault

dPidP

=1

P (0, Ti)e−

r Ti0 rsds (12.1)

of the forward measure Pi. The following proposition will allow us to pricecontingent claims using the forward measure Pi, it is a direct consequence ofProposition 10.1, noting that here we have P (Tt, Ti) = 1.

Proposition 12.1. For all sufficiently integrable random variables F we have

IE[Fe−

r Tit rsds

∣∣∣ Ft] = P (t, Ti)IEi[F | Ft], 0 ≤ t ≤ T, i = 1, . . . , n.

(12.2)

Recall that for all Ti, Tj ≥ 0, the process

t 7−→ P (t, Tj)

P (t, Ti), 0 ≤ t ≤ min(Ti, Tj),

is an Ft-martingale under Pi, cf. Proposition 10.2.

Dynamics under the forward measure

In order to apply Proposition 12.1 and to compute the price

IE[e−

r TtrsdsF

∣∣∣ Ft] = P (t, Ti)IEi[F | Ft],

it can be useful to determine the dynamics of the underlying processes rt,f(t, T, S), and P (t, T ) under the forward measure Pi.

Let us assume that the dynamics of the bond price P (t, Ti) is given by

dP (t, Ti)

P (t, Ti)= rtdt+ ζi(t)dWt, (12.3)

for i = 1, . . . , n, where (Wt)t∈R+ is a standard Brownian motion under P and(rt)t∈R+

and (ζi(t))t∈R+are adapted processes with respect to the filtration

(Ft)t∈R+generated by (Wt)t∈R+

.

By the Girsanov theorem,

W it := Wt −

w t

0ζi(s)ds, 0 ≤ t ≤ Ti, (12.4)

is a standard Brownian motion under Pi for all i = 1, . . . , n, cf. e.g. (10.10),hence we have

dW jt = dWt − ζj(t)dt, dW i

t = dWt − ζi(t)dt, ,

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anddW j

t = dW it − (ζj(t)− ζi(t))dt.

Hence the dynamics of t 7−→ P (t, Ti) under Pi is given by

dP (t, Ti)

P (t, Ti)= rtdt+ |ζi(t)|2dt+ ζtdW

it , (12.5)

where (W it )t∈R+ is a standard Brownian motion under Pi.

Note that (W it )t∈R+

has drift (ζj(t)− ζi(t))t∈R+under Pj , since we have

dW it = dWt − ζi(t)dt = dW j

t + (ζj(t)− ζi(t))dt.

In case the short rate process (rt)t∈R+is Markovian and solution of

drt = µ(t, rt)dt+ σ(t, rt)dWt,

its dynamics will be given under Pi by

drt = µ(t, rt)dt+ σ(t, rt)ζi(t)dt+ σ(t, rt)dWit . (12.6)

In the Vasicek case we have

drt = (a− brt)dt+ σdWt,

andζi(t) = −σ

b(1− e−b(Ti−t)), 0 ≤ t ≤ Ti,

by (11.12), hence from (12.6) we have

drt = (a− brt)dt−σ2

b(1− e−b(Ti−t))dt+ σdW i

t (12.7)

and we obtain

dP (t, Ti)

P (t, Ti)= rtdt+

σ2

b2(1− e−b(Ti−t))2dt− σ

b(1− e−b(Ti−t))dW i

t ,

from (11.12).

12.2 Bond Options

The next proposition can be obtained as an application of the Margrabeformula (10.26) of Proposition 10.8 by taking Xt = P (t, Tj), Nt = P (t, Ti),

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N. Privault

and Xt = Xt/Nt = P (t, Tj)/P (t, Ti). In the Vasicek model, this formula hasbeen first obtained in [38].

Proposition 12.2. Assume that the dynamics of the bond prices P (t, Ti),P (t, Tj) are given by

dP (t, Ti)

P (t, Ti)= rtdt+ ζi(t)dWt,

dP (t, Tj)

P (t, Tj)= rtdt+ ζj(t)dWt,

where (Wt)t∈R+is a standard Brownian motion under P, (rt)t∈R+

is anadapted processes with respect to the filtration (Ft)t∈R+

generated by (Wt)t∈R+,

and (ζi(t))t∈R+ , (ζj(t))t∈R+ are deterministic functions. Then the price of abond call option on P (Ti, Tj) with payoff F = (P (Ti, Tj)−κ)+ can be writtenas

IE[e−

r Tit rsds(P (Ti, Tj)− κ)+

∣∣∣ Ft]= P (t, Tj)Φ

(v

2+

1

vlog

P (t, Tj)

κP (t, Ti)

)− κP (t, Ti)Φ

(−v

2+

1

vlog

P (t, Tj)

κP (t, Ti)

),

with

v2 =w T

t|ζj(s)− ζi(s)|2ds.

Proof. First we note that using Nt = P (t, Ti) as a numeraire the price of abond call option on P (Ti, Tj) with payoff F = (P (Ti, Tj)−κ)+ can be written

from Proposition 10.4 using the forward measure P, or directly by (10.5), as

IE[e−

r Tit rsds(P (Ti, Tj)− κ)+

∣∣∣ Ft] = P (t, Ti)IEi

[(P (Ti, Tj)− κ)+

∣∣∣ Ft] .Next we use the Black-Scholes formula and the martingale property of theforward price P (t, Tj)/P (t, Ti), which can be written as the geometric Brow-nian motion

P (Ti, Tj) =P (t, Tj)

P (t, Ti)exp

(w Ti

t(ζi(s)− ζj(s))dW i

s −1

2

w Ti

t|ζi(s)− ζj(s)|2ds

),

under the forward measure P when (ζi(s))s∈[0,Ti] and (ζj(s))s∈[0,Tj ] in (12.3)are deterministic functions. The above relation can be obtained by solving(10.12) in Proposition 10.3.

In the Vasicek case the above bond option price could also be computed

from the joint law of(rT ,

r Ttrsds

), which is Gaussian, or from the dynamics

(12.5)-(12.7) of P (t, T ) and rt under Pi, cf. § 7.3 of [60].

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12.3 Caplet Pricing

The caplet on the spot forward rate f(T, T, Ti) with strike κ is a contractwith payoff

(f(T, T, Ti)− κ)+,

priced at time t ∈ [0, T ] from Proposition 10.4 using the forward measure Pias

IE[e−

r Tit rsds(f(T, T, Ti)− κ)+

∣∣∣ Ft] = P (t, Ti)IEi[(f(T, T, Ti)− κ)+ | Ft

],

(12.8)by taking Nt = P (t, Ti) as a numeraire.

Next we consider the caplet with payoff

(L(Ti, Ti, Ti+1)− κ)+

on the LIBOR rate

L(t, Ti, Ti+1) =1

Ti+1 − Ti

(P (t, Ti)

P (t, Ti+1)− 1

), 0 ≤ t ≤ Ti < Ti+1,

which is a martingale under Pi+1 defined in (12.1), from Proposition 10.2.

We assume that L(t, Ti, Ti+1) is modeled in the BGM model of Sec-tion 11.7, i.e. we have

dL(t, Ti, Ti+1)

L(t, Ti, Ti+1)= γi(t)dB

i+1t ,

0 ≤ t ≤ Ti, i = 1, . . . , n − 1, where t 7−→ γi(t) is a deterministic function,i = 1, . . . , n− 1.

The next formula (12.9) is known as the Black caplet formula.

Proposition 12.3. The caplet on L(Ti, Ti, Ti+1) is priced as time t ∈ [0, Ti]as

IE[e−

r Ti+1t rsds(L(Ti, Ti, Ti+1)− κ)+

∣∣∣ Ft] (12.9)

= P (t, Ti+1)L(t, Ti, Ti+1)Φ(d+)− κP (t, Ti+1)Φ(d−),

where

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N. Privault

d+ =log(L(t, Ti, Ti+1)/κ) + σ2

i (t)(Ti − t)/2σi(t)

√Ti − t

,

and

d− =log(L(t, Ti, Ti+1)/κ)− σ2

i (t)(Ti − t)/2σi(t)

√Ti − t

,

and

|σi(t)|2 =1

Ti − tw Ti

t|γi|2(s)ds.

Proof. By (12.8) we have

IE[e−


∣∣∣ Ft]= P (t, Ti+1) IEi+1

[(L(Ti, Ti, Ti+1)− κ)+ | Ft

]= P (t, Ti+1)BS(κ, L(t, Ti, Ti+1), σi(t), 0, Ti − t),

t ∈ [0, Ti], where

BS(κ, x, σ, r, τ) = xΦ(d+)− κe−rτΦ(d−)

is the Black-Scholes function with

|σi(t)|2 =1

Ti − tw Ti

t|γi|2(s)ds,

since t 7−→ L(t, Ti, Ti+1) is a geometric Brownian motion with volatility γi(t)

under Pi+1.

We may also write

(Ti+1 − Ti) IE[e−


∣∣∣ Ft]= P (t, Ti+1)

(P (t, Ti)

P (t, Ti+1)− 1

)Φ(d+)− κ(Ti+1 − Ti)P (t, Ti+1)Φ(d−)

= (P (t, Ti)− P (t, Ti+1))Φ(d+)− κ(Ti+1 − Ti)P (t, Ti+1)Φ(d−),

and this yields a self-financing hedging strategy

(Φ(d+),− (1 + κ(Ti+1 − Ti))Φ(d−))

in the bonds (P (t, Ti), P (t, Ti+1)) with maturities Ti and Ti+1, cf. Corol-lary 10.2 and [61]. Proposition 12.3 can also be proved by taking P (t, Ti+1)as numeraire and letting

Xt = P (t, Ti)/P (t, Ti+1) = 1 + (Ti+1 − Ti)L(Ti, Ti, Ti+1).

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Floorlets

Similarly, a floorlet on f(T, T, Ti) with strike κ is a contract with payoff(κ− f(T, T, Ti))

+, priced at time t ∈ [0, T ] as

IE[e−

r Tit rsds(κ− f(T, T, Ti))

+∣∣∣ Ft] = P (t, Ti)IEi

[(κ− f(T, T, Ti))

+ | Ft].

Floorlets are analog to put options and can be similarly priced by the call/putparity in the Black-Scholes formula.

Cap Pricing

More generally one can consider caps that are relative to a given tenor struc-ture T1, . . . , Tn, with discounted payoff

n−1∑k=1

(Tk+1 − Tk)e−r Tk+1t rsds(f(Tk, Tk, Tk+1)− κ)+.

Pricing formulas for caps are easily deduced from analog formulas for caplets,since the payoff of a cap can be decomposed into a sum of caplet payoffs. Thusthe price of a cap at time t ∈ [0, T1] is given by

IE

[n−1∑k=1

(Tk+1 − Tk)e−r Tk+1t rsds(f(Tk, Tk, Tk+1)− κ)+

∣∣∣ Ft]

=n−1∑k=1

(Tk+1 − Tk) IE

[e−

r Tk+1t rsds(f(Tk, Tk, Tk+1)− κ)+

∣∣∣ Ft]

=

n−1∑k=1

(Tk+1 − Tk)P (t, Tk+1)IEk+1

[(f(Tk, Tk, Tk+1)− κ)+

∣∣∣ Ft] .In the above BGM model, the cap with payoff

n−1∑k=1

(Tk+1 − Tk)(L(Tk, Tk, Tk+1)− κ)+

can be priced at time t ∈ [0, T1] as

n−1∑k=1

(Tk+1 − Tk)P (t, Tk+1)BS(κ, L(t, Tk, Tk+1), σk(t), 0, Tk − t).

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N. Privault

12.4 Forward Swap Measures

In this section we introduce the forward measures to be used for the pricingof swaptions, and we study their properties. We start with the definition ofthe annuity numeraire

P (t, Ti, Tj) =

j−1∑k=i

(Tk+1 − Tk)P (t, Tk+1), 0 ≤ t ≤ Ti, (12.10)

with in particular

P (t, Ti, Ti+1) = (Ti+1 − Ti)P (t, Ti+1), 0 ≤ t ≤ Ti.

1 ≤ i < n. The annuity numeraire satisfies the following martingale property,which can be proved by linearity and the fact that t 7−→ e−

r t0rsdsP (t, Tk) is

a martingale for all k = 1, . . . , n.

Proposition 12.4. The discounted annuity numeraire

t 7−→ e−r t0rsdsP (t, Ti, Tj) = e−

r t0rsds

j−1∑k=i

(Tk+1−Tk)P (t, Tk+1), 0 ≤ t ≤ Ti,

is a martingale under P.

The forward swap measure Pi,j is defined by

dPi,jdP

= e−r Ti0 rsds

P (Ti, Ti, Tj)

P (0, Ti, Tj), (12.11)

1 ≤ i < j ≤ n. We have

IE

[dPi,jdP

∣∣∣ Ft] =1

P (0, Ti, Tj)IE[e−

r Ti0 rsdsP (Ti, Ti, Tj)

∣∣∣ Ft]=P (t, Ti, Tj)

P (0, Ti, Tj)e−

r t0rsds,

0 ≤ t ≤ Ti, by Proposition 12.4, and

dPi,j|FtdP|Ft

= e−r Tit rsds

P (Ti, Ti, Tj)

P (t, Ti, Tj), 0 ≤ t ≤ Ti+1, (12.12)

by Proposition 10.3. We also know that the process

t 7−→ vi,jk (t) :=P (t, Tk)

P (t, Ti, Tj)

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is an Ft-martingale under Pi,j by Proposition 10.2. It follows that the swaprate

S(t, Ti, Tj) :=P (t, Ti)− P (t, Tj)

P (t, Ti, Tj)= vi,ji (t)− vi,jj (t), 0 ≤ t ≤ Ti,

defined in Proposition 11.5 is also a martingale under Pi,j .

Using the forward swap measure we obtain the following pricing formulafor a given integrable claim with payoff of the form P (Ti, Ti, Tj)F :

IE[e−

r Tit rsdsP (Ti, Ti, Tj)F

∣∣∣ Ft] = P (t, Ti, Tj) IE

[FdPi,j|FtdP|Ft

∣∣∣ Ft]= P (t, Ti, Tj)IEi,j

[F∣∣∣ Ft] , (12.13)

after applying (12.11) and (12.12) on the last line, or Proposition 10.1.

12.5 Swaption Pricing on the LIBOR

A swaption on the forward rate f(T1, Tk, Tk+1) is a contract meant to protectoneself against a risk based on an interest rate swap, and has payoff(

j−1∑k=i

(Tk+1 − Tk)e−

r Tk+1Ti

rsds(f(Ti, Tk, Tk+1)− κ)

)+

,

at time Ti.

This swaption can be priced at time t ∈ [0, Ti] under a risk-neutral measureas

IE

e− r Tit rsds

(j−1∑k=i

(Tk+1 − Tk)e−

r Tk+1Ti

rsds(f(Ti, Tk, Tk+1)− κ)

)+ ∣∣∣ Ft .

(12.14)In the sequel and in practice the price (12.14) of the swaption will be evalu-ated as

IE

e− r Tit rsds

(j−1∑k=i

(Tk+1 − Tk)P (Ti, Tk+1)(f(Ti, Tk, Tk+1)− κ)

)+ ∣∣∣ Ft ,

(12.15)

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N. Privault

where we approximate the discount factor e−

r Tk+1Ti

rsds by its conditional ex-pectation P (Ti, Tk+1) given FTi .

Note that when j = i+ 1, the swaption price (12.15) coincides with the priceat time t of a caplet on [Ti, Ti+1] up to a factor Ti+1 − Ti since

IE[e−

r Tit rsds ((Ti+1 − Ti)P (Ti, Ti+1)(f(Ti, Ti, Ti+1)− κ))

+∣∣∣ Ft]

= (Ti+1 − Ti) IE[e−

r Tit rsdsP (Ti, Ti+1) ((f(Ti, Ti, Ti+1)− κ))

+∣∣∣ Ft]

= (Ti+1 − Ti) IE

[e−

r Tit rsds IE

[e−

r Ti+1Ti

rsds∣∣∣ FTi] ((f(Ti, Ti, Ti+1)− κ))

+∣∣∣ Ft]

= (Ti+1 − Ti) IE

[IE

[e−

r Tit rsdse

−r Ti+1Ti

rsds ((f(Ti, Ti, Ti+1)− κ))+∣∣∣ FTi] ∣∣∣ Ft]

= (Ti+1 − Ti) IE[e−

r Ti+1t rsds (f(Ti, Ti, Ti+1)− κ)

+∣∣∣ Ft] ,

0 ≤ t ≤ Ti.

In case we replace the forward rate f(t, T, S) with the LIBOR rateL(t, T, S) defined in Proposition 11.5, the payoff of the swaption can berewritten as in the following lemma which is a direct consequence of thedefinition of the swap rate S(Ti, Ti, Tj).

Lemma 12.1. The payoff of the swaption in (12.15) can be rewritten as(j−1∑k=i

(Tk+1 − Tk)P (Ti, Tk+1)(L(Ti, Tk, Tk+1)− κ)

)+

= (P (Ti, Ti)− P (Ti, Tj)− κP (Ti, Ti, Tj))+

= P (Ti, Ti, Tj) (S(Ti, Ti, Tj)− κ)+. (12.16)

Proof. The relation

j−1∑k=i

(Tk+1 − Tk)P (t, Tk+1)(L(t, Tk, Tk+1)− S(t, Ti, Tj)) = 0

that defines the forward swap rate S(t, Ti, Tj) shows that

j−1∑k=i

(Tk+1 − Tk)P (t, Tk+1)L(t, Tk, Tk+1)

= S(t, Ti, Tj)

j−1∑k=i

(Tk+1 − Tk)P (t, Tk+1)

= P (t, Ti, Tj)S(t, Ti, Tj)

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= P (t, Ti)− P (t, Tj),

by the definition (12.10) of P (t, Ti, Tj), hence

j−1∑k=i

(Tk+1 − Tk)P (t, Tk+1)(L(t, Tk, Tk+1)− κ)

= P (t, Ti)− P (t, Tj)− κP (t, Ti, Tj)

= P (t, Ti, Tj) (S(t, Ti, Tj)− κ) ,

and for t = Ti we get(j−1∑k=i


)+

= P (Ti, Ti, Tj) (S(Ti, Ti, Tj)− κ)+.

The next proposition simply states that a swaption on the LIBOR rate canbe priced as a European call option on the swap rate S(Ti, Ti, Tj) under the

forward swap measure Pi,j .

Proposition 12.5. The price (12.15) of the swaption with payoff(j−1∑k=i


)+

(12.17)

on the LIBOR market can be written under the forward swap measure Pi,j as

P (t, Ti, Tj)IEi,j

[(S(Ti, Ti, Tj)− κ)

+∣∣∣ Ft] , 0 ≤ t ≤ Ti.

Proof. As a consequence of (12.13) and Lemma 12.1 we find

IE

e− r Tit rsds

(j−1∑k=i


)+ ∣∣∣ Ft

= IE[e−

r Tit rsds (P (Ti, Ti)− P (Ti, Tj)− κP (Ti, Ti, Tj))

+∣∣∣ Ft]

= IE[e−

r Tit rsdsP (Ti, Ti, Tj) (S(Ti, Ti, Tj)− κ)

+∣∣∣ Ft]

=1

P (t, Ti, Tj)IE

[dPi,j|FtdP|Ft

(S(Ti, Ti, Tj)− κ)+∣∣∣ Ft]

= P (t, Ti, Tj)IEi,j

[(S(T1, T1, Tn)− κ)

+∣∣∣ Ft] . (12.18)

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N. Privault

In the next proposition we price a swaption with payoff (12.17) or equivalently(12.16).

Proposition 12.6. Assume that the LIBOR swap rate (11.30) is modeled asa geometric Brownian motion under Pi,j, i.e.

dS(t, Ti, Tj) = S(t, Ti, Tj)σ(t)dW i,jt ,

where (σ(t))t∈R+ is a deterministic function. Then the swaption with payoff

(P (T, Ti)− P (T, Tj)− κP (Ti, Ti, Tj))+ = P (Ti, Ti, Tj) (S(Ti, Ti, Tj)− κ)

+

can be priced using the Black-Scholes formula as

IE[e−


+∣∣∣ Ft]

= (P (t, Ti)− P (t, Tj))Φ+(t, S(t, Ti, Tj))

−κΦ−(t, S(t, Ti, Tj))

j−2∑k=i

(Tk+1 − Tk)P (t, Tk+1),

where

d+ =log(S(t, Ti, Tj)/κ) + σ2

i,j(t)(Ti − t)/2σi,j(t)

√Ti − t

,

and

d− =log(S(t, Ti, Tj)/κ)− σ2

i,j(t)(Ti − t)/2σi,j(t)

√Ti − t

,

and

|σi,j(t)|2 =1

Ti − tw Ti

t|σ|2(s)ds.

Proof. Since S(t, Ti, Tj) is a geometric Brownian motion with variance

(σ(t))t∈R+under Pi,j , by (12.18) we have

IE[e−


+∣∣∣ Ft]

= IE[e−

r Ttrsds(P (T, Ti)− P (T, Tj)− κP (Ti, Ti, Tj))

+∣∣∣ Ft]

= P (t, Ti, Tj)IEi,j


+∣∣∣ Ft]

= P (t, Ti, Tj)BS(κ, S(Ti, Ti, Tj), σi,j(t), 0, Ti − t)= P (t, Ti, Tj) (S(t, Ti, Tj)Φ+(t, S(t, Ti, Tj))− κΦ−(t, S(t, Ti, Tj)))

= (P (t, Ti)− P (t, Tj))Φ+(t, S(t, Ti, Tj))− κP (Ti, Ti, Tj)Φ−(t, S(t, Ti, Tj))

= (P (t, Ti)− P (t, Tj))Φ+(t, S(t, Ti, Tj))

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−κΦ−(t, S(t, Ti, Tj))

j−2∑k=i

(Tk+1 − Tk)P (t, Tk+1).

In addition the hedging strategy

(Φ+(t, S(t, Ti, Tj)),−κΦ−(t, S(t, Ti, Tj))(Ti+1 − Ti), . . .. . . ,−κΦ−(t, S(t, Ti, Tj))(Tj−1 − Tj−2),−Φ+(t, S(t, Ti, Tj)))

based on the assets (P (t, Ti), . . . , P (t, Tj)) is self-financing by Corollary 10.2,cf. also [61].

Swaption prices can also be computed by an approximation formula, from theexact dynamics of the swap rate S(t, Ti, Tj) under Pi,j , based on the bondprice dynamics of the form (12.3), cf. [69], page 17.

Exercises

Exercise 12.1 Given two bonds with maturities T , S and prices P (t, T ),P (t, S), consider the LIBOR rate

L(t, T, S) =P (t, T )− P (t, S)

(S − T )P (t, S)

at time t, modeled as

dL(t, T, S) = µtL(t, T, S)dt+ σL(t, T, S)dWt, 0 ≤ t ≤ T, (12.19)

where (Wt)t∈[0,T ] is a standard Brownian motion under the risk-neutral mea-sure P∗, σ > 0 is a constant, and (µt)t∈[0,T ] is an adapted process. Let

Ft = IE∗[e−

r Strsds(κ− L(T, T, S))+

∣∣∣ Ft]denote the price at time t of a floorlet option with strike κ, maturity T , andpayment date S.

1. Rewrite the value of Ft using the forward measure PS with maturity S.2. What is the dynamics of L(t, T, S) under the forward measure PS ?3. Write down the value of Ft using the Black-Scholes formula.

Hint. Given X a centered Gaussian random variable with variance v2 wehave

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N. Privault

IE[(κ− em+X)+] = κΦ(−(m− log κ)/v)− em+ v2

2 Φ(−v − (m− log κ)/v),

where Φ denotes the Gaussian cumulative distribution function.

Exercise 12.2 We work in the short rate model

drt = σdBt,

where (Bt)t∈R+ is a standard Brownian motion under P, and P2 is the forwardmeasure defined by

dP2

dP=

1

P (0, T2)e−

r T20 rsds.

1. State the expressions of ζ1t and ζ2t in

dP (t, Ti)

P (t, Ti)= rtdt+ ζitdBt, i = 1, 2,

and the dynamics of the P (t, T1)/P (t, T2) under P2, where P (t, T1) andP (t, T2) are bond prices with maturities T1 and T2.

2. State the expression of the forward rate f(t, T1, T2).3. Compute the dynamics of f(t, T1, T2) under the forward measure P2 with

dP2

dP=

1

P (0, T2)e−

r T20 rsds.


(T2 − T1) IE[e−

r T2t rsds(f(T1, T1, T2)− κ)+

∣∣∣ Ft]of a cap at time t ∈ [0, T1], using the expectation under the forwardmeasure P2.

5. Compute the dynamics of the swap rate process

S(t, T1, T2) =P (t, T1)− P (t, T2)

(T2 − T1)P (t, T2), t ∈ [0, T1],

under P2.6. Compute the swaption price

(T2 − T1) IE[e−

r T1t rsdsP (T1, T2)(S(T1, T1, T2)− κ)+

∣∣∣ Ft]on the swap rate S(T1, T1, T2) using the expectation under the forwardswap measure P1,2.

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Exercise 12.3 Consider three zero-coupon bonds P (t, T1), P (t, T2) andP (t, T3) with maturities T1 = δ, T2 = 2δ and T3 = 3δ respectively, andthe forward LIBOR L(t, T1, T2) and L(t, T2, T3) defined by

L(t, Ti, Ti+1) =1

δ

(P (t, Ti)

P (t, Ti+1)− 1

), i = 1, 2.

Assume that L(t, T1, T2) and L(t, T2, T3) are modeled in the BGM model by

dL(t, T1, T2)

L(t, T1, T2)= e−atdW 2

t , 0 ≤ t ≤ T1, (12.20)

and L(t, T2, T3) = b, 0 ≤ t ≤ T2, for some constants a, b > 0, where W 2t is a

standard Brownian motion under the forward rate measure P2 defined by

dP2

dP=e−

r T20 rsds

P (0, T2).

1. Compute L(t, T1, T2), 0 ≤ t ≤ T2 by solving Equation (12.20).2. Show that the price at time t of the caplet with strike κ can be written

as

E[e−

r T2t rsds(L(T1, T1, T2)− κ)+

∣∣∣Ft] = P (t, T2)E2

[(L(T1, T1, T2)− κ)+ | Ft

],

where E2 denotes the expectation under the forward measure P2.3. Using the hint below, compute the price at time t of the caplet with strikeκ on L(T1, T1, T2).

4. Compute

P (t, T1)

P (t, T1, T3), 0 ≤ t ≤ T1, and

P (t, T3)

P (t, T1, T3), 0 ≤ t ≤ T2,

in terms of b and L(t, T1, T2), where P (t, T1, T3) is the annuity numeraire

P (t, T1, T3) = δP (t, T2) + δP (t, T3), 0 ≤ t ≤ T2.

5. Compute the dynamics of the swap rate

t 7−→ S(t, T1, T3) =P (t, T1)− P (t, T3)

P (t, T1, T3), 0 ≤ t ≤ T1,

i.e. show that we have

dS(t, T1, T3) = σ1.3(t)S(t, T1, T3)dW 2t ,

where σ1,3(t) is a process to be determined.

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N. Privault

6. Using the Black-Scholes formula, compute an approximation of the swap-tion price

E[e−

r T1t rsdsP (T1, T1, T3)(S(T1, T1, T3)− κ)+

∣∣∣ Ft]= P (t, T1, T3)E2

[(S(T1, T1, T3)− κ)+ | Ft

],

at time t ∈ [0, T1]. You will need to approximate σ1,3(s), s ≥ t, by “freez-ing” all random terms at time t.

Hint. Given X a centered Gaussian random variable with variance v2 wehave

E[(em+X − κ)+] = em+ v2

2 Φ(v + (m− log κ)/v)− κΦ((m− log κ)/v),


Exercise 12.4 Consider a portfolio (ξTt , ξSt )t∈[0,T ] made of two bonds with

maturities T , S, and value

Vt = ξTt P (t, T ) + ξSt P (t, S), 0 ≤ t ≤ T,

at time t. We assume that the portfolio is self-financing, i.e.

dVt = ξTt dP (t, T ) + ξSt dP (t, S), 0 ≤ t ≤ T, (12.21)

and that it hedges the claim (P (T, S)− κ)+, so that

Vt = IE[e−


+∣∣∣ Ft]

= P (t, T ) IET

[(P (T, S)− κ)

+∣∣∣ Ft] , 0 ≤ t ≤ T.

1. Show that

IE[e−

r Ttrsds (P (T, S)−K)

+∣∣∣ Ft]

= P (0, T ) IET

[(P (T, S)−K)

+]

+w t

0ξTs dP (s, T ) +

w t

0ξSs dP (s, S).

2. Show that under the self-financing condition (12.21), the discounted port-

folio price Vt = e−r t0rsdsVt satisfies

dVt = ξTt dP (t, T ) + ξSt dP (t, S),

where P (t, T ) = e−r t0rsdsP (t, T ) and P (t, S) = e−

r t0rsdsP (t, S) denote

the discounted bond prices.3. Show that

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IET

[(P (T, S)−K)

+ |Ft]

= IET

[(P (T, S)−K)

+]

+w t

0

∂C

∂x(Xu, T − u, v(u, T ))dXu.

Hint: use the martingale property and the Ito formula.4. Show that the discounted portfolio price Vt = Vt/P (t, T ) satisfies

dVt =∂C

∂x(Xt, T − t, v(t, T ))dXt

=P (t, S)

P (t, T )

∂C

∂x(Xt, T − t, v(t, T ))(σSt − σTt )dBTt .

5. Show that

dVt = P (t, S)∂C

∂x(Xt, T − t, v(t, T ))(σSt − σTt )dBt + VtdP (t, T ).

6. Show that

dVt = P (t, S)∂C


7. Compute the hedging strategy (ξTt , ξSt )t∈[0,T ] of the bond option.

8. Show that∂C

∂x(x, τ, v) = Φ

(log(x/K) + τv2/2√

τv

).

Exercise 12.5 Given n bonds with maturities T1, . . . , Tn, consider the an-nuitry numeraire P (t, Ti, Tj) =

∑j−1k=i(Tk+1 − Tk)P (t, Tk+1) and the swap

rate

S(t, Ti, Tj) =P (t, Ti)− P (t, Tj)

P (t, Ti, Tj)

at time t ∈ [0, Ti], modeled as dS(t, Ti, Tj) = µtS(t, Ti, Tj)dt+σS(t, Ti, Tj)dWt,0 ≤ t ≤ Ti, where (Wt)t∈[0,Ti] is a standard Brownian motion under the risk-neutral measure P∗, (µt)t∈[0,T ] is an adapted process and σ > 0 is a constant.Let

IE∗[e−

r Tit rsdsP (Ti, Ti, Tj)(κ− S(Ti, Ti, Tj))

+∣∣∣Ft]

denote the price at time t ∈ [0, Ti] of a put swaption with strike κ.

1. Rewrite the above swaption price using the forward swap measure Pi,jdefined from the annuity numeraire P (t, Ti, Tj).

2. What is the dynamics of S(t, Ti, Tj) under the forward swap measure Pi,j?

3. Write down the value of the above swaption price using the Black-Scholesformula.

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N. Privault

Hint. Given X a centered Gaussian random variable with variance v2 we have

IE[(κ− em+X)+] = κΦ(−(m− log κ)/v)− em+ v2

2 Φ(−v − (m− log κ)/v),


Exercise 12.6 Consider a bond market with tenor structure Ti, . . . , Tj andbonds with maturities Ti, . . . , Tj , whose prices P (t, Ti), . . . P (t, Tj) at time tare given by

dP (t, Tk)

P (t, Tk)= rtdt+ ζk(t)dBt, k = i, . . . , j,

where (rt)t∈R+is a short term interest rate process and (Bt)t∈R+

de-notes a standard Brownian motion generating a filtration (Ft)t∈R+

, andζi(t), . . . , ζj(t) are volatility processes.

The swap rate S(t, Ti, Tj) is defined by

S(t, Ti, Tj) =P (t, Ti)− P (t, Tj)

P (t, Ti, Tj),

where

P (t, Ti, Tj) =

j−1∑k=i

(Tk+1 − Tk)P (t, Tk+1)

is the annuity numeraire. Recall that a swaption on the LIBOR market canbe priced at time t ∈ [0, Ti] as

IE

e− r Tit rsds

(j−1∑k=i


)+ ∣∣∣ Ft

= P (t, Ti, Tj) IEi,j


+∣∣∣ Ft] , (12.22)

under the forward swap measure Pi,j defined by

dPi,jdP

= e−r Ti0 rsds

P (Ti, Ti, Tj)

P (0, Ti, Tj), 1 ≤ i < j ≤ n,

under which

Bi,jt := Bt −j−1∑k=i

(Tk+1 − Tk)P (t, Tk+1)

P (t, Ti, Tj)ζk+1(t)dt (12.23)

is a standard Brownian motion. We assume that the swap rate is modeled asa geometric Brownian motion

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dS(t, Ti, Tj) = S(t, Ti, Tj)σi,j(t)dBi,jt , 0 ≤ t ≤ Ti, (12.24)

where the swap rate volatility is a deterministic function σi,j(t). In the sequelwe denote St = S(t, Ti, Tj) for simplicity of notation.

1. Solve the equation (12.24) on the interval [t, Ti], and compute S(Ti, Ti, Tj)from the initial condition S(t, Ti, Tj).

2. Show that the price (12.18) of the swaption can be written as

P (t, Ti, Tj)C(St, v(t, Ti)),

where

v2(t, Ti) =w Ti

t|σi,j(s)|2ds,

and C(x, v) is a function to be specified using the Black-Scholes formulaBS(K,x, σ, r, τ), with

IE[(xem+X −K)+] = Φ(v+ (m+ log(x/K))/v)−KΦ((m+ log(x/K))/v),

whereX is a centered Gaussian random variable with meanm = rτ−v2/2and variance v2.

3. Consider a portfolio (ξit, . . . , ξjt )t∈[0,Ti] made of bonds with maturities

Ti, . . . , Tj and value

Vt =

j∑k=i

ξkt P (t, Tk),

at time t ∈ [0, Ti]. We assume that the portfolio is self-financing, i.e.

dVt =

j∑k=i

ξkt dP (t, Tk), 0 ≤ t ≤ Ti, (12.25)

and that it hedges the claim (S(Ti, Ti, Tj)− κ)+, so that

Vt = IE

e− r Tit rsds

(j−1∑k=i


)+ ∣∣∣ Ft



+∣∣∣ Ft] ,

0 ≤ t ≤ Ti. Show that

IE

e− r Tit rsds

(j−1∑k=i


)+ ∣∣∣ Ft

= P (0, Ti, Tj) IEi,j


+]

+

j∑k=i

w t

0ξks dP (s, Ti),

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N. Privault

0 ≤ t ≤ Ti.4. Show that under the self-financing condition (12.25), the discounted port-

folio price Vt = e−r t0rsdsVt satisfies

dVt =

j∑k=i

ξkt dP (t, Tk),

where P (t, Tk) = e−r t0rsdsP (t, Tk), k = i, . . . , j, denote the discounted

bond prices.5. Show that

IEi,j


+∣∣∣ Ft]

= IEi,j


+]

+w t

0

∂C

∂x(Su, v(u, Ti))dSu.

Hint: use the martingale property and the Ito formula.6. Show that the discounted portfolio price Vt = Vt/P (t, Ti, Tj) satisfies

dVt =∂C

∂x(St, v(t, Ti))dSt = St

∂C

∂x(St, v(t, Ti))σ

i,jt dBi,jt .

7. Show that

dVt = (P (t, Ti)− P (t, Tj))∂C

∂x(St, v(t, Ti))σ

i,jt dBt + VtdP (t, Ti, Tj).

8. Show that

dVt = Stζi(t)∂C

∂x(St, v(t, Ti))

j−1∑k=i

(Tk+1 − Tk)P (t, Tk+1)dBt

+(Vt − St∂C

∂x(St, v(t, Ti)))

j−1∑k=i

(Tk+1 − Tk)P (t, Tk+1)ζk+1(t)dBt

+∂C

∂x(St, v(t, Ti))P (t, Tj)(ζi(t)− ζj(t))dBt.

9. Show that

dVt =∂C

∂x(St, v(t, Ti))d(P (t, Ti)− P (t, Tj))

+(Vt − St∂C

∂x(St, v(t, Ti)))dP (t, Ti, Tj).

10. Show that

∂C

∂x(x, v(t, Ti)) = Φ

(log(x/K)

v(t, Ti)+v(t, Ti)

2

).

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11. Show that we have

dVt = Φ

(log(St/K)

v(t, Ti)+v(t, Ti)

2

)d(P (t, Ti)− P (t, Tj))

−κΦ(

log(St/K)

v(t, Ti)− v(t, Ti)

2

)dP (t, Ti, Tj).

12. Show that the hedging strategy is given by

ξit = Φ

(log(St/K)

v(t, Ti)+v(t, Ti)

2

),

ξjt = −Φ(

log(St/K)

v(t, Ti)+v(t, Ti)

2

)−κ(Tj+1−Tj)Φ

(log(St/K)


2

),

and

ξkt = −κ(Tk+1 − Tk)Φ

(log(St/K)


2

), i+ 1 ≤ k ≤ j − 1.

" 357



Chapter 13

Default Risk in Bond Markets

The bond pricing model of Chapter 11 is based on the terminal conditionP (T, T ) = $1, i.e. the bond payoff at maturity is always equal to $1, anddefault never occurs. In this chapter we allow for the possibility of defaultat a random time τ , in which case the terminal payoff of a bond vanishes atmaturity. We also consider the credit default options (swaps) that can act asa protection against default.

13.1 Survival Probabilities and failure rate

Given t > 0, let P(τ ≥ t) denote the probability that a random systemwith lifetime τ survives at least t years. Assuming that survival probabilitiesP(τ ≥ t) are strictly positive for all t > 0, we can compute the conditionalprobability for that system to survive up to time T , given that it was stillfunctioning at time t ∈ [0, T ], as

P(τ > T | τ > t) =P(τ > T and τ > t)

P(τ > t)=

P(τ > T )

P(τ > t), 0 ≤ t ≤ T,

with

P(τ < T | τ > t) = 1− P(τ > T | τ > t)

=P(τ > t)− P(τ > T )

P(τ > t)

=P(τ < T )− P(τ < t)

P(τ > t)

=P(t < τ < T )

P(τ > t), 0 ≤ t ≤ T,

and the conditional survival probability law

"



N. Privault

P(τ ∈ dx | τ > t) = P(x < τ < x+ dx | τ > t)

= P(τ < x+ dx | τ > t)− P(τ < x | τ > t)

=P(τ < x+ dx)− P(τ < x)

P(τ > t)

=1

P(τ > t)dP(τ < x)

= − 1

P(τ > t)dP(τ > x), x > t.

From this we can deduce the failure rate function

λ(t) :=P(τ < t+ dt | τ > t)

dt

=1

P(τ > t)

P(t < τ < t+ dt)

dt

=1

P(τ > t)

P(τ > t)− P(τ > t+ dt)

dt

= − d

dtlogP(τ > t)

= − 1

P(τ > t)

d

dtP(τ > t), t > 0,

which satisfies the differential equation

d

dtP(τ > t) = −λ(t)P(τ > t),

which can be solved as

P(τ > t) = exp

(−

w t

0λ(u)du

), t ∈ R+, (13.1)

under the initial condition P(τ > 0) = 1. This allows us to rewrite the survivalprobability as

P(τ > T | τ > t) =P(τ > T )

P(τ > t)= exp

(−

w T

tλ(u)du

), 0 ≤ t ≤ T,

withP(τ > t+ h | τ > t) = e−λ(t)h ' 1− λ(t)h, (13.2)

andP(τ < t+ h | τ > t) = 1− e−λ(t)h ' λ(t)h, (13.3)

as h tends to 0. When the failure rate λ(t) = λ > 0 is a constant function oftime, Relation (13.1) shows that

360


"



P(τ > T ) = e−λT , T ≥ 0,

i.e. τ has the exponential distribution with parameter λ. Note that given(τn)n≥1 a sequence of i.i.d. exponentially distributed random variables, let-ting

Tn = τ1 + · · ·+ τn, n ≥ 1,

defines the sequence of jump times of a standard Poisson process with inten-sity λ > 0, cf. Section 14.1 below for details.

13.2 Stochastic Default

We now model the failure rate function (λt)t∈R+as a random process adapted

to a filtration (Ft)t∈R+.

In case the random time τ is a stopping time with respect to (Ft)t∈R+ , i.e.the knowledge of whether default already occurred at time t is contained inFt, t ∈ R+, and we have

τ > t ∈ Ft, t ∈ R+,

cf. Section 9.3, we have

P(τ > t | Ft) = IE[1τ>t | Ft

]= 1τ>t, t ∈ R+.

In the sequel we will not assume that τ is an Ft-stopping time, and by analogywith (13.1) we will write P(τ > t | Ft) as

P(τ > t | Ft) = exp

(−

w t

0λudu

), t > 0. (13.4)

This is the case in particular in [48] when λu has the form λu = h(Xu), andτ is given by

τ = inf

t ∈ R+ :

w t

0h(Xu)du ≥ L

,

where h is a non-negative function, (Xt)t∈R+is a process generating a fil-

tration (Ft)t∈R+ , and L is an independent exponentially distributed randomvariable.

We let (Gt)t∈R+be the filtration defined by

Gt = Ft ∨ σ(τ ≤ u : 0 ≤ u ≤ t), t ∈ R+,

" 361



N. Privault

i.e. Gt contains the additional information on whether default at time τ hasoccurred or not before time t. The process λt can also be chosen amongthe classical mean-reverting diffusion processes, including jump-diffusion pro-cesses.

Taking F = 1 in the next Lemma 13.1 shows that the survival probabilityup to time T , given information known up to t, is given by

P(τ > T | Gt) = IE[1τ>T | Gt

](13.5)

= 1τ>t IE

[exp

(−

w T

tλudu

) ∣∣∣ Ft] , 0 ≤ t ≤ T.

Lemma 13.1. ([32]) For any FT -measurable integrable random variable Fwe have

IE[F1τ>T | Gt

]= 1τ>t IE

[F exp

(−

w T

tλudu

) ∣∣∣ Ft] .Proof. By (13.4) we have

P(τ > T | FT )

P(τ > t | Ft)=

exp

(−

w T

0λudu

)exp

(−

w t

0λudu

) = exp

(−

w T

tλudu

),

hence, since F is FT -measurable,

1τ>t IE

[F exp

(−

w T

tλudu

) ∣∣∣ Ft] = 1τ>t IE

[FP(τ > T | FT )

P(τ > t | Ft)∣∣∣ Ft]

=1τ>t

P(τ > t | Ft)IE[F IE[1τ>T | FT ]

∣∣∣ Ft]=

1τ>t

P(τ > t | Ft)IE[IE[F1τ>T | FT ]

∣∣∣ Ft]= 1τ>t

IE[F1τ>T

∣∣∣ Ft]P(τ > t | Ft)

= 1τ>t IE[F1τ>T

∣∣∣ Ft ∨ τ > t]

= 1τ>t IE[F1τ>T

∣∣∣ Gt] , 0 ≤ t ≤ T.

In the above argument we used the identity

IE[G∣∣∣ Ft ∨ τ > t

]=

1

P(τ > t | Ft)IE[G∣∣∣ Ft] ,

362


"



which follows from e.g. (16.22) in the appendix with G the integrable randomvariable given by G = F1τ>T, under the probability measure P|Ft , 0 ≤ t ≤T .

The computation of P(τ > T | Gt) according to (13.5) is then similar to thatof a bond price, by considering the failure rate λ(t) as a virtual short terminterest rate. In particular the failure rate λ(t, T ) can be modeled in the HJMframework of Chapter 11.4, and

P(τ > T | Gt) = IE

[exp

(−

w T

tλ(t, u)du

) ∣∣∣ Ft]can then be computed by applying HJM bond pricing techniques.

The computation of expectations given Gt as in Lemma 13.1 can be usefulfor pricing under insider trading, in which the insider has access to the aug-mented filtration Gt while the ordinary trader has only access to Ft, thereforegenerating two different prices IE∗[F | Ft] and IE∗[F | Gt] for the same claimF under the same risk-neutral measure P∗. This leads to the issue of com-puting the dynamics of the underlying asset price by decomposing it using aFt-martingale vs a Gt-martingale instead of using different forward measuresas in in § 12.1. This can be obtained by the technique of enlargement offiltration, cf. [42], [22], [36], [78].

13.3 Defaultable Bonds

The price of a default bond with maturity T , (random) default time τ and(possibly random) recovery rate ξ ∈ [0, 1] is given by

P (t, T ) = IE

[1τ>T exp

(−

w T

trudu

) ∣∣∣ Gt]+ IE

[ξ1τ≤T exp

(−

w T

trudu

) ∣∣∣ Gt] , 0 ≤ t ≤ T,

Taking F = exp(−

r Ttrudu

)in Lemma 13.1, we get

IE

[1τ>T exp

(−

w T

trudu

) ∣∣∣ Gt] = 1τ>t IE

[exp

(−

w T

t(ru + λu)du

) ∣∣∣ Ft] ,cf. e.g. [48], [32], [19], hence

P (t, T ) = 1τ>t IE

[exp

(−

w T

t(ru + λu)du

) ∣∣∣ Ft]

" 363



N. Privault

+ IE

[ξ1τ≤T exp

(−

w T

trudu

) ∣∣∣ Gt] , 0 ≤ t ≤ T.

In the case of complete default (zero-recovery) we have ξ = 0 and

P (t, T ) = 1τ>t IE

[exp

(−

w T

t(rs + λs)ds

) ∣∣∣ Ft] , 0 ≤ t ≤ T. (13.6)

From the above expression (13.6) we note that the effect of the presence ofa default time τ is to decrease the bond price, which can be viewed as anincrease of the short rate by the amount λu.

This treatment of default risk has some similarity with that of couponbonds which can be priced as

P (t, T ) = ec(T−t) IE

[exp

(−

w T

trsds

) ∣∣∣ Gt] ,where c > 0 is a continuous-time deterministic coupon rate.

Finally, from Proposition 12.1 the bond price (13.6) can also be expressedunder the forward measure P with maturity T , as

P (t, T ) = 1τ>t IE

[exp

(−

w T

t(rs + λs)ds

) ∣∣∣ Ft]= 1τ>t IE

[exp

(−

w T

trsds

) ∣∣∣ Ft] IEP

[exp

(−

w T

tλsds

) ∣∣∣ Ft]= 1τ>t IE

[exp

(−

w T

trsds

) ∣∣∣ Ft]Q(t, T ),

where

Q(t, T ) = IEP

[exp

(−

w T

tλsds

) ∣∣∣ Ft]denotes the survival probability under the forward measure P, cf. [11], [10].

13.4 Credit Default Swaps

We work with a tenor structure t = Ti < · · · < Tj = T.

A Credit Default Swap (CDS) is a contract consisting in

- a premium leg: the buyer is purchasing protection at time t against de-fault at time Tk, k = i+ 1, . . . , j, and has to make a fixed payment St at

364


"



times Ti+1, . . . , Tj between t and T in compensation.

The discounted value at time t of the premium leg is

V (t, T ) = IE

[j−1∑k=i

Stδk1τ>Tk+1 exp

(−

w Tk+1

trsds

) ∣∣∣ Gt]

=

j−1∑k=i

Stδk IE

[1τ>Tk+1 exp

(−

w Tk+1

trsds

) ∣∣∣ Gt]

= St

j−1∑k=i

δkP (t, Tk+1)

= StP (t, Ti, Tj),

where δk = Tk+1 − Tk, P (t, Ti, Tj) is the annuity numeraire (12.10), and

P (t, Tk) = 1τ>t IE

[exp

(−

w Tk

t(rs + λs)ds

) ∣∣∣ Ft] , 0 ≤ t ≤ Tk,

is the defaultable bond price with maturity Tk, k = i, . . . , j−1. For simplic-ity we have ignored a possible accrual interest term over the time period[Tk, τ ] when τ ∈ [Tk, Tk+1] in the above value of the premium leg.

- a protection leg: the seller or issuer of the contract makes a payment1− ξk+1 to the buyer in case default occurs at time Tk+1, k = i, . . . , j− 1.

The value at time t of the protection leg is

IE

[j−1∑k=i

1(Tk,Tk+1](τ)(1− ξk+1) exp

(−

w Tk+1

trsds

) ∣∣∣ Gt] ,where ξk+1 is the recovery rate associated with the maturity Tk+1, k =i, . . . , j − 1.

In the case of a non-random recovery rate ξk the value of the protectionleg becomes

j−1∑k=i

δk(1− ξk+1) IE

[1(Tk,Tk+1](τ) exp

(−

w Tk+1

trsds

) ∣∣∣ Gt] .The spread St is computed by equating the values of the protection andpremium legs, i.e. from the relation

V (t, T ) = StP (t, Ti, Tj)

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N. Privault

= IE

[j−1∑k=i

δk1(Tk,Tk+1](τ)(1− ξk+1) exp

(−

w Tk+1

trsds

) ∣∣∣ Gt] ,which yields

St =1

P (t, Ti, Tj)

j−1∑k=i

δk IE

[1(Tk,Tk+1](τ)(1− ξk+1) exp

(−

w Tk+1

trsds

) ∣∣∣ Gt] .In the case of a constant recovery rate ξ we find

St =1− ξ

P (t, Ti, Tj)

j−1∑k=i

δk IE

[1(Tk,Tk+1](τ) exp

(−

w Tk+1

trsds

) ∣∣∣ Gt] ,and if τ is constrained to take values in the tenor structure t = Ti, . . . , Tjwith δk = δ, k = i, . . . , j − 1, we get

St = δ1− ξ

P (t, Ti, Tj)IE[1[t,T ](τ) exp

(−

w τ

trsds

) ∣∣∣ Gt] .

13.5 Exercises

Exercise 13.1 Defaultable bonds. Consider a (random) default time τ withlaw

P(τ > t | Ft) = exp

(−

w t

0λudu

),

where λt is a (random) default rate process which is adapted to the filtra-tion (Ft)t∈R+

. Recall that the probability of survival up to time T , giveninformation known up to time t, is given by

P(τ > T | Gt) = 1τ>tE

[exp

(−

w T

tλudu

) ∣∣∣Ft] ,where Gt = Ft ∨ σ(τ < u : 0 ≤ u ≤ t), t ∈ R+, is the filtration de-fined by adding the default time information to the history (Ft)t∈R+

. In thisframework, the price P (t, T ) of defaultable bond with maturity T , short terminterest rate rt and (random) default time τ is given by

P (t, T ) = E

[1τ>T exp

(−

w T

trudu

) ∣∣∣Gt] (13.7)

= 1τ>tE

[exp

(−

w T

t(ru + λu)du

) ∣∣∣Ft] .366


"



In the sequel we assume that the processes (rt)t∈R+and (λt)t∈R+

are modeledaccording to the Vasicek processes

drt = −artdt+ σdB(1)t ,

dλt = −bλtdt+ ηdB(2)t ,

where (B(1)t )t∈R+

and (B(2)t )t∈R+

are two standard Ft-Brownian motions with

correlation ρ ∈ [−1, 1], and dB(1)t dB

(2)t = ρdt.

1. Give a justification for the fact that

E

[exp

(−

w T

t(ru + λu)du

) ∣∣∣Ft]can be written as a function F (t, rt, λt) of t, rt and λt, t ∈ [0, T ].

2. Show that

t 7−→ exp

(−

w t

0(rs + λs)ds

)E

[exp

(−

w T

t(ru + λu)du

) ∣∣∣Ft]is an Ft-martingale under P.

3. Use the Ito formula with two variables to derive a PDE on R2 for thefunction F (t, x, y).

4. Show that we have

w T

trsds = C(a, t, T )rt + σ

w T

tC(a, s, T )dB(1)

s ,

and w T

tλsds = C(b, t, T )λt + η

w T

tC(b, s, T )dB(2)

s ,

where

C(a, t, T ) = −1

a(e−a(T−t) − 1).

5. Show that the random variable

w T

trsds+

w T

tλsds

is Gaussian and compute its conditional mean

IE

[w T

trsds+

w T

tλsds

∣∣∣Ft]and variance

Var

[w T

trsds+

w T

tλsds

∣∣∣Ft] ," 367



N. Privault

conditionally to Ft.6. Compute P (t, T ) from its expression (13.7) as a conditional expectation.7. Show that the solution F (t, x, y) to the 2-dimensional PDE of Question 3

is

F (t, x, y) = exp (−C(a, t, T )x− C(b, t, T )y)

× exp

(σ2

2

w T

tC2(a, s, T )ds+

η2

2

w T

tC2(b, s, T )ds

)× exp

(ρση

w T

tC(a, s, T )C(b, s, T )ds

).

8. Show that the defaultable bond price P (t, T ) can also be written as

P (t, T ) = eU(t,T )P(τ > T | Gt) IE

[exp

(−

w T

trsds

) ∣∣∣Ft] ,where

U(t, T ) = ρση

ab(T − t− C(a, t, T )− C(b, t, T ) + C(a+ b, t, T )) .

9. By partial differentiation of logP (t, T ) with respect to T , compute the

corresponding instantaneous short rate f(t, T ) = −∂ logP (t, T )

∂T.

10. Show that P(τ > T | Gt) can be written using an HJM type default rateas

P(τ > T | Gt) = 1τ>t exp

(−

w T

tf2(t, u)du

),

where

f2(t, u) = λte−b(u−t) − η2

2C2(b, t, u).

11. Show how the result of Question 8 can be simplified when (B(1)t )t∈R+ and

(B(2)t )t∈R+

are independent.

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"


Chapter 14

Stochastic Calculus for Jump Processes

The modelling of risky asset by stochastic processes with continuous paths,based on Brownian motions, suffers from several defects. First, the path con-tinuity assumption does not seem reasonable in view of the possibility ofsudden price variations (jumps) resulting of market crashes. Secondly, themodeling of risky asset prices by Brownian motion relies on the use of theGaussian distribution which tends to underestimate the probabilities of ex-treme events.

A solution is to use stochastic processes with jumps, that will account forsudden variations of the asset prices. On the other hand, such jump modelsare generally based on the Poisson distribution which has a slower tail decaythan the Gaussian distribution. This allows one to assign higher probabilitiesto extreme events, resulting in a more realistic modeling of asset prices.

14.1 The Poisson Process

The most elementary and useful jump process is the standard Poisson processwhich is a stochastic process (Nt)t∈R+ with jumps of size +1 only, and whosepaths are constant in between two jumps, i.e. at time t, the value Nt of theprocess is given by1

Nt =∞∑k=1

1[Tk,∞)(t), t ∈ R+, (14.1)

where

1 The notation Nt is not to be confused with the same notation used for numeraireprocesses in Chapter 10.

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N. Privault

1[Tk,∞)(t) =

1 if t ≥ Tk,

0 if 0 ≤ t < Tk,

k ≥ 1, and (Tk)k≥1 is the increasing family of jump times of (Nt)t∈R+such

thatlimk→∞

Tk = +∞.

In addition, (Nt)t∈R+ satisfies the following conditions:

1. Independence of increments: for all 0 ≤ t0 < t1 < · · · < tn and n ≥ 1 therandom variables

Nt1 −Nt0 , . . . , Ntn −Ntn−1,

are independent.

2. Stationarity of increments: Nt+h − Ns+h has the same distribution asNt −Ns for all h > 0 and 0 ≤ s ≤ t.

The meaning of the above stationarity condition is that for all fixed k ∈ Nwe have

P(Nt+h −Ns+h = k) = P(Nt −Ns = k),

for all h > 0, i.e. the value of the probability

P(Nt+h −Ns+h = k)

does not depend on h > 0, for all fixed 0 ≤ s ≤ t and k ∈ N.

The next figure represents a sample path of a Poisson process.

0

1

2

3

4

5

6

7

0 2 4 6 8 10

Nt

t

Fig. 14.1: Sample path of a Poisson process (Nt)t∈R+ .

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Based on the above assumption, given a time value T > 0 a natural questionarises:

what is the probability distribution of the random variable NT ?

We already know that Nt takes values in N and therefore it has a discretedistribution for all t ∈ R+.

It is a remarkable fact that the distribution of the increments of (Nt)t∈R+,

can be completely determined from the above conditions, as shown in thefollowing theorem.

As seen in the next result, cf. [6], Nt − Ns has the Poisson distributionwith parameter λ(t− s).Theorem 14.1. Assume that the counting process (Nt)t∈R+

satisfies theabove Conditions 1 and 2. Then for all fixed 0 ≤ s ≤ t we have

P (Nt −Ns = k) = e−λ(t−s)(λ(t− s))k

k!, k ∈ N, (14.2)

for some constant λ > 0.

The parameter λ > 0 is called the intensity of the Poisson process (Nt)t∈R+

and it is given by

λ := limh→0

1

hP(Nh = 1). (14.3)

The proof of the above Theorem 14.1 is technical and not included here,cf. e.g. [6] for details, and we could in fact take this distribution property(14.2) as one of the hypotheses that define the Poisson process.

Precisely, we could restate the definition of the standard Poisson process(Nt)t∈R+

with intensity λ > 0 as being a process defined by (14.1), which isassumed to have independent increments distributed according to the Poissondistribution, in the sense that for all 0 ≤ t0 ≤ t1 < · · · < tn,

(Nt1 −Nt0 , . . . , Ntn −Ntn−1)

is a vector of independent Poisson random variables with respective param-eters

(λ(t1 − t0), . . . , λ(tn − tn−1)).

In particular, Nt has the Poisson distribution with parameter λt, i.e.

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http://en.wikipedia.org/wiki/Simeon_Denis_Poisson

http://en.wikipedia.org/wiki/Poisson_distribution


N. Privault

P(Nt = k) =(λt)k

k!e−λt, t > 0.

The expected value E[Nt] of Nt can be computed as

IE[Nt] = λt, (14.4)

cf. Exercise 16.1.

Short Time Behaviour

From (14.3) above we deduce the short time asymptotics 2

P(Nh = 1) = hλe−hλ ' hλ, h→ 0,

andP(Nh = 0) = e−hλ ' 1− hλ, h→ 0.

By stationarity of the Poisson process we find more generally that

P(Nt+h −Nt = 1) = hλe−hλ ' hλ, h→ 0,

andP(Nt+h −Nt = 0) = e−hλ ' 1− hλ, h→ 0,

for all t > 0.

This means that within a “short” interval [t, t + h] of length h, the in-crement Nt+h−Nt behaves like a Bernoulli random variable with parameterλh. This fact can be used for the random simulation of Poisson process paths.

We also find that

P(Nt+h −Nt = 2) ' h2λ2

2, h→ 0, t > 0,

and more generally

P(Nt+h −Nt = k) ' hk λk

k!, h→ 0, t > 0.

The intensity of the Poisson process can in fact be made time-dependent (e.g.by a time change), in which case we have

2 We use the notation f(h) ' hk to mean that limh→0 f(h)/hk = 1.

372


"



P(Nt −Ns = k) = exp

(−

w t

sλ(u)du

) (r tsλ(u)du

)kk!

, k ≥ 0.

In particular,

P(Nt+dt −Nt = k) =

e−λ(t)dt ' 1− λ(t)dt, k = 0,

λ(t)e−λ(t)dtdt ' λ(t)dt, k = 1,

o(dt), k ≥ 2,

and P(Nt+dt −Nt = 0), P(Nt+dt −Nt = 1) coincide respectively with (13.2)and (13.3) above. The intensity process (λ(t))t∈R+

can also be made randomin the case of Cox processes.

Poisson Process Jump Times

In order to prove the next proposition we note that we have the equivalence

T1 > t ⇐⇒ Nt = 0,

and more generallyTn > t ⇐⇒ Nt ≤ n− 1,

for all n ≥ 1.

In the next proposition we compute the distribution of Tn with its density.It coincides with the gamma distribution with integer parameter n ≥ 1, alsoknown as the Erlang distribution in queueing theory.

Proposition 14.1. For all n ≥ 1 the probability distribution of Tn has thedensity function

t 7−→ λne−λttn−1

(n− 1)!

on R+, i.e. for all t > 0 the probability P(Tn ≥ t) is given by

P(Tn ≥ t) = λnw∞t

e−λssn−1

(n− 1)!ds.

Proof. We have

P(T1 > t) = P(Nt = 0) = e−λt, t ∈ R+,

and by induction, assuming that

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N. Privault

P(Tn−1 > t) = λw∞t

e−λs(λs)n−2

(n− 2)!ds, n ≥ 2,

we obtain

P(Tn > t) = P(Tn > t ≥ Tn−1) + P(Tn−1 > t)

= P(Nt = n− 1) + P(Tn−1 > t)

= e−λt(λt)n−1

(n− 1)!+ λ

w∞t

e−λs(λs)n−2

(n− 2)!ds

= λw∞t

e−λs(λs)n−1

(n− 1)!ds, t ∈ R+,

where we applied an integration by parts to derive the last line.

In particular, for all n ∈ Z and t ∈ R+, we have

P(Nt = n) = pn(t) = e−λt(λt)n

n!,

i.e. pn−1 : R+ → R+, n ≥ 1, is the density function of Tn.

Similarly we could show that the time

τk := Tk+1 − Tk

spent in state k ∈ N, with T0 = 0, forms a sequence of independent identi-cally distributed random variables having the exponential distribution withparameter λ > 0, i.e.

P(τ0 > t0, . . . , τn > tn) = e−λ(t0+t1+···+tn), t0, . . . , tn ∈ R+.

Since the expectation of the exponentially distributed random variable τkwith parameter λ > 0 is given by

IE[τk] =1

λ,

we can check that the higher the intensity λ (i.e. the higher the probabilityof having a jump within a small interval), the smaller is the time spent ineach state k ∈ N on average.

In addition, given that NT = n, the n jump times on [0, T ] of the Poissonprocess (Nt)t∈R+ are independent uniformly distributed random variables on[0, T ]n.

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Compensated Poisson Martingale

From (14.4) above we deduce that

IE[Nt − λt] = 0, (14.5)

i.e. the compensated Poisson process (Nt−λt)t∈R+has centered increments.

Since in addition (Nt − λt)t∈R+also has independent increments we get

the following proposition.

Proposition 14.2. The compensated Poisson process

(Nt − λt)t∈R+

is a martingale with respect to its own filtration (Ft)t∈R+ .

Extensions of the Poisson process include Poisson processes with time-dependent intensity, and with random time-dependent intensity (Cox pro-cesses). Renewal processes are counting processes

Nt =∑n≥1

1[Tn,∞)(t), t ∈ R+,

in which τk = Tk+1 − Tk, k ∈ N, is a sequence of independent identicallydistributed random variables. In particular, Poisson processes are renewalprocesses.

14.2 Compound Poisson Processes

The Poisson process itself appears to be too limited to develop realistic pricemodels as its jumps are of constant size. Therefore there is some interest inconsidering jump processes that can have random jump sizes.

Let (Zk)k≥1 denote an i.i.d. sequence of square-integrable random vari-ables with probability distribution ν(dy) on R, independent of the Poissonprocess (Nt)t∈R+

. We have

P(Zk ∈ [a, b]) = ν([a, b]) =w b

aν(dy), −∞ < a ≤ b <∞.

Definition 14.1. The process

Yt =

Nt∑k=1

Zk, t ∈ R+, (14.6)

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N. Privault

is called a compound Poisson process.

The next figure represents a sample path of a compound Poisson process,with here Z1 = 0.9, Z2 = −0.7, Z3 = 1.4, Z4 = 0.6, Z5 = −2.5, Z6 = 1.5,Z7 = −1.2.

-0.5

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10

Yt

t

Fig. 14.2: Sample path of a compound Poisson process (Yt)t∈R+.

Given that NT = n, the n jump sizes of (Yt)t∈R+on [0, T ] are independent

random variables which are distributed on R according to ν(dx). Based onthis fact, the next proposition allows us to compute the characteristic functionof the increment YT − Yt.Proposition 14.3. For any t ∈ [0, T ] we have

IE [exp (iα(YT − Yt))] = exp(λ(T − t)

w∞−∞

(eiyα − 1)ν(dy)),

α ∈ R.

Proof. Since Nt has a Poisson distribution with parameter t > 0 and isindependent of (Zk)k≥1, for all α ∈ R we have by conditioning:

IE [exp (iα(YT − Yt))] = IE

[exp

(iα

NT∑k=Nt+1

Zk

)]

= IE

[exp

(iα

NT−Nt∑k=1

Zk

)]

=∞∑n=0

IE

[exp

(iα

n∑k=1

Zk

)]P(NT −Nt = n)

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= e−λ(T−t)∞∑n=0

λn

n!(T − t)n IE

[exp

(iα

n∑k=1

Zk

)]

= e−λ(T−t)∞∑n=0

λn

n!(T − t)n (IE [exp (iαZ1)])

n

= exp (λ(T − t) IE [exp (iαZ1)])

= exp(λ(T − t)

w∞−∞

(eiαy − 1)ν(dy)),

since ν(dy) is the probability distribution of Z1 andr∞−∞ ν(dy) = 1.

From the characteristic function we can compute the expectation and vari-ance of Yt for fixed t, as

IE[Yt] = λt IE[Z1] and Var [Yt] = λt IE[|Z1|2].

For the expectation we have

IE[Yt] = −i ddα

IE[eiαYt ]|α=0 = λtw∞−∞

yν(dy) = λt IE[Z1].

This relation can also be directly recovered as

IE[Yt] = IE

[IE

[Nt∑k=1

Zk

∣∣∣Nt]]

= e−λt∞∑n=0

λntn

n!IE

[n∑k=1

Zk

∣∣∣Nt = n

]

= e−λt∞∑n=0

λntn

n!IE

[n∑k=1

Zk

]

= λte−λt IE[Z1]

∞∑n=1

(λt)n−1

(n− 1)!

= λt IE[Z1].

More generally one can show that for all 0 ≤ t0 ≤ t1 · · · ≤ tn andα1, . . . , αn ∈ R we have

IE

[n∏k=1

eiαk(Ytk−Ytk−1)

]= exp

(λ

n∑k=1

(tk − tk−1)w∞−∞

(eiαky − 1)ν(dy)

)

=n∏k=1

exp(λ(tk − tk−1)

w∞−∞

(eiαky − 1)ν(dy))

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N. Privault

=

n∏k=1

IE[eiα(Ytk−Ytk−1

)].

This shows in particular that the compound Poisson process (Yt)t∈R+ hasindependent increments, as the standard Poisson process (Nt)t∈R+ .

Since the compensated Poisson process also has centered increments by(14.5), we have the following proposition.

Proposition 14.4. The compensated compound Poisson process

Mt := Yt − λt IE[Z1], t ∈ R+,

is a martingale.

By construction, compound Poisson processes only have a finite numberof jumps on any interval. They belong to the family of Levy processes whichmay have an infinite number of jumps on any finite time interval, cf. [12].

14.3 Stochastic Integrals with Jumps

Given (φt)t∈R+ a stochastic process we let the stochastic integral of (φt)t∈R+

with respect to (Yt)t∈R+be defined by

w T

0φtdYt :=

NT∑k=1

φTkZk.

Note that this expressionw T

0φtdYt has a natural financial interpretation as

the value at time T of a portfolio containing a (possibly fractional) quantityφt of a risky asset at time t, whose price evolves according to random returnsZk at random times Tk.

In particular the compound Poisson process (Yt)t∈R+in (14.1) admits the

stochastic integral representation

Yt = Y0 +w t

0ZNsdNs.

Next, given (Wt)t∈R+ a standard Brownian motion independent of (Yt)t∈R+

and (Xt)t∈R+ a jump-diffusion process of the form

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Xt =w t

0usdWs +

w t

0vsds+ Yt, t ∈ R+,

where (φt)t∈R+is a process which is adapted to the filtration (Ft)t∈R+

gen-erated by (Wt)t∈R+ and (Yt)t∈R+ , and such that

IE[w∞

0φ2s|us|2ds

]<∞ and IE

[w∞0|φsvs|ds

]<∞,

we let the stochastic integral of (φs)s∈R+with respect to (Xs)s∈R+

be definedby

w T

0φsdXs :=

w T

0φsusdWs +

w T

0φsvsds+

NT∑k=1

φTkZk, T > 0.

The coumpound Poisson compensated stochastic integral can be shown tosatisfy the Ito isometry

IE

[(w T

0φs−(dYs − λ IE[Z1]dt)

)2]

= λ IE[|Z1|2] IE

[w T

0|φ|2sds

],

(14.7)

provided the process (φs)s∈R+is adapted to the filtration generated by

(Yt)t∈R+, which makes the left limit process (φs−)s∈R+

predictable. The proofof (14.7) is similar to that of Proposition 4.2 in the case of simple predictableprocesses.

For the mixed continuous-jump martingale

Xt =w t

0usdWs + Yt − λt IE[Z1], t ∈ R+,

we have the isometry

IE

[(w T

0φs−dXs

)2]

= IE

[w T

0|φs|2|us|2ds

]+ λ IE[|Z1|2] IE

[w T

0|φs|2ds

].

(14.8)

provided (φs)s∈R+is adapted to the filtration (Ft)t∈R+

generated by (Wt)t∈R+

and (Yt)t∈R+.

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N. Privault

This isometry formula will be used in Section 15.5 for the computation ofhedging strategies in jump models.

When (Xt)t∈R+takes the form

Xt = X0 +w t

0usdWs +

w t

0vsds+

w t

0ηsdYs, t ∈ R+,

the stochastic integral of (φt)t∈R+with respect to (Xt)t∈R+

satisfies

w T

0φsdXs :=

w T

0φsusdWs +

w T

0φsvsds+

w T

0ηsφsdYs

=w T

0φsusdWs +

w T

0φsvsds+

NT∑k=1

φTkηTkZk, T > 0.

14.4 Ito Formula with Jumps

Let us first consider the case of a standard Poisson process (Nt)t∈R+ withintensity λ. We have the telescoping sum

f(Nt) = f(0) +

Nt∑k=1

(f(k)− f(k − 1))

= f(0) +w t

0(f(1 +Ns−)− f(Ns−))dNs

= f(0) +w t

0(f(Ns)− f(Ns−))dNs.

Here, Ns− denotes the left limit of the Poisson process at time s, i.e.

Ns− = limh0

Ns−h.


k = NTk = 1 +NT−k, k ≥ 1.

By the same argument we find, in the case of the compound Poisson process(Yt)t∈R+ ,

f(Yt) = f(0) +

Nt∑k=1

(f(YT−k+ Zk)− f(YT−k

))

= f(0) +w t

0(f(ZNs + Ys−)− f(Ys−))dNs

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"



= f(0) +w t

0(f(Ys)− f(Ys−))dNs,

which can be decomposed using a compensated Poisson stochastic integralas

f(Yt) = f(0) +w t

0(f(Ys)− f(Ys−))(dNs − λds) + λ

w t

0(f(Ys)− f(Ys−))ds.

More generally, for a process of the form

Xt = X0 +w t

0usdWs +

w t

0vsds+

w t

0ηsdYs, t ∈ R+,

we find, by combining the Ito formula for Brownian motion with the aboveargument we get

f(Xt) = f(X0) +w t

0usf

′(Xs)dWs +1

2

w t

0f ′′(Xs)|us|2ds

+w t

0vsf′(Xs)ds+

NT∑k=1

(f(XT−k+ ηTkZk)− f(XT−k

))

= f(X0) +w t

0usf

′(Xs)dWs +1

2

w t

0f ′′(Xs)|us|2ds+

w t

0vsf′(Xs)ds

+w t

0(f(Xs− + ηsZNs)− f(Xs−))dNs t ∈ R+.

i.e.

f(Xt) = f(X0) +w t

0usf

′(Xs)dWs +1

2

w t


w t

0vsf′(Xs)ds

+w t

0(f(Xs)− f(Xs−))dNs, t ∈ R+. (14.9)

For example, in case

Xt =w t

0usdWs +

w t

0vsds+

w t

0ηsdNs, t ∈ R+,

we get

f(Xt) = f(0) +w t

0usf

′(Xs)dWs +1

2

w t

0|us|2f ′′(Xs)dWs

+w t

0vsf′(Xs)ds+

w t

0(f(Xs− + ηs)− f(Xs−))dNs

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N. Privault

= f(0) +w t

0usf

′(Xs)dWs +1

2

w t

0|us|2f ′′(Xs)dWs (14.10)

+w t

0vsf′(Xs)ds+

w t

0(f(Xs)− f(Xs−))dNs.

Given two processes (Xt)t∈R+and (Yt)t∈R+

written as

Xt =w t

0usdWs +

w t

0vsds+

w t

0ηsdNs, t ∈ R+,

and

Yt =w t

0asdWs +

w t

0bsds+

w t

0csdNs, t ∈ R+,

the Ito formula for jump processes also shows that

d(XtYt) = XtdYt + YtdXt + dXt · dYt

where the product dXt · dYt is computed according to the extension

· dt dBt dNtdt 0 0 0dBt 0 dt 0

dNt 0 0 dNt

of the Ito multiplication table (4.19), i.e. we have

dXt · dYt = (vtdt+ utdBt + ηtdNt)(btdt+ atdBt + ctdNt)

= btvt(dt)2 + btutdt · dBt + btηtdt · dNt

+atvtdtdBt + atut(dBt)2 + atηtdBt · dNt

+ctvtdNt · dBt + ctut(dBt)2 + ctηtdNt · dNt

= atutdt+ ctηtdNt,

and in particular

(dXt)2 = (vtdt+ utdBt + ηtdNt)

2 = u2tdt+ η2t dNt.

For a process of the form

Xt = X0 +w t

0usdWs +

w t

0ηsdYt, t ∈ R+,

the Ito formula with jumps (14.10) can be rewritten as

f(Xt) = f(X0) +w t

0vsf′(Xs−)ds+

w t

0usf

′(Xs−)dWs

+1

2

w t


w t

0ηsf′(Xs−)dYs

+w t

0(f(Xs)− f(Xs−)−∆Xsf

′(Xs−)) d(Ns − s)

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"



+w t

0(f(Xs)− f(Xs−)−∆Xsf

′(Xs−)) ds, t ∈ R+,

where we used the relation dYs = ∆Xsf′(Xs−)dNs, which implies

w t

0ηsf′(Xs−)dYs =

w t

0∆Xsf

′(Xs−)dNs, t ≥ 0.

This above formulation is at the basis of the extension of Ito’s formula toLevy processes with an infinite number of jumps on any interval, using thebound

|f(x+ y)− f(x)− yf ′(x)| ≤ Cy2,for f a C2b (R) function. Such processes, also called “infinite activity Levy

processes” [12] are also useful in financial modeling and include the gammaprocess, stable processes, variance gamma processes, inverse Gaussian pro-cesses, etc, as in the following illustrations.

1. Gamma process, d = 1.

0

t

Fig. 14.3: Sample trajectories of a gamma process.

2. Stable process, d = 1.

0

t

Fig. 14.4: Sample trajectories of a stable process.

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N. Privault

3. Variance Gamma process, d = 1.

0

t

Fig. 14.5: Sample trajectories of a variance gamma process.

4. Inverse Gaussian process, d = 1.

0

t

Fig. 14.6: Sample trajectories of an inverse Gaussian process.

5. Negative Inverse Gaussian process, d = 1.

0

t

Fig. 14.7: Sample trajectories of a negative inverse Gaussian process.

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14.5 Stochastic Differential Equations with Jumps

Let us start with the simplest example

dSt = ηSt−dNt, (14.11)

of a stochastic differential equation with respect to the standard Poisson pro-cess, with constant coefficient η ∈ R.

When∆Nt = Nt −Nt− = 1,

i.e. when the Poisson process has a jump at time t, the equation (14.11) reads

dSt = St − St− = ηSt− , t > 0.

which can be solved to yield

St = (1 + η)St− , t > 0.

By induction, applying this procedure for each jump time gives us the solution

St = S0(1 + η)Nt , t ∈ R+.

Next, consider the case where η is time-dependent, i.e.

dSt = ηtSt−dNt. (14.12)

At each jump time Tk, Relation (14.12) reads

dSTk = STk − ST−k = ηTkST−k,

i.e.STk = (1 + ηTk)ST−k

,

and repeating this argument for all k = 1, . . . , Nt yields the product solution

St = S0

Nt∏k=1

(1 + ηTk) = S0

∏∆Ns=10≤s≤t

(1 + ηs), t ∈ R+.

The equationdSt = µtStdt+ ηtSt−(dNt − λdt), (14.13)

is then solved as

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N. Privault

St = S0 exp

(w t

0µsds− λ

w t

0ηsds

) Nt∏k=1

(1 + ηTk), t ∈ R+.

A random simulation of the numerical solution of the above equation (14.13)is given in Figure 14.8.

Fig. 14.8: Geometric Poisson process.

The above simulation can be compared to the real sales ranking data ofFigure 14.9.

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"



Fig. 14.9: Ranking data.

A random simulation of the geometric compound Poisson process

St = S0 exp

(w t

0µsds− λ IE[Z1]

w t

0ηsds

) Nt∏k=1

(1 + ηTkZk) t ∈ R+,

solution ofdSt = µtStdt+ ηtSt−(dYt − λ IE[Z1]dt),

is given in Figure 14.10.

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N. Privault

Fig. 14.10: Geometric compound Poisson process.

In the case of a jump-diffusion stochastic differential equation of the form

dSt = µtStdt+ ηtSt−(dYt − λ IE[Z1]dt) + σtStdWt,

we get

St = S0 exp

(w t

0µsds− λ IE[Z1]

w t

0ηsds+

w t

0σsdWs −

1

2

w t

0|σs|2ds

)×

Nt∏k=1

(1 + ηTkZk),

t ∈ R+. A random simulation of the geometric Brownian motion with com-pound Poisson jumps is given in Figure 14.11.

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"



Fig. 14.11: Geometric Brownian motion with compound Poisson jumps.

By rewriting St as

St = S0 exp

(w t

0µsds+

w t

0ηs(dYs − λ IE[Z1]ds) +

w t

0σsdWs −

1

2

w t

0|σs|2ds

)×

Nt∏k=1

(e−ηTk (1 + ηTkZk)),

t ∈ R+, one can extend this jump model to processes with an infinite numberof jumps on any finite time interval, cf. [12].

14.6 Girsanov Theorem for Jump Processes

Recall that in its simplest form, the Girsanov theorem for Brownian motionfollows from the calculation

IE[f(WT − µT )] =1√2πT

w∞−∞

f(x− µT )e−x2/(2T )dx

=1√2πT

w∞−∞

f(x)e−(x+µT )2/(2T )dx

=1√2πT

w∞−∞

f(x)e−µx−µ2T/2e−x

2/(2T )dx

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N. Privault

= IE[f(WT )e−µWT−µ2T/2]

= IE[f(WT )], (14.14)

for any bounded measurable function f on R, which shows that WT is aGaussian random variable with mean −µT under the probability measure Pdefined by

dP = e−µWT−µ2T/2dP,

cf. Section 6.2. Equivalently we have

IE[f(WT )] = IE[f(WT + µT )], (14.15)

hence

under the probability measure

dP = e−µWT−µ2T/2dP,

the random variable WT +µT has a centered Gaussian distribution.

More generally, the Girsanov theorem states that (Wt + µt)t∈[0,T ] is a stan-

dard Brownian motion under P.

When Brownian motion is replaced with a standard Poisson process(Nt)t∈R+

, the above space shift

Wt 7−→Wt + µt

may not be used because Nt + µt cannot be a Poisson process, whatever thechange of probability applied, since by construction, the paths of the stan-dard Poisson process has jumps of unit size and remain constant betweenjump times.

The correct way to proceed in order to extend (14.15) to the Poisson caseis to replace the space shift with a time contraction (or dilation) by a certainfactor 1 + c with c > −1, i.e.

Nt 7−→ Nt/(1+c).

By analogy with (14.14) we write

IE[f(NT (1+c))] =

∞∑k=0

f(k)P(NT (1+c) = k) (14.16)

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"



= e−λ(1+c)T∞∑k=0

f(k)(λT (1 + c))k

k!

= e−λT e−λcT∞∑k=0

f(k)(1 + c)k(λT )k

k!

= e−λcT∞∑k=0

f(k)(1 + c)kP(NT = k)

= e−λcT IE[f(NT )(1 + c)NT ]

= e−λcTwΩ

(1 + c)NT f(NT )dP

=wΩf(NT )dP

= IE[f(NT )],

for f any bounded function on N, where the probability measure P is definedby

dPλ = e−λcT (1 + c)NT dP.

Consequently,


dP = e−λcT (1 + c)NT dP,

the law of the random variable NT is that of NT (1+c) under P, i.e. it is aPoisson random variable with intensity λ(1 + c)T .

Equivalently we have

IE[f(NT )] = IE[f(NT/(1+c))],

i.e. under P the law of NT/(1+c) is that of a standard Poisson random variablewith parameter λT .

In addition we have

Nt/(1+c) =∑n≥1

1[Tn,∞)(t/(1 + c))

=∑n≥1

1[(1+c)Tn,∞)(t), t ∈ R+,

which shows that under P, the jump times of (Nt/(1+c))t∈[0,T ] are given by

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N. Privault

((1 + c)Tn)n≥1,

and we know that they are distributed as the jump times of a Poisson processwith intensity λ.

Next taking λ > 0 and

c := −1 +λ

λ,

we can rewrite the above by saying that


dPλ = e−λcT (1 + c)NT dP = e−(λ−λ)T

(λ

λ

)NTdP,

the law of NT is that of a Poisson random variable with intensity

λT = λ(1 + c)T.

Consequently, since both (Nt−λt)t∈R+and (Nt−(1+c)λt)t∈R+

are processeswith independent increments, the compensated Poisson process

Nt − (1 + c)λt = Nt − λt

is a martingale under Pλ by (6.2), although when c 6= 0 it is not a martingaleunder P.

In the case of compound Poisson processes the Girsanov theorem can beextended to variations in jump sizes in addition to time variations, and wehave the following more general result.

Theorem 14.2. Let (Yt)t≥0 be a compound Poisson process with inten-sity λ > 0 and jump distribution ν(dx). Consider another jump distributionν(dx), and let

φ(x) =λ

λ

dν

dν(x)− 1, x ∈ R.

Then,

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"




dPλ,ν := e−(λ−λ)TNT∏k=1

(1 + φ(Zk))dPλ,ν ,

the process

Yt =

Nt∑k=1

Zk, t ∈ R+,

is a compound Poisson process with

- modified intensity λ > 0, and

- modified jump distribution ν(dx).

Proof. For any bounded measurable function f on R, we extend (14.16) tothe following change of variable

IEλ,ν [f(YT )] = e−(λ−λ)T IEλ,ν

[f(YT )

NT∏i=1

(1 + φ(Zi))

]

= e−(λ−λ)T∞∑k=0

IEλ,ν

[f

(k∑i=1

Zi

)k∏i=1

(1 + φ(Zi))∣∣∣NT = k

]P(NT = k)

= e−λT∞∑k=0

(λT )k

k!IEλ,ν

[f

(k∑i=1

Zi

)k∏i=1

(1 + φ(Zi))

]

= e−λT∞∑k=0

(λT )k

k!

w∞−∞· · ·

w∞−∞

f(z1 + · · ·+ zk)

k∏i=1

(1 + φ(zi))ν(dz1) · · · ν(dzk)

= e−λT∞∑k=0

(λT )k

k!

w∞−∞· · ·

w∞−∞

f(z1 + · · ·+ zk)

(k∏i=1

dν

dν(zi)

)ν(dz1) · · · ν(dzk)

= e−λT∞∑k=0

(λT )k

k!

w∞−∞· · ·

w∞−∞

f(z1 + · · ·+ zk)ν(dz1) · · · ν(dzk).

This shows that under Pλ,ν , YT has the distribution of a compound Poisson

process with intensity λ and jump distribution ν. We refer to Proposition 9.6of [12] for the independence of increments of (Yt)t∈R+

under Pλ,ν .

Note that the compound Poisson process with intensity λ > 0 and jumpdistribution ν can be built as

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N. Privault

Xt :=

Nλt/λ∑k=1

h(Zk),

provided ν is the image measure of ν by the function h : R→ R, i.e.

P(h(Zk) ∈ A) = P(Zk ∈ h−1(A)) = ν(h−1(A)) = ν(A),

for all measurable subset A of R.

Compensated Compound Poisson Martingale

As a consequence of Theorem 14.2, the compensated process

Yt − λt IEν [Z1]

becomes a martingale under the probability measure Pλ,ν defined by

dPλ,ν = e−(λ−λ)TNT∏k=1

(1 + φ(Zk))dPλ,ν .

Finally, the Girsanov theorem can be extended to the linear combinationof a standard Brownian motion (Wt)t∈R+ and an independent compoundPoisson process (Yt)t∈R+ , as in the following result which is a particular caseof Theorem 33.2 of [68].

Theorem 14.3. Let (Yt)t≥0 be a compound Poisson process with intensityλ > 0 and jump distribution ν(dx). Consider another jump distribution ν(dx)and intensity parameter λ > 0, and let

φ(x) =λ

λ

dν

dν(x)− 1, x ∈ R,

and let (ut)t∈R+ be a bounded adapted process. Then the process(Wt +

w t

0usds+ Yt − λ IEν [Z1]t

)t∈R+

is a martingale under the probability measure

dPu,λ,ν = exp

(−(λ− λ)T −

w T

0usdWs −

1

2

w T

0|us|2ds

) NT∏k=1

(1+φ(Zk))dPλ,ν .

(14.17)

As a consequence of Theorem 14.3, if

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Wt +w t

0vsds+ Yt

is not a martingale under Pλ,ν , it will become a martingale under Pu,λ,νprovided u, λ and ν are chosen in such a way that

vs = us − λ IEν [Z1], s ∈ R, (14.18)

in which case we will have the martingale decomposition

dWt + utdt+ dYt − λ IEν [Z1]dt,

in which both

(Wt +

w t

0usds

)t∈R+

and(Yt − λt IEν [Z1]

)t∈R+

are both mar-

tingales under Pu,λ,ν

When λ = λ = 0, Theorem 14.3 coincides with the usual Girsanov theoremfor Brownian motion, in which case (14.18) admits only one solution givenby u = v and there is uniqueness of Pu,0,0. Note that uniqueness occurs alsowhen u = 0 in the absence of Brownian motion with Poisson jumps of fixedsize a (i.e. ν(dx) = ν(dx) = δa(dx)) since in this case (14.18) also admitsonly one solution λ = v and there is uniqueness of P0,λ,δa

. These remarks willbe of importance for arbitrage pricing in jump models in Chapter 15.

Exercises

Exercise 14.1 Let (Nt)t∈R+be a standard Poisson process with intensity

λ > 0, started at N0 = 0.

1. Solve the stochastic differential equation

dSt = ηSt−dNt − ηλStdt = ηSt−(dNt − λdt).

2. Using the first Poisson jump time T1, solve the stochastic differentialequation

dSt = −ηStdt+ dNt

for t ∈ (0, T2).

Exercise 14.2 Consider the compound Poisson process Yt :=

Nt∑k=1

Zk, where

(Nt)t∈R+ is a standard Poisson process with intensity λ > 0, (Zk)k≥1 is an

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N. Privault

i.i.d. sequence of N (0, 1) Gaussian random variables. Solve the stochasticdifferential equation

dSt = rStdt+ ηSt−dYt,

where η, r ∈ R.

Exercise 14.3 Show, by direct computation or using the characteristic func-tion, that the variance of the compound Poisson process Yt with intensityλ > 0 satisfies

Var [Yt] = λt IE[|Z1|2] = λtw∞−∞

x2ν(dx).

Exercise 14.4 Consider an exponential compound Poisson process of the form

St = S0eµt+σWt+Yt , t ∈ R+,

where (Yt)t∈R+ is a compound Poisson process of the form (14.6).

1. Derive the stochastic differential equation with jumps satisfied by (St)t∈R+.

2. Let r > 0. Find a family (Pu,λ,ν) of probability measures under which

the discounted asset price e−rtSt is a martingale.

Exercise 14.5 Consider (Nt)t∈R+a standard Poisson process with intensity

λ > 0, independent of (Wt)t∈R+, under a probability measure P. Let (St)t∈R+

be defined by the stochastic differential equation

dSt = µStdt+ YNtSt−dNt, (14.19)

where (Yk)k≥1 is an i.i.d. sequence of random variables of the form Yk =eXk − 1 where Xk ' N (0, σ2), k ≥ 1.

1. Solve the equation (14.19).2. We assume that µ and the risk-free rate r > 0 are chosen such that the

discounted process (e−rtSt)t∈R+is a martingale under P. What relation

does this impose on µ and r ?3. Under the relation of Question (2), compute the price at time t of a

European call option on ST with strike κ and maturity T , using a seriesexpansion of Black-Scholes functions.

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Chapter 15

Pricing and Hedging in Jump Models

In this chapter we consider the problem of option pricing and hedging injump-diffusion models. In comparison with the continuous case the situationis further complicated by the existence of multiple risk-neutral measures. Asa consequence, perfect replicating hedging strategies cannot be computed ingeneral.

15.1 Risk-Neutral Measures

Consider an asset price modeled by the equation,

dSt = µStdt+ σStdWt + St−dYt, (15.1)

where (Yt)t∈R+ is the compound Poisson process defined in Section 14.2, withjump size distribution ν(dx) under Pν . The equation (15.1) has for solution

St = S0 exp

(µt+ σWt −

1

2σ2t

) Nt∏k=1

(1 + Zk), (15.2)

t ∈ R+. An important issue for non-abitrage pricing is to determine arisk-neutral probability measure P∗ under which the discounted process(e−rtSt)t∈R+

is a martingale, and this goal can be achieved using the Gir-sanov theorem for jump processes, cf. Section 14.6.

We have

d(e−rtSt) = −re−rtStdt+ e−rtdSt

= (µ− r)e−rtStdt+ σe−rtStdWt + e−rtSt−dYt

= (µ− r + λ IEν [Z1])e−rtStdt+ σe−rtStdWt + e−rtSt−(dYt − λ IEν [Z1]dt),

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N. Privault

which yields a martingale under P provided

µ− r + λ IEν [Z1] = 0,

however that condition may not be satisfied under Pν by the market param-eters.

In that case a change of measure might be needed. In order for the dis-counted process (e−rtSt)t∈R+

to be a martingale, we may choose a drift pa-

rameter u ∈ R, and intensity λ > 0, and a jump distribution ν satisfying

µ− r = σu− λ IEν [Z1]. (15.3)

The Girsanov theorem for jump processes then shows that

dWt + udt+ dYt − λ IEν [Z1]dt

is a martingale under the probability measure Pu,λ,ν defined in (14.17). Con-sequently the discounted asset price

d(e−rtSt) = (µ− r)e−rtStdt+ σe−rtStdWt + e−rtSt−dYt

= σe−rtSt(dWt + udt) + e−rtSt−(dYt − λ IEν [Z1]dt),

is a martingale under Pu,λ,ν .

In this setting the non-uniqueness of the risk neutral measure is apparentsince additional degrees of freedom are involved in the choices of u, λ andthe measure ν, whereas in the continuous case the choice of u = (µ− r)/σ in(6.4) was unique.

15.2 Pricing in Jump Models

Recall that a market is without arbitrage if and only it admits at least onerisk-neutral measure.

Consider the probability measure Pu,λ,ν built in the previous section, underwhich the discounted asset price

d(e−rtSt) = e−rtSt−(dYt − λ IEν [Z1]dt) + σe−rtStdWt,

is a martingale, and Wt = Wt + udt is a standard Brownian motion underPu,λ,ν .

Then the arbitrage price of a claim with payoff C is given by

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e−r(T−t) IEu,λ,ν [C | Ft] (15.4)

under Pu,λ,ν .

Clearly the price (15.4) of C is no longer unique in the presence of jumpsdue to the infinity of choices satisfying the martingale condition (15.3), andsuch a market is not complete, except if either λ = λ = 0, or (σ = 0 andν = ν = δ1).

Pricing of Vanilla Options

The price of a vanilla option with payoff of the form φ(ST ) on the underlyingasset ST can be written from (15.4) as

e−r(T−t) IEu,λ,ν [φ(ST ) | Ft], (15.5)

where the expectation can be computed as

IEu,λ,ν [φ(ST ) | Ft]

= IEu,λ,ν

[φ

(S0 exp

(µT + σWT −

1

2σ2T

) NT∏k=1

(1 + Zk)

) ∣∣∣Ft]

= IEu,λ,ν

φSt exp

(µ(T − t) + σ(WT −Wt)−

1

2σ2(T − t)

) NT∏k=Nt

(1 + Zk)

∣∣∣Ft

= IEu,λ,ν

φx exp

(µ(T − t) + σ(WT −Wt)−

1

2σ2(T − t)

) NT∏k=Nt

(1 + Zk)

x=St

=

∞∑n=0

Pu,λ,ν(NT −Nt = n)

IEu,λ,ν

φxeµ(T−t)+σ(WT−Wt)− 1

2σ2(T−t)

NT∏k=Nt

(1 + Zk)

∣∣∣NT −Nt = n

x=St

= e−λ(T−t)∞∑n=0

(λ(T − t))nn!

× IEu,λ,ν

[φ

(xeµ(T−t)+σ(WT−Wt)− 1

2σ2(T−t)

n∏k=1

(1 + Zk)

)]x=St

= e−λ(T−t)∞∑n=0

(λ(T − t))nn!

w∞−∞· · ·

w∞−∞

IEu,λ,ν

[φ

(xeµ(T−t)+σ(WT−Wt)− 1

2σ2(T−t)

n∏k=1

(1 + zk)

)]x=St

ν(dz1) · · · ν(dzn),

hence

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N. Privault

e−r(T−t) IE0,λ,ν [φ(ST ) | Ft]

=1√

2π(T − t)e−(r+λ)(T−t)

∞∑n=0

(λ(T − t))nn!

w∞−∞· · ·

w∞−∞

φ

(Ste

µ(T−t)+σx−σ2(T−t)/2n∏k=1

(1 + zk)

)e−

x2

2(T−t) ν(dz1) · · · ν(dzn)dx.

15.3 Black-Scholes PDE with Jumps

Recall that by the Markov property of (St)t∈R+the price (15.5) at time t of

the option with payoff φ(ST ) can be written as a function f(t, St) of t andSt, i.e.

f(t, St) = e−r(T−t) IEu,λ,ν [φ(ST ) | Ft], (15.6)

with the terminal condition f(T, x) = φ(x). In addition,

t 7−→ er(T−t)f(t, St)

is a martingale under Pu,λ,ν by the same argument as in (6.1).

In this section we derive a partial integro-differential equation (PIDE) forthe function (t, x) 7−→ f(t, x). We have

dSt = rStdt+ σStdWt + St−(dYt − λ IEν [Z1]dt), (15.7)

where Wt = Wt + ut is a standard Brownian motion under Pu,λ,ν .

Hence the Ito formula with jumps (14.9) shows that

df(t, St) =∂f

∂t(t, St)dt+ rSt

∂f

∂x(t, St)dt+ σSt

∂f

∂x(t, St)dWt +

1

2σ2S2

t

∂2f

∂x2(t, St)dt

−λ IEν [Z1]St∂f

∂x(t, St)dt+ (f(t, St−(1 + ZNt))− f(t, St−))dNt

= σSt∂f

∂x(t, St)dWt

+(f(t, St−(1 + ZNt))− f(t, St−))dNt − λ IEν [(f(t, x(1 + Z1))− f(t, x))]x=Stdt

+∂f

∂t(t, St)dt+ rSt

∂f

∂x(t, St)dt+

1

2σ2S2

t

∂2f

∂x2(t, St)dt

+λ IEν [(f(t, x(1 + Z1))− f(t, x))]x=Stdt− λ IEν [Z1]St∂f

∂x(t, St)dt.

Based on the relation

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d(er(T−t)f(t, St)) = −rer(T−t)f(t, St)dt+ er(T−t)df(t, St)

and the facts that the Brownian motion (Wt)t∈R+ , the differential

(f(t, St−(1 + ZNt))− f(t, St−))dNt − λ IEν [(f(t, x(1 + Z1))− f(t, x))]x=Stdt

and the process t 7→ er(T−t)f(t, St) all represent martingales under Pu,λ,ν ,we conclude to the vanishing

−rf(t, St) +∂f

∂t(t, St) + rSt

∂f

∂x(t, St) +

1

2σ2S2

t

∂2f

∂x2(t, St)

+λ IEν [(f(t, x(1 + Z1))− f(t, x))]x=St − λ IEν [Z1]St∂f

∂x(t, St) = 0,

or

−rf(t, x) +∂f

∂t(t, x) + rx

∂f

∂x(t, x) +

1

2σ2x2

∂2f

∂x2(t, x)

+λ IEν [(f(t, x(1 + Z1))− f(t, x))]− λ IEν [Z1]x∂f

∂x(t, x) = 0,

which leads to the Partial Integro-Differential Equation (PIDE)

rf(t, x) =∂f

∂t(t, x) + rx

∂f

∂x(t, x) +

1

2σ2x2

∂2f

∂x2(t, x)

+λw∞−∞

(f(t, x(1 + y))− f(t, x)− yx∂f

∂x(t, x)

)ν(dy),

(15.8)

under the terminal condition f(T, x) = φ(x).

In addition we found that the change df(t, St) in the portfolio price (15.6)is given by

df(t, St) = σSt∂f

∂x(t, St)dWt + rf(t, St)dt (15.9)

+(f(t, St−(1 + ZNt))− f(t, St−))dNt − λ IEν [(f(t, x(1 + Z1))− f(t, x))]x=Stdt.

In the case of Poisson jumps with fixed size a, i.e. Yt = aNt, ν(dx) = δa(dx),the PIDE (15.8) reads

rf(t, x) =∂f

∂t(t, x) + rx

∂f

∂x(t, x) +

1

2σ2x2

∂2f

∂x2(t, x)

+λ

(f(t, x(1 + a))− f(t, x)− ax∂f

∂x(t, x)

),

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N. Privault

and we have

df(t, St) = σSt∂f

∂x(t, St)dWt + rf(t, St)dt

+(f(t, St−(1 + a))− f(t, St−))dNt − λ(f(t, St(1 + a))− f(t, St))dt.

15.4 Exponential Models

Instead of modeling the asset price (St)t∈R+ through a stochastic exponential(15.2) solution of the stochastic differential equation with jumps of the form(15.1), we may consider an exponential price process of the form

St = S0eµt+σWt+Yt = S0 exp

(µt+ σWt +

Nt∑k=1

Zk

), t ∈ R+,

and choose a risk-neutral measure Pu,λ,ν under which (e−rtSt)t∈R+is a mar-

tingale. Then the expectation

e−r(T−t) IEu,λ,ν [φ(ST ) | Ft]

also becomes a (non-unique) arbitrage price at time t ∈ [0, T ] for the contin-gent claim with payoff φ(ST ).

Such an arbitrage price can be expressed as

e−r(T−t) IEu,λ,ν [φ(ST ) | Ft] = e−r(T−t) IEu,λ,ν [φ(S0eµT+σWT+YT ) | Ft]

= e−r(T−t) IEu,λ,ν [φ(Steµ(T−t)+σ(WT−Wt)+YT−Yt) | Ft]

= e−r(T−t) IEu,λ,ν [φ(xeµ(T−t)+σ(WT−Wt)+YT−Yt)]x=St

= e−r(T−t) IEu,λ,ν

[φ

(x exp

(µ(T − t) + σ(WT −Wt) +

NT∑k=Nt+1

Zk

))]x=St

= e−r(T−t)−λ(T−t)

×∞∑n=0

(λ(T − t))nn!

IEu,λ,ν

[φ

(xeµ(T−t)+σ(WT−Wt) exp

(n∑k=1

Zk

))]x=St

.

In the exponential model

St = S0eµt+σWt+Yt = S0e

(µ+σ2/2)t+σWt−σ2t/2+Yt

the process St satisfies

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dSt =

(µ+

1

2σ2

)Stdt+ σStdWt + St−(e∆Yt − 1)dNt,

hence St has jumps of size ST−k(eZk − 1), k ≥ 1, and (15.3) reads

µ+1

2σ2 − r = σu− λ IEν [eZ1 − 1].

The Merton Model

We assume that (Zk)k≥1 is a family of independent identically distributedGaussian N (δ, η2) random variables under Pu,λ,ν with

µ+1

2σ2 − r = σu− λ IEν [eZ1 − 1] = σu− λ(eδ+η

2/2 − 1),

from (15.3), hence is a standard Brownian motion under Pu,λ,ν . For simplicitywe choose u = 0, i.e.

µ = r − 1

2σ2 − λ(eδ+η

2/2 − 1),

Hence we have

e−r(T−t) IEλ,ν [φ(ST ) | Ft]

= e−r(T−t)−λ(T−t)∞∑n=0

(λ(T − t))nn!

× IEλ,ν

[φ

(xeµ(T−t)+σ(WT−Wt) exp

(n∑k=1

Zk

))]x=St

= e−r(T−t)−λ(T−t)∞∑n=0

(λ(T − t))nn!

IE[φ(xeµ(T−t)+nδ+X)

]x=St

,

where

X = σ(WT −Wt) +

n∑k=1

(Zk − δ) ' N (0, σ2(T − t) + nη2)

is a centered Gaussian random variable with variance

v2 = σ2(T − t) +n∑k=1

VarZk = σ2(T − t) + nη2.

Hence when φ(x) = (x− κ)+, using the relation

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N. Privault

BS(x, κ, v2/τ, r, τ) = e−rτ IE[(xeX−v2/2+rτ −K)+]

we get

e−r(T−t)−λ(T−t) IEλ,ν [(ST − κ)+ | Ft]

= e−r(T−t)−λ(T−t)∞∑n=0

(λ(T − t))nn!

IE[(xeµ(T−t)+nδ+X − κ)+

]x=St

= e−r(T−t)−λ(T−t)∞∑n=0

(λ(T − t))nn!

× IE[(xe(r−

12σ

2−λ(eδ+η2/2−1))(T−t)+nδ+X − κ)+

]x=St

= e−r(T−t)−λ(T−t)∞∑n=0

(λ(T − t))nn!

× IE[(xenδ+nη

2/2−λ(eδ+η2/2−1)(T−t)+X−v2/2+r(T−t) − κ)+

]x=St

= e−λ(T−t)∞∑n=0

(λ(T − t))nn!

×BS(Stenδ+ 1

2nη2−λ(eδ+η

2/2−1)(T−t), κ, σ2 + nη2/(T − t), r, T − t).

We may also write

e−r(T−t)−λ(T−t) IEλ,ν [(ST − κ)+ | Ft]

= e−λ(T−t)∞∑n=0

(λ(T − t))nn!

enδ+12nη

2−λ(eδ+η2/2−1)(T−t)

×BS(St, κe

−nδ− 12nη

2+λ(eδ+η2/2−1)(T−t), σ2 + nη2/(T − t), r, T − t

)= e−λe

δ+η2/2(T−t)∞∑n=0

(λeδ+12nη

2

(T − t))nn!

×BS

(St, κ, σ

2 + nη2/(T − t), r + nδ + η2/2

T − t − λ(eδ+η2/2 − 1), T − t

).

15.5 Self-Financing Hedging with Jumps

Consider a portfolio with value

Vt = ηtert + ξtSt

at time t, and satisfying the self-financing condition

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dVt = rηtertdt+ ξtdSt,

cf. Relation (5.1). When the portfolio hedges the claim φ(ST ) we must haveVt = f(t, St) for all times t ∈ [0, T ] hence, by (15.7) we have

dVt = df(t, St)

= rηtertdt+ ξtdSt

= rηtertdt+ ξt(rStdt+ σStdWt + St−(dYt − λ IEν [Z1]dt))

= rVtdt+ σξtStdWt + ξtSt−(dYt − λ IEν [Z1]dt)

= rf(t, St)dt+ σξtStdWt + ξtSt−(dYt − λ IEν [Z1]dt), (15.10)

has to match

df(t, St) = rf(t, St)dt+ σSt∂f

∂x(t, St)dWt (15.11)

+(f(t, St−(1 + ZNt))− f(t, St−))dNt − λ IEν [(f(t, x(1 + Z1))− f(t, x))]x=Stdt,

which is obtained from (15.9).

In such a situation we say that the claim C can be exactly replicated.

Exact replication is possible in essentially only two situations:

(i) Continuous market, λ = λ = 0. In this case we find the usual Black-Scholes Delta:

ξt =∂f

∂x(t, St). (15.12)

(ii) Poisson jump market, σ = 0 and Yt = aNt, ν(dx) = δa(dx). In thiscase we find

ξt =1

aSt−(f(t, St−(1 + a))− f(t, St−)). (15.13)

Note that in the limit a → 0 this expression recovers the Black-ScholesDelta formula (15.12).

When Conditions (i) or (ii) above are not satisfied, exact replication is notpossible and this results into an hedging error given from (15.10) and (15.11)by

VT − φ(ST ) = VT − f(T, ST )

= V0 − f(0, S0) +w T

0dVt −

w T

0df(t, St)

= V0 − f(0, S0) + σw T

0St

(ξt −

∂f

∂x(t, St)

)dWt

+w T

0ξtSt−(ZNtdNt − λ IEν [Z1]dt)

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N. Privault

−w T

0(f(t, St−(1 + ZNt))− f(t, St−))dNt

+λw T

0IEν [(f(t, x(1 + Z1))− f(t, x))]x=Stdt.

Assuming for simplicity that Yt = aNt, i.e. ν(dx) = δa(dx), we get

VT − f(T, ST ) = V0 − f(0, S0) + σw T

0St

(ξt −

∂f

∂x(t, St)

)dWt

−w T

0(f(t, St−(1 + a))− f(t, St−)− aξtSt−)(dNt − λdt),

hence the mean square hedging error is given from the Ito isometry (14.8) by

IEu,λ[(VT − f(T, ST ))2]

= (V0 − f(0, S0))2 + σ2 IEu,λ

[(w T

0St

(ξt −

∂f

∂x(t, St)

)dWt

)2]

+ IEu,λ

[(w T

0(f(t, St−(1 + a))− f(t, St−)− aξtSt−)(dNt − λdt)

)2]

= (V0 − f(0, S0))2 + σ2 IEu,λ

[w T

0S2t

(ξt −

∂f

∂x(t, St)

)2

dt

]

+λ IEu,λ

[w T

0((f(t, St(1 + a))− f(t, St)− aξtSt))2 dt

].

Clearly, the initial portfolio value V0 that minimizes the above quantity is

V0 = f(0, S0) = e−rT IEu,λ,ν [φ(ST )].

When hedging only the risk generated by the Brownian part we let

ξt =∂f

∂x(t, St)

as in the Black-Scholes model, and in this case the hedging error due to thepresence of jumps becomes

IEu,λ[(VT−f(T, ST ))2] = λ IEu,λ

[w T

0((f(t, St(1 + a))− f(t, St)− aξtSt))2 dt

].

Next, let us find the optimal strategy (ξt)t∈R+that minimizes the remaining

hedging error

IEu,λ

[w T

0

(σ2S2

t

(ξt −

∂f

∂x(t, St)

)2

+ λ ((f(t, St(1 + a))− f(t, St)− aξtSt))2)dt

].

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"



For all t ∈ [0, T ], the almost-sure minimum of

ξt 7−→ σ2S2t

(ξt −

∂f

∂x(t, St)

)2

+ λ ((f(t, St(1 + a))− f(t, St)− aξtSt))2

is given by differentiation with respect to ξt, as the solution of

σ2S2t

(ξt −

∂f

∂x(t, St)

)− aλSt ((f(t, St(1 + a))− f(t, St)− aξtSt)) = 0,

i.e.

ξt =σ2 ∂f

∂x(t, St) +

aλ

St(f(t, St−(1 + a))− f(t, St−))

σ2 + a2λ, t ∈ [0, T ].

(15.14)

We note that the optimal strategy (15.14) is a weighted average of theBrownian and jump hedging strategies (15.12) and (15.13) according to therespective variance parameters σ2 and a2λ of the continuous and jump com-ponents.

Clearly, if aλ = 0 we get

ξt =∂f

∂x(t, St), t ∈ [0, T ],

which is the Black-Scholes perfect replication strategy, and when σ = 0 werecover

ξt =f(t, St−(1 + a))− f(t, St−)

aSt−, t ∈ [0, T ].

which is (15.13).

Note that the fact that perfect replication is not possible in a jump-diffusion model can be interpreted as a more realistic feature of the model,as perfect replication is not possible in the real world.

See [40] for an example of a complete market model with jumps, in whichcontinuous and jump noise are mutually excluding each other over time.

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N. Privault

Exercises

Exercise 15.1 Consider a standard Poisson process (Nt)t∈R+with intensity

λ > 0 under a probability measure P. Let (St)t∈R+be defined by the stochas-

tic differential equation

dSt = rStdt+ ηSt−(dNt − αdt),

where η > 0.

1. Find the value of α ∈ R such that the discounted process (e−rtSt)t∈R+is

a martingale under P.2. Compute the price at time t of a power option with payoff |ST |2 at ma-

turity T .

Exercise 15.2 Consider a long forward contract with payoff ST − K on ajump diffusion risky asset (St)t∈R+

given by

dSt = µStdt+ σStdWt + St−dYt.

1. Show that the forward claim admits a unique arbitrage price to be com-puted in a market with risk-free rate r > 0.

2. Show that the forward claim admits an exact replicating portfolio strat-egy based on the two assets St and ert.

3. Show that the portfolio strategy of Question 2 coincides with the optimalportfolio strategy (15.14).

Exercise 15.3 Consider (Wt)t∈R+ a standard Brownian motion and (Nt)t∈R+

a standard Poisson process with intensity λ > 0, independent of (Wt)t∈R+ ,under a probability measure P∗. Let (St)t∈R+

be defined by the stochasticdifferential equation

dSt = µStdt+ ηSt−dNt + σStdWt. (15.15)

1. Solve the equation (15.15).2. We assume that µ, η and the risk-free rate r > 0 are chosen such that the

discounted process (e−rtSt)t∈R+is a martingale under P∗. What relation

does this impose on µ, η, λ and r ?3. Under the relation of Question (2), compute the price at time t of a

European call option on ST with strike κ and maturity T , using a seriesexpansion of Black-Scholes functions.

Exercise 15.4 Consider (Nt)t∈R+a standard Poisson process with intensity



408


"



dSt = rStdt+ YNtSt−dNt,

where (Yk)k≥1 is an i.i.d. sequence of uniformly distributed random variableson [−1, 1].

1. Show that the discounted process (e−rtSt)t∈R+ is a martingale under P.2. Compute the price at time 0 of a European call option on ST with strikeκ and maturity T , using a series of multiple integrals.

Exercise 15.5 Consider a standard Poisson process (Nt)t∈R+with intensity



dSt = rStdt+ YNtSt−(dNt − αdt),

where (Yk)k≥1 is an i.i.d. sequence of uniformly distributed random variableson [0, 1].

(a) Find the value of α ∈ R such that the discounted process (e−rtSt)t∈R+

is a martingale under P.(b) Compute the price at time t of a forward call contract with maturity T

and payoff ST − κ.

Exercise 15.6 Consider (Nt)t∈R+ a standard Poisson process with intensityλ > 0 under a risk-neutral probability measure P. Let (St)t∈R+

be defined bythe stochastic differential equation

dSt = rStdt+ αSt−(dNt − λdt),

where α > 0. Consider a portfolio with value

Vt = ηtert + ξtSt

at time t, and satisfying the self-financing condition

dVt = rηtertdt+ ξtdSt.

We assume that the portfolio hedges the claim φ(ST ), i.e. we have Vt =f(t, St) for all times t ∈ [0, T ].

1. Show that under self-financing the portfolio value Vt satisfies

dVt = rf(t, St)dt+ αξtSt−(dNt − λdt). (15.16)

2. Show that the claim C can be exactly replicated by the hedging strategy

ξt =1

αSt−(f(t, St−(1 + α))− f(t, St−)).

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Chapter 16

Basic Numerical Methods

This chapter is an elementary introduction to finite difference methods forthe resolution of PDEs and stochastic differential equations. We cover theexplicit and implicit finite difference schemes for the heat equations and theBlack-Scholes PDE, as well as the Euler and Milshtein schemes for stochasticdifferential equations.

16.1 The Heat Equation

Consider the heat equation

∂φ

∂t(t, x) =

∂2φ

∂x2(t, x) (16.1)

with initial conditionφ(0, x) = f(x)

on a compact interval [0, T ]× [0, X] divided into the grid points

(ti, xj) = (i∆t, j∆x), i = 0, . . . , N, j = 0, . . . ,M,

with ∆t = T/N and ∆x = X/M . Our goal is to obtain a discrete approxi-mation

(φ(ti, xj))0≤i≤N, 0≤j≤M

of the solution to (16.1), by evaluating derivatives using finite differences.

Explicit method

Using the forward time difference approximation of (16.1) we get

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N. Privault

φ(ti+1, xj)− φ(ti, xj)

∆t=φ(ti, xj+1) + φ(ti, xj−1)− 2φ(ti, xj)

(∆x)2

and letting ρ = (∆t)/(∆x)2 this yields

φ(ti+1, xj) = ρφ(ti, xj+1) + (1− 2ρ)φ(ti, xj) + ρφ(ti, xj−1),

1 ≤ j ≤M − 1, i.e

Φi+1 = AΦi + ρ

φ(ti, x0)

0...0

φ(ti, xM )

, i = 0, 1, . . . , N − 1,

with

Φi =

φ(ti, x1)...

φ(ti, xM−1)

, i = 0, 1, . . . , N,

and

A =

1− 2ρ ρ 0 · · · 0 0 0ρ 1− 2ρ ρ · · · 0 0 00 ρ 1− 2ρ · · · 0 0 0...

......

. . ....

......

0 0 0 · · · 1− 2ρ ρ 00 0 0 · · · ρ 1− 2ρ ρ0 0 0 · · · 0 ρ 1− 2ρ

.

The vector φ(ti, x0)

0...0

φ(ti, xM )

, i = 0, . . . , N,

can be given by the lateral boundary conditions φ(t, 0) and φ(t,X).

Implicit method

Using the backward time difference approximation

φ(ti, xj)− φ(ti−1, xj)

∆t=φ(ti, xj+1) + φ(ti, xj−1)− 2φ(ti, xj)

(∆x)2

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and letting ρ = (∆t)/(∆x)2 we get

φ(ti−1, xj) = −ρφ(ti, xj+1) + (1 + 2ρ)φ(ti, xj)− ρφ(ti, xj−1),

1 ≤ j ≤M − 1, i.e.

Φi−1 = BΦi + ρ

φ(ti, x0)

0...0

φ(ti, xM )

, i = 1, 2, . . . , N,

with

B =

1 + 2ρ −ρ 0 · · · 0 0 0−ρ 1 + 2ρ −ρ · · · 0 0 00 −ρ 1 + 2ρ · · · 0 0 0...

......

. . ....

......

0 0 0 · · · 1 + 2ρ −ρ 00 0 0 · · · −ρ 1 + 2ρ −ρ0 0 0 · · · 0 −ρ 1 + 2ρ

.

By inversion of the matrix B, Φi is given in terms of Φi−1 as

Φi = B−1Φi−1 − ρB−1

φ(ti, x0)

0...0

φ(ti, xM )

, i = 1, . . . , N.

16.2 The Black-Scholes PDE

Consider the Black-Scholes PDE

rφ(t, x) =∂φ

∂t(t, x) + rx

∂φ

∂x(t, x) +

1

2x2σ2 ∂

2φ

∂x2(t, x), (16.2)

under the terminal condition φ(T, x) = (x−K)+, resp. φ(T, x) = (K − x)+,for a European call, resp. put, option.

Note that here time runs backwards as we start from a terminal conditionat time T . Thus here the explicit method uses backward differences while theimplicit method uses forward differences.

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N. Privault

Explicit method

Using the backward time difference approximation of (16.2) we get

rφ(ti, xj) =φ(ti, xj)− φ(ti−1, xj)

∆t+ rxj

φ(ti, xj+1)− φ(ti, xj−1)

2∆x

+1

2x2jσ

2φ(ti, xj+1) + φ(ti, xj−1)− 2φ(ti, xj)

(∆x)2,

1 ≤ j ≤M − 1, i.e.

φ(ti−1, xj) =1

2∆t(σ2j2 − rj)φ(ti, xj−1) + (1−∆t(σ2j2 + r))φ(ti, xj)

+1

2∆t(σ2j2 + rj)φ(ti, xj+1),

1 ≤ j ≤ M − 1, where the boundary conditions φ(ti, x0) and φ(ti, xM ) aregiven by

φ(ti, x0) = 0, φ(ti, xM ) = xM −Ke−r(T−ti), 0 ≤ i ≤ N,

for a European call option, and

φ(ti, x0) = Ke−r(T−ti), φ(ti, xM ) = 0 0 ≤ i ≤ N,

for a European put option.

The explicit finite difference method is known to have a divergent be-haviour when time runs backwards, as illustrated in Figure 16.1.

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"



Explicit method

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

time to maturity

0 20 40 60 80 100 120 140 160 180 200

strike

-100

-50

0

50

100

Fig. 16.1: Divergence of the explicit finite difference method.

Implicit method

Using the forward time difference approximation of (16.2) we get

rφ(ti, xj) =φ(ti+1, xj)− φ(ti, xj)

∆t+ rxj

φ(ti, xj+1)− φ(ti, xj−1)

2∆x

+1

2x2jσ

2φ(ti, xj+1) + φ(ti, xj−1)− 2φ(ti, xj)

(∆x)2,

1 ≤ j ≤M − 1, i.e.

φ(ti+1, xj) = −1

2∆t(σ2j2 − rj)φ(ti, xj−1) + (1 +∆t(σ2j2 + r))φ(ti, xj)

−1

2∆t(σ2j2 + rj)φ(ti, xj+1),

1 ≤ j ≤M − 1, i.e.

Φi+1 = BΦi +

12∆t

(r − σ2

)φ(ti, x0)

0...0

− 12∆t

(r(M − 1) + σ2(M − 1)2

)φ(ti, xM )

,

i = 0, 1, . . . , N − 1, with

Bj,j−1 =1

2∆t(rj − σ2j2

), Bj,j = 1 + σ2j2∆t+ r∆t,

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N. Privault

and

Bj,j+1 = −1

2∆t(rj + σ2j2

),

for j = 1, . . . ,M − 1, and B(i, j) = 0 otherwise.

By inversion of the matrix B, Φi is given in terms of Φi+1 as

Φi = B−1Φi+1 −B−1

12∆t

(r − σ2

)φ(ti, x0)

0...0

− 12∆t

(r(M − 1) + σ2(M − 1)2

)φ(ti, xM )

,

i = 0, 1, . . . , N − 1, where the boundary conditions φ(ti, x0) and φ(ti, xM )can be provided as in the case of the explicit method.

Note that for all j = 1, . . . ,M − 1 we have

Bj,j−1 +Bj,j +Bj,j+1 = 1 + r∆t,

hence when the terminal condition is a constant φ(T, x) = c > 0 we get

φ(ti, x) = c(1 + r∆t)−(N−i) = c(1 + rT/N)−(N−i), i = 0, . . . , N,

hence for all s ∈ [0, T ],

φ(s, x) = limN→∞

φ(t[Ns/T ], x)

= c limN→∞

(1 + rT/N)−(N−[Ns/T ])

= c limN→∞

(1 + rT/N)−[N(T−s)/T ]

= c limN→∞

(1 + rT/N)−(T−s)/T

= ce−r(T−s),

as expected, where [x] denotes the integer part of x ∈ R. The implicit finitedifference method is known to be more stable than the explicit method, asillustrated in Figure 16.2, in which the discretization parameters have beentaken to be the same as in Figure 16.1.

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Implicit method

0 1

2 3

4 5

6 7

8 9

10

time to maturity

0 20

40 60

80 100

120 140

160 180

200

strike

0

20

40

60

80

100

120

140

Fig. 16.2: Stability of the implicit finite difference method.

16.3 Euler Discretization

In order to apply the Monte Carlo method in option pricing, we need togenerate random samples whose empirical means are used for the evaluationof expectations. This can be done by discretizing the solutions of stochas-tic differential equations. Despite its apparent simplicity, the Monte Carlomethod can be delicate to implement and the optimization of Monte Carloalgorithms and random number generation have been the object of numerousworks which are outside the scope of this text, cf. e.g. [31], [46].

The Euler discretization scheme for the stochastic differential equation

dXt = b(Xt)dt+ a(Xt)dWt

is given by

XNtk+1

= XNtk

+w tk+1

tkb(Xs)ds+

w tk+1

tka(Xs)dWs

' XNtk

+ b(XNtk

)(tk+1 − tk) + a(XNtk

)(Wtk+1−Wtk).

In particular, when Xt is the geometric Brownian motion given by

dXt = rXtdt+ σXtdWt

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N. Privault

we getXNtk+1

= XNtk

+ rXNtk

(tk+1 − tk) + σXNtk

(Wtk+1−Wtk),

which can be computed as

XNtk

= XNt0

k∏i=1

(1 + r(ti − ti−1) + σ(Wti −Wti−1

)).

16.4 Milshtein Discretization

In the Milshtein scheme we expand a(Xs) as

a(Xs) ' a(Xtk) + a′(Xtk)b(Xtk)(s− tk) + a′(Xtk)a(Xtk)(Ws −Wtk).

As a consequence we get

XNtk+1

= XNtk

+w tk+1

tkb(Xs)ds+

w tk+1

tka(Xs)dWs

' XNtk

+w tk+1

tkb(Xs)ds+ a(Xtk)(Wtk+1

−Wtk)

+a′(Xtk)b(Xtk)w tk+1

tk(s− tk)dWs + a′(Xtk)a(Xtk)

w tk+1

tk(Ws −Wtk)dWs

= XNtk

+w tk+1


−Wtk)

+1

2a′(Xtk)b(Xtk)(tk+1 − tk)2 + a′(Xtk)a(Xtk)

w tk+1

tk(Ws −Wtk)dWs

' XNtk

+w tk+1


−Wtk)

+a′(Xtk)a(Xtk)w tk+1

tk(Ws −Wtk)dWs.

Next using Ito’s formula we note that

(Wtk+1−Wtk)2 = 2

w tk+1

tk(Ws −Wtk)dWs +

w tk+1

tkds,

hence

w tk+1

tk(Ws −Wtk)dWs =

1

2((Wtk+1

−Wtk)2 − (tk+1 − tk)),

and

XNtk+1' XN

tk+

w tk+1


−Wtk)

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"



+1

2a′(Xtk)a(Xtk)((Wtk+1

−Wtk)2 − (tk+1 − tk))

= XNtk

+w tk+1


−Wtk)

+1


−Wtk)2 − (tk+1 − tk))

= XNtk

+ b(Xtk)(tk+1 − tk) + a(Xtk)(Wtk+1−Wtk)

+1


−Wtk)2 − (tk+1 − tk)).

As a consequence the Milshtein scheme is written as

XNtk+1' XN

tk+ b(XN

tk)(tk+1 − tk) + a(XN

tk)(Wtk+1

−Wtk)

+1

2a′(XN

tk)a(XN

tk)((Wtk+1

−Wtk)2 − (tk+1 − tk)),

i.e. in the Milshtein scheme we take into account the “small” difference

(Wtk+1−Wtk)2 − (tk+1 − tk)

existing between (∆Wt)2 and ∆t. Taking (∆Wt)

2 equal to ∆t brings us backto the Euler scheme.

When Xt is the geometric Brownian motion given by

dXt = rXtdt+ σXtdWt

we get

XNtk+1

= XNtk

+(r−σ2/2)XNtk

(tk+1−tk)+σXNtk

(Wtk+1−Wtk)+

1

2σ2XN

tk(Wtk+1

−Wtk)2,

which can be computed as

XNtk

= XNt0

k∏i=1

(1 + (r − σ2/2)(ti − ti−1) + σ(Wti −Wti−1

) +1

2σ2(Wti −Wti−1)2

).

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Appendix: Background on ProbabilityTheory

In this appendix we review a number of basic probabilistic tools that areneeded in option pricing and hedging. We refer to [37], [16], [57] for more onthe needed probability background.

Probability Spaces and Events

We will need the following notation coming from set theory. Given A and Bto abstract sets, “A ⊂ B” means that A is contained in B, and the propertythat ω belongs to the set A is denoted by “ω ∈ A”. The finite set made of nelements ω1, . . . , ωn is denoted by ω1, . . . , ωn, and we will usually make adistinction between the element ω and its associated singleton set ω.

A probability space is an abstract set Ω that contains the possible out-comes of a random experiment.

Examples:

i) Coin tossing: Ω = H,T.

ii) Rolling one die: Ω = 1, 2, 3, 4, 5, 6.

iii) Picking on card at random in a pack of 52: Ω = 1, 2, 3, , . . . , 52.

iv) An integer-valued random outcome: Ω = N.

In this case the outcome ω ∈ N can be the random number of trialsneeded until some event occurs.

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N. Privault

v) A non-negative, real-valued outcome: Ω = R+.

In this case the outcome ω ∈ R+ may represent the (non-negative) valueof a continuous random time.

vi) A random continuous parameter (such as time, weather, price or wealth,temperature, ...): Ω = R.

vii) Random choice of a continuous path in the space Ω = C(R+) of all con-tinuous functions on R+.

In this case, ω ∈ Ω is a function ω : R+ −→ R and a typical example isthe graph t 7−→ ω(t) of a stock price over time.

Product spaces:

Probability spaces can be built as product spaces and used for the modelingof repeated random experiments.

i) Rolling two dice: Ω = 1, 2, 3, 4, 5, 6 × 1, 2, 3, 4, 5, 6.

In this case a typical element of Ω is written as ω = (k, l) withk, l ∈ 1, 2, 3, 4, 5, 6.

ii) A finite number n of real-valued samples: Ω = Rn.

In this case the outcome ω is a vector ω = (x1, . . . , xn) ∈ Rn with ncomponents.

Note that to some extent, the more complex Ω is, the better it fits a practicaland useful situation, e.g. Ω = H,T corresponds to a simple coin tossingexperiment while Ω = C(R+) the space of continuous functions on R+ can beapplied to the modeling of stock markets. On the other hand, in many casesand especially in the most complex situations, we will not attempt to specifyΩ explicitly.

Events

An event is a collection of outcomes, which is represented by a subset of Ω.

The collections G of events that we will consider are called σ-algebras, andassumed to satisfy the following conditions.

(i) ∅ ∈ G,

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"



(ii) For all countable sequences An ∈ G, n ≥ 1, we have⋃n≥1

An ∈ G,

(iii) A ∈ G =⇒ (Ω \A) ∈ G,

where Ω \A := ω ∈ Ω : ω /∈ A.

The collection of all events in Ω will often be denoted by F . The empty set∅ and the full space Ω are considered as events but they are of less importancebecause Ω corresponds to “any outcome may occur” while ∅ corresponds toan absence of outcome, or no experiment.

In the context of stochastic processes, two σ-algebras G and F such thatG ⊂ F will refer to two different amounts of information, the amount of in-formation associated to G being here lower than the one associated to F .

The formalism of σ-algebras helps in describing events in a short and preciseway.

Examples:

i) Ω = 1, 2, 3, 4, 5, 6.

The event A = 2, 4, 6 corresponds to

“the result of the experiment is an even number”.ii) Taking again Ω = 1, 2, 3, 4, 5, 6,

F := Ω, ∅, 2, 4, 6, 1, 3, 5

defines a σ-algebra on Ω which corresponds to the knowledge of parityof an integer picked at random from 1 to 6.

Note that in the set-theoretic notation, an event A is a subset of Ω, i.e.A ⊂ Ω, while it is an element of F , i.e. A ∈ F . For example, we haveΩ ⊃ 2, 4, 6 ∈ F , while 2, 4, 6, 1, 3, 5 ⊂ F .

Taking

G := Ω, ∅, 2, 4, 6, 2, 4, 6, 1, 2, 3, 4, 5, 1, 3, 5, 6, 1, 3, 5 ⊃ F ,

defines a σ-algebra on Ω which is bigger than F and corresponds to theknowledge whether the outcome is equal to 6 or not, in addition to theparity information contained in F .

iii) Take

Ω = H,T × H,T = (H,H), (H.T ), (T,H), (T, T ).

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N. Privault

In this case, the collection F of all possible events is given by

F = ∅, (H,H), (T, T ), (H,T ), (T,H), (16.3)

(T, T ), (H,H), (H,T ), (T,H), (H,T ), (T, T ),(T,H), (T, T ), (H,T ), (H,H), (T,H), (H,H),(H,H), (T, T ), (T,H), (H,H), (T, T ), (H,T ),(H,T ), (T,H), (H,H), (H,T ), (T,H), (T, T ), Ω .

Note that the set F of all events considered in (16.3) above has altogether

1 =

(n

0

)event of cardinal 0,

4 =

(n

1

)events of cardinal 1,

6 =

(n

2


4 =

(n

3


1 =

(n

4

)event of cardinal 4,

with n = 4, for a total of

16 = 2n =4∑k=0

(4

k

)= 1 + 4 + 6 + 4 + 1

events.

The collection of events

G := ∅, (T, T ), (H,H), (H,T ), (T,H), Ω

defines a sub σ-algebra of F , associated to the information “the results oftwo coin tossings are different”.

Exercise: Write down the set of all events on Ω = H,T.

Note also that (H,T ) is different from (T,H), whereas (H,T ), (T,H) isequal to (T,H), (H,T ).

In addition we will usually make a distinction between the outcome ω ∈ Ωand its associated event ω ∈ F , which satisfies ω ⊂ Ω.

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Probability Measures

A probability measure is a mapping P : F −→ [0, 1] that assigns a probabilityP(A) ∈ [0, 1] to any event A, with the properties

a) P(Ω) = 1, and

b) P

( ∞⋃n=1

An

)=∞∑n=1

P(An), whenever Ak ∩Al = ∅, k 6= l.

A property or event is said to hold P-almost surely (also written P-a.s.) if itholds with probability equal to one.


P(A1 ∪ · · · ∪An) = P(A1) + · · ·+ P(An)

when the subsets A1, . . . , An of Ω are disjoints, and

P(A ∪B) = P(A) + P(B)

if A ∩B = ∅. In the general case we can write

P(A ∪B) = P(A) + P(B)− P(A ∩B).

The triple(Ω,F ,P) (16.4)

was introduced by A.N. Kolmogorov (1903-1987), and is generally referredto as the Kolmogorov framework.

In addition we have the following convergence properties.

1. Let (An)n∈N be a nondecreasing sequence of events, i.e. An ⊂ An+1,n ∈ N. Then we have

P

(⋃n∈N

An

)= limn→∞

P(An). (16.5)

2. Let (An)n∈N be a nonincreasing sequence of events, i.e. An+1 ⊂ An,n ∈ N. Then we have

P

(⋂n∈N

An

)= limn→∞

P(An). (16.6)

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Conditional Probabilities and Independence

Given any two events A,B ⊂ Ω with P(B) 6= 0, we call

P(A | B) :=P(A ∩B)

P(B)

the probability of A given B, or conditionally to B. Note that if P(B) = 1we have P(A∩Bc) ≤ P(Bc) = 0 hence P(A∩B) = P(A) and P(A | B) = P(A).

We also recall the following property:

P

(B ∩

∞⋃n=1

An

)=∞∑n=1

P(B∩An) =∞∑n=1

P(B | An)P(An) =∞∑n=1

P(An | B)P(B),

for any family of events (An)n≥1, B, provided Ai ∩ Aj = ∅, i 6= j, andP(An) > 0, n ≥ 1. This also shows that conditional probability measures areprobability measures, in the sense that whenever P(B) > 0 we have

a) P(Ω | B) = 1, and

b) P

( ∞⋃n=1

An

∣∣∣B) =

∞∑n=1

P(An | B), whenever Ak ∩Al = ∅, k 6= l.

In particular if∞⋃n=1

An = Ω, (An)n≥1 becomes a partition of Ω and we get

the law of total probability

P(B) =

∞∑n=1

P(B ∩An) =

∞∑n=1

P(An | B)P(B) =

∞∑n=1

P(B | An)P(An), (16.7)

provided Ai ∩ Aj = ∅, i 6= j, and P(An) > 0, n ≥ 1. However we have ingeneral

P

(A∣∣∣ ∞⋃n=1

Bn

)6=∞∑n=1

P(A | Bn),

even when Bk ∩Bl = ∅, k 6= l. Indeed, taking for example A = Ω = B1 ∪B2

with B1 ∩B2 = ∅ and P(B1) = P(B2) = 1/2, we have

1 = P(Ω | B1 ∪B2) 6= P(Ω | B1) + P(Ω | B2) = 2.

Finally, two events A and B are said to be independent if

P(A | B) = P(A),

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i.e. ifP(A ∩B) = P(A)P(B).

In this case we findP(A | B) = P(A).

Random Variables

A real-valued random variable is a mapping

X : Ω −→ Rω 7−→ X(ω)

from a probability space Ω into the state space R.

Given X : Ω −→ R a random variable and A a (measurable)1 subset of R,we denote by X ∈ A the event

X ∈ A = ω ∈ Ω : X(ω) ∈ A.

Given G a σ-algebra on G, the mapping X : Ω −→ R is said to be G-measurable if

X ≤ x = ω ∈ Ω : X(ω) ≤ x ∈ G,for all x ∈ R. In this case we will also say that the knowledge of X dependsonly on the information contained in G.

Examples:

i) Let Ω = 1, 2, 3, 4, 5, 6 × 1, 2, 3, 4, 5, 6, and consider the mapping

X : Ω −→ R(k, l) 7−→ k + l.

Then X is a random variable giving the sum of the two numbers appear-ing on each die.

ii) the time needed everyday to travel from home to work or school is arandom variable, as the precise value of this time may change from dayto day under unexpected circumstances.

iii) the price of a risky asset is a random variable.

1 Measurability of subsets of R refers to Borel measurability, a concept which will notbe defined in this text.

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In the sequel we will often use the notion of indicator function 1A of an eventA. The indicator function 1A is the random variable

1A : Ω −→ 0, 1ω 7−→ 1A(ω)

defined by

1A(ω) =

1 if ω ∈ A,0 if ω /∈ A,

with the property1A∩B(ω) = 1A(ω)1B(ω), (16.8)

since

ω ∈ A ∩B ⇐⇒ ω ∈ A and ω ∈ B⇐⇒ 1A(ω) = 1 and 1B(ω) = 1⇐⇒ 1A(ω)1B(ω) = 1.

We also have

1A∪B = 1A + 1B − 1A∩B = 1A + 1B − 1A1B ,

and1A∪B = 1A + 1B , (16.9)

if A ∩ B = ∅. In addition, any Bernoulli random variable X : Ω −→ 0, 1can be written as an indicator function

X = 1A

on Ω with A = X = 1 = ω ∈ Ω : X(ω) = 1. For example if Ω = N andA = k, for all l ∈ N we have

1k(l) =

1 if k = l,0 if k 6= l.

If X is a random variable we also let

1X=n =

1 if X = n,0 if X 6= n,

and

1X<n =

1 if X < n,0 if X ≥ n.

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Probability Distributions

The probability distribution of a random variable X : Ω −→ R is the collec-tion

P(X ∈ A) : A measurable subset of R.In fact the distributions of X can be reduced to the knowledge of either

P(a < X ≤ b) = P(X ≤ b)− P(X ≤ a) : a < b ∈ R,

orP(X ≤ a) : a ∈ R,

orP(X ≥ a) : a ∈ R.

Two random variables X and Y are said to be independent under theprobability P if their probability distributions satisfy

P(X ∈ A , Y ∈ B) = P(X ∈ A)P(Y ∈ B)

for all (measurable) subsets A and B of R.

Distributions Admitting a Density

In this case the distribution of X is given by

P(a ≤ X ≤ b) =w b

af(x)dx

where the function f : R −→ R+ is called the density of the distribution ofX. We also say that the distribution of X is absolutely continuous, or that Xis an absolutely continuous random variable. This, however, does not implythat the density function f : R −→ R+ is continuous.

In particular we always have

w∞−∞

f(x)dx = P(−∞ ≤ X ≤ ∞) = 1

for all probability density functions f : R −→ R+.

The density fX can be recovered from the distribution functions

x 7−→ P(X ≤ x) =w x

−∞fX(s)ds, x ∈ R,

and

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N. Privault

x 7−→ P(X ≥ x) =w∞xfX(s)ds, x ∈ R,

as

fX(x) =∂

∂x

w x

−∞fX(s)ds = − ∂

∂x

w∞xfX(s)ds, x ∈ R.

Examples:

i) The uniform distribution on an interval.

The density of the uniform distribution on the interval [a, b], a < b, isgiven by

f(x) =1

b− a1[a,b](x), x ∈ R.

ii) The Gaussian distribution.

The density of the standard normal distribution is given by

f(x) =1√2πe−x

2/2, x ∈ R.

More generally, X has a Gaussian distribution with mean µ ∈ R andvariance σ2 > 0 (in this case we write X ' N (µ, σ2)) if

f(x) =1√

2πσ2e−(x−µ)

2/(2σ2), x ∈ R.

iii) The exponential distribution with parameter λ > 0.

In this case we have

f(x) = λ1[0,∞)(x)e−λx =

λe−λx, x ≥ 0

0, x < 0.(16.10)

We also haveP(X > t) = e−λt, t ∈ R+. (16.11)

In addition, if X1, . . . , Xn are independent exponentially distributed ran-dom variables with parameters λ1, . . . , λn we have

P(min(X1, . . . , Xn) > t) = P(X1 > t, . . . ,Xn > t)

= P(X1 > t) · · ·P(Xn > t)

= e−t(λ1+···+λn), t ∈ R+, (16.12)

hence min(X1, . . . , Xn) is an exponentially distributed random variablewith parameter λ1 + · · ·+ λn.

We also have

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P(X1 < X2) = P(X1 ≤ X2) = λ1λ2w∞0

w y

0e−λ1x−λ2ydxdy =

λ1λ1 + λ2

,

(16.13)and we note that

P(X1 = X2) = λ1λ2w(x,y)∈R2

+ : x=ye−λ1x−λ2ydxdy = 0.

iv) The gamma distribution.


f(x) =aλ

Γ (λ)1[0,∞)(x)xλ−1e−ax =

aλ

Γ (λ)xλ−1e−ax, x ≥ 0

0, x < 0,

where a > 0 and λ > 0 are parameters and

Γ (λ) =w∞0xλ−1e−xdx, λ > 0,

is the Gamma function.

v) The Cauchy distribution.


f(x) =1

π(1 + x2), x ∈ R.

vi) The lognormal distribution.

In this case,

f(x) = 1[0,∞)(x)1

xσ√

2πe−

(µ−log x)2

2σ2 =

1

xσ√

2πe−

(µ−log x)2

2σ2 , x ≥ 0

0, x < 0.

Exercise: For each of the above probability density functions, check that thecondition w∞

−∞f(x)dx = 1

is satisfied.

Remark 16.1. Note that if the distribution of X admits a density then forall a ∈ R, we have

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N. Privault

P(X = a) =w a

af(x)dx = 0, (16.14)

and this is not a contradiction.

In particular, Remark 16.1 shows that

P(a ≤ X ≤ b) = P(X = a) + P(a < X ≤ b) = P(a < X ≤ b) = P(a < X < b),

for a ≤ b.

In practice, Property (16.14) appears for example in the framework oflottery games with a large number of participants, in which a given number“a” selected in advance has a very low (almost zero) probability to be chosen.

Given two absolutely continuous random variables X : Ω −→ R and Y :Ω −→ R we can form the R2-valued random variable (X,Y ) defined by

(X,Y ) : Ω −→ R2

ω 7−→ (X(ω), Y (ω)).

We say that (X,Y ) admits a joint probability density

f(X,Y ) : R2 −→ R+

whenP((X,Y ) ∈ A×B) =

wA

wBf(X,Y )(x, y)dxdy

for all measurable subsets A, B of R. The density f(X,Y ) can be recoveredfrom the distribution functions

(x, y) 7−→ P(X ≤ x, Y ≤ y) =w x

−∞

w y

−∞f(X,Y )(s, t)dsdt,

and(x, y) 7−→ P(X ≥ x, Y ≥ y) =

w∞x

w∞yf(X,Y )(s, t)dsdt,

as

f(X,Y )(x, y) =∂2

∂x∂y

w x

−∞

w y

−∞f(X,Y )(s, t)dsdt (16.15)

=∂2

∂x∂y

w∞x

w∞yf(X,Y )(s, t)dsdt,

x, y ∈ R.

The probability densities fX : R −→ R+ and fY : R −→ R+ of X : Ω −→R and Y : Ω −→ R are called the marginal densities of (X,Y ) and are givenby

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fX(x) =w∞−∞

f(X,Y )(x, y)dy, x ∈ R, (16.16)

andfY (y) =

w∞−∞

f(X,Y )(x, y)dx, y ∈ R.

The conditional density fX|Y=y : R −→ R+ of X given Y = y is defined by

fX|Y=y(x) :=f(X,Y )(x, y)

fY (y), x, y ∈ R, (16.17)

provided fY (y) > 0.

Discrete Distributions

We only consider integer-valued random variables, i.e. the distribution of Xis given by the values of P(X = k), k ∈ N.

Examples:

i) The Bernoulli distribution.

We have

P(X = 1) = p and P(X = 0) = 1− p, (16.18)

where p ∈ [0, 1] is a parameter.

ii) The binomial distribution.

We have

P(X = k) =

(n

k

)pk(1− p)n−k, k = 0, 1, . . . , n,

where n ≥ 1 and p ∈ [0, 1] are parameters.

iii) The geometric distribution.

We haveP(X = k) = (1− p)pk, k ∈ N, (16.19)

where p ∈ (0, 1) is a parameter.

Note that if (Xk)k∈N is a sequence of independent Bernoulli randomvariables with distribution (16.18), then the random variable

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N. Privault

X := infk ∈ N : Xk = 0

has the geometric distribution (16.19).

iv) The negative binomial distribution (or Pascal distribution).

We have

P(X = k) =

(k + r − 1

r − 1

)(1− p)rpk, k ∈ N,

where p ∈ (0, 1) and r ≥ 1 are parameters. Note that the negative bino-mial distribution recovers the geometric distribution when r = 1.

v) The Poisson distribution.

We have

P(X = k) =λk

k!e−λ, k ∈ N,

where λ > 0 is a parameter.

Remark 16.2. The distribution of a discrete random variable cannot admita density. If this were the case, by Remark 16.1 we would have P(X = k) = 0for all k ∈ N and

1 = P(X ∈ R) = P(X ∈ N) =∞∑k=0

P(X = k) = 0,

which is a contradiction.

Given two discrete random variables X and Y , the conditional distributionof X given Y = k is given by

P(X = n | Y = k) =P(X = n and Y = k)

P(Y = k), n ∈ N,

provided P(Y = k) > 0, k ∈ N.

Expectation of a Random Variable

The expectation of a random variable X is the mean, or average value, ofX. In practice, expectations can be even more useful than probabilities. Forexample, knowing that a given equipment (such as a bridge) has a failureprobability of 1.78493 out of a billion can be of less practical use than know-

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ing the expected lifetime (e.g. 200000 years) of that equipment.

For example, the time T (ω) to travel from home to work/school can be arandom variable with a new outcome and value every day, however we usu-ally refer to its expectation IE[T ] rather than to its sample values that maychange from day to day.

In general, the expectation of the indicator function 1A is defined as

IE[1A] = P(A),

for any event A. For a Bernoulli random variable X : Ω −→ 0, 1 withparameter p ∈ [0, 1], written as X = 1A with A = X = 1, we have

p = P(X = 1) = P(A) = IE[1A] = IE[X].

Discrete Distributions

Next, let X : Ω −→ N be a discrete random variable. The expectation IE[X]of X is defined as the sum

IE[X] =∞∑k=0

kP(X = k),

in which the possible values k ∈ N of X are weighted by their probabilities.More generally we have

IE[φ(X)] =∞∑k=0

φ(k)P(X = k),

for all sufficiently summable functions φ : N −→ R.

Given a non-negative random variable X, the finiteness of IE[X] < ∞implies P(X < ∞) < 1, however the converse is not true. For example theexpectation IE[φ(X)] may be infinite even when φ(X) is always finite, takefor example

φ(X) = 2X and P(X = k) = 1/2k, k ≥ 1. (16.20)

The expectation of the indicator function X = 1A can be recovered as

IE[1A] = 0× P(Ω \A) + 1× P(A) = P(A).

Note that the expectation is a linear operation, i.e. we have

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N. Privault

IE[aX + bY ] = a IE[X] + b IE[Y ], a, b ∈ R, (16.21)

providedIE[|X|] + IE[|Y |] <∞.

The notion of expectation takes its full meaning under conditioning. For ex-ample, the expected return of a random asset usually depends on informationsuch as economic data, location, etc. In this case, replacing the expectation bya conditional expectation will provide a better estimate of the expected value.

For instance, life expectancy is a natural example of a conditional expec-tation since it typically depends on location, gender, and other parameters.

The conditional expectation of X : Ω −→ N given an event A is definedby

IE[X | A] =

∞∑k=0

kP(X = k | A).

Lemma 16.1. Given an event A such that P(A) > 0, we have

IE[X | A] =1

P(A)IE [X1A] . (16.22)

Proof. By Relation (16.8) we have

IE[X | A] =1

P(A)

∞∑k=0

kP(X = k ∩A) =1

P(A)

∞∑k=0

k IE[1X=k∩A

]=

1

P(A)

∞∑k=0

k IE[1X=k1A

]=

1

P(A)IE

[1A

∞∑k=0

k1X=k

]

=1

P(A)IE [1AX] , (16.23)

where we used the relation

X =

∞∑k=0

k1X=k

which holds since X takes only integer values.

If X is independent of A (i.e. P(X = k ∩ A) = P(X = k)P(A), k ∈ N)we have IE[X1A] = IE[X]P(A) and we naturally find

IE[X | A] = IE[X].

If X = 1B we also have in particular

IE[1B | A] = 0× P(X = 0 | A) + 1× P(X = 1 | A)

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= P(X = 1 | A)

= P(B | A).

One can also define the conditional expectation of X given that Y = k, as

IE[X | Y = k] =

∞∑n=0

nP(X = n | Y = k),

where Y : Ω −→ N is a discrete random variable. In general we have

IE[IE[X | Y ]] =

∞∑k=0

IE[X | Y = k]P(Y = k)

=∞∑k=0

∞∑n=0

nP(X = n | Y = k)P(Y = k)

=

∞∑n=0

n

∞∑k=0

P(X = n and Y = k)

=∞∑n=0

nP(X = n) = IE[X],

where we used the marginal distribution

P(X = n) =∞∑k=0

P(X = n and Y = k), n ∈ N,

that follows from the law of total probability (16.7) by taking Ak = Y = k.

Hence we have the relation

IE[X] = IE[IE[X | Y ]], (16.24)

which is sometimes referred to as the tower property. Taking

Y =∞∑k=0

k1Ak ,

i.e. Ak = Y = k, k ∈ N, we also get the law of total expectation

IE[X] = IE[IE[X | Y ]] =∞∑k=0

IE[X | Ak]P(Ak), (16.25)

whenever (Ak)k∈N is a partition of Ω.

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Random sums

Based on the tower property or ordinary conditioning, the expectation of a

random sum

Y∑k=1

Xk, where (Xk)k∈N is a sequence of random variables, can

be computed from the tower property (16.24) as

IE

[Y∑k=1

Xk

]= IE

[IE

[Y∑k=1

Xk

∣∣∣Y ]]

=∞∑n=0

IE

[Y∑k=1

Xk

∣∣∣Y = n

]P(Y = n)

=∞∑n=0

IE

[n∑k=1

Xk

∣∣∣Y = n

]P(Y = n),

and if Y is independent of (Xk)k∈N this yields

IE

[Y∑k=1

Xk

]=

∞∑n=0

IE

[n∑k=1

Xk

]P(Y = n).

Similarly, for a random product we will have

IE

[Y∏k=1

Xk

]=

∞∑n=0

IE

[n∏k=1

Xk

]P(Y = n). (16.26)

Example:

The life expectancy in Singapore is IE[T ] = 80 years overall, where Tdenotes the lifetime of a given individual chosen at random. Let G ∈m,w denote the gender of that individual. The statistics show that

IE[T | G = w] = 78 and IE[T | G = m] = 81.9,

and we have

80 = IE[T ]

= IE[IE[T |G]]

= P(G = w) IE[T | G = w] + P(G = m) IE[T | G = m]

= 81.9× P(G = w) + 78× P(G = m)

= 81.9× (1− P(G = m)) + 78× P(G = m),

showing that

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80 = 81.9× (1− P(G = m)) + 78× P(G = m),

i.e.

P(G = m) =81.9− 80

81.9− 78=

1.9

3.9= 0.487.

Distributions Admitting a Density

Given a random variable X whose distribution admits a density fX : R −→R+ we have

IE[X] =w∞−∞

xfX(x)dx,

and more generally,

IE[φ(X)] =w∞−∞

φ(x)fX(x)dx,

for all sufficiently integrable function φ on R. For example, ifX has a standardnormal distribution we have

IE[φ(X)] =w∞−∞

φ(x)e−x2/2 dx√

2π.

In case X has a Gaussian distribution with mean µ ∈ R and variance σ2 > 0we get

IE[φ(X)] =1√

2πσ2

w∞−∞

φ(x)e−(x−µ)2/(2σ2)dx.

In case (X,Y ) : Ω −→ R2 is a R2-valued couple of random variables whosedistribution admits a density fX,Y : R −→ R+ we have

IE[φ(X,Y )] =w∞−∞

w∞−∞

φ(x, y)fX,Y (x, y)dxdy,

for all sufficiently integrable function φ on R2.

The expectation of an absolutely continuous random variable satisfies thesame linearity property (16.21) as in the discrete case.

The variance of the random variable X is defined by

Var[X] := IE[X2]− (IE[X])2,

provided IE[|X|2] < ∞. If (Xk)k∈N is a sequence of independent randomvariables we have

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N. Privault

Var

[n∑k=1

Xk

]= IE

( n∑k=1

Xk

)2−(IE

[n∑k=1

Xk

])2

= IE

[n∑k=1

Xk

n∑l=1

Xl

]− IE

[n∑k=1

Xk

]IE

[n∑l=1

Xl

]

= IE

[n∑k=1

n∑l=1

XkXl

]−

n∑k=1

n∑l=1

IE[Xk] IE[Xl]

=

n∑k=1

IE[X2k ] +

∑1≤k 6=l≤n

IE[XkXl]−n∑k=1

(IE[Xk])2 −∑

1≤k 6=l≤n

IE[Xk] IE[Xl]

=n∑k=1

(IE[X2k ]− (IE[Xk])2)

=n∑k=1

Var [Xk]. (16.27)

Exercise: In caseX has a Gaussian distribution with mean µ ∈ R and varianceσ2 > 0, check that

µ = IE[X] and σ2 = IE[X2]− (IE[X])2.

The conditional expectation of an absolutely continuous random variable canbe defined as

IE[X | Y = y] =w∞−∞

xfX|Y=y(x)dx

where the conditional density fX|Y=y(x) is defined in (16.17), with the rela-tion

IE[X] = IE[IE[X | Y ]] (16.28)

as in the discrete case, since

IE[IE[X | Y ]] =w∞−∞

IE[X | Y = y]fY (y)dy =w∞−∞

w∞−∞

xfX|Y=y(x)fY (y)dxdy

=w∞−∞

xw∞−∞

f(X,Y )(x, y)dydx =w∞−∞

xfX(x)dx = IE[X],

where we used Relation (16.16) between the density of (X,Y ) and its marginalX.

For example, an exponentially distributed random variable X with prob-ability density function (16.10) has the expected value

IE[X] = λw∞0xe−λxdx =

1

λ.

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If X and Y are independent exponentially distributed random variables withparameters λ and µ, using (16.22) and (16.13) we can also compute theconditional expectation

IE[min(X,Y ) | X < Y ] = IE[X | X < Y ] =1

P(X < Y )IE[X1X<Y

]= µ(λ+ µ)

w∞0e−µy

w y

0xe−λxdxdy

= (λ+ µ)µ

λ2

w∞0e−µy(1− (1 + λy)e−λy)dy

= (λ+ µ)µ

λ2

(w∞0e−µydy −

w∞0e−µye−λydy − λ

w∞0e−µyye−λy)dy

)= (λ+ µ)

µ

λ2

(1

µ− 1

λ+ µ− λ

(λ+ µ)2

)= (λ+ µ)

µ

λ2

((λ+ µ)2

µ(λ+ µ)2− µ(λ+ µ)

µ(λ+ µ)2− λµ

µ(λ+ µ)2

)= (λ+ µ)

µ

λ2

((λ+ µ)2 − µ(λ+ µ)− λµ

µ(λ+ µ)2

)=

1

λ+ µ= IE[min(X,Y )]. (16.29)

Conditional Expectation

The construction of conditional expectation given above for discrete and ab-solutely continuous random variables can be generalized to σ-algebras.

For any p ≥ 1 we let

Lp(Ω) = F : Ω −→ R : IE[|F |p] <∞ (16.30)

denote the space of p-integrable random variables F : Ω −→ R. Given G ⊂ Fa sub σ-algebra of F and F ∈ L2(Ω,F ,P), the conditional expectation of Fgiven G, and denoted

IE[F | G],

can be defined to be the orthogonal projection of F onto L2(Ω,G,P).

That is, IE[F | G] is characterized by the relation

〈G,F − IE[F | G]〉 = IE[G(F − IE[F | G])] = 0,

i.e.

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N. Privault

IE[GF ] = IE[G IE[F | G]], (16.31)

for all bounded and G-measurable random variables G, where 〈·, ·〉 is thescalar product in L2(Ω,F ,P).

In other words, anytime the relation

IE[GF ] = IE[GX]

holds for all bounded and G-measurable random variables G, and a givenG-measurable random variable X, we can claim that

X = IE[F | G]

by uniqueness of the orthogonal projection onto the subspace L2(Ω,G,P) ofL2(Ω,F ,P).

The conditional expectation operator has the following properties.

i) IE[FG | G] = G IE[F | G] if G depends only on the information containedin G.

Proof: By the characterization (16.31) it suffices to show that

IE[HFG] = IE[HG IE[F |G]], (16.32)

for all bounded and G-measurable random variables G, H, which impliesIE[FG | G] = G IE[F | G].

Relation (16.32) holds from (16.31) because the product HG is G-measurable hence G can be replaced with HG in (16.31).

ii) IE[G|G] = G when G depends only on the information contained in G.

Proof: This is a consequence of point (i) above by taking F = 1.

iii) IE[IE[F |G] | H] = IE[F |H] if H ⊂ G, called the tower property.

Proof: First we note that (iii) holds when H = ∅, Ω because takingG = 1 in (16.31) yields

IE[F ] = IE[IE[F | G]]. (16.33)

Next, by the characterization (16.31) it suffices to show that

IE[H IE[F |G]] = IE[H IE[F |H]], (16.34)

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for all bounded and G-measurable random variables H, which will imply(iii) from (16.31).

In order to prove (16.34) we check that by (16.33) and point (i) abovewe have

IE[H IE[F |G]] = IE[IE[HF |G]] = IE[HF ]

= IE[IE[HF |H]] = IE[H IE[F |H]],

and we conclude by the characterization (16.31).

iv) IE[F |G] = IE[F ] when F “does not depend” on the information containedin G or, more precisely stated, when the random variable F is indepen-dent of the σ-algebra G.

Proof: It suffices to note that for all bounded G-measurable G we have

IE[FG] = IE[F ] IE[G] = IE[G IE[F ]],

and we conclude again by (16.31).

v) If G depends only on G and F is independent of G, then

IE[h(F,G)|G] = IE[h(x, F )]x=G.

The notion of conditional expectation can be extended from square-integrablerandom variables in L2(Ω) to integrable random variables in L1(Ω), cf. e.g.[43], Theorem 5.1.

Moment Generating Functions

The characteristic function of a random variable X is the function ΨX : R −→C defined by

ΨX(t) = IE[eitX ], t ∈ R.

The Laplace transform (or moment generating function) of a random vari-able X is the function ΦX : R −→ R defined by

ΦX(t) = IE[etX ], t ∈ R,

provided the expectation is finite.


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N. Privault

IE[Xn] =∂n

∂tΦX(0), n ≥ 1,

provided IE[|X|n] < ∞. The Laplace transform ΦX of a random variable Xwith density f : R −→ R+ satisfies

ΦX(t) =w∞−∞

etxf(x)dx, t ∈ R.

Note that in probability we are using the bilateral Laplace transform for whichthe integral is from −∞ to +∞.

The characteristic function ΨX of a random variable X with density f :R −→ R+ satisfies

ΨX(t) =w∞−∞

eitxf(x)dx, t ∈ R.

On the other hand, if X : Ω −→ N is a discrete random variable we have

ΨX(t) =

∞∑n=0

eitnP(X = n), t ∈ R.

The main applications of characteristic functions lie in the following theorems:

Theorem 16.1. Two random variables X : Ω −→ R and Y : Ω −→ R havesame distribution if and only if

ΨX(t) = ΨY (t), t ∈ R.

Theorem 16.1 is used to identify or to determine the probability distributionof a random variable X, by comparison with the characteristic function ΨYof a random variable Y whose distribution is known.

The characteristic function of a random vector (X,Y ) is the functionΨX,Y : R2 −→ C defined by

ΨX,Y (s, t) = IE[eisX+itY ], s, t ∈ R.

Theorem 16.2. Given two independent random variables X : Ω −→ R andY : Ω −→ R are independent if and only if

ΨX,Y (s, t) = ΨX(s)ΨY (t), s, t ∈ R.

A random variable X is Gaussian with mean µ and variance σ2 if and onlyif its characteristic function satisfies

IE[eiαX ] = eiαµ−α2σ2/2, α ∈ R. (16.35)

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In terms of Laplace transforms we have, replacing iα by α,

IE[eαX ] = eαµ+α2σ2/2, α ∈ R. (16.36)

From Theorems 16.1 and 16.2 we deduce the following proposition.

Proposition 16.1. Let X ' N (µ, σ2X) and Y ' N (ν, σ2

Y ) be independentGaussian random variables. Then X + Y also has a Gaussian distribution

X + Y ' N (µ+ ν, σ2X + σ2

Y ).

Proof. Since X and Y are independent, by Theorem 16.2 the characteristicfunction ΨX+Y of X + Y is given by

ΦX+Y (t) = ΦX(t)ΦY (t)

= eitµ−t2σ2X/2eitν−t

2σ2Y /2

= eit(µ+ν)−t2(σ2

X+σ2Y )/2, t ∈ R,

where we used (16.35). Consequently, the characteristic function of X + Y isthat of a Gaussian random variable with mean µ+ ν and variance σ2

X + σ2Y

and we conclude by Theorem 16.1.

Exercises

Exercise 1 Compute the expected value IE[X] of a Poisson random variableX with parameter λ > 0.

Exercise 2 Let X denote a centered Gaussian random variable with varianceη2, η > 0. Show that the probability P (eX > c) is given by

P (eX > c) = Φ(−(log c)/η),

where log = ln denotes the natural logarithm and

Φ(x) =1√2π

w x

−∞e−y

2/2dy, x ∈ R,

denotes the Gaussian cumulative distribution function.

Exercise 3 Let X ' N (µ, σ2) be a Gaussian random variable with parametersµ > 0 and σ2 > 0, and density function

f(x) =1√

2πσ2e−

(x−µ)2

2σ2 , x ∈ R.

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N. Privault

1. Write down IE[X] as an integral and show that

µ = IE[X].

2. Write down IE[X2] as an integral and show that

σ2 = IE[(X − IE[X])2].

3. Consider the function x 7→ (x−K)+ from R to R+, defined as

(x−K)+ =

x−K if x ≥ K,

0 if x ≤ K,

where K ∈ R be a fixed real number. Write down IE[(X − K)+] as anintegral and compute this integral.

Hints: (x − K)+ is zero when x < K, and when µ = 0 and σ = 1 theresult is

IE[(X −K)+] =1√2πe−

K2

2 −KΦ(−K),

where

Φ(x) :=w x

−∞e−

y2

2dy√2π, x ∈ R.

4. Write down IE[eX ] as an integral, and compute IE[eX ].

Exercise 4 Let X be a centered Gaussian random variable with varianceα2 > 0 and density x 7→ 1√

2πα2e−x

2/(2α2) and let β ∈ R.

1. Write down IE[(β−X)+] as an integral. Hint: (β−x)+ is zero when x > β.2. Compute this integral to show that

IE[(β −X)+] =α√2πe−

β2

2α2 + βΦ(β/α),

where

Φ(x) =w x

−∞e−

y2

2dy√2π, x ∈ R.

Exercise 5 Let X be a centered Gaussian random variable with varianceα2 > 0 and density x 7→ 1√

2πα2e−x

2/(2α2) and let β ∈ R.

1. Write down IE[(β + X)+] as an integral. Hint: (β + x)+ is zero whenx < −β.

2. Compute this integral to show that

IE[(β +X)+] =α√2πe−

β2

2α2 + βΦ(β/α),

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where

Φ(x) =w x

−∞e−

y2

2dy√2π, x ∈ R.

Exercise 6 Let X be a centered Gaussian random variable with variance v2.

1. Compute

IE[eσX1[K,∞[(xe

σX)]

=1√

2πv2

w∞1σ log Kx

eσy−y2/(2v2)dy.

Hint: use the decomposition

σy − y2

v2=v2σ2

4−(yv− vσ

2

)2.

2. Compute

IE[(em+X −K)+] =1√

2πv2

w∞−∞

(em+x −K)+e−x2

2v2 dx.

3. Compute the expectation (16.36) above.

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Exercise Solutions

Chapter 1

Exercise 1.1

1. The possible values of R are a and b.2. We have

IE∗[R] = aP∗(R = a) + bP∗(R = b)

= ab− rb− a + b

r − ab− a

= r.

3. By Theorem 1.1, there do not exist arbitrage opportunities in this marketsince there exists a risk-neutral measure P∗ from Question 2.

4. The risk-neutral measure is unique hence the market model is completeby Theorem 1.2.

5. Taking

η =α(1 + b)− β(1 + a)

π1(b− a)and ξ =

β − αS0(b− a)

,

we check that ηπ1 + ξS0(1 + a) = α

ηπ1 + ξS0(1 + b) = β,

which shows thatηπ1 + ξS1 = C.

6. We have

π(C) = ηπ0 + ξS0

=α(1 + b)− β(1 + a)

(1 + r)(b− a)+α− βa− b

449

N. Privault

=α(1 + b)− β(1 + a)− (1 + r)(α− β)

(1 + r)(b− a)

=αb− βa− r(α− β)

(1 + r)(b− a). (16.37)

7. We have

IE∗[C] = αP∗(R = a) + βP∗(R = b)

= αb− rb− a + β

r − ab− a . (16.38)

8. Comparing (16.37) and (16.38) above we do obtain

π(C) =1

1 + rIE∗[C]

9. The initial value π(C) of the portfolio is interpreted as the arbitrage priceof the option contract and it equals the expected value of the discountedpayoff.

10. We have

C = (K − S1)+ = (11− S1)+ =

11− S1 if K > S1,

0 if K ≤ S1.

11. We have

ξ =−(11− (1 + a))

b− a = −2

3, η =

(1 + b)(11− (1 + a))

(1 + r)(b− a)=

8

1.05.

12. The arbitrage price π(C) of the contingent claim C is

π(C) = ηπ0 + ξS0 = 6.952.

Chapter 2

Exercise 2.1

1. The possible values of Rt are a and b.2. We have

IE∗[Rt+1 | Ft] = aP∗(Rt+1 = a | Ft) + bP∗(Rt+1 = b | Ft)

= ab− rb− a + b

r − ab− a = r.

3. We have

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IE∗[St+k | Ft] =

k∑i=0

(r − ab− a

)i(b− rb− a

)k−i(k

i

)(1 + b)i(1 + a)k−iSt

= St

k∑i=0

(k

i

)(r − ab− a (1 + b)

)i(b− rb− a (1 + a)

)k−i= St

(r − ab− a (1 + b) +

b− rb− a (1 + a)

)k= (1 + r)kSt.

Assuming that the formula holds for k = 1, its extension to k ≥ 2 can alsobe proved recursively from the “tower property” (16.24) of conditionalexpectations, as follows:

IE∗[St+k | Ft] = IE∗[IE∗[St+k | Ft+k−1] | Ft]= (1 + r) IE∗[St+k−1 | Ft]= (1 + r) IE∗[IE∗[St+k−1 | Ft+k−2] | Ft]= (1 + r)2 IE∗[St+k−2 | Ft]= (1 + r)2 IE∗[IE∗[St+k−2 | Ft+k−3] | Ft]= (1 + r)3 IE∗[St+k−3 | Ft]= · · ·= (1 + r)k−2 IE∗[St+2 | Ft]= (1 + r)k−2 IE∗[IE∗[St+2 | Ft+1] | Ft]= (1 + r)k−1 IE∗[St+1 | Ft]= (1 + r)kSt.

Chapter 3

Exercise 3.1

1. The condition VN = C reads ηNπN + ξN (1 + a)SN−1 = (1 + a)SN−1 −K

ηNπN + ξN (1 + b)SN−1 = (1 + b)SN−1 −K

from which we deduce ξN = 1 and ηN = −K(1 + r)−N/π0.2. We have ηN−1πN−1 + ξN−1(1 + a)SN−1 = ηNπN−1 + ξN (1 + a)SN−1

ηN−1πN−1 + ξN−1(1 + b)SN−1 = ηNπN−1 + ξN (1 + b)SN−1,

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N. Privault

which yields ξN−1 = ξN = 1 and ηN−1 = ηN = −K(1 + r)−N/π0.Similarly, solving the self-financing condition ηtπt + ξt(1 + a)St = ηt+1πt + ξt+1(1 + a)St

ηtπt + ξt(1 + b)St = ηt+1πt + ξt+1(1 + b)St,

at time t yields ξt = 1 and ηt = −K(1 + r)−N/π0, t = 1, 2, . . . , N .3. We have

πt(C) = Vt = ηtπt+ξtSt = St−K(1+r)−Nπt/π0 = St−K(1+r)−(N−t).

4. For all t = 0, 1, . . . , N we have

(1 + r)−(N−t) IE∗[C | Ft] = (1 + r)−(N−t) IE∗[SN −K | Ft],= (1 + r)−(N−t) IE∗[SN | Ft]− (1 + r)−(N−t) IE∗[K | Ft]= (1 + r)−(N−t)(1 + r)N−tSt −K(1 + r)−(N−t)

= St −K(1 + r)−(N−t)

= Vt = πt(C).

Exercise 3.2

1. This model admits a unique risk-neutral measure P∗ because we havea < r < b. We have

P∗(Rt = a) =b− rb− a =

0.07− 0.05

0.07− (−0.02),

and

P(Rt = b) =r − ab− a =

0.05− (−0.02)

0.07− (−0.02),

t = 1, . . . , N .2. There are no arbitrage opportunities in this model, due to the existence

of a risk-neutral measure.3. This market model is complete because the risk-neutral measure is

unique.4. We have

C = (SN )2,

henceH = (SN )2/(1 + r)N = h(XN ),

withh(x) = x2(1 + r)−N .

Now we haveVt = vt(Xt),

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where the function vt(x) is given by

vt(x) =

N−t∑k=0

(N − t)!k!(N − t− k)!

×(r − ab− a

)k (b− rb− a

)N−t−kh

(x

(1 + b

1 + r

)k (1 + a

1 + r

)N−t−k)

= x2(1 + r)−NN−t∑k=0

(N − t)!k!(N − t− k)!

×(r − ab− a

)k (b− rb− a

)N−t−k (1 + b

1 + r

)2k (1 + a

1 + r

)2(N−t−k)

= x2(1 + r)−NN−t∑k=0

(N − t)!k!(N − t− k)!

×(

(r − a)(1 + b)2

(b− a)(1 + r)2

)k ((b− r)(1 + a)2

(b− a)(1 + r)2

)N−t−k= x2(1 + r)−N

((r − a)(1 + b)2

(b− a)(1 + r)2+

(b− r)(1 + a)2

(b− a)(1 + r)2

)N−t=x2((r − a)(1 + b)2 + (b− r)(1 + a)2

)N−t(1 + r)N−2t(b− a)N−t

=x2((r − a)(1 + 2b+ b2) + (b− r)(1 + 2a+ a2)

)N−t(1 + r)N−2t(b− a)N−t

=x2(r(1 + 2b+ b2)− a(1 + 2b+ b2) + b(1 + 2a+ a2)− r(1 + 2a+ a2)

)N−t(1 + r)N−2t(b− a)N−t

= x2(1 + r(2 + a+ b)− ab)N−t

(1 + r)N−2t.

5. We have

ξ1t =vt

(1+b1+rXt−1

)− vt

(1+a1+rXt−1

)Xt−1(b− a)/(1 + r)

= Xt−1

(1+b1+r

)2−(

1+a1+r

)2(b− a)/(1 + r)

(1 + r(2 + a+ b)− ab)N−t(1 + r)N−2t

= St−1(2 + b+ a)(1 + r(2 + a+ b)− ab)N−t

(1 + r)N−t, t = 1, . . . , N,

representing the quantity of the risky asset to be present in the portfolioat time t. On the other hand we have

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N. Privault

ξ0t =Vt − ξ1tXt

X0t

=Vt − ξ1tXt

π0

= Xt(1 + r(2 + a+ b)− ab)N−tXt −Xt−1(2 + b+ a)/(1 + r)

π0(1 + r)N−2t

= St(1 + r(2 + a+ b)− ab)N−tSt − St−1(2 + b+ a)

π0(1 + r)N

= −(St−1)2(1 + r(2 + a+ b)− ab)N−t (1 + a)(1 + b)

π0(1 + r)N,

t = 1, . . . , N .6. Let us check that the portfolio is self-financing. We have

ξt+1 · St = ξ0t+1S0t + ξ1t+1S

1t

= −(St)2(1 + r(2 + a+ b)− ab)N−t−1 (1 + a)(1 + b)

π0(1 + r)NS0t

+(St)2(2 + b+ a)

(1 + r(2 + a+ b)− ab)N−t−1(1 + r)N−t−1

= (St)2 (1 + r(2 + a+ b)− ab)N−t−1

(1 + r)N−t

× ((2 + b+ a)(1 + r)− (1 + a)(1 + b))

= (Xt)2(1 + r(2 + a+ b)− ab)N−t 1

(1 + r)N−3t

= (1 + r)tVt

= ξt · St, t = 1, . . . , N.

Exercise 3.3

1. We have

Vt = ξtSt + ηtπt

= ξt(1 +Rt)St−1 + ηt(1 + r)πt−1.

2. We have

IE∗[Rt|Ft−1] = aP∗(Rt = a | Ft−1) + bP∗(Rt = b | Ft−1)

= ab− rb− a + b

r − ab− a

= br

b− a − ar

b− a= r.

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3. By the result of Question 1 we have

IE∗[Vt | Ft−1] = IE∗[ξt(1 +Rt)St−1 | Ft−1] + IE∗[ηt(1 + r)πt−1 | Ft−1]

= ξtSt−1 IE∗[1 +Rt | Ft−1] + (1 + r) IE∗[ηtπt−1 | Ft−1]

= (1 + r)ξtSt−1 + (1 + r)ηtπt−1

= (1 + r)ξtSt + (1 + r)ηtπt

= (1 + r)Vt−1,

where we used the self-financing condition.4. We have

Vt−1 =1

1 + rIE∗[Vt | Ft−1]

=3

1 + rP∗(Rt = a | Ft−1) +

8

1 + rP∗(Rt = b | Ft−1)

=1

1 + 0.15

(3

0.25− 0.15

0.25− 0.05+ 8

0.15− 0.05

0.25− 0.05

)=

1

1.15

(3

2+

8

2

)= 4.78.

Chapter 4

Exercise 4.1

1. We need to check whether the four properties of the definition of Brow-nian motion are satisfied. Checking Conditions (i) to (iii) does not poseany particular problem since the time changes t 7→ c + t, t 7→ t/c2 andt 7→ ct2 are deterministic, continuous, and increasing. As for Condi-tion (iv), Bc+t −Bc+s clearly has a centered Gaussian distribution withvariance t, and the same property holds for cBt/c2 since

Var (c(Bt/c2 −Bs/c2)) = c2Var (Bt/c2 −Bs/c2) = c2(t− s)/c2 = t− s.

As a consequence, (a) and (b) are standard Brownian motions.

Concerning (c), we note that Bct2 is a centered Gaussian random variablewith variance ct2 - not t, hence (Bct2)t∈R+ is not a standard Brownianmotion.

2. We havew T

02dBt = 2(BT −B0) = 2BT , which has a Gaussian law with

mean 0 and variance 4T . On the other hand,

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N. Privault

w T

0(2×1[0,T/2](t)+1(T/2,T ](t))dBt = 2(BT/2−B0)+(BT−BT/2) = BT+BT/2,

which has a Gaussian law with mean 0 and variance 4(T/2)+T/2 = 5T/2.

3. The stochastic integralw 2π

0sin(t) dBt has a Gaussian distribution with

mean 0 and variance

w 2π

0sin2(t)dt =

w 2π

0

1− cos(2t)

2dt = π.

4. If 0 ≤ s ≤ t we have

IE[BtBs] = IE[(Bt−Bs)Bs]+IE[B2s ] = IE[(Bt−Bs)] IE[Bs]+IE[B2

s ] = 0+s = s,

and similarly we obtain IE[BtBs] = t when 0 ≤ t ≤ s, hence in generalwe have IE[BtBs] = min(s, t), s, t ≥ 0.

5. We have

d(f(t)Bt) = f(t)dBt +Btdf(t) + df(t) · dBt= f(t)dBt +Btf

′(t)dt+ f ′(t)dt · dBt= f(t)dBt +Btf

′(t)dt,

and by integration on both sides we get

0 = f(T )BT − f(0)B0

=w T

0d(f(t)Bt)

=w T

0f(t)dBt +

w T

0Btf

′(t)dt,

hence the conclusion.

Exercise 4.2 Let f ∈ L2([0, T ]). We have

E[er T0f(s)dBs

∣∣∣Ft] = exp

(w t

0f(s)dBs +

1

2

w T

0|f(s)|2ds

), 0 ≤ t ≤ T.

Exercise 4.3 We have

E

[exp

(β

w T

0BtdBt

)]= E

[exp

(β(B2

T − T )/2)]

= e−βT/2E[exp

(eβ(BT )

2/2)]

=e−βT/2√

2πT

w∞−∞

e(β−1T ) x

2

2 dx

=e−βT/2√1− βT .

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for all β < 1/T .

Exercise 4.4 We have f(t) = f(0)ect (interest rate compounding) and

St = S0eσBt−σ2t/2+rt, t ∈ R+, (geometric Brownian motion).

Exercise 4.5

1. By (4.25) we have

d(XTt /(T − t)) =

dXTt

T − t +XTt

(T − t)2 dt = σdBtT − t ,

hence by integration using the initial condition X0 = 0 we have

XTt

T − t = σw t

0

1

T − sdBs, t ∈ [0, T ].

2. We have

IE[XTt ] = σ(T − t) IE

[w t

0

1

T − sdBs]

= 0.

3. Using the Ito isometry we have

Var[XTt ] = σ2(T − t)2 Var

[w t

0

1

T − sdBs]

= σ2(T − t)2w t

0

1

(T − s)2 ds

= σ2(T − t)2(

1

T − t −1

T

)= σ2(1− t/T ).

4. We have Var[XTT ] = 0 hence XT

T = IE[XTT ] = 0 by Question 2.

Exercise 4.6 Exponential Vasicek model.

1. We have zt = e−atz0 + σw t

0e−a(t−s)dBs.

2. We have yt = e−aty0 +θ

a(1− e−at) + σ

w t

0e−a(t−s)dBs.

3. We have dxt = xt

(θ +

σ2

2− a log xt

)dt+ σxtdBt.

4. We have rt = exp

(e−at log r0 +

θ

a(1− e−at) + σ

w t

0e−a(t−s)dBs

), with

η = θ + σ2/2.5. We have

IE[rt] = exp

(e−at log r0 +

θ

a(1− e−at) +

σ2

4a(1− e−2at)

).

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N. Privault

6. We have limt→∞

IE[rt] = exp

(θ

a+σ2

4a

).

Exercise 4.7 Cox-Ingersoll-Ross (CIR) model.

1. We have rt = r0 +w t

0(α− βrs)ds+ σ

w t

0

√rsdBs.

2. Using the fact that the expectation of the stochastic integral with respectto Brownian motion is zero, we get, taking expectations on both sides ofthe above integral equation: u′(t) = α− βu(t).

3. Apply Ito’s formula to

r2t = f

(r0 +

w t

0(α− βrs)ds+ σ

w t

0

√rsdBs

),

with f(x) = x2, to obtain

d(rt)2 = rt(σ

2 + 2α− 2βrt)dt+ 2rtσ√rtdBt. (16.39)

4. Taking again the expectation on both sides of (16.39) we get

IE[r2t ] = IE[r20] +w t

0(σ2 IE[rt] + 2α IE[rt]− 2β IE[r2t ])dt,

and after differentiation with respect to t this yields

v′t = (σ2 + 2α)u(t)− 2βv(t).

Exercise 4.8

1. We have

St = eXt

= eX0 +w t

0use

XsdBs +w t

0vse

Xsds+1

2

w t

0u2se

Xsds

= eX0 + σw t

0eXsdBs + ν

w t

0eXsds+

σ2

2

w t

0eXsds

= S0 + σw t

0SsdBs + ν

w t

0Ssds+

σ2

2

w t

0Ssds.

2. Let r > 0. The process (St)t∈R+satisfies the stochastic differential equa-

tiondSt = rStdt+ σStdBt

when r = ν + σ2/2.3. Let the process (St)t∈R+ be defined by St = S0e

σBt+νt, t ∈ R+. Using

the decomposition ST = Steσ(BT−Bt)+ντ , we have

P(ST > K | St = x) = P(Steσ(BT−Bt)+ν(T−t) > K | St = x)

458


"



= P(xeσ(BT−Bt)+ν(T−t) > K)

= P(eσ(BT−Bt) > Ke−ν(T−t)/x)

= Φ

(− log(Ke−ν(T−t)/x)

σ√τ

)= Φ

(log(x/K) + ντ

σ√τ

),

where τ = T − t.4. We have

η2 = Var[X] = Var[σ(BT −Bt)] = σ2 Var[BT −Bt] = σ2(T − t),

hence η = σ√T − t.

Chapter 5

Exercise 5.1

1. We have

St = S0eαt + σ

w t

0eα(t−s)dBs.

2. We have αM = r.3. After computing the conditional expectation

C(t, x) = e−r(T−t) exp

(xer(T−t) +

σ2

4r(e2r(T−t) − 1)

).

4. Here we need to note that the usual Black-Scholes argument applies andyields ζt = ∂C(t, St)/∂x, that is

ζt = exp

(Ste

r(T−t) +σ2

4r(e2r(T−t) − 1)

).

Exercise 5.2

1. We have, counting approximately 46 days to maturity,

d− =(r − 1

2σ2)(T − t) + log St

K

σ√T − t

=(0.04377− 1

2 (0.9)2)(46/365) + log 17.236.08

0.9√

46/365

= −2.46,

and

" 459



N. Privault

d+ = d− + 0.9√

46/365 = −2.14.

From the attached table we get

Φ(d+) = Φ(−2.14) = 0.0162

andΦ(d−) = Φ(−2.46) = 0.0069,

hence

f(t, St) = StΦ(d+)−Ke−r(T−t)Φ(d−)

= 17.2× 0.0162− 36.08 ∗ e−0.04377×46/365 × 0.0069

= HK$ 0.031.

2. We have

ηt =∂f

∂x(t, St) = Φ(d+) = Φ(−2.14) = 0.0162, (16.40)

hence one should only hold a fractional quantity 16.2 of the risky assetin order to hedge 1000 such call options when σ = 0.90.

3. From the curve it turns out that when f(t, St) = 10× 0.023 = HK$ 0.23,the volatility σ is approximately equal to σ = 122%.

This approximate value of implied volatility can be found under the col-umn “Implied Volatility (IV.)” on this set of market data from the HongKong Stock Exchange:Derivative Warrant Search

Link to Relevant Exchange Traded Options

Updated: 6 November 2008 Basic Data Market Data

DWCode

Issuer UL Call/Put

DWType

Listing(D-M-Y)

Maturity(D-M-Y)

Strike Entitle-ment

Ratio^

TotalIssueSize

O/S(%)

Delta(%)

IV.(%)

DayHigh($)

DayLow($)

ClosingPrice #

($)

T/O('000)

ULPrice($)

Base ListingDocument

Supplemental&

Announcements

01897 FB 00066 Call Standard 18-12-2007 23-12-2008 36.08 10 138,000,000 16.43 0.780 125.375 0.000 0.000 0.023 0 17.200

04348 BP 00066 Call Standard 18-12-2007 23-02-2009 38.88 10 300,000,000 0.25 0.767 88.656 0.000 0.000 0.024 0 17.200

04984 AA 00066 Call Standard 02-06-2005 22-12-2008 12.88 10 300,000,000 0.36 8.075 128.202 0.000 0.000 0.540 0 17.200

05931 SB 00066 Call Standard 27-03-2008 29-12-2008 27.868 10 200,000,000 0.04 2.239 126.132 0.000 0.000 0.086 0 17.200

09133 CT 00066 Call Standard 31-01-2008 08-12-2008 36.88 10 200,000,000 0.15 0.416 133.443 0.000 0.000 0.010 0 17.200

13436 SG 00066 Call Standard 14-05-2008 30-04-2009 32 10 200,000,000 0.10 1.059 61.785 0.000 0.000 0.031 0 17.200

13562 BP 00066 Call Standard 26-05-2008 08-12-2008 30 10 150,000,000 0.00 0.987 127.080 0.000 0.000 0.026 0 17.200

13688 RB 00066 Call Standard 04-06-2008 20-02-2009 26.6 10 200,000,000 7.17 0.706 49.625 0.000 0.000 0.013 0 17.200

13764 SG 00066 Call Standard 13-06-2008 26-02-2009 28 10 300,000,000 0.31 0.549 49.880 0.010 0.010 0.010 6 17.200

13785 ML 00066 Call Standard 17-06-2008 19-01-2009 27.38 10 100,000,000 0.50 0.714 63.598 0.000 0.000 0.014 0 17.200

13821 JP 00066 Call Standard 18-06-2008 18-12-2008 28.8 10 100,000,000 0.81 0.670 91.664 0.000 0.000 0.014 0 17.200

14111 UB 00066 Call Standard 09-07-2008 16-02-2009 23.88 10 500,000,000 0.88 2.288 66.247 0.000 0.000 0.068 0 17.200

14264 BI 00066 Call Standard 16-07-2008 25-02-2009 26.38 10 200,000,000 0.03 1.250 58.172 0.000 0.000 0.030 0 17.200

14305 DB 00066 Call Standard 22-07-2008 09-03-2009 27 10 300,000,000 0.00 0.000 0.000 0.000 0.000 99,999,999.000 0 17.200

14489 FB 00066 Call Standard 06-08-2008 29-06-2009 28.08 10 175,000,000 0.15 1.681 52.209 0.000 0.000 0.053 0 17.200

14512 MB 00066 Call Standard 08-08-2008 26-02-2009 26 10 300,000,000 0.86 1.273 56.527 0.000 0.000 0.030 0 17.200

14531 UB 00066 Call Standard 08-08-2008 11-05-2009 26.88 10 500,000,000 0.00 3.261 84.055 0.000 0.000 0.162 0 17.200

14548 CT 00066 Call Standard 12-08-2008 25-05-2009 26.88 10 400,000,000 0.03 2.749 70.622 0.000 0.000 0.117 0 17.200

14571 CT 00066 Put Standard 15-08-2008 22-06-2009 26 10 240,000,000 0.06 (6.751) 71.243 0.000 0.000 1.020 0 17.200

14925 CT 00066 Call Standard 18-09-2008 12-10-2009 28.88 10 330,000,000 0.00 0.000 0.000 0.000 0.000 99,999,999.000 0 17.200

15159 JP 00066 Call Standard 29-09-2008 12-10-2009 28.08 10 100,000,000 2.00 3.270 66.344 0.000 0.000 0.169 0 17.200

15257 RB 00066 Call Standard 15-10-2008 30-07-2009 22 10 100,000,000 0.41 4.158 60.731 0.280 0.201 0.201 234 17.200

15339 CT 00066 Call Standard 17-10-2008 19-10-2009 20 10 200,000,000 0.07 5.085 58.805 0.290 0.290 0.285 12 17.200

^ The entitlement ratio in general represents the number of derivative warrants required to be exercised into one share or one unit of the underlying asset (subject to any adjustments as may benecessary to reflect any capitalization, rights issue, distribution or the like).# Closing price value = 99999999 when it is not available.Delayed data on Delta and Implied Volatility of Derivative Warrants are provided by Reuters.Implied Volatility is displayed as 200 even if its value is greater than 200.Users should not use such data provided by Reuters for commercial purposes without its prior written consent.

For Underlying Stock price, please refer to Today's Stock Quote of Investment Service Centre.If you are interested in real time data subscription, please click to view the list of Realtime Data Information Vendors.

Print http://www.hkex.com.hk/dwrc/search/dwlist.asp

1 of 1 11/08/2008 01:57 PM

Derivative Warrant Search Link to Relevant Exchange Traded Options

Updated: 6 November 2008 Basic Data Market Data

DWCode

Issuer UL Call/Put

DWType

Listing(D-M-Y)

Maturity(D-M-Y)

Strike Entitle-ment

Ratio^

TotalIssueSize

O/S(%)

Delta(%)

IV.(%)

DayHigh($)

DayLow($)

ClosingPrice #

($)

T/O('000)

ULPrice($)

Base ListingDocument

Supplemental&

Announcements

01897 FB 00066 Call Standard 18-12-2007 23-12-2008 36.08 10 138,000,000 16.43 0.780 125.375 0.000 0.000 0.023 0 17.200

04348 BP 00066 Call Standard 18-12-2007 23-02-2009 38.88 10 300,000,000 0.25 0.767 88.656 0.000 0.000 0.024 0 17.200

04984 AA 00066 Call Standard 02-06-2005 22-12-2008 12.88 10 300,000,000 0.36 8.075 128.202 0.000 0.000 0.540 0 17.200

05931 SB 00066 Call Standard 27-03-2008 29-12-2008 27.868 10 200,000,000 0.04 2.239 126.132 0.000 0.000 0.086 0 17.200

09133 CT 00066 Call Standard 31-01-2008 08-12-2008 36.88 10 200,000,000 0.15 0.416 133.443 0.000 0.000 0.010 0 17.200

13436 SG 00066 Call Standard 14-05-2008 30-04-2009 32 10 200,000,000 0.10 1.059 61.785 0.000 0.000 0.031 0 17.200

13562 BP 00066 Call Standard 26-05-2008 08-12-2008 30 10 150,000,000 0.00 0.987 127.080 0.000 0.000 0.026 0 17.200

13688 RB 00066 Call Standard 04-06-2008 20-02-2009 26.6 10 200,000,000 7.17 0.706 49.625 0.000 0.000 0.013 0 17.200

13764 SG 00066 Call Standard 13-06-2008 26-02-2009 28 10 300,000,000 0.31 0.549 49.880 0.010 0.010 0.010 6 17.200

13785 ML 00066 Call Standard 17-06-2008 19-01-2009 27.38 10 100,000,000 0.50 0.714 63.598 0.000 0.000 0.014 0 17.200

13821 JP 00066 Call Standard 18-06-2008 18-12-2008 28.8 10 100,000,000 0.81 0.670 91.664 0.000 0.000 0.014 0 17.200

14111 UB 00066 Call Standard 09-07-2008 16-02-2009 23.88 10 500,000,000 0.88 2.288 66.247 0.000 0.000 0.068 0 17.200

14264 BI 00066 Call Standard 16-07-2008 25-02-2009 26.38 10 200,000,000 0.03 1.250 58.172 0.000 0.000 0.030 0 17.200

14305 DB 00066 Call Standard 22-07-2008 09-03-2009 27 10 300,000,000 0.00 0.000 0.000 0.000 0.000 99,999,999.000 0 17.200

14489 FB 00066 Call Standard 06-08-2008 29-06-2009 28.08 10 175,000,000 0.15 1.681 52.209 0.000 0.000 0.053 0 17.200

14512 MB 00066 Call Standard 08-08-2008 26-02-2009 26 10 300,000,000 0.86 1.273 56.527 0.000 0.000 0.030 0 17.200

14531 UB 00066 Call Standard 08-08-2008 11-05-2009 26.88 10 500,000,000 0.00 3.261 84.055 0.000 0.000 0.162 0 17.200

14548 CT 00066 Call Standard 12-08-2008 25-05-2009 26.88 10 400,000,000 0.03 2.749 70.622 0.000 0.000 0.117 0 17.200

14571 CT 00066 Put Standard 15-08-2008 22-06-2009 26 10 240,000,000 0.06 (6.751) 71.243 0.000 0.000 1.020 0 17.200

14925 CT 00066 Call Standard 18-09-2008 12-10-2009 28.88 10 330,000,000 0.00 0.000 0.000 0.000 0.000 99,999,999.000 0 17.200

15159 JP 00066 Call Standard 29-09-2008 12-10-2009 28.08 10 100,000,000 2.00 3.270 66.344 0.000 0.000 0.169 0 17.200

15257 RB 00066 Call Standard 15-10-2008 30-07-2009 22 10 100,000,000 0.41 4.158 60.731 0.280 0.201 0.201 234 17.200

15339 CT 00066 Call Standard 17-10-2008 19-10-2009 20 10 200,000,000 0.07 5.085 58.805 0.290 0.290 0.285 12 17.200

^ The entitlement ratio in general represents the number of derivative warrants required to be exercised into one share or one unit of the underlying asset (subject to any adjustments as may benecessary to reflect any capitalization, rights issue, distribution or the like).# Closing price value = 99999999 when it is not available.Delayed data on Delta and Implied Volatility of Derivative Warrants are provided by Reuters.Implied Volatility is displayed as 200 even if its value is greater than 200.Users should not use such data provided by Reuters for commercial purposes without its prior written consent.

For Underlying Stock price, please refer to Today's Stock Quote of Investment Service Centre.If you are interested in real time data subscription, please click to view the list of Realtime Data Information Vendors.

Print http://www.hkex.com.hk/dwrc/search/dwlist.asp

1 of 1 11/08/2008 01:57 PM

Remark: a typical value for the volatility in standard market conditionswould be around 20%. The observed volatility value σ = 1.22 per year isactually quite high.

Exercise 5.3

1. We find h(x) = x−K.

460


"



2. Letting g(t, x), the PDE rewrites as

r(x− α(t)) = −α′(t) + rx,

hence α(t) = α(0)ert and g(t, x) = x− α(0)ert. The final condition

g(T, x) = h(x) = x−K

yields α(0) = KerT and g(t, x) = x−Ke−r(T−t).3. We have

ξt =∂g

∂x(t, St) = 1,

hence

ηt =Vt − ξtSt

At=g(t, St)− St

At=St −Ke−r(T−t) − St

At= −Ke−rT .

Note that we could also have directly used the identification

Vt = g(St, t) = St −Ke−r(T−t) = St −Ke−rTAt = ξtSt + ηtAt,

which immediately yields ξt = 1 and ηt = −Ke−rT .

Exercise 5.4

1. We have

Ct = e−r(T−t) IE∗[ST −K | Ft]= e−r(T−t) IE∗[ST | Ft]−Ke−r(T−t)= ert IE∗[e−rTST | Ft]−Ke−r(T−t)= erte−rtSt −Ke−r(T−t)= St −Ke−r(T−t).

We can check that the function g(x, t) = x − Ke−r(T−t) satisfies theBlack-Scholes PDE

rg(x, t) =∂g

∂t(x, t) + rx

∂g

∂x(x, t) +

σ2

2x2∂2g

∂x2(x, t)

with terminal condition g(x, T ) = x−K, since ∂g(x, t)/∂t = −rKe−r(T−t)and ∂g(x, t)/∂x = 1.

2. We simply take ξt = 1 and ηt = −Ke−rT in order to have

Ct = ξtSt + ηtert = St −Ke−r(T−t), t ∈ [0, T ].

Note again that this hedging strategy is constant over time, and therelation ξt = ∂g(St, t)/∂x for the Delta, cf. (16.40), is satisfied.

" 461



N. Privault

Exercise 5.5 Using Ito’s formula and the fact that the expectation of thestochastic integral with respect to (Wt)t∈R+

is zero, cf. Relation (4.11), wehave

C(x, T ) = e−rT IE [φ(ST )]

= φ(x)− IE

[w T

0re−rsφ(St)dt

]+ IE

[rw T

0e−rtφ′(St)Stdt

]+ IE

[w T

0e−rtφ′(St)bs(St)dWt

]+

1

2IE

[w T

0e−rtφ′′(St)σ

2(St)dt

]= φ(x)−

w T

0re−rt IE [φ(St)] dt+ r

w T

0e−rt IE [φ′(St)St] dt

+1

2

w T

0e−rt IE

[φ′′(St)σ

2(St)]dt.

Chapter 6

Exercise 6.1

1. For all t ∈ [0, T ] we have

C(t, St) = e−r(T−t)S2t IE

[S2T

S2t

]= e−r(T−t)S2

t IE[e2σ(BT−Bt)−σ

2(T−t)+2r(T−t)]

= S2t e

(r+σ2)(T−t).

2. For all t ∈ [0, T ] we have

ξt =∂C

∂x(t, x)|x=St = 2Ste

(r+σ2)(T−t),

and

ηt =C(t, St)− ξtSt

At=e−rt

A0(S2t e

(r+σ2)(T−t) − 2S2t e

(r+σ2)(T−t))

= −S2t

A0eσ

2(T−t)+r(T−2t).

3. We have

dC(t, St) = d(S2t e

(r+σ2)(T−t))

= −(r + σ2)e(r+σ2)(T−t)S2

t dt+ e(r+σ2)(T−t)d(S2

t )

462


"



= −(r + σ2)e(r+σ2)(T−t)S2

t dt+ e(r+σ2)(T−t)(2StdSt + σ2S2

t dt)

= −re(r+σ2)(T−t)S2t dt+ 2Ste

(r+σ2)(T−t)dSt,

and

ξtdSt + ηtdAt = 2Ste(r+σ2)(T−t)dSt − r

S2t

A0eσ

2(T−t)+r(T−2t)Atdt

= 2Ste(r+σ2)(T−t)dSt − rS2

t eσ2(T−t)+r(T−t)dt,

hence we can check that the strategy is self-financing since dC(t, St) =ξtdSt + ηtdAt.

Exercise 6.2

1. We have

St = S0ert + σ

w t

0er(t−s)dBs.

2. We have

St = S0 + σw t

0e−rsdBs,

which is a martingale, being a stochastic integral with respect to Brow-nian motion.

This fact can also be proved directly by computing the conditional ex-pectation E[St | Fs] and showing it is equal to Ss:

E[St | Fs] = E

[S0 + σ

w t

0e−rudBu | Fs

]= E[S0] + σE

[w t

0e−rudBu | Fs

]= S0 + σE

[w s

0e−rudBu | Fs

]+ σE

[w t

se−rudBu | Fs

]= S0 + σ

w s

0e−rudBu + σE

[w t

se−rudBu

]= S0 + σ

w s

0e−rudBu

= Ss.

3. We have

C(t, St) = e−r(T−t)E[exp(ST )|Ft]

= e−r(T−t)E

[exp

(erTS0 + σ

w T

0er(T−u)dBu

) ∣∣∣Ft]= e−r(T−t)E

[exp

(erTS0 + σ

w t

0er(T−u)dBu + σ

w T

ter(T−u)dBu

) ∣∣∣Ft]" 463



N. Privault

= exp(−r(T − t) + er(T−t)St

)E

[exp

(σ

w T

ter(T−u)dBu

) ∣∣∣Ft]= exp

(−r(T − t) + er(T−t)St

)E

[exp

(σ

w T

ter(T−u)dBu

)]= exp

(−r(T − t) + er(T−t)St

)exp

(σ2

2

w T

t(er(T−u))2du

)= exp

(−r(T − t) + er(T−t)St +

σ2

4r(e2r(T−t) − 1)

).

4. We have

ξt =∂C

∂x(t, St) = exp

(Ste

r(T−t) +σ2

4r(e2r(T−t) − 1)

)and

ηt =C(t, St)− ξtSt

At

=e−r(T−t)

Atexp

(Ste

r(T−t) +σ2

4r(e2r(T−t) − 1)

)− StAt

exp

(Ste

r(T−t) +σ2

4r(e2r(T−t) − 1)

).

5. We have

dC(t, St) = re−r(T−t) exp

(Ste

r(T−t) +σ2

4r(e2r(T−t) − 1)

)dt

−rSt exp

(Ste

r(T−t) +σ2

4r(e2r(T−t) − 1)

)dt

−σ2

2er(T−t) exp

(Ste

r(T−t) +σ2

4r(e2r(T−t) − 1)

)dt

+ exp

(Ste

r(T−t) +σ2

4r(e2r(T−t) − 1)

)dSt

+1

2er(T−t) exp

(Ste

r(T−t) +σ2

4r(e2r(T−t) − 1)

)σ2dt

= re−r(T−t) exp

(Ste

r(T−t) +σ2

4r(e2r(T−t) − 1)

)dt

−rSt exp

(Ste

r(T−t) +σ2

4r(e2r(T−t) − 1)

)dt

+ξtdSt.

On the other hand we have

464


"



ξtdSt + ηtdAt = ξtdSt

+re−r(T−t) exp

(Ste

r(T−t) +σ2

4r(e2r(T−t) − 1)

)dt

−rSt exp

(Ste

r(T−t) +σ2

4r(e2r(T−t) − 1)

)dt,

showing thatdC(t, St) = ξtdSt + ηtdAt,

and confirming that the strategy (ξt, ηt)t∈R+ is self-financing.

Exercise 6.3

1. We have

∂f

∂t(t, x) = (r − σ2/2)f(t, x),

∂f

∂x(t, x) = σf(t, x),

and∂2f

∂x2(t, x) = σ2f(t, x),

hence

dSt = df(t, Bt)

=∂f

∂t(t, Bt)dt+

∂f

∂x(t, Bt)dBt +

1

2

∂2f

∂x2(t, Bt)dt

=

(r − 1

2σ2

)f(t, Bt)dt+ σf(t, Bt)dBt +

1

2σ2f(t, Bt)dt

= rf(t, Bt)dt+ σf(t, Bt)dBt

= rStdt+ σStdBt.

2. We have

E[eσBT |Ft] = E[eσ(BT−Bt+Bt)|Ft]= eσBtE[eσ(BT−Bt)|Ft]= eσBtE[eσ(BT−Bt)]

= eσBt+σ2(T−t)/2.

3. We have

E[ST |Ft] = E[eσBT+rT−σ2T/2|Ft]

= erT−σ2T/2E[eσBT |Ft]

= erT−σ2T/2eσBt+σ

2(T−t)/2

= erT+σBt−σ2t/2

= er(T−t)+σBt+rt−σ2t/2

" 465



N. Privault

= er(T−t)St.

4. We have

Vt = e−r(T−t)E[C|Ft]= e−r(T−t)E[ST −K|Ft]= e−r(T−t)E[ST |Ft]− e−r(T−t)E[K|Ft]= St − e−r(T−t)K.

5. We take ξt = 1 and ηt = −Ke−rT /A0, t ∈ [0, T ].6. We have

VT = E[C | FT ] = C.

Exercise 6.4 Digital options.

1. By definition of the indicator functions 1[K,∞) and 1[0,K] we have

1[K,∞)(x) =

1 if x ≥ K,

0 if x < K,resp. 1[0,K](x) =

1 if x ≤ K,

0 if x > K,

which shows the claimed result by the definition of Cd and Pd.2. We have

πt(Cd) + πt(Pd) = e−r(T−t) IE[Cd | Ft] + e−r(T−t) IE[Pd | Ft]= e−r(T−t) IE[Cd + Pd | Ft]= e−r(T−t) IE[1[K,∞)(ST ) + 1[0,K](ST ) | Ft]= e−r(T−t) IE[1[0,∞)(ST ) | Ft]= e−r(T−t) IE[1 | Ft]= e−r(T−t), 0 ≤ t ≤ T,

since P(ST = K) = 0.3. We have

πt(Cd) = e−r(T−t) IE[Cd | Ft]= e−r(T−t) IE[1[K,∞)(ST ) | St]= e−r(T−t)P (ST ≥ K | St)= Cd(t, St).

4. We have

Cd(t, x) = e−r(T−t)P (ST > K | St = x)

= e−r(T−t)Φ

(rτ − σ2τ/2 + log(x/K)

σ√τ

),

466


"



where τ = T − t.5. We have

πt(Cd) = Cd(t, St)

= e−r(T−t)Φ

(rτ − σ2τ/2 + log(St/K)

σ√τ

)= e−r(T−t)Φ (d−) ,

where

d− =(r − σ2/2)τ + log(St/K)

σ√τ

.

6. We have

πt(Pd) = e−r(T−t) − πt(Cd)

= e−r(T−t) − e−r(T−t)Φ(rτ − σ2τ/2 + log(x/K)

σ√τ

)= e−r(T−t)(1− Φ(d−))

= e−r(T−t)Φ(−d−).

7. We have

ξt =∂Cd∂x

(t, St)

= e−r(T−t)∂

∂xΦ

(rτ − σ2τ/2 + log(x/K)

σ√τ

)x=St

= e−r(T−t)1

σ√

2πτSte−(d−)

2/2

> 0.

The Black-Scholes hedging strategy of such a call option does not involveshort-selling because ξt > 0 for all t.

8. Here we have

ξt =∂Pd∂x

(t, St)

= e−r(T−t)∂

∂xΦ

(−rτ − σ

2τ/2 + log(x/K)

σ√τ

)x=St

= −e−r(T−t) 1

σ√

2πτSte−(d−)

2/2

< 0.

The Black-Scholes hedging strategy of such a call option does involveshort-selling because ξt < 0 for all t.

" 467



N. Privault

Chapter 8

Exercise 8.1

1. We have St = S0eσBt , t ∈ R+.

2. We haveIE[ST ] = S0 IE[eσBT ] = S0e

σ2T/2.

3. We have

P

(supt∈[0,T ]

Bt ≥ a)

= 2w∞ae−x

2/(2T ) dx√2πT

, a > 0,

i.e. the probability density function ϕ of supt∈[0,T ]

Bt is given by

ϕ(a) =

√2

πTe−a

2/(2T )1[0,∞)(a), a ∈ R.

4. We have

E[ST ] = S0E

[exp

(σ supt∈[0,T ]

Bt

)]= S0

w∞0eσxϕ(x)dx

=2S0√2πT

w∞0eσx−x

2/(2T )dx =2S0√2πσ2T

w∞0e−(x−σT )2/(2T )+σ2T/2dx

=2S0√2πT

eσ2T/2

w∞−σT

e−x2/(2T )dx =

2S0√2πeσ

2T/2w∞−σ√Te−x

2/2dx

= 2S0eσ2T/2

w σ√T

−∞e−x

2/2dx = 2S0eσ2T/2Φ(σ

√T ) = 2 IE[ST ]Φ(σ

√T ).

Remark: We note that the ratio between the expected gains by selling at

the maximum and selling at time T is given by 2Φ(σ√T ), which cannot

be greater than 2.

468


"



0

0.5

1

1.5

2

0 0.5 1 1.5 2 2.5 3 3.5 4

ratio

time T

2 Φ(σT1/2

)

Fig. 16.3: Average return by selling at the maximum vs selling at maturityT as a function of T .

Exercise 8.2

1. We have

P (τa ≥ t) = P (Xt > a) =w∞aϕXt(x)dx =

√2

πt

w∞ye−x

2/(2t)dx, y > 0.

2. We have

ϕτa(t) =d

dtP (τa ≤ t)

=d

dt

w∞aϕXt(x)dx

= −1

2

√2

πt−3/2

w∞ae−x

2/(2t)dx+1

2

√2

πt−3/2

w∞a

x2

te−x

2/(2t)dx

=1

2

√2

πt−3/2

(−

w∞ae−x

2/(2t)dx+ ae−a2/(2t) +

w∞ae−x

2/(2t)dx)

=a√2πt3

e−a2/(2t), t > 0.

3. We have

E[(τa)−2] =a√2π

w∞0t−5/2e−a

2/(2t)dt

=2a√2π

w∞0x2e−a

2x2/2dx

=1

a2,

" 469



N. Privault

by the change of variable x = t−1/2, x2 = 1/t, t = x−2, dt = −2x−3dx.

Remark: We have

E[τa] =a√2π

w∞0t−1/2e−a

2/(2t)dt = +∞.

Exercise 8.3 Barrier options.

1. By (8.33) and (8.21) we find

ξt =∂g

∂y(t, St) = Φ

(δT−t+

(StK

))− Φ

(δT−t+

(StB

))+K

Be−r(T−t)

(1− 2r

σ2

)(StB

)−2r/σ2 (Φ

(δT−t−

(B2

KSt

))− Φ

(δT−t−

(B

St

)))+

2r

σ2

(StB

)−1−2r/σ2 (Φ

(δT−t+

(B2

KSt

))− Φ

(δT−t+

(B

St

)))− 2

σ√

2π(T − t)

(1− K

B

)exp

(−1

2

(δT−t+

(StB

))2),

0 < St ≤ B, 0 ≤ t ≤ T , cf. also Exercise 7.1-(ix) of [71] and Figure 8.13above.

2. We find

P(YT ≤ a & BT ≥ b) = P(BT ≤ 2a− b), a b ∧ 0.

3. We find

fYT ,BT (a, b) = 1(−∞,b∧0](a)1

T

√2

πT(b− 2a)e−µ

2T/2+µb−(2a−b)2/(2T )

470


"



=

1

T

√2

πT(2a− b)e−µ2T/2+µb−(2a−b)2/(2T ), a b ∧ 0.

4. The function g(t, x) is given in Relations (8.14) and (8.15) above.

Exercise 8.4 Barrier forward contracts.

1. Up-and-in barrier long forward contract. We have

e−r(T−t) IE[C | Ft] = e−r(T−t) IE

(ST −K) 1max

0≤u≤TSu > B

∣∣∣∣∣ Ft

= 1

max0≤u≤t

Su > B(St −Ke−r(T−t)) + 1

max0≤u≤t

Su ≤ Bφ(t, St),

(16.41)

where the function

φ(t, x) := xΦ(δT−t+ (x/B)

)−Ke−r(T−t)Φ

(δT−t− (x/B)

)+B(B/x)2r/σ

2

Φ(−δT−t+ (B/x)

)−Ke−r(T−t)(B/x)−1+2r/σ2

Φ(−δT−t− (B/x)

)solves the Black-Scholes PDE with the terminal condition

φ(T, x) = (x−K + (B/x)2r/σ2

(B −Kx/B))1[B,∞)(x),

as in the proof of Proposition 8.2. Note that only the values of φ(t, x)with x ∈ [0, B] are used for pricing.

" 471



N. Privault

40 45

50 55

60 65

70 75

80

100 120

140 160

180 200

220

0

2

4

6

8

10

12

14

16

18

Up-and-in barrier long forward contract price

underlying

Time in days

Fig. 16.4: Graph of the down-and-in long forward contract price with K < B = 80.

As for the hedging strategy we find

ξt =∂φ

∂x(t, St) = Φ

(δT−t+ (x/B)

)+

1√2πe−(δ

T−t+ (x/B))2/2

− 1

x√

2πKe−r(T−t)−(δ

T−t− (x/B))2/2 − 2r

σ2(B/x)1+2r/σ2


)+

1√2π

(B/x)1+2r/σ2

e−(δT−t+ (B/x))2/2

−K(1− 2r/σ2)

Be−r(T−t)(B/x)2r/σ

2


)− K

B√

2π(B/x)2r/σ

2

e−r(T−t)−(δT−t− (B/x))2/2

= Φ(δT−t+ (x/B)

)− 2r

σ2(B/x)1+2r/σ2


)+

1√2π

(1−K/B)

(e−(δ

T−t+ (x/B))2/2 +

B

xe−r(T−t)−(δ

T−t− (x/B))2/2

)−KB

(1− 2r/σ2)e−r(T−t)(B/x)2r/σ2


),

since by (8.38) we have

e−(δT−t− (B/x))2/2 = er(T−t)(x/B)2r/σ

2

e−(δT−t+ (x/B))2/2

ande−(δ

T−t− (x/B))2/2 = er(T−t)(B/x)2r/σ

2

e−(δT−t+ (B/x))2/2.

472


"



40 45

50 55

60 65

70 75

80

100 120

140 160

180 200

220

0

0.05

0.1

0.15

0.2

0.25

0.3

Delta of the Up-and-in barrier long forward contract

underlying

Time in days

Fig. 16.5: Delta of the down-and-in long forward contract with K < B = 80.

2. Up-and-out barrier long forward contract. We have


(ST −K) 1max

0≤u≤TSu < B

∣∣∣∣∣ Ft

= 1

max0≤u≤t

Su ≤ Bφ(t, St), (16.42)

where the function

φ(t, x) := xΦ(−δT−t+ (x/B)

)−Ke−r(T−t)Φ

(−δT−t− (x/B)

)−B(B/x)2r/σ

2


)+Ke−r(T−t)(B/x)−1+2r/σ2



φ(T, x) = (x−K)1[0,B](x)− (B/x)2r/σ2

(B −Kx/B)1[B,∞)(x).

Note that only the values of φ(t, x) with x ∈ [B,∞) are used for pricing.

" 473



N. Privault

60 65

70 75

80

100 120

140 160

180 200

220

-10

-5

0

5

10

15

20

Up-and-in barrier long forward contract price

underlying

Time in days

Fig. 16.6: Graph of the up-and-out long forward contract price with K < B = 80.


ξt =∂φ

∂x(t, St) = Φ

(−δT−t+ (x/B)

)− 1√

2πe−(δ

T−t+ (x/B))2/2

+1

x√

2πKe−r(T−t)−(δ

T−t− (x/B))2/2 +

2r

σ2(B/x)1+2r/σ2


)− 1√

2π(B/x)1+2r/σ2

e−(δT−t+ (B/x))2/2

+K(1− 2r/σ2)

Be−r(T−t)(B/x)2r/σ

2


)+

K

B√

2π(B/x)2r/σ

2

e−r(T−t)−(δT−t− (B/x))2/2

= Φ(−δT−t+ (x/B)

)+

2r

σ2(B/x)1+2r/σ2


)− 1√

2πe−(δ

T−t+ (x/B))2/2 − 1√

2π

B

xe−r(T−t)−(δ

T−t− (x/B))2/2

+K

B√

2πe−(δ

T−t+ (x/B))2/2 +

1√2π

K

xe−r(T−t)−(δ

T−t− (x/B))2/2

+K

B(1− 2r/σ2)e−r(T−t)(B/x)2r/σ

2


)= Φ

(−δT−t+ (x/B)

)+

2r

σ2(B/x)1+2r/σ2


)− 1√

2π(1−K/B)

(e−(δ

T−t+ (x/B))2/2 +

B

xe−r(T−t)−(δ

T−t− (x/B))2/2

)+K

B(1− 2r/σ2)e−r(T−t)(B/x)2r/σ

2


),

by (8.38).

474


"



60 65

70 75

80 100 120 140 160 180 200 220

0.7

0.75

0.8

0.85

0.9

0.95

1

Delta of the up-and-out barrier long forward contract

underlying Time in days

Fig. 16.7: Graph of the up-and-out long forward contract price with K < B = 80.

3. Down-and-in barrier long forward contract. We have


(ST −K) 1min

0≤u≤TSu < B

∣∣∣∣∣ Ft

= 1

min0≤u≤t

Su < B(St −Ke−r(T−t)) + 1

min0≤u≤t

Su ≥ Bφ(t, St)

(16.43)

where the function

φ(t, x) := xΦ(−δT−t+ (x/B)

)−Ke−r(T−t)Φ

(−δT−t− (x/B)

)+B(B/x)2r/σ

2

Φ(δT−t+ (B/x)

)−Ke−r(T−t)(B/x)−1+2r/σ2

Φ(δT−t− (B/x)


φ(T, x) = (x−K + (B/x)2r/σ2

(B −Kx/B))1[0,B](x).

" 475



N. Privault

80 85

90 95

100

100 120

140 160

180 200

220

0

2

4

6

8

10

12

14

16

18

Down-and-in barrier long forward contract price

underlyingTime in days

Fig. 16.8: Graph of the down-and-in long forward contract price with K < B = 80.


ξt =∂φ

∂x(t, St)

= Φ(−δT−t+ (x/B)

)+

2r

σ2(B/x)1+2r/σ2

Φ(δT−t+ (B/x)

)− 1√

2π(1−K/B)

(e−(δ

T−t+ (x/B))2/2 +

B

xe−r(T−t)−(δ

T−t− (x/B))2/2

)+K

B(1− 2r/σ2)e−r(T−t)(B/x)2r/σ

2

Φ(δT−t− (B/x)

).

80 85

90 95

100

100 120

140 160

180 200

220

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

Delta of the own-and-in barrier long forward contract


Fig. 16.9: Delta of the down-and-in long forward contract with K B

∣∣∣∣∣ Ft

= 1

min0≤u≤t

Su ≥ Bφ(t, St) (16.44)

where the function

φ(t, x) = xΦ(δT−t+ (x/B)

)−Ke−r(T−t)Φ

(δT−t− (x/B)

)−B(B/x)2r/σ

2

Φ(δT−t+ (B/x)

)+Ke−r(T−t)(B/x)−1+2r/σ2

Φ(δT−t− (B/x)

))

solves the Black-Scholes PDE with the terminal condition

φ(T, x) = (x−K)1[B,∞)(x)− (B −Kx/B)(B/x)2r/σ2

1[0,B](x).

Note that φ(t, x) above coincides with the price of (8.15) of the standarddown-and-out barrier call option in the case K < B, cf. Exercise 8.3-(4).

80 85

90 95

100

100 120 140 160 180 200 220

0

5

10

15

20

25

30

35

40

Down-and-out barrier long forward contract price


Fig. 16.10: Graph of the down-and-out long forward contract price with K < B = 80.


ξt =∂φ

∂x(t, St)

= Φ(δT−t+ (x/B)

)− 2r

σ2(B/x)1+2r/σ2

Φ(δT−t+ (B/x)

)+

1√2π

(1−K/B)

(e−(δ

T−t+ (x/B))2/2 +

B

xe−r(T−t)−(δ

T−t− (x/B))2/2

)−KB

(1− 2r/σ2)e−r(T−t)(B/x)2r/σ2

Φ(δT−t− (B/x)

).

" 477



N. Privault

80 85

90 95

100

100 120

140 160

180 200

220

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Delta of the down-and-out barrier long forward contract


Fig. 16.11: Delta of the down-and-out long forward contract with K < B = 80.

5. Up-and-in barrier short forward contract. The price of the up-and-in bar-rier short forward contract is identical to (16.41) with a negative sign.

6. Up-and-out barrier short forward contract. The price of the up-and-outbarrier short forward contract is identical to (16.42) with a negative sign.Note that φ(t, x) coincides with the price of (8.12) of the standard up-and-out barrier put option in the case B < K.

7. Down-and-in barrier short forward contract. The price of the down-and-in barrier short forward contract is identical to (16.43) with a negativesign.

8. Down-and-out barrier short forward contract. The price of the down-and-out barrier short forward contract is identical to (16.44) with a negativesign.

Exercise 8.5

1. We have

P

(mint∈[0,T ]

Bt ≤ a)

= 2w a

−∞e−x

2/(2T ) dx√2πT

, a < 0,

i.e. the probability density function ϕ of supt∈[0,T ]

Bt is given by

ϕ(a) =

√2

πTe−a

2/(2T )1(−∞,0](a), a ∈ R.

2. We have

478


"



IE

[mint∈[0,T ]

St

]= S0 IE

[exp

(σ mint∈[0,T ]

Bt

)]=

2S0√2πT

w 0

−∞eσx−x

2/(2T )dx =2S0√2πσ2T

w 0

−∞e−(x−σT )2/(2T )+σ2T/2dx

=2S0√2πT

eσ2T/2

w −σT−∞

e−x2/(2T )dx =

2S0√2πeσ

2T/2w −σ√T−∞

e−x2/2dx

= 2S0eσ2T/2Φ

(−σ√T)

= 2 IE[ST ](

1− Φ(σ√T))

,

hence

IE

[ST − min

t∈[0,T ]St

]= IE[ST ]− IE

[mint∈[0,T ]

St

]= IE[ST ]− 2 IE[ST ]

(1− Φ

(σ√T))

= IE[ST ](

2Φ(σ√T)− 1)

= 2S0eσ2T/2

(Φ(σ√T)− 1/2

),

and

e−σ2T/2 IE

[ST − min

t∈[0,T ]St

]= S0

(2Φ(σ√T)− 1)

= S0

(1− 2Φ

(−σ√T))

.

Remark: We note that as T goes to infinity, the price of the lookbackoption converges to S0.

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

price

time T

2 (Φ(σT1/2

)-1)

Fig. 16.12: Price of the lookback call option as a function of T with S0 = 1.

Exercise 8.6 Lookback options. By (8.24) and (8.25) we find

ξt =∂f

∂x(t, St,M

t0)

= −1 +

(1 +

2r

σ2

)Φ

(δT−t+

(StM t

0

))" 479



N. Privault

+e−r(T−t)(M t

0

St

)2r/σ2 (1− σ2

2r

)Φ

(−δT−t−

(M t

0

St

)),

t ∈ [0, T ], and

ηtAt = f(t, St,Mt0)− ξtSt


(−δT−t−

(StM t

0

))− e−r(T−t)

(M t

0

St

)−1+2r/σ2

Φ

(−δT−t−

(M t

0

St

)).

Exercise 8.7

1. The integralr T0rsds is centered Gaussian with variance

IE

[(w T

0rsds

)2]

= σ2 IE

[w T

0

w T

0BsBtdsdt

]= σ2

w T

0

w T

0IE[BsBt]dsdt

= σ2w T

0

w T

0min(s, t)dsdt

= 2σ2w T

0

w t

0sdsdt

= σ2w T

0t2dt

= σ2T 3/3.

2. Since the integralr T0rsds is a random variable with probability density

ϕ(x) =1√

2πT 3/3e−3x

2/(2πT 3),

we have

e−rT IE

[(w T

0rudu− κ

)+]

= e−rTw∞−∞

(x− κ)+ϕ(x)dx

=e−rT√

2πσ2T 3/3

w∞κ

(x− κ)e−3x2/(2σ2T 3)dx

=e−rT√

2π

w∞κ/√σ2T 3/3

(x√σ2T 3/3− κ)e−x

2/2dx

=e−rT

√σ2T 3/3√2π

w∞κ/√σ2T 3/3

xe−x2/2dx− κe

−rT√

2π

w∞κ/√σ2T 3/3

e−x2/2dx

= −e−rT

√σ2T 3/3√2π

[e−x

2/2]∞κ/√σ2T 3/3

− κe−rT√

2π(1− Φ(κ/

√σ2T 3/3))

480


"



=e−rT

√σ2T 3/3√2π

e−3κ2/(2σ2T 3) − κe

−rT√

2π(1− Φ(κ/

√σ2T 3/3))

= e−rT√σ2T 3

6πe−3κ

2/(2σ2T 3) − κe−rT√

2πΦ

(−κ√

3

σ2T 3

).


e−r(T−t) IE

[(1

T

w T

0Sudu− κ

)+ ∣∣∣Ft] = e−r(T−t) IE

[1

T

w T

0Sudu− κ

∣∣∣Ft]= e−r(T−t) IE

[1

T

w T

0Sudu

∣∣∣Ft]− κe−r(T−t)= e−r(T−t)

1

TIE

[w t

0Sudu

∣∣∣Ft]+ e−r(T−t)1

TIE

[w T

tSudu


1

T

w t

0Sudu+ e−r(T−t)

1

TIE

[w T

tSudu


1

T

w t

0Sudu+ e−r(T−t)

1

T

w T

tIE[Su | Ft]du− κe−r(T−t)

= e−r(T−t)1

T

w t

0Sudu+ e−r(T−t)

1

T

w T

tSte

r(u−t)du− κe−r(T−t)

= e−r(T−t)1

T

w t

0Sudu+ e−r(T−t)

StT

w T−t

0erudu− κe−r(T−t)

= e−r(T−t)1

T

w t

0Sudu+ e−r(T−t)

StrT

(er(T−t) − 1)− κe−r(T−t)

= e−r(T−t)1

T

w t

0Sudu+ St

1− e−r(T−t)rT

− κe−r(T−t),

t ∈ [0, T ], cf. [29] page 361. We check that the function f(t, x, y) =e−r(T−t)(y/T − κ) + x(1− e−r(T−t))/(rT ) satisfies the PDE

rf(t, x, y) =∂f

∂t(t, x, y) + x

∂f

∂y(t, x, y) + rx

∂f

∂x(t, x, y) +

1

2x2σ2 ∂

2f

∂x2(t, x, y),

t, x > 0, and the boundary conditions f(t, 0, y) = e−r(T−t)(y/T − κ),0 ≤ t ≤ T , y ∈ R+, and f(T, x, y) = y/T − κ, x, y ∈ R+. However, thecondition limy→−∞ f(t, x, y) = 0 is not satisfied because we need to takey > 0 in the above calculation.

Exercise 8.9 The Asian option price can be written as

e−r(T−t) IE∗

[(1

T

w T

0Sudu−K

)+ ∣∣∣Ft] = StIE[(UT )+ | Ut

]= Sth(t, Ut) = Stg(t, Zt),

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N. Privault

which shows thatg(t, Zt) = h(t, Ut),

and it remains to use the relation

Ut =1− e−r(T−t)

rT+ e−r(T−t)Zt, t ∈ [0, T ].

Chapter 9

Exercise 9.1 Stopping times.

1. When 0 ≤ t < 1 the question “is ν > t ?” cannot be answered at time twithout waiting to know the value of B1 at time 1. Therefore ν is not astopping time.

2. For any t ∈ R+, the question “is τ > t ?” can be answered based on theobservation of the paths of (Bs)0≤s≤t and of the (deterministic) curve(αe−s/2)0≤s≤t up to the time t. Therefore τ is a stopping time.Since τ is a stopping time and (Bt)t∈R+ is a martingale, the stopping time

theorem shows that (eBt∧τ−(t∧τ)/2)t∈R+ is also a martingale and in partic-

ular its expectation IE[eBt∧τ−(t∧τ)/2] = IE[eB0∧τ−(0∧τ)/2] = IE[eB0−0/2] =1 is constantly equal to 1 for all t. This shows that

IE[eBτ−τ/2] = IE[

limt→∞

eBt∧τ−(t∧τ)/2]

= limt→∞

IE[eBt∧τ−(t∧τ)/2] = 1.

Next, we note that we have eBτ = αe−τ/2, hence

α IE[e−τ ] = IE[eBτ−τ/2] = 1,

i.e.α IE[e−τ ] = 1/α ≤ 1.

Remark: note that this argument fails when α < 1 because in that caseτ is not a.s. finite.

3. For any t ∈ R+, the question “is τ > t ?” can be answered based on theobservation of the paths of (Bs)0≤s≤t and of the (deterministic) curve(1 + αs)0≤s≤t up to the time t. Therefore τ is a stopping time.

Since τ is a stopping time and (Bt)t∈R+is a martingale, the stopping

time theorem shows that (B2t∧τ − (t ∧ τ))t∈R+

is also a martingale andin particular its expectation IE[B2

t∧τ − (t ∧ τ)] = IE[B20∧τ − (0 ∧ τ)] =

IE[B20 − 0] = 0 is constantly equal to 0 for all t. This shows that

IE[B2τ − τ ] = IE

[limt→∞

(B2t∧τ − (t ∧ τ))

]= limt→∞

IE[(B2t∧τ − (t ∧ τ))] = 0.

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"



Next, we note that we have B2τ = 1 + ατ , hence

1 + α IE[τ ] = IE[1 + ατ ] = IE[B2τ ] = IE[τ ] = 0,

i.e.IE[τ ] = 1/(1− α).

Remark: Note that this argument is valid whenever α ≤ 1 and yieldsIE[τ ] = +∞ when α = 1, however it fails when α > 1 because in thatcase τ is not a.s. finite.

Exercise 9.2

1. Letting A0 = 0,

An+1 = An + IE[Mn+1 −Mn | Fn], n ≥ 0,

andNn = Mn −An, n ∈ N, (16.45)

we have,

(i) for all n ∈ N,

IE[Nn+1 | Fn] = IE[Mn+1 −An+1 | Fn]

= IE[Mn+1 −An − IE[Mn+1 −Mn | Fn] | Fn]

= IE[Mn+1 −An | Fn]− IE[IE[Mn+1 −Mn | Fn] | Fn]

= IE[Mn+1 −An | Fn]− IE[Mn+1 −Mn | Fn]

= − IE[An | Fn] + IE[Mn | Fn]

= Mn −An= Nn,

hence (Nn)n∈N is a martingale with respect to (Fn)n∈N.(ii) We have

An+1 −An = IE[Mn+1 −Mn | Fn]

= IE[Mn+1 | Fn]− IE[Mn | Fn]

= IE[Mn+1 | Fn]−Mn ≥ 0, n ∈ N,

since (Mn)n∈N is a submartingale.(iii) By induction we have

An = An−1 + IE[Mn −Mn−1 | Fn−1], n ≥ 1,

which is Fn−1-measurable provided An is Fn−1-measurable, n ≥ 1.(iv) This property is obtained by construction in (16.45).

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N. Privault

2. For all bounded stopping times σ and τ such that σ ≤ τ a.s., we have

IE[Mσ] = IE[Nσ] + IE[Aσ]

≤ IE[Nσ] + IE[Aτ ]

= IE[Nτ ] + IE[Aτ ]

= IE[Mτ ],

by (9.11), since (Mn)n∈N is a martingale and (An)n∈N is non-decreasing.

Exercise 9.3 American digital options.

1. The optimal strategy is as follows:

(i) if St ≥ K, then exercise immediately.(ii) if St < K, then wait.

2. The optimal strategy is as follows:

(i) if St > K, then wait.(ii) if St ≤ K, exercise immediately.

3. Based on the answers to Question 1 we set

CAmd (t,K) = 1, 0 ≤ t < T,

andCAmd (T, x) = 0, 0 ≤ x < K.

4. Based on the answers to Question 2, we set

PAmd (t,K) = 1, 0 ≤ t < T,

andPAmd (T, x) = 0, x > K.

5. Starting from St ≤ K, the maximum possible payoff is clearly reachedas soon as St hits the level K before the expiration date T , hence thediscounted optimal payoff of the option is e−r(τK−t)1τK<T.

6. From Relation (8.7) we find

P(τa ≤ u) = Φ

(a− µu√

u

)− e2µaΦ

(−a− µu√u

),

and by differentiation with respect to u this yields the probability densityfunction

fτa(u) =∂

∂uP(τa ≤ u) =

a√2πu3

e−(a−µu)2

2u 1[0,∞)(u)

484


"



of the first hitting time of level a by Brownian motion with drift µ. Giventhe relation

Su = Steσ(Bu−Bt)−σ2(u−t)/2+µ(u−t), u ≥ t,

we find that the probability density function of the first hitting time oflevel K after time t by (Su)u∈[t,∞) is given by

u 7−→ a√2π(u− t)3

e−(a−µ(u−t))2

2(u−t) , u ≥ t,

with µ = σ−1(r − σ2/2) and

a =1

σlog

K

x,

given that St = x. Hence for x ∈ (0,K) we have, letting τ = T − t,

CAmd (t, x) = IE[e−r(τK−t)1τK<T | St = x]

=w T

te−r(s−t)

a√2π(s− t)3

e−(a−µ(s−t))2

2(s−t) ds

=w τ

0e−rs

a√2πs3

e−(a−µs)2

2s ds

=w τ

0

log(K/x)

σ√

2πs3exp

(−rs− 1

2σ2s

(−(r − σ2

2

)s+ log

K

x

)2)ds

=

(K

x

)( rσ2− 1

2 )±( rσ2

+ 12 )

×w τ

0

log(K/x)

σ√

2πs3exp

(− 1

2σ2s

(±(r +

σ2

2

)s+ log

K

x

)2)ds

=1√2π

x

K

w∞y−e−y

2/2dy +1√2π

(K

x

)2r/σ2 w∞y+e−y

2/2dy

=x

KΦ

((r + σ2/2)τ + log(x/K)

σ√τ

)+( xK

)−2r/σ2

Φ

(−(r + σ2/2)τ + log(x/K)

σ√τ

), 0 < x < K,

where

y± =1

σ√τ

(±(r +

σ2

2

)τ + log

K

x

),

and used the decomposition

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N. Privault

logK

x=

1

2

((r +

σ2

2

)s+ log

K

x

)+

1

2

(−(r +

σ2

2

)s+ log

K

x

).

We check thatCAmd (t,K) = Φ(∞) + Φ(−∞) = 1,

and

CAmd (T, x) =

x

KΦ (−∞) +

( xK

)−2r/σ2

Φ (−∞) = 0, x < K,

since τ = 0, which is consistent with the answers to Question 3.7. Starting from St ≥ K, the maximum possible payoff is clearly reached

as soon as St hits the level K before the expiration date T , hence thediscounted optimal payoff of the option is e−r(τK−t)1τK<T.

8. Using the notation and answer to Question 6, for x > K we find, lettingτ = T − t,

PAmd (t, x) = IE[e−r(τK−t)1τK<T | St = x]

=w τ

0e−rs

a√2πs3

e−(a−µs)2

2s ds

=w τ

0

log(x/K)

σ√

2πs3exp

(−rs− 1

2σ2s

((r − σ2

2

)s+ log

x

K

)2)ds

=

(K

x

)( rσ2− 1

2 )±( rσ2

+ 12 )

×w τ

0

log(x/K)

σ√

2πs3exp

(− 1

2σ2s

(∓(r +

σ2

2

)s+ log

x

K

)2)ds

=1√2π

x

K

w∞y−e−y

2/2dy +1√2π

( xK

)2r/σ2 w∞y+e−y

2/2dy

=x

KΦ

(−(r + σ2/2)τ − log(x/K)

σ√τ

)+( xK

)−2r/σ2

Φ

((r + σ2/2)τ − log(x/K)

σ√τ

), x > K,

with

y± =1

σ√τ

(∓(r +

σ2

2

)τ + log

x

K

),

We check thatPAmd (t,K) = Φ(−∞) + Φ(∞) = 1,

and

PAmd (T, x) =

x

KΦ (−∞) +

( xK

)−2r/σ2

Φ (−∞) = 0, 0 < x < K,

486


"



since τ = 0, which is consistent with the answers to Question 3.9. The call-put parity does not hold for American digital options since forx ∈ (0,K) we have

CAmd (t, x) + PAm

d (t, x) = 1 +x

KΦ

((r + σ2/2)τ + log(x/K)

σ√τ

)+( xK

)−2r/σ2

Φ

(−(r + σ2/2)τ + log(x/K)

σ√τ

),

while for x > K we find

CAmd (t, x) + PAm

d (t, x) = 1 +x

KΦ

(−(r + σ2/2)τ − log(x/K)

σ√τ

)+( xK

)−2r/σ2

Φ

((r + σ2/2)τ − log(x/K)

σ√τ

).

Exercise 9.4

1. For all stopping times τ such that t ≤ τ ≤ T we have


∣∣∣St] = K IE∗[e−r(τ−t)

∣∣∣St]− IE∗[e−r(τ−t)Sτ

∣∣∣St]= e−r(τ−t)K − St,

since τ ∈ [t, T ] is bounded and (e−rtSt)t∈R+is a martingale, and the

above quantity is clearly maximized by taking τ = t. Hence we have


τ stopping time


∣∣∣St] = K − St,

and the optimal strategy is to exercise immediately at time t.2. Similarly we have


∣∣∣St] = IE∗[e−r(τ−t)Sτ

∣∣∣St]−K IE∗[e−r(τ−t)

∣∣∣St]= St −K IE∗

[e−r(τ−t)

∣∣∣St] ,since τ ∈ [t, T ] is bounded and (e−rtSt)t∈R+ is a martingale, and theabove quantity is clearly maximized by taking τ = T . Hence we have

f(t, St) = sup t≤τ≤Tτ stopping time


∣∣∣St] = St − e−r(T−t)K,

and the optimal strategy is to wait until the maturity time T in order toexercise.

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N. Privault

3. Concerning the perpetual American long forward contract, since u 7→e−r(u−t)Su is a martingale, for all stopping times τ we have2


∣∣∣St] = IE∗[e−r(τ−t)Sτ

∣∣∣St]−K IE∗[e−r(τ−t)

∣∣∣St]= St −K IE∗

[e−r(τ−t)

∣∣∣St]≤ St, t ≥ 0.

On the other hand, for all fixed T > 0 we have

IE∗[e−r(T−t)(ST −K)

∣∣∣St] = IE∗[e−r(T−t)ST

∣∣∣St]−K IE∗[e−r(T−t)

∣∣∣St]= St − e−r(T−t)K, t ≥ 0,

hence

supτ≥t

τ stopping time


∣∣∣St] ≥ (St − e−r(T−t)K), T ≥ t,

and letting T →∞ we get

supτ≥t

τ stopping time


∣∣∣St] ≥ limT→∞

(St − e−r(T−t)K)

= St,

hence we have


τ stopping time


∣∣∣St] = St,

and the optimal strategy τ∗ = +∞ is to wait indefinitely.

Concerning the perpetual American short forward contract we have


τ stopping time


∣∣∣St]≤ sup

t≤τ≤Tτ stopping time


∣∣∣St]= fL∗(St).

On the other hand, for τ = τL∗ we have

(K − SτL∗ ) = (K − L∗) = (K − L∗)∗

2 by Fatou’s Lemma.

488


"



since 0 < L∗ = 2Kr/(2r + σ2) < K, hence

fL∗(St) = IE∗[e−r(τ−t)(K − SτL∗ )+

∣∣∣St]= IE∗

[e−r(τ−t)(K − SτL∗ )

∣∣∣St]≤ sup

t≤τ≤Tτ stopping time


∣∣∣St]= f(t, St),

which shows thatf(t, St) = fL∗(St),

i.e. the perpetual American short forward contract has same price andexercise strategy as the perpetual American put option.

Exercise 9.5

1. We have

Yt = e−rt(S0ert+σBt−σ2t/2)−2r/σ

2

= S−2r/σ2

0 e−rt−2r2t/σ2+2rBt/σ+rt

= S−2r/σ2

0 e2rBt/σ−(2r/σ)2t/2

andZt = e−rtSt = S0e

σBt−σ2t/2,

which are both martingales under P∗ because they are standard geometricBrownian motions with respective volatilities σ and 2r/σ.

2. Since Yt and Zt are both martingales and τL is a stopping time we have

S−2r/σ2

0 = IE∗[Y0]

= IE∗[YτL ]

= IE∗[e−rτLS−2r/σ2

τL ]

= IE∗[e−rτLL−2r/σ2

]

= L−2r/σ2

IE∗[e−rτL ],

henceIE∗[e−rτL ] = (x/L)−2r/σ

2

if S0 = x ≥ L (note that in this case YτL∧t remains bounded by L−2r/σ2

),and

S0 = IE∗[Z0] = IE∗[ZτL ] = IE∗[e−rτLSτL ] = IE∗[e−rτLL] = L IE∗[e−rτL ],

" 489



N. Privault

henceIE∗[e−rτL ] = x/L

if S0 = x ≤ L (note that in this case ZτL∧t remains bounded by L).3. We find

IE[e−rτL(K − SτL) | S0 = x

]= (K − L) IE∗

[e−rτL | S0 = x

]=

xK − LL

, 0 < x ≤ L,

(K − L)( xL

)−2r/σ2

, x ≥ L.(16.46)

4. By differentiating

∂

∂LIE[e−rτL(K − SτL) | S0 = x

]

=

(x/L)−2r/σ

2

(2r

σ2

(K

L− 1

)− 1

), 0 < L < x,

−KxL2

, L > x,

and check that the minimum occurs for L∗ = x.5. The value L∗ = x shows that the optimal strategy for the American finite

expiration short forward contract is to exercize immediately starting fromS0 = x, which is consistent with the result of Exercise 9.4-(1), since givenany stopping time τ upper bounded by T we have

IE[e−rτ (K − Sτ )] = K IE[e−rτ ]− IE[e−rτSτ ] = K IE[e−rτ ]− S0 ≤ K − S0.

Exercise 9.6

1. The option payoff equals (κ− St)p if St ≤ L.2. We have

fL(St) = IE∗[e−r(τL−t)((κ− SτL)+)p

∣∣∣St]= IE∗

[e−r(τL−t)((κ− L)+)p

∣∣∣St]= (κ− L)p IE∗

[e−r(τL−t)

∣∣∣St] .3. We have

fL(x) = IE∗[e−r(τL−t)(κ− SτL)+

∣∣∣St = x]

490


"



=

(κ− x)p, 0 < x ≤ L,

(κ− L)p( xL

)−2r/σ2

, x ≥ L.(16.47)

4. By differentiating ddx (κ− x)p = −p(κ− x)p−1 we find

f ′L∗(L∗) = − 2r

σ2(κ− L∗)p (L∗)−2r/σ

2−1

(L∗)−2r/σ2 = −p(κ− L∗)p−1,

i.e.2r

σ2(κ− L∗) = pL∗,

or

L∗ =2r

2r + pσ2κ < κ.

5. By (16.47) the price can be computed as

f(t, St) = fL∗(St) =

(κ− St)p, 0 < St ≤ L∗,

(pσ2κ

2r + pσ2

)p(2r + pσ2

2r

Stκ

)−2r/σ2

, St ≥ L∗,

using (9.12) as in the proof of Proposition 9.4, since


is a nonnegative supermartingale.

Exercise 9.7

1. The payoff will be κ− (St)p.

2. We have

fL(St) = E∗[e−r(τL−t)(κ− (SτL)p)

∣∣∣St]= E∗

[e−r(τL−t)(κ− Lp)

∣∣∣St]= (κ− Lp)E∗

[e−r(τL−t)

∣∣∣St] .3. We have

fL(x) = E∗[e−r(τL−t)(κ− (SτL)p)

∣∣∣St = x]

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N. Privault

=

κ− xp, 0 < x ≤ L,

(κ− Lp)( xL

)−2r/σ2

, x ≥ L.

4. We have

f ′L∗(L∗) = − 2r

σ2(κ− (L∗)p)

(L∗)−2r/σ2−1

(L∗)−2r/σ2 = −p(L∗)p−1,

i.e.2r

σ2(κ− (L∗)p) = p(L∗)p,

or

L∗ =

(2rκ

2r + pσ2

)1/p

< (κ)1/p. (16.48)

Remark: We may also compute L∗ by maximizing L 7→ fL(x) for all fixedx. The derivative ∂fL(x)/∂L can be computed as

∂fL(x)

∂L=

∂

∂L

((κ− Lp)

(L

x

)2r/σ2)

= −pLp−1(L

x

)2r/σ2

+2r

σ2L−1(κ− Lp)

(L

x

)2r/σ2

,

and equating ∂fL(x)/∂L to 0 at L = L∗ yields

−p(L∗)p−1 +2r

σ2(L∗)−1(κ− (L∗)p) = 0,

which recovers (16.48).5. We have

fL∗(St) =

κ− (St)

p, 0 < St ≤ L∗,

(κ− (L∗)p)(St)

−2r/σ2

(L∗)−2r/σ2 , St ≥ L∗

=

κ− (St)

p, 0 < St ≤ L∗,

σ2

2rp(St)

−2r/σ2

(L∗)p+2r/σ2

, St ≥ L∗,

=

κ− (St)

p, 0 < St ≤ L∗,

pσ2κ

2r + pσ2

(2r + pσ2

2r

Sptκ

)−2r/(pσ2)

< κ, St ≥ L∗,

492


"



however we cannot conclude as in Exercise 9.6-(5) since the process


does not remain nonnegative when p > 1, so that (9.12) cannot be ap-plied as in the proof of Proposition 9.4.

Exercise 9.8

1. We have that

Zt :=

(StS0

)λe−(r−a)λt+λσ

2t/2−λ2σ2t/2 = eλσBt−λ2σ2t/2, t ∈ R+,

is a geometric Brownian motion without drift under the risk-neutral prob-ability measure P∗, hence it is a martingale.

2. By the stopping time theorem we have

IE∗[ZτL ] = IE∗[Z0] = 1,

which rewrites as

IE∗

[(SτLS0

)λe−((r−a)λ−λσ

2/2+λ2σ2/2)τL

]= 1,

or, given the relation SτL = L,(L

S0

)λIE∗[e−((r−a)λ−λσ

2/2+λ2σ2/2)τL]

= 1,

i.e.

IE∗[e−rτL

]=

(S0

L

)λ,

provided we choose λ such that

− ((r − a)λ− λσ2/2 + λ2σ2/2) = −r, (16.49)

i.e.0 = λ2σ2/2 + λ(r − a− σ2/2)− r.

This equation admits two solutions

λ =−(r − a− σ2/2)±

√(r − a− σ2/2)2 + 4rσ2/2

σ2,

and we choose the negative solution

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N. Privault

λ =−(r − a− σ2/2)−

√(r − a− σ2/2)2 + 4rσ2/2

σ2

since S0/L = x/L > 1 and the expectation IE∗[e−rτL ] < 1 is lower than1 as r ≥ 0.

3. Noting that τL = 0 if S0 ≤ L, for all L ∈ (0,K) we have

IE∗[e−rτL(K − SτL)+

∣∣∣S0 = x]

=

K − x, 0 < x ≤ L,

E[e−rτL(K − L)+

∣∣∣S0 = x], x ≥ L.

=

K − x, 0 < x ≤ L,

(K − L)E[e−rτL

∣∣∣S0 = x], x ≥ L.

=

K − x, 0 < x ≤ L,

(K − L)( xL

)−(r−a−σ2/2)−√

(r−a−σ2/2)2+4rσ2/2

σ2

, x ≥ L.

4. In order to compute L∗ we observe that, geometrically, the slope of fL(x)at x = L∗ is equal to −1, i.e.

f ′L∗(L∗) = λ(K − L∗) (L∗)λ−1

(L∗)λ= −1,

orλ(K − L∗) = L∗,

or

L∗ =λ

λ− 1K < K.

5. For x ≥ L we have

fL∗(x) = (K − L∗)( xL∗

)λ=

(K − λ

λ− 1K

)(xλλ−1K

)λ

=

(− K

λ− 1

)(x(λ− 1)

λK

)λ=

(− K

λ− 1

)(x

−λ

)λ(λ− 1

−K

)λ

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"



=

(x

−λ

)λ(λ− 1

−K

)λ−1=( xK

)λ(λ− 1

λ

)λK

1− λ. (16.50)

6. Let us check that the relation

fL∗(x) ≥ (K − x)+ (16.51)

holds. For all x ≤ K we have

fL∗(x)− (K − x) =( xK

)λ(λ− 1

λ

)λK

1− λ + x−K

= K

(( xK

)λ(λ− 1

λ

)λ1

1− λ +x

K− 1

).

Hence it suffices to take K = 1 and to show that for all

L∗ =λ

λ− 1≤ x ≤ 1

we have

fL∗(x)− (1− x) =xλ

1− λ

(λ− 1

λ

)λ+ x− 1

≥ 0.

Equality to 0 holds for x = λ/(λ− 1). By differentiation of this relationwe get

f ′L∗(x)− (1− x)′ = λxλ−1(λ− 1

λ

)λ1

1− λ + 1

= xλ−1(λ− 1

λ

)λ−1+ 1

≥ 0,

hence the function fL∗(x)− (1− x) is non-decreasing and the inequalityholds throughout the interval [λ/(λ− 1),K].

On the other hand, using (16.49) it can be checked by hand that fL∗given by (16.50) satisfies the equality

(r − a)xf ′L∗(x) +1

2σ2x2f ′′L∗(x) = rfL∗(x) (16.52)

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N. Privault

for x ≥ L∗ =λ

λ− 1K. In case

0 ≤ x ≤ L∗ =λ

λ− 1K < K,

we havefL∗(x) = K − x = (K − x)+,

hence the relation(rfL∗(x)− (r − a)xf ′L∗(x)− 1

2σ2x2f ′′L∗(x)

)(fL∗(x)− (K − x)+) = 0

always holds. On the other hand, in that case we also have

(r − a)xf ′L∗(x) +1

2σ2x2f ′′L∗(x) = −(r − a)x,

and to conclude we need to show that

(r − a)xf ′L∗(x) +1

2σ2x2f ′′L∗(x) ≤ rfL∗(x) = r(K − x), (16.53)

which is true ifax ≤ rK.

Indeed by (16.49) we have

(r − a)λ = r + λ(λ− 1)σ2/2

≥ r,

hence

aλ

λ− 1≤ r,

since λ < 0, which yields

ax ≤ aL∗ ≤ a λ

λ− 1K ≤ rK.

7. By Ito’s formula and the relation

dSt = (r − a)Stdt+ σStdBt

we have

d(fL∗(St)) = −re−rtfL∗(St)dt+ e−rtdfL∗(St)

= −re−rtfL∗(St)dt+ e−rtf ′L∗(St)dSt +1

2e−rtσ2S2

t f′′L∗(St)

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"



= e−rt(−rfL∗(St) + (r − a)Stf

′L∗(St) +

1

2σ2S2

t f′′L∗(St)

)dt

+e−rtσStf′L∗(St)dBt,

and from Equations (16.52) and (16.53) we have

(r − a)xf ′L∗(x) +1

2σ2x2f ′′L∗(x) ≤ rfL∗(x),

hencet 7→ e−rtfL∗(St)

is a supermartingale.8. By the supermartingale property of

t 7→ e−rtfL∗(St),

for all stopping times τ we have

fL∗(S0) ≥ IE∗[e−rτfL∗(Sτ )

∣∣∣S0

]≥ IE∗

[e−rτ (K − Sτ )+

∣∣∣S0

],

by (16.51), hence

fL∗(S0) ≥ supτ stopping time


∣∣∣S0

]. (16.54)

9. The stopped process

t 7→ e−rt∧τL∗ fL∗(St∧τL∗ )

is a martingale since it has vanishing drift up to time τL∗ by (16.52),and it is constant after time τL∗ , hence by the martingale stopping timeTheorem (9.1) we find

fL∗(S0) = IE∗[e−rτfL∗(SτL∗ )

∣∣∣S0

]= IE∗

[e−rτfL∗(L

∗)∣∣∣S0

]= IE∗

[e−rτ (K − SτL∗ )+

∣∣∣S0

]≤ sup

τ stopping time


∣∣∣S0

].

10. By combining the above results and conditioning at time t instead of time0 we deduce that


∣∣∣St]

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N. Privault

=

K − St, 0 < St ≤

λ

λ− 1K,

(λ− 1

−K

)λ−1(−Stλ

)λ, St ≥

λ

λ− 1K,

for all t ∈ R+, where

τL∗ = infu ≥ t : Su ≤ L.

We note that the perpetual put option price does not depend on the valueof t ≥ 0.

Exercise 9.9

1. By the definition (9.35) of S1(t) and S2(t) we have

Zt = e−rtS2(t)

(S1(t)

S2(t)

)α= e−rtS1(t)αS2(t)1−α

= S1(0)αS2(0)1−αe(ασ1+(1−α)σ2)Wt−σ22t/2,

which is a martingale when

σ22 = (ασ1 + (1− α)σ2)2,

i.e.ασ1 + (1− α)σ2 = ±σ2,

which yields either α = 0 or

α =2σ2

σ2 − σ1> 1,

since 0 ≤ σ1 < σ2.2. We have

IE[e−rτL(S1(τL)− S2(τL))+] = IE[e−rτL(LS2(τL)− S2(τL))+]

= (L− 1)+ IE[e−rτLS2(τL)]. (16.55)

3. Since τL ∧ t is a bounded stopping time we can write

S2(0)

(S1(0)

S2(0)

)α= IE

[e−r(τL∧t)S2(τL ∧ t)

(S1(τL ∧ t)S2(τL ∧ t)

)α](16.56)

= IE

[e−rτLS2(τL)

(S1(τL)

S2(τL)

)α1τL<t

]+ IE

[e−rtS2(t)

(S1(t)

S2(t)

)α1τL>t

]

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We have

e−rtS2(t)

(S1(t)

S2(t)

)α1τL>t ≤ e−rtS2(t)Lα1τL>t ≤ e−rtS2(t)Lα,

hence by a uniform integrability argument,

limt→∞

IE

[e−rtS2(t)

(S1(t)

S2(t)

)α1τL>t

]= 0,

and letting t go to infinity in (16.56) shows that

S2(0)

(S1(0)

S2(0)

)α= IE

[e−rτLS2(τL)

(S1(τL)

S2(τL)

)α]= Lα IE

[e−rτLS2(τL)

],

since S1(τL)/S2(τL) = L/L = 1. The conclusion

IE[e−rτL(S1(τL)− S2(τL))+] = (L− 1)+L−αS2(0)

(S1(0)

S2(0)

)α(16.57)

then follows by an application of (16.55).4. In order to maximize (16.57) as a function of L we consider the derivative

∂

∂L

L− 1

Lα=

1

Lα− α(L− 1)L−α−1 = 0,

which vanishes forL∗ =

α

α− 1,

and we substitute L in (16.57) with the value of L∗.5. In addition to r = σ2

2/2 it is sufficient to let S1(0) = κ and σ1 = 0 whichyields α = 2, L∗ = 2, and we find

supτ stopping time

IE[e−rτ (κ− S2(τ))+] =1

S2(0)

(κ2

)2,

which coincides with the result of Proposition 9.4.

Chapter 10

Exercise 10.1

1. We have

dXt = d

(Xt

Nt

)=X0

N0d(e(σ−η)Bt−(σ

2−η2)t/2)

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N. Privault

=X0

N0(σ − η)e(σ−η)Bt−(σ

2−η2)t/2dBt +X0

2N0(σ − η)2e(σ−η)Bt−(σ

2−η2)t/2dt

− X0

2N0(σ2 − η2)e(σ−η)Bt−(σ

2−η2)t/2dt

= − Xt

2Nt(σ2 − η2)dt+

Xt

Nt(σ − η)dBt +

Xt

2Nt(σ − η)2dt

= −Xt

Ntη(σ − η)dt+

Xt

Nt(σ − η)dBt

=Xt

Nt(σ − η)(dBt − ηdt)

= (σ − η)Xt

NtdBt = (σ − η)XtdBt,

where dBt = dBt − ηdt is a standard Brownian motion under P.2. By the result of Question 1, Xt is a driftless geometric Brownian motion

with volatility σ − η under P, hence

IE[(XT −λ)+] = X0Φ

(log(X0/λ)

σ√T

+σ√T

2

)−λΦ

(log(X0/λ)

σ√T

− σ√T

2

)

is given by the Black-Scholes formula with zero interest rate and volatilityparameter σ = σ − η, which shows (10.30) by multiplication by N0 and

the relation X0 = N0X0, i.e.

e−rt IE[(XT − λNT )+] = IE

[N0

NT(XT − λNT )+

]= N0IE

[(XT − λ)+

]= N0X0Φ(d+)− λN0Φ(d−)

= X0Φ(d+)− λN0Φ(d−).

3. We have σ = σ − η.

Exercise 10.2 Bond options.

1. Ito’s formula yields

d

(P (t, S)

P (t, T )

)=P (t, S)

P (t, T )(ζS(t)− ζT (t))(dWt − ζT (t)dt)

=P (t, S)

P (t, T )(ζS(t)− ζT (t))dWt, (16.58)

where (Wt)t∈R+is a standard Brownian motion under P by the Girsanov

theorem.2. From (16.58) we have

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"



P (t, S)

P (t, T )=P (0, S)

P (0, T )exp

(w t

0(ζS(s)− ζT (s))dWs −

1

2

w t

0|ζS(s)− ζT (s)|2ds

),

hence

P (u, S)

P (u, T )=P (t, S)

P (t, T )exp

(w u


1

2

w u


),

u ≥ t, and for u = T this yields

P (T, S) =P (t, S)

P (t, T )exp

(w T


1

2

w T


),

since P (T, T ) = 1. Let P denote the forward measure associated to thenumeraire

Nt := P (t, T ), 0 ≤ t ≤ T.3. For all S ≥ T > 0 we have

IE[e−

r Ttrsds(P (T, S)−K)+

∣∣∣Ft]= P (t, T )IE

[(P (t, S)

P (t, T )exp

(X − 1

2

w T


)−K

)+ ∣∣∣Ft]

= P (t, T )IE

[(eX+m(t,T,S) −K

)+ ∣∣∣Ft] ,where X is a centered Gaussian random variable with variance

v2(t, T, S) =w T


given Ft, and

m(t, T, S) = −1

2v2(t, T, S) + log

P (t, S)

P (t, T ).

Recall that when X is a centered Gaussian random variable with variancev2, the expectation of (em+X −K)+ is given, as in the standard Black-Scholes formula, by

IE[(em+X −K)+] = em+ v2

2 Φ(v + (m− logK)/v)−KΦ((m− logK)/v),

where

Φ(z) =w z

−∞e−y

2/2 dy√2π, z ∈ R,

denotes the Gaussian cumulative distribution function and for simplic-ity of notation we dropped the indices t, T, S in m(t, T, S) and v2(t, T, S).

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N. Privault

Consequently we have

IE[e−

r Ttrsds(P (T, S)−K)+

∣∣∣Ft]= P (t, S)Φ

(v

2+

1

vlog

P (t, S)

KP (t, T )

)−KP (t, T )Φ

(−v

2+

1

vlog

P (t, S)

KP (t, T )

).

4. The self-financing hedging strategy that hedges the bond option is ob-tained by holding a (possibly fractional) quantity

Φ

(v

2+

1

vlog

P (t, S)

KP (t, T )

)of the bond with maturity S, and by shorting a quantity

KΦ

(−v

2+

1

vlog

P (t, S)

KP (t, T )

)of the bond with maturity T .

Exercise 10.3

1. The processe−rtS2(t) = S2(0)eσ2Wt+(µ−r)t

is a martingale if

r − µ =1

2σ22 .

2. We note that

e−rtXt = e−rte(r−µ)t−σ21t/2S1(t)

= e−rte(σ22−σ

21)t/2S1(t)

= e−µt−σ21t/2S1(t)

= S1(0)eµt−σ21t/2eσ1Wt+µt

= S1(0)eσ1Wt−σ21t/2

is a martingale, where

Xt = e(r−µ)t−σ21t/2S1(t) = e(σ

22−σ

21)t/2S1(t).

3. By (10.32) we have

X(t) =Xt

Nt

= e(σ22−σ

21)t/2

S1(t)

S2(t)

502


"



=S1(0)

S2(0)e(σ

22−σ

21)t/2+(σ1−σ2)Wt

=S1(0)

S2(0)e(σ

22−σ

21)t/2+(σ1−σ2)Wt+σ2(σ1−σ2)t

=S1(0)

S2(0)e(σ1−σ2)Wt+σ2σ1t−(σ2

2+σ21)t/2

=S1(0)

S2(0)e(σ1−σ2)Wt−(σ1−σ2)

2t/2,

whereWt := Wt − σ2t

is a standard Brownian motion under the forward measure P defined by

dPdP

= e−r T0rsds

NTN0

= e−rTS2(T )

S2(0)

= e−rT eσ2WT+µT

= eσ2WT+(µ−r)T

= eσ2WT−σ22t/2.

4. Given that Xt = e(σ22−σ

21)t/2S1(t) and X(t) = Xt/Nt = Xt/S2(t), we

have

e−rT IE[(S1(T )− κS2(T ))+] = e−rT IE[(e−(σ22−σ

21)T/2XT − κS2(T ))+]

= e−rT e−(σ22−σ

21)T/2 IE[(XT − κe(σ

22−σ

21)T/2S2(T ))+]

= S2(0)e−(σ22−σ

21)T/2IE[(XT − κe(σ

22−σ

21)T/2)+]

= S2(0)e−(σ22−σ

21)T/2IE[(X0e

(σ1−σ2)WT−(σ1−σ2)2T/2 − κe(σ2

2−σ21)T/2)+]

= S2(0)e−(σ22−σ

21)T/2

(X0Φ

0+(T, X0)− κe(σ2

2−σ21)T/2Φ0

−(T, X0))

= S2(0)e−(σ22−σ

21)T/2X0Φ

0+(T, X0)

−κS2(0)e−(σ22−σ

21)T/2e(σ

22−σ

21)T/2Φ0

−(T, X0)

= e−(σ22−σ

21)T/2X0Φ

0+(T, X0)− κS2(0)Φ0

−(T, X0)

= S1(0)Φ0+(T, X0)− κS2(0)Φ0

−(T, X0),

where

Φ0+(T, x) = Φ

(log(x/κ)

|σ1 − σ2|√T

+(σ1 − σ2)2 − (σ2

2 − σ21)

2|σ1 − σ2|√T

)

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N. Privault

=

Φ

(log(x/κ)

|σ1 − σ2|√T

+ σ1√T

), σ1 > σ2,

Φ

(log(x/κ)

|σ1 − σ2|√T− σ1

√T

), σ1 < σ2,

and

Φ0−(T, x) = Φ

(log(x/κ)

|σ1 − σ2|√T− (σ1 − σ2)2 + (σ2

2 − σ21)

2|σ1 − σ2|√T

)

=

Φ

(log(x/κ)

|σ1 − σ2|√T

+ σ2√T

), σ1 > σ2,

Φ

(log(x/κ)

|σ1 − σ2|√T− σ2

√T

), σ1 < σ2,

if σ1 6= σ2. In case σ1 = σ2 we find

e−rT IE[(S1(T )− κS2(T ))+] = e−rT IE[S1(T )(1− κS2(0)/S1(0))+]

= (1− κS2(0)/S1(0))+e−rT IE[S1(T )]

= (S1(0)− κS2(0))1S1(0)>κS2(0).

Exercise 10.4

1. It suffices to check that the definition of (WNt )t∈R+

implies the correlationidentity dWS

t · dWNt = ρdt by Ito’s calculus.

2. We let

σt =√

(σSt )2 − 2ρσRt σSt + (σRt )2

and

dWXt =

σSt − ρσNtσt

dWSt −

√1− ρ2σ

Nt

σtdWt, t ∈ R+,

which defines a standard Brownian motion under P∗ due to the definitionof σt.

Exercise 10.5

1. We have σ =√

(σS)2 − 2ρσRσS + (σR)2.

2. Letting Xt = e−rtXt = e(a−r)tSt/Rt, t ∈ R+, we have

IE∗

[(STRT− κ)+ ∣∣∣Ft] = e−aT IE∗

[(XT − eaTκ

)+ ∣∣∣Ft]= e−(a−r)T IE∗

[(XT − e(a−r)Tκ

)+ ∣∣∣Ft]

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"



= e−(a−r)T(XtΦ

((r − a+ σ2/2)(T − t)

σ√T − t

+1

σ√T − t

logStκRt

)−κe(a−r)TΦ

((r − a− σ2/2)(T − t)

σ√T − t +

1

σ√T − t log

StκRt

))=StRte(r−a)(T−t)Φ

((r − a+ σ2/2)(T − t)

σ√T − t

+1

σ√T − t

logStκRt

)−κΦ

((r − a− σ2/2)(T − t)

σ√T − t +

1

σ√T − t log

StκRt

),

hence the price of the quanto option is

e−r(T−t) IE∗

[(STRT− κ)+ ∣∣∣Ft]

=StRte−a(T−t)Φ

((r − a+ σ2/2)(T − t)

σ√T − t

+1

σ√T − t

logStκRt

)−κe−r(T−t)Φ

((r − a− σ2/2)(T − t)

σ√T − t

+1

σ√T − t

logStκRt

).

Chapter 11

Exercise 11.1 Letting Yt = ebtXt we have

dYt = d(ebtXt) = bebtXtdt+ebtdXt = bebtXtdt+e

bt(−bXtdt+σe−btdBt) = σdBt,

hence

Yt = Y0 +w t

0dYs = Y0 + σ

w t

0dBs = Y0 + σBt,

andXt = e−btYt = e−btY0 + σe−btBt = e−btX0 + σe−btBt.

Exercise 11.2

1. We have rt = r0 + at+ σBt, and

F (t, rt) = F (t, r0 + at+ σBt),

hence by Proposition 11.2 the PDE satisfied by F (t, x) is

− xF (t, x) +∂F

∂t(t, x) + a

∂F

∂x(t, x) +

1

2σ2 ∂

2F

∂x2(t, x) = 0, (16.59)

with terminal condition F (T, x) = 1.2. We have rt = r0 + at+ σBt and

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N. Privault

F (t, rt) = IE∗[exp

(−

w T

trsds

) ∣∣∣Ft]= IE∗

[exp

(−r0(T − t)− a

w T

tsds− σ

w T

tBsds

) ∣∣∣Ft]= IE∗

[e−r0(T−t)−a(T

2−t2)/2 exp

(−(T − t)Bt − σ

w T

t(T − s)dBs

) ∣∣∣Ft]= e−r0(T−t)−a(T

2−t2)/2−σ(T−t)Bt IE∗[exp

(−σ

w T

t(T − s)dBs

) ∣∣∣Ft]= e−r0(T−t)−a(T−t)(T+t)/2−σ(T−t)Bt IE∗

[exp

(−σ

w T

t(T − s)dBs

)]= exp

(−(T − t)rt − a(T − t)2/2 +

σ2

2

w T

t(T − s)2ds

)= exp

(−(T − t)rt − a(T − t)2/2 + σ2(T − t)3/6

),

hence F (t, x) = exp(−(T − t)x− a(T − t)2/2 + σ2(T − t)3/6

).

Note that the PDE (16.59) can also be solved by looking for a solutionof the form F (t, x) = eA(T−t)+xC(T−t), in which case one would findA(s) = −as2/2 + s3/6 and C(s) = −s.

3. We check that the function F (t, x) of Question 2 satisfies the PDE (16.59)of Question 1, since F (T, x) = 1 and

−xF (t, x) +

(x+ a(T − t)− σ2

2(T − t)2

)F (t, x)− a(T − t)F (t, x)

+1

2σ2(T − t)2F (t, x) = 0.

4. We have

f(t, T, S) =1

S − T (logP (t, T )− logP (t, S))

=1

S − T

((−(T − t)rt +

σ2

6(T − t)3

)−(−(S − t)rt +

σ2

6(S − t)3

))= rt +

1

S − Tσ2

6((T − t)3 − (S − t)3).

5. We have

f(t, T ) = − ∂

∂TlogP (t, T ) = rt −

σ2

2(T − t)2.

6. We havedtf(t, T ) = σ2(T − t)dt+ adt+ σdBt.

7. The HJM condition (11.36) is satisfied since the drift of dtf(t, T ) equals

σr Ttσds.

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Exercise 11.3

1. We have σ(t, s) = σs and we check that

α(t, T ) = σ2T (T 2 − t2)/2 = σTw T

tσsds = σ(t, T )

w T

tσ(t, s)ds.

2. We have

f(t, T ) = f(0, T ) +w t

0dsf(s, T )

= f(0, T ) +σ2

2T

w t

0(T 2 − s2)ds+ σT

w t

0dBs

= f(0, T ) +σ2

2T 3

w t

0ds− σ2

2T

w t

0s2ds+ σT

w t

0dBs

= f(0, T ) + σ2T 3t/2− σ2Tt3/6 + σTBt

= f(0, T ) + σ2Tt(T 2/2− t2/6) + σTBt.

3. We have

rt = f(t, t) = f(0, t) +σ2t2(t2/2− t2/6) +σtBt = f(0, t) +σ2t4/3 +σtBt.

4. We have

drt =4

3σ2t3dt+ σBtdt+ σtdBt

= 4σ2t3/3dt+1

t(rt − f(0, t)− σ2t4/3)dt+ σtdBt

=1

t(rt − f(0, t) + σ2t4)dt+ σtdBt

= σ2t3dt+1

t(rt − f(0, t))dt+ σtdBt,

which is a Hull-White type short term model with the time-dependentdeterministic coefficients η(t) = σ2t3, ψ(t) = 1/t and ξ(t) = σt. Notethat t 7→ f(0, t) is the initial rate curve data.

Exercise 11.4

1. We have

P (t, T ) = P (s, T ) exp

(w t

srudu+

w t

sσTu dBu −

1

2

w t

s|σTu |2du

),

0 ≤ s ≤ t ≤ T .2. We have

d(e−

r t0rsdsP (t, T )

)= e−

r t0rsdsσTt P (t, T )dBt,

which gives a martingale after integration, from the properties of the Itointegral.

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N. Privault

3. By the martingale property of the previous question we have

IE[e−

r T0rsds

∣∣∣Ft] = IE[P (T, T )e−

r T0rsds

∣∣∣Ft]= P (t, T )e−


4. By the previous question we have

P (t, T ) = er t0rsds IE

[e−

r T0rsds

∣∣∣Ft]= IE

[er t0rsdse−

r T0rsds

∣∣∣Ft]= IE

[e−

r Ttrsds

∣∣∣Ft] , 0 ≤ t ≤ T,

since e−r t0rsds is an Ft-measurable random variable.

5. We have

P (t, S)

P (t, T )=P (s, S)

P (s, T )exp

(w t

s(σSu − σTu )dBu −

1

2

w t

s(|σSu |2 − |σTu |2)du

)=P (s, S)

P (s, T )exp

(w t

s(σSu − σTu )dBTu −

1

2

w t

s(σSu − σTu )2du

),

0 ≤ t ≤ T , hence letting s = t and t = T in the above expression we have

P (T, S) =P (t, S)

P (t, T )exp

(w T


1

2

w T


).

6. We have

P (t, T ) IET

[(P (T, S)− κ)

+]

= P (t, T ) IET

[(P (t, S)

P (t, T )er Tt

(σSs −σTs )dBTs − 1

2

r Tt

(σSs −σTs )2ds − κ

)+]

= P (t, T ) IE[(eX − κ)+ | Ft]

= P (t, T )emt+v2t /2Φ

(vt2

+1

vt(mt + v2t /2− log κ)

)−κP (t, T )Φ

(−vt

2+

1


),

with

mt = log(P (t, S)/P (t, T ))− 1

2

w T


and

v2t =w T

t(σSs − σTs )2ds,

508


"



i.e.

P (t, T ) IET

[(P (T, S)− κ)

+]

= P (t, S)Φ

(vt2

+1

vtlog

P (t, S)

κP (t, T )

)− κP (t, T )Φ

(−vt

2+

1

vtlog

P (t, S)

κP (t, T )

).

Exercise 11.5

1. We check that P (T, T ) = eXTT = 1.

2. We have

f(t, T, S) = − 1

S − T(XSt −XT

t − µ(S − T ))

= µ− σ 1

S − T

((S − t)

w t

0

1

S − sdBs − (T − t)w t

0

1

T − sdBs)

= µ− σ 1

S − Tw t

0

(S − tS − s −

T − tT − s

)dBs

= µ− σ 1

S − Tw t

0

(T − s)(S − t)− (T − t)(S − s)(S − s)(T − s) dBs

= µ+σ

S − Tw t

0

(s− t)(S − T )

(S − s)(T − s)dBs.

3. We have

f(t, T ) = µ− σw t

0

t− s(T − s)2 dBs.

4. We note that

limTt

f(t, T ) = µ− σw t

0

1

t− sdBs

does not exist in L2(Ω).5. By Ito’s calculus we have

dP (t, T )

P (t, T )= σdBt +

1

2σ2dt+ µdt− XT

t

T − tdt

= σdBt +1


T − t dt, t ∈ [0, T ].

6. Let

rSt = µ+1

2σ2 − XS

t

S − t= µ+

1

2σ2 − σ

w t

0

1

S − sdBs,

and apply the result of Exercise 11.11.7-(4).

" 509



N. Privault

7. We have

IE

[dPTdP

∣∣∣Ft] = eσBt−σ2t/2.

8. By the Girsanov theorem, the process Bt := Bt− σt is a standard Brow-nian motion under PT .

9. We have

logP (T, S) = −µ(S − T ) + σ(S − T )w T

0

1

S − sdBs

= −µ(S − T ) + σ(S − T )w t

0

1

S − sdBs + σ(S − T )w T

t

1

S − sdBs

=S − TS − t logP (t, S) + σ(S − T )

w T

t

1

S − sdBs

=S − TS − t logP (t, S) + σ(S − T )

w T

t

1

S − sdBs + σ2(S − T )w T

t

1

S − sds

=S − TS − t logP (t, S) + σ(S − T )

w T

t

1

S − sdBs + σ2(S − T ) logS − tS − T ,

0 < T < S.10. We have

P (t, T ) IET

[(P (T, S)−K)+

∣∣∣Ft]= P (t, T ) IE[(eX − κ)+ | Ft]

= P (t, T )emt+v2t /2Φ

(vt2

+1


)−κP (t, T )Φ

(−vt

2+

1


)= P (t, T )emt+v

2t /2Φ

(vt +

1

vt(mt − log κ)

)− κP (t, T )Φ

(1

vt(mt − log κ)

),

with

mt =S − TS − t logP (t, S) + σ2(S − T ) log

S − tS − T

and

v2t = σ2(S − T )2w T

t

1

(S − s)2 ds

= σ2(S − T )2(

1

S − T −1

S − t

)= σ2(S − T )

(T − t)(S − t) ,

hence

510


"



P (t, T ) IET

[(P (T, S)−K)+

∣∣∣Ft]= P (t, T ) (P (t, S))

(S−T )(S−t)(S − tS − T

)σ2(S−T )

ev2t /2

×Φ(vt +

1

vtlog

((P (t, S))

(S−T )(S−t)

κ

(S − tS − T

)σ2(S−T )))

−κP (t, T )Φ

(1

vtlog

((P (t, S))

(S−T )(S−t)

κ

(S − tS − T

)σ2(S−T )))

.

Exercise 11.6 From Proposition 11.2 the bond pricing PDE is∂F

∂t(t, x) = xF (t, x)− (α− βx)

∂F

∂x(t, x)− 1

2σ2x2

∂2F

∂x2(t, x)

F (T, x) = 1.

Let us search for a solution of the form

F (t, x) = eA(T−t)−xB(T−t),

with A(0) = B(0) = 0, which impliesA′(s) = 0

B′(s) + βB(s) + 12σ

2B2(s) = 1.

hence in particular A(s) = 0, s ∈ R, and B(s) solves a Riccatti equation,whose solution is easily checked to be

B(s) =2(eγs − 1)

2γ + (β + γ)(eγs − 1),

with γ =√β2 + 2σ2.

Chapter 12

Exercise 12.1

1. The forward measure PS is defined from the numeraire Nt := P (t, S) andthis gives

Ft = P (t, S)IE[(κ− L(T, T, S))+ | Ft].2. The LIBOR rate L(t, T, S) is a driftless geometric Brownian motion with

volatility σ under the forward measure PS . Indeed, the LIBOR rate

" 511



N. Privault

L(t, T, S) can be written as the forward price L(t, T, S) = Xt = Xt/Ntwhere Xt = (P (t, T )−P (t, S))/(S−T ) and Nt = P (t, S). Since both dis-

counted bond prices e−r t0rsdsP (t, T ) and e−

r t0rsdsP (t, S) are martingales

under P∗, the same is true of Xt. Hence L(t, T, S) = Xt/Nt becomes a

martingale under the forward measure PS by Proposition 2.1, and com-puting its dynamics under PS amounts to removing any “dt” term in(12.19), i.e.

dL(t, T, S) = σL(t, T, S)dWt, 0 ≤ t ≤ T,

hence L(t, T, S) = L(0, T, S)eσWt−σ2t/2, where (Wt)t∈R+is a standard

Brownian motion under PS .3. We find

Ft = P (t, S)IE[(κ− L(T, T, S))+ | Ft]= P (t, S)IE[(κ− L(t, T, S)e−σ

2(T−t)/2+σ(WT−Wt))+ | Ft]= P (t, S)(κΦ(−d−)− XtΦ(−d+))

= κP (t, S)Φ(−d−)− P (t, S)L(t, T, S)Φ(−d+)

= κP (t, S)Φ(−d−)− (P (t, T )− P (t, S))Φ(−d+)/(S − T ),

where em = L(t, T, S)e−σ2(T−t)/2, v2 = (T − t)σ2, and

d+ =log(L(t, T, S)/κ)

σ√T − t +

σ√T − t2

,

and

d− =log(L(t, T, S)/κ)

σ√T − t − σ

√T − t2

,

because L(t, T, S) is a driftless geometric Brownian motion with volatility

σ under the forward measure PS .

Exercise 12.2

1. We havedP (t, Ti)

P (t, Ti)= rtdt+ ζitdBt, i = 1, 2,

and

P (T, Ti) = P (t, Ti) exp

(w T

trsds+

w T

tζisdBs −

1

2

w T

t(ζis)

2ds

),

0 ≤ t ≤ T ≤ Ti, i = 1, 2, hence

logP (T, Ti) = logP (t, Ti) +w T

trsds+

w T

tζisdBs −

1

2

w T

t(ζis)

2ds,

512


"



0 ≤ t ≤ T ≤ Ti, i = 1, 2, and

d logP (t, Ti) = rtdt+ ζitdBt −1

2(ζit)

2dt, i = 1, 2.

In the present modeldrt = σdBt,

where (Bt)t∈R+is a standard Brownian motion under P, we have

ζit = −σ(Ti − t), 0 ≤ t ≤ Ti, i = 1, 2.

LettingdBit = dBt − ζitdt,

defines a standard Brownian motion under Pi, i = 1, 2, and we have

P (T, T1)

P (T, T2)=P (t, T1)

P (t, T2)exp

(w T

t(ζ1s − ζ2s )dBs −

1

2

w T

t((ζ1s )2 − (ζ2s )2)ds

)=P (t, T1)

P (t, T2)exp

(w T

t(ζ1s − ζ2s )dB2

s −1

2

w T

t(ζ1s − ζ2s )2ds

),

which is an Ft-martingale under P2 and under P1,2, and

P (T, T2)

P (T, T1)=P (t, T2)

P (t, T1)exp

(−

w T


s −1

2

w T


),

which is an Ft-martingale under P1.2. We have

f(t, T1, T2) = − 1

T2 − T1(logP (t, T2)− logP (t, T1))

= rt +1

T2 − T1σ2

6((T1 − t)3 − (T2 − t)3).

3. We have

df(t, T1, T2) = − 1

T2 − T1d log (P (t, T2)/P (t, T1))

= − 1

T2 − T1

((ζ2t − ζ1t )dBt −

1

2((ζ2t )2 − (ζ1t )2)dt

)= − 1

T2 − T1

((ζ2t − ζ1t )(dB2

t + ζ2t dt)−1

2((ζ2t )2 − (ζ1t )2)dt

)= − 1

T2 − T1

((ζ2t − ζ1t )dB2

t −1

2(ζ2t − ζ1t )2dt

).

4. We have

" 513



N. Privault

f(T, T1, T2) = − 1

T2 − T1log (P (T, T2)/P (T, T1))

= f(t, T1, T2)− 1

T2 − T1

(w T


1

2((ζ2s )2 − (ζ1s )2)ds

)= f(t, T1, T2)− 1

T2 − T1

(w T


s −1

2

w T


)= f(t, T1, T2)− 1

T2 − T1

(w T


s +1

2

w T


).

Hence f(T, T1, T2) has a Gaussian distribution given Ft with conditionalmean

m = f(t, T1, T2) +1

2

w T


under P2, resp.

m = f(t, T1, T2)− 1

2

w T


under P1, and variance

v2 =1

(T2 − T1)2

w T

t(ζ2s − ζ1s )2ds.

Hence

(T2 − T1) IE[e−

r T2t rsds(f(T1, T1, T2)− κ)+

∣∣∣Ft]= (T2 − T1)P (t, T2) IE2

[(f(T1, T1, T2)− κ)+

∣∣∣Ft]= (T2 − T1)P (t, T2) IE2

[(m+X − κ)+

∣∣∣Ft]= (T2 − T1)P (t, T2)

(v√2πe−

(κ−m)2

2v2 + (m− κ)Φ((m− κ)/v)

).

5. We have

L(T, T1, T2) = S(T, T1, T2)

=1

T2 − T1

(P (T, T1)

P (T, T2)− 1

)=

1

T2 − T1

(P (t, T1)

P (t, T2)exp

(w T


1

2

w T

t((ζ1s )2 − (ζ2s )2)ds

)− 1

)=

1

T2 − T1

(P (t, T1)

P (t, T2)exp

(w T


s −1

2

w T


)− 1

)=

1

T2 − T1

(P (t, T1)

P (t, T2)exp

(w T


s +1

2

w T


)− 1

),

514


"



and by Ito calculus,

dS(t, T1, T2) =1

T2 − T1d

(P (t, T1)

P (t, T2)

)=

1

T2 − T1P (t, T1)

P (t, T2)

((ζ1t − ζ2t )dBt +

1

2(ζ1t − ζ2t )2dt− 1

2((ζ1t )2 − (ζ2t )2)dt

)=

(1

T2 − T1+ S(t, T1, T2)

)((ζ1t − ζ2t )dBt + ζ2t (ζ2t − ζ1t )dt)dt

)=

(1

T2 − T1+ S(t, T1, T2)

)((ζ1t − ζ2t )dB1

t + ((ζ2t )2 − (ζ1t )2)dt)

=

(1

T2 − T1+ S(t, T1, T2)

)(ζ1t − ζ2t )dB2

t , t ∈ [0, T1],

hence 1T2−T1

+ S(t, T1, T2) is a geometric Brownian motion, with

1

T2 − T1+ S(T, T1, T2)

=

(1

T2 − T1+ S(t, T1, T2)

)exp

(w T


s −1

2

w T


),

0 ≤ t ≤ T ≤ T1.6. We have

(T2 − T1) IE[e−


∣∣∣Ft]= (T2 − T1) IE

[e−

r T1t rsdsP (T1, T2)(L(T1, T1, T2)− κ)+

∣∣∣Ft]= P (t, T1, T2) IE1,2

[(S(T1, T1, T2)− κ)+

∣∣∣Ft] .The forward measure P2 is defined by

IE

[dP2

dP

∣∣∣Ft] =P (t, T2)

P (0, T2)e−

r t0rsds, 0 ≤ t ≤ T2,

and the forward swap measure is defined by

IE

[dP1,2

dP

∣∣∣Ft] =P (t, T2)

P (0, T2)e−

r t0rsds, 0 ≤ t ≤ T1,

hence P2 and P1,2 coincide up to time T1 and (B2t )t∈[0,T1] is a standard

Brownian motion until time T1 under P2 and under P1,2, consequentlyunder P1,2 we have

L(T, T1, T2) = S(T, T1, T2)

" 515



N. Privault

= − 1

T2 − T1+

(1

T2 − T1+ S(t, T1, T2)

)er Tt

(ζ1s−ζ2s )dB

2s− 1

2

r Tt

(ζ1s−ζ2s )

2ds,

has same law as

1

T2 − T1

(P (t, T1)

P (t, T2)eX−

12Var [X] − 1

),

where X is a centered Gaussian random variable with variance

w T1


given Ft. Hence

(T2 − T1) IE[e−


∣∣∣Ft]= P (t, T1, T2)

×BS

(1

T2 − T1+ S(t, T1, T2),

r T1


T1 − t, κ+

1

T2 − T1, T1 − t

).

Exercise 12.3

1. We have

L(t, T1, T2) = L(0, T1, T2)er t0γ1(s)dW

2s− 1

2

r t0|γ1(s)|2ds, 0 ≤ t ≤ T1,

and L(t, T2, T3) = b. Note that we have P (t, T2)/P (t, T3) = 1 + δb henceP2 = P3 = P1,2 up to time T1.

2. We use change of numeraire under the forward measure P2.3. We have

E[e−


∣∣∣Ft]= P (t, T2)E2

[(L(T1, T1, T2)− κ)+ | Ft

]= P (t, T2)E2

[(L(t, T1, T2)e

r T1t γ1(s)dW

2s− 1

2

r T1t |γ1(s)|2ds − κ)+ | Ft

]= P (t, T2)BS(κ, L(t, T1, T2), σ1(t), 0, T1 − t),

where

σ21(t) =

1

T1 − tw T1

t|γ1(s)|2ds.

4. We have

P (t, T1)

P (t, T1, T3)=

P (t, T1)

δP (t, T2) + δP (t, T3)

=P (t, T1)

δP (t, T2)

1

1 + P (t, T3)/P (t, T2)

516


"



=1 + δb

δ(δb+ 2)(1 + δL(t, T1, T2)), 0 ≤ t ≤ T1,

and

P (t, T3)

P (t, T1, T3)=

P (t, T3)

P (t, T2) + P (t, T3)

=1

1 + P (t, T2)/P (t, T3)

=1

δ

1

2 + δb, 0 ≤ t ≤ T2. (16.60)

5. We have

S(t, T1, T3) =P (t, T1)

P (t, T1, T3)− P (t, T3)

P (t, T1, T3)

=1 + δb

δ(2 + δb)(1 + δL(t, T1, T2))− 1

δ(2 + δb)

=1

2 + δb(b+ (1 + δb)L(t, T1, T2)), 0 ≤ t ≤ T2.

We have

dS(t, T1, T3) =1 + δb

2 + δbL(t, T1, T2)γ1(t)dW 2

t

=

(S(t, T1, T3)− b

2 + δb

)γ1(t)dW 2

t

= S(t, T1, T3)σ1,3(t)dW 2t , 0 ≤ t ≤ T2,

with

σ1,3(t) =

(1− b

S(t, T1, T3)(2 + δb)

)γ1(t)

=

(1− b

b+ (1 + δb)L(t, T1, T2)

)γ1(t)

=(1 + δb)L(t, T1, T2)

b+ (1 + δb)L(t, T1, T2)γ1(t)

=(1 + δb)L(t, T1, T2)

(2 + δb)S(t, T1, T3)γ1(t).

6. The process (W 2)t∈R+is a standard Brownian motion under P2 and

P (t, T1, T3)E1,3

[(S(T1, T1, T3)− κ)+ | Ft

]= P (t, T2)BS(κ, S(t, T1, T2), σ1,3(t), 0, T1 − t),

" 517



N. Privault

where |σ1,3(t)|2 is the approximation of the volatility

1

T1 − tw T1

t|σ1,3(s)|2ds =

1

T1 − tw T1

t

((1 + δb)L(s, T1, T2)

(2 + δb)S(s, T1, T3)

)2

γ1(s)ds

obtained by freezing the random component of σ1,3(s) at time t, i.e.

σ21,3(t) =

1

T1 − t

((1 + δb)L(t, T1, T2)

(2 + δb)S(t, T1, T3)

)2 w T1

t|γ1(s)|2ds.

Exercise 12.4

1. We have

IE[e−


+∣∣∣Ft] = VT = V0 +

w T

0dVt

= P (0, T ) IET

[(P (T, S)− κ)

+]

+w t

0ξTs dP (s, T ) +

w t

0ξSs dP (s, S).

2. We have

dVt = d(e−

r t0rsdsVt

)= −rte−

r t0rsdsVtdt+ e−

r t0rsdsdVt

= −rte−r t0rsds(ξTt P (t, T )

+ξSt P (t, S))dt+ e−r t0rsdsξTt dP (t, T ) + e−

r t0rsdsξSt dP (t, S)

= ξTt dP (t, T ) + ξSt dP (t, S).

3. By Ito’s formula we have

IET

[(P (T, S)− κ)

+ |Ft]

= C(XT , 0, 0)

= C(X0, T, v(0, T )) +w t

0

∂C

∂x(Xs, T − s, v(s, T ))dXs

= IET

[(P (T, S)− κ)

+]

+w t

0

∂C

∂x(Xs, T − s, v(s, T ))dXs,

since the process

t 7→ IET

[(P (T, S)− κ)

+ |Ft]

is a martingale under P.4. We have

dVt = d(Vt/P (t, T ))

= d IET

[(P (T, S)− κ)

+ |Ft]

=∂C

∂x(Xt, T − t, v(t, T ))dXt

518


"



=P (t, S)

P (t, T )

∂C

∂x(Xt, T − t, v(t, T ))(σSt − σTt )dBTt .

5. We have

dVt = d(P (t, T )Vt)

= P (t, T )dVt + VtdP (t, T ) + dVt · dP (t, T )

= P (t, S)∂C

∂x(Xt, T − t, v(t, T ))(σSt − σTt )dBTt + VtdP (t, T )

+P (t, S)∂C

∂x(Xt, T − t, v(t, T ))(σSt − σTt )σTt dt

= P (t, S)∂C


6. We have

dVt = d(e−r t0rsdsVt)

= −rte−r t0rsdsVtdt+ e−

r t0rsdsdVt

= P (t, S)∂C


7. We have

dVt = P (t, S)∂C

∂x(Xt, T − t, v(t, T ))(σSt − σTt )dBt + VtdP (t, T )

=∂C

∂x(Xt, T − t, v(t, T ))dP (t, S)

−P (t, S)

P (t, T )

∂C

∂x(Xt, T − t, v(t, T ))dP (t, T ) + VtdP (t, T )

=

(Vt −

P (t, S)

P (t, T )

∂C

∂x(Xt, T − t, v(t, T ))

)dP (t, T )

+∂C

∂x(Xt, T − t, v(t, T ))dP (t, S),

hence the hedging strategy (ξTt , ξSt )t∈[0,T ] of the bond option is given by

ξTt = Vt −P (t, S)

P (t, T )

∂C

∂x(Xt, T − t, v(t, T ))

= C(Xt, T − t, v(t, T ))− P (t, S)

P (t, T )

∂C

∂x(Xt, T − t, v(t, T )),

and

ξSt =∂C

∂x(Xt, T − t, v(t, T )),

t ∈ [0, T ].

" 519



N. Privault

8. We have

∂C

∂x(x, τ, v)

=∂

∂x

[xΦ

(v√τ

2+

1

v√τ

logx

κ

)− κΦ

(−v√τ

2+

1

v√τ

logx

κ

)]= x

∂

∂xΦ

(v√τ

2+

1

v√τ

logx

κ

)− κ ∂

∂xΦ

(−v√τ

2+

1

v√τ

logx

κ

)+Φ

(v√τ

2+

1

v√τ

logx

κ

)

= xe− 1

2

(v√τ

2 + 1v√τlog x

κ

)2

√2π

(1

v√τx

)− κe

− 12

(− v√τ

2 + 1v√τlog x

κ

)2

√2π

(1

v√τx

)+Φ

(v√τ

2+

1

v√τ

logx

κ

)= Φ

(log(x/κ) + τv2/2√

τv

).

As a consequence we get

ξTt = C(Xt, T − t, v(t, T ))− P (t, S)

P (t, T )

∂C

∂x(Xt, T − t, v(t, T ))

=P (t, S)

P (t, T )Φ

((T − t)v2(t, T )/2 + logXt√

T − tv(t, T )

)−κΦ

(−v(t, T )

2+

1

v(t, T )log

P (t, S)

κP (t, T )

)−P (t, S)

P (t, T )Φ

(log(Xt/κ) + (T − t)v2(t, T )/2√

T − tv(t, T )

)= −κΦ

(log(Xt/κ)− (T − t)v2(t, T )/2

v(t, T )√T − t

),

and

ξSt =∂C

∂x(Xt, T − t, v(t, T )) = Φ

(log(Xt/κ) + (T − t)v2(t, T )/2

v(t, T )√T − t

),

t ∈ [0, T ], and the hedging strategy is given by

VT = IE[e−


+∣∣∣Ft]

= V0 +w t

0ξTs dP (s, T ) +

w t

0ξSs dP (s, S)

= V0 − κw t

0Φ

(log(Xt/κ)− (T − t)v2(t, T )/2

v(t, T )√T − t

)dP (s, T )

520


"



+w t

0Φ

(log(Xt/κ) + (T − t)v2(t, T )/2

v(t, T )√T − t

)dP (s, S).

Consequently the bond option can be hedged by shortselling a bond withmaturity T for the amount

κΦ

(log(Xt/κ)− (T − t)v2(t, T )/2

v(t, T )√T − t

),

and by buying a bond with maturity S for the amount

Φ

(log(Xt/κ) + (T − t)v2(t, T )/2

v(t, T )√T − t

).

Exercise 12.5

1. Choosing the annuity numeraire Nt = P (Ti, Ti, Tj) we have

IE∗[e−

r Tit rsdsP (Ti, Ti, Tj)(κ− S(Ti, Ti, Tj))

+∣∣∣Ft]

= NtIEi,j

[P (Ti, Ti, Tj)

NTi(κ− S(Ti, Ti, Tj))

+∣∣∣Ft]

= P (t, Ti, Tj)IEi,j [(κ− S(Ti, Ti, Tj))+ | Ft].

2. Since S(t, Ti, Tj) is a forward price for the numeraire P (t, Ti, Tj), it is a

martingale under the forward swap measure Pi,j and we have

S(t, Ti, Tj) = σS(t, Ti, Tj)dWi,jt , 0 ≤ t ≤ Ti,

where (W i,jt )t∈R+

is a standard Brownian motion under the forward swap

measure Pi,j .3. We find

P (t, Ti, Tj)IEi,j [(κ− S(Ti, Ti, Tj))+ | Ft]

= P (t, Ti, Tj)IEi,j [(κ− S(t, Ti, Tj)e−σ2(Ti−t)/2+σ(WTi

−Wt))+ | Ft]= P (t, Ti, Tj)(κΦ(−d−)− XtΦ(−d+))

= P (t, Ti, Tj)κΦ(−d−)− P (t, Ti, Tj)S(t, Ti, Tj)Φ(−d+)

= P (t, Ti, Tj)κΦ(−d−)− (P (t, Ti)− P (t, Tj))Φ(−d+)/(Tj − Ti),

where em = S(t, Ti, Tj)e−σ2(T−t)/2, v2 = (T − t)σ2, and

d+ =log(S(t, Ti, Tj)/κ)

σ√Ti − t

+σ√Ti − t2

,

and

" 521



N. Privault

d− =log(S(t, Ti, Tj)/κ)

σ√Ti − t

− σ√Ti − t2

,

because S(t, Ti, Tj) is a driftless geometric Brownian motion with volatil-

ity σ under the forward measure Pi,j .

Exercise 12.6

1. We have

S(Ti, Ti, Tj) = S(t, Ti, Tj) exp

(w Ti

tσi,j(s)dB

i,js −

1

2

w Ti

t|σi,j |2(s)ds

).

2. We have

P (t, Ti, Tj) IEi,j


+∣∣∣Ft]


[(S(t, Ti, Tj)e

r Tit σi,j(s)dB

i,js − 1

2

r Tit |σi,j |

2(s)ds − κ)+ ∣∣∣Ft]

= P (t, Ti, Tj)BS(κ, v(t, Ti)/√Ti − t, 0, Ti − t)

= P (t, Ti, Tj)

×(S(t, Ti, Tj)Φ

(log(x/K)

v(t, Ti)+v(t, Ti)

2

)− κΦ

(log(x/K)


2

)),

where

v2(t, Ti) =w Ti

t|σi,j |2(s)ds.

3. Integrate the self-financing condition (12.25) between 0 and t.4. We have

dVt = d(e−

r t0rsdsVt

)= −rte−

r t0rsdsVtdt+ e−

r t0rsdsdVt

= −rte−r t0rsds

j∑k=i

ξkt P (t, Tk), dt+ e−r t0rsds

j∑k=i

ξkt dP (t, Tk)

=

j∑k=i

ξkt dP (t, Tk), 0 ≤ t ≤ Ti.

sincedP (t, Tk)

P (t, Tk)= ζk(t)dt, k = i, . . . , j.

5. We apply the Ito formula and the fact that

t 7→ IEi,j


+∣∣∣Ft]

522


"



and (St)t∈R+are both martingales under Pi,j .

6. Use the fact that

Vt = IEi,j


+∣∣∣Ft] ,

and apply the result of Question 5.7. Apply the Ito rule to Vt = P (t, Ti, Tj)Vt using Relation (12.23) and the

result of Question 6.8. We have

dVt = St∂C

∂x(St, v(t, Ti))

×(j−1∑k=i

(Tk+1 − Tk)P (t, Tk+1)(ζi(t)− ζk+1(t)) + P (t, Tj)(ζi(t)− ζj(t)))dBt

+VtdP (t, Ti, Tj)

= St∂C

∂x(St, v(t, Ti))

×(j−1∑k=i

(Tk+1 − Tk)P (t, Tk+1)(ζi(t)− ζk+1(t)) + P (t, Tj)(ζi(t)− ζj(t)))dBt

+Vt

j−1∑k=i

(Tk+1 − Tk)ζk+1(t)P (t, Tk+1)dBt

= St∂C

∂x(St, v(t, Ti))

j−1∑k=i

(Tk+1 − Tk)P (t, Tk+1)(ζi(t)− ζk+1(t))dBt

+∂C

∂x(St, v(t, Ti))P (t, Tj)(ζi(t)− ζj(t))dBt

+Vt

j−1∑k=i


= Stζi(t)∂C

∂x(St, v(t, Ti))

j−1∑k=i


−St∂C

∂x(St, v(t, Ti))

j−1∑k=i


+∂C


+Vt

j−1∑k=i


" 523



N. Privault

= Stζi(t)∂C

∂x(St, v(t, Ti))

j−1∑k=i


+

(Vt − St

∂C

∂x(St, v(t, Ti))

) j−1∑k=i


+∂C

∂x(St, v(t, Ti))P (t, Tj)(ζi(t)− ζj(t))dBt.

9. We have

dVt = Stζi(t)∂C

∂x(St, v(t, Ti))

j−1∑k=i


+

(Vt − St

∂C

∂x(St, v(t, Ti))

) j−1∑k=i


+∂C


= (P (t, Ti)− P (t, Tj))ζi(t)∂C

∂x(St, v(t, Ti))dBt

+

(Vt − St

∂C

∂x(St, v(t, Ti))

)dP (t, Ti, Tj)

+∂C


= (ζi(t)P (t, Ti)− ζj(t)P (t, Tj))∂C

∂x(St, v(t, Ti))dBt

+

(Vt − St

∂C

∂x(St, v(t, Ti))

)dP (t, Ti, Tj)

=∂C


+

(Vt − St

∂C

∂x(St, v(t, Ti))

)dP (t, Ti, Tj).

10. We have

∂C

∂x(x, τ, v) =

∂

∂x

[xΦ

(v

2+

1

vlog

x

κ

)− κΦ

(−v

2+

1

vlog

x

κ

)]= x

∂

∂xΦ

(v

2+

1

vlog

x

κ

)− κ ∂

∂xΦ

(−v

2+

1

vlog

x

κ

)+ Φ

(v

2+

1

vlog

x

κ

)= x

e−12 ( v2+

1v log x

κ )2

√2π

(1

vx

)− κe

− 12 (− v2+

1v log x

κ )2

√2π

(1

vx

)+Φ

(v

2+

1

vlog

x

κ

)

524


"



= Φ

(log(x/κ)

v+v

2

).

11. We have

dVt =∂C


+

(Vt − St

∂C

∂x(St, v(t, Ti))

)dP (t, Ti, Tj)

= Φ

(log(St/K)

v(t, Ti)+v(t, Ti)

2

)d(P (t, Ti)− P (t, Tj))

−κΦ(

log(St/K)


2

)dP (t, Ti, Tj).

12. We compare the results of Questions 4x and 11.

Chapter 13

Exercise 13.1 Defaultable bonds.

1. Use the fact that (rt, λt)t∈[0,T ] is a Markov process.2. Use the “tower property” (16.24) for the conditional expectation givenFt.

3. We have

d(e−

r t0(rs+λs)dsP (t, T )

)= −(rt + λt)e

−r t0(rs+λs)dsP (t, T )dt+ e−

r t0(rs+λs)dsdP (t, T )

= −(rt + λt)e−

r t0(rs+λs)dsP (t, T )dt+ e−

r t0(rs+λs)dsdF (t, rt, λt)

= −(rt + λt)e−

r t0(rs+λs)dsP (t, T )dt+ e−

r t0(rs+λs)ds

∂F

∂x(t, rt, λt)drt

+e−r t0(rs+λs)ds

∂F

∂y(t, rt, λt)dλt +

1

2e−

r t0(rs+λs)ds

∂2F

∂x2(t, rt, λt)σ

21(t, rt)dt

+1

2e−

r t0(rs+λs)ds

∂2F

∂y2(t, rt, λt)σ

22(t, λt)dt

+e−r t0(rs+λs)dsρ

∂2F

∂x∂y(t, rt, λt)σ1(t, rt)σ2(t, λt)dt+ e−

r t0(rs+λs)ds

∂F

∂t(t, rt, λt)dt

= e−r t0(rs+λs)ds

∂F

∂x(t, rt, λt)σ1(t, rt)dB

(1)t + e−

r t0(rs+λs)ds

∂F

∂y(t, rt, λt)σ2(t, λt)dB

(2)t

+e−r t0(rs+λs)ds

(−(rt + λt)P (t, T ) +

∂F

∂x(t, rt, λt)µ1(t, rt)

" 525



N. Privault

+∂F

∂y(t, rt, λt)µ2(t, λt) +

1

2

∂2F

∂x2(t, rt, λt)σ

21(t, rt) +

1

2

∂2F

∂y2(t, rt, λt)σ

22(t, λt)

+ρ∂2F

∂x∂y(t, rt, λt)σ1(t, rt)σ2(t, λt) +

∂F

∂t(t, rt, λt)

)dt,

hence the bond pricing PDE is

−(x+ y)F (t, x, y) + µ1(t, x)∂F

∂x(t, x, y)

+µ2(t, y)∂F

∂y(t, x, y) +

1

2σ21(t, x)

∂2F

∂x2(t, x, y)

+1

2σ22(t, y)

∂2F

∂y2(t, x, y) + ρσ1(t, x)σ2(t, y)

∂2F

∂x∂y(t, x, y) +

∂F

∂t(t, rt, λt) = 0.

4. We have

w t

0rsds =

1

a

(σB

(1)t − rt

)=σ

a

(B

(1)t −

w t

0e−a(t−s)dB(1)

s

)=σ

a

w t

0(1− e−a(t−s))dB(1)

s ,

hence

w T

trsds =

w T

0rsds−

w t

0rsds

=σ

a

w T

0(1− e−a(T−s))dB(1)

s −σ

a

w t

0(1− e−a(t−s))dB(1)

s

= −σa

(w t

0(e−a(T−s) − e−a(t−s))dB(1)

s +w T

t(e−a(T−s) − 1)dB(1)

s

)= −σ

a(e−a(T−t) − 1)

w t

0e−a(t−s)dB(1)

s −σ

a

w T

t(e−a(T−s) − 1)dB(1)

s

= −1

a(e−a(T−t) − 1)rt −

σ

a

w T

t(e−a(T−s) − 1)dB(1)

s .

The answer for λt is similar.5. As a consequence of the previous question we have

IE

[w T

trsds+

w T

tλsds

∣∣∣Ft] = C(a, t, T )rt + C(b, t, T )λt,

and

Var

[w T

trsds+

w T

tλsds

∣∣∣Ft] =

526


"



= Var

[w T

trsds

∣∣∣Ft]+ Var

[w T

tλsds

∣∣∣Ft]+2 Cov

(w T

tXsds,

w T

tYsds

∣∣∣Ft)=σ2

a2

w T

t(e−a(T−s) − 1)2ds

+2ρση

ab

w T

t(e−a(T−s) − 1)(e−b(T−s) − 1)ds

+η2

b2

w T

t(e−b(T−s) − 1)2ds

= σ2w T

tC2(a, s, T )ds+ 2ρση

w T


+η2w T

tC2(b, sT )ds,

from the Ito isometry.6. We have

P (t, T ) = 1τ>t IE

[exp

(−

w T

trsds−

w T

tλsds

) ∣∣∣Ft]= 1τ>t exp

(− IE

[w T

trsds

∣∣∣Ft]− IE

[w T

tλsds

∣∣∣Ft])× exp

(1

2Var

[w T

trsds+

w T

tλsds

∣∣∣Ft])= 1τ>t exp (−C(a, t, T )rt − C(b, t, T )λt)

× exp

(σ2

2

w T

tC2(a, s, T )ds+

η2

2

w T

tC2(b, s, T )e−b(T−s)ds

)× exp

(ρση

w T


).

7. This is a direct consequence of the answers to Questions 3 and 6.8. The above analysis shows that

P(τ > T | Gt) = 1τ>t IE

[exp

(−

w T

tλsds

) ∣∣∣Ft]= 1τ>t exp

(−C(b, t, T )λt +

η2

2

w T

tC2(b, s, T )ds

),

for a = 0 and

IE

[exp

(−

w T

trsds

) ∣∣∣Ft] = exp

(−C(a, t, T )rt +

σ2

2

w T

tC2(a, s, T )ds

),

for b = 0, and this implies

" 527



N. Privault

U(t, T ) = exp

(ρση

w T


)= exp

(ρση

ab(T − t− C(a, t, T )− C(b, t, T ) + C(a+ b, t, T ))

).

9. We have

f(t, T ) = −1τ>t∂ logP (t, T )

∂T

= 1τ>t

(rte−a(T−t) − σ2

2C2(a, t, T ) + λte

−b(T−t) − η2

2C2(b, t, T )

)−1τ>tρσηC(a, t, T )C(b, t, T ).

10. We use the relation

P(τ > T | Gt) = 1τ>t IE

[exp

(−

w T

tλsds

) ∣∣∣Ft]= 1τ>t exp

(−C(b, t, T )λt +

η2

2

w T

tC2(b, s, T )ds

)= 1τ>te

−r Ttf2(t,u)du,

where f2(t, T ) is the Vasicek forward rate corresponding to λt, i.e.

f2(t, u) = λte−b(u−t) − η2

2C2(b, t, u).

11. In this case we have ρ = 0 and

P (t, T ) = P(τ > T | Gt) IE

[exp

(−

w T

trsds

) ∣∣∣Ft] ,since U(t, T ) = 0.

Chapter 14

Exercise 14.1

1. When t ∈ [0, T1) the equation reads

dSt = −ηλSt−dt = −ηλStdt,

which is solved as St = S0e−ηλt, 0 ≤ t < T1. Next, at the first jump time

t = T1 we have

dSt = St − St− = ηSt−dNt = ηSt− ,

528


"



which yields St = (1+η)St− , hence ST1= (1+η)ST−1

= S0(1+η)e−ηλT1 .

Repeating this procedure over the Nt jump times contained in the interval[0, t] we get

St = S0(1 + η)Nte−ληt, t ∈ R+.

2. When t ∈ [0, T1) the equation reads

dSt = −ηλSt−dt = −ηλStdt,

which is solved as St = S0e−ηλt, 0 ≤ t < T1. Next, at the first jump time

t = T1 we havedSt = St − St− = dNt = 1,

which yields St = 1 + St− , hence ST1= 1 + ST−1

= 1 + S0e−ηλT1 , and for

t ∈ [T1, T2) we will find

St = (1 + S0e−ληT1)e−λη(t−T1), t ∈ [T1, T2).

Exercise 14.2 We have St = S0ert

Nt∏k=1

(1 + ηZk), t ∈ R+.


Var [YT ] = IE

(NT∑k=1

Zk − IE[YT ]

)2

=∞∑n=0

IE

(NT∑k=1

Zk − λt IE[Z1]

)2 ∣∣∣NT = k

P(NT = k)

= e−λt∞∑n=0

λntn

n!IE

( n∑k=1

Zk − λt IE[Z1]

)2

= e−λt∞∑n=0

λntn

n!IE

( n∑k=1

Zk

)2

− 2λt IE[Z1]n∑k=1

Zk + λ2t2(IE[Z1])2

= e−λt

∞∑n=0

λntn

n!

× IE

2∑

1≤k<l≤n

ZkZl +n∑k=1

|Zk|2 − 2λt IE[Z1]n∑k=1

Zk + λ2t2(IE[Z1])2

= e−λt

∞∑n=0

λntn

n!

×(n(n− 1)(IE[Z1])2 + n IE[|Z1|2]− 2nλt(IE[Z1])2 + λ2t2(IE[Z1])2)

" 529



N. Privault

= e−λt(IE[Z1])2∞∑n=2

λntn

(n− 2)!+ e−λt IE[|Z1|2]

∞∑n=1

λntn

(n− 1)!

−2e−λtλt(IE[Z1])2∞∑n=1

λntn

(n− 1)!+ λ2t2(IE[Z1])2)

= λt IE[|Z1|2],

or, using the characteristic function of Proposition 14.3,

Var [YT ] = IE[|YT |2]− (IE[YT ])2

= − d2

dα2IE[eiαYT ]|α=0 − λ2t2(IE[Z1])2

= λtw∞−∞|y|2µ(dy) = λt IE[|Z1|2].

Exercise 14.4

1. We have

dSt =

(µ+

1

2σ2

)Stdt+ σStdWt + (St − St−)dNt

=

(µ+

1

2σ2

)Stdt+ σStdWt + (S0e

µt+σWt+Yt − S0eµt+σWt+Yt−)dNt

=

(µ+

1

2σ2

)Stdt+ σStdWt + (S0e

µt+σWt+Yt−+ZNt − eµt+σWt+Yt−)dNt

=

(µ+

1

2σ2

)Stdt+ σStdWt + St−(eZNt − 1)dNt,

hence the jumps of St are given by the sequence (eZk − 1)k≥1.2. The discounted process e−rtSt satisfies

d(e−rtSt) = e−rt(µ− r +

1

2σ2

)Stdt+σe

−rtStdWt+e−rtSt−(eZNt−1)dNt.

Hence by the Girsanov theorem, choosing u, λ, ν such that

µ− r +1

2σ2 = u− λ IEν [eZ1 − 1],

shows that

d(e−rtSt) = σe−rtSt(dWt+udt)+e−rtSt−((eZNt−1)dNt−λ IEν [eZ1−1]dt)

is a martingale under (Pu,λ,ν).

Exercise 14.5

530


"



1. We have

St = S0eµt

Nt∏k=1

(1 + Yk) = S0 exp

(µt+

Nt∑k=1

Xk

), t ∈ R+.

2. We have

e−rtSt = S0 exp

((µ− r)t+

Nt∑k=1

Xk

), t ∈ R+,

which is a martingale if

0 = µ− r + λ IE[Yk] = µ− r + λ IE[eXk − 1] = µ− r + λ(eσ2/2 − 1).

3. We have

e−r(T−t) IE[(ST − κ)+ | St]

= e−r(T−t) IE

(S0 exp

(µT +

Nt∑k=1

Xk

)− κ)+ ∣∣∣St

= e−r(T−t)

∞∑n=0

IE

[(Ste

µ(T−t)+∑nk=1Xk − κ

)+ ∣∣∣St]P(NT −Nt = n)

= e−(r+λ)(T−t)∞∑n=0

IE

[(Ste

µ(T−t)+∑nk=1Xk − κ

)+ ∣∣∣St] (λ(T − t))nn!

= e−λ(T−t)∞∑n=0

BS(Ste(µ−r)(T−t)+nσ2/2, r, nσ2/(T − t), κ, T − t) (λ(T − t))n

n!

= e−λ(T−t)∞∑n=0

(Ste

(µ−r)(T−t)+nσ2/2Φ(d+)− κe−r(T−t)Φ(d−)) (λ(T − t))n

n!,

with

d+ =log(Ste

(µ−r)(T−t)+nσ2/2/κ) + r(T − t) + nσ2/2√nσ

=log(St/κ) + µ(T − t) + nσ2

√nσ

,

d− =log(Ste

(µ−r)(T−t)+nσ2/2/κ) + r(T − t)− nσ2/2√nσ

=log(St/κ) + µ(T − t)√

nσ,

" 531



N. Privault

and µ = r + λ(1− eσ2/2).

Chapter 15

Exercise 15.1

1. We have IE[Nt−αt] = IE[Nt]−αt = λt−αt, hence Nt−αt is a martingaleif and only if α = λ. Given that

d(e−rtSt) = ηe−rtSt−(dNt − αdt),

we conclude that the discounted price process e−rtSt is a martingale ifand only if α = λ.

2. Since we are pricing under the risk neutral measure we take α = λ. Nextwe note that

ST = S0e(r−ηλ)T (1 + η)NT = Ste

(r−ηλ)(T−t)(1 + η)NT−Nt , 0 ≤ t ≤ T,

hence the price at time t of the option is

e−r(T−t) IE[|ST |2 | Ft]= e−r(T−t) IE[|St|2e2(r−ηλ)(T−t)(1 + η)2(NT−Nt) | Ft]= |St|2e(r−2ηλ)(T−t) IE[(1 + η)2(NT−Nt) | Ft]= |St|2e(r−2ηλ)(T−t) IE[(1 + η)2(NT−Nt)]

= |St|2e(r−2ηλ)(T−t)∞∑n=0

(1 + η)2nP(NT −Nt = n)

= |St|2e(r−2ηλ−λ)(T−t)∞∑n=0

(1 + η)2n(λ(T − t))n

n!

= |St|2e(r−2ηλ−λ)(T−t)+(1+η)2λ(T−t)

= |St|2e(r+η2λ)(T−t), 0 ≤ t ≤ T.

Exercise 15.2

1. Independently of the choice of a risk-neutral measure Pu,λ,ν we have

e−r(T−t) IEu,λ,ν [ST −K | Ft] = ert IEu,λ,ν [e−rTST | Ft]−Ke−r(T−t)

= erte−rtSt −Ke−r(T−t)= St −Ke−r(T−t)= f(t, St),

for

532


"



f(t, x) = x−Ke−r(T−t), t, x > 0.

2. Clearly, holding one unit of the risky asset and shorting a (possibly frac-tional) quantity Ke−rT of the riskless asset will hedge the payoff ST −K,and this hedging strategy is self-financing because it is constant in time.

3. Since∂f

∂x(t, x) = 1 we have

ξt =

σ2 ∂f

∂x(t, St−) +

aλ

St−(f(t, St−(1 + a))− f(t, St−))

σ2 + a2λ

=

σ2 +aλ

St−(St−(1 + a)− St−)

σ2 + a2λ= 1, t ∈ [0, T ],

which coincides with the result of Question 2.

Exercise 15.3

1. We have

St = S0 exp

(µt+ σBt −

1

2σ2t

)(1 + η)Nt .

2. We have

St = S0 exp

((µ− r)t+ σBt −

1

2σ2t

)(1 + η)Nt ,

anddSt = (µ− r + λη)Stdt+ ηSt−(dNt − λdt) + σStdWt,

hence we need to takeµ− r + λη = 0,

since the compensated Poisson process (Nt − λt)t∈R+is a martingale.

3. We have

e−r(T−t)E∗[(ST − κ)+ | St]

= e−r(T−t)E∗[(

S0 exp

(µT + σBT −

1

2σ2T

)(1 + η)NT − κ

)+ ∣∣∣St]

= e−r(T−t)E∗[(Ste

µ(T−t)+σ(BT−Bt)− 12σ

2(T−t)(1 + η)NT−Nt − κ)+ ∣∣∣St]

= e−r(T−t)∞∑n=0

P(NT −Nt = n)

×E∗[(Ste

µ(T−t)+σ(BT−Bt)− 12σ

2(T−t)(1 + η)n − κ)+ ∣∣∣St]

" 533



N. Privault

= e−(r+λ)(T−t)∞∑n=0

(λ(T − t))nn!

×E∗[(Ste

(r−λη)(T−t)+σ(BT−Bt)− 12σ

2(T−t)(1 + η)n − κ)+ ∣∣∣St]

= e−λ(T−t)∞∑n=0

BS(Ste−λη(T−t)(1 + η)n, r, σ2, T − t, κ)

(λ(T − t))nn!

= e−λ(T−t)∞∑n=0

(Ste−λη(T−t)(1 + η)nΦ(d+)− κe−r(T−t)Φ(d−)

) (λ(T − t))nn!

,

with

d+ =log(Ste

−λη(T−t)(1 + η)n/κ) + (r + σ2/2)(T − t)σ√T − t

=log(St(1 + η)n/κ) + (r − λη + σ2/2)(T − t)

σ√T − t ,

and

d− =log(Ste

−λη(T−t)(1 + η)n/κ) + (r − σ2/2)(T − t)σ√T − t

=log(St(1 + η)n/κ) + (r − λη − σ2/2)(T − t)

σ√T − t .

Exercise 15.4

1. The discounted process St = e−rtSt satisfies the equation

dSt = YNt St−dNt, ,

and it is a martingale since the compound Poisson process YNtdNt iscentered with independent increments as IE[Y1] = 0.

2. We have

ST = S0erT

NT∏k=1

(1 + Yk),

hence

e−rT IE[(ST − κ)] = e−rT IE

(S0erT

NT∏k=1

(1 + Yk)− κ)+

= e−rT∞∑n=0

IE

(S0erT

NT∏k=1

(1 + Yk)− κ)+ ∣∣∣NT = n

P(NT = n)

534


"



= e−rT−λT∞∑k=0

IE

(S0erT

n∏k=1

(1 + Yk)− κ)+ (λT )n

n!

= e−rT−λT∞∑k=0

(λT )n

2nn!

w 1

−1· · ·

w 1

−1

(S0e

rTn∏k=1

(1 + yk)− κ)+

dy1 · · · dyn.

Exercise 15.5

1. We find α = λ where λ is the intensity of the Poisson process (Nt)t∈R+ .2. We have

e−r(T−t) IE[ST − κ | Ft] = ert IE[e−rTST | Ft]− e−r(T−t) IE[κ | Ft]= ert IE[e−rtSt | Ft]− e−r(T−t)κ= St − e−r(T−t)κ,

since the process (e−rtSt)t∈R+is a martingale.

Exercise 15.6

1. We have

dVt = df(t, St)

= rηtertdt+ ξtdSt

= rηtertdt+ ξt(rStdt+ αSt−(dNt − λdt))

= rVtdt+ αξtSt−(dNt − λdt)= rf(t, St)dt+ αξtSt−(dNt − λdt).

2. We apply the Ito formula with jumps and the martingale property oft 7→ ertf(t, St) to get

df(t, St) = rf(t, St)dt

+(f(t, St−(1 + α))− f(t, St−))dNt − λ(f(t, St(1 + α))− f(t, St))dt,

and we identify the terms in the above formula with those appearing in(15.16).

Background on Probability Theory

Exercise 1 We have

IE[X] =

∞∑k=0

kP(X = k) = e−λ∞∑k=0

kλk

k!

" 535



N. Privault

= e−λ∞∑k=1

λk

(k − 1)!= λe−λ

∞∑k=0

λk

k!= λ.

Exercise 2 We have

P(eX > c) = P(X > log c) =w∞log c

e−y2/(2η2) dy√

2πη2

=w∞(log c)/η

e−y2/2 dy√

2π= 1− Φ((log c)/η) = Φ(−(log c)/η).

Exercise 3

1. If µ = 0 we have

IE[X] =w∞−∞

xf(x)dx =1√

2πσ2

w∞−∞

xe−x2

2σ2 dx

=1√2π

w∞−∞

ye−y2

2 dy =1√2π

limA→+∞

w A

−Aye−

y2

2 dy = 0,

by symmetry of the function y 7→ ye−y2

2 . Note that we have

w∞−∞|y|e− y

2

2 dy = limA→+∞

w A

−A|y|e− y

2

2 dy = 2 limA→+∞

w A

0ye−

y2

2 dy

= −2 limA→+∞

[e−

y2

2

]A0

= 2 limA→+∞

(1− e−A2

2 ) = 2 <∞,

hence the function y 7→ ye−y2

2 is integrable on R and the above compu-tation of IE[X] is valid. Next, for all µ ∈ R we have

IE[X] =w∞−∞

xf(x)dx =1√

2πσ2

w∞−∞

xe−(x−µ)2

2σ2 dx

=1√

2πσ2

w∞−∞

(y + µ)e−y2

2σ2 dy

=1√

2πσ2

w∞−∞

ye−y2

2σ2 dy +µ√

2πσ2

w∞−∞

e−y2

2σ2 dy

=µ√

2πσ2

w∞−∞

e−y2

2σ2 dy = µw∞−∞

f(y)dy = µP(X ∈ R) = µ.

536


"


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Index

σ-algebra, 422

absence of arbitrage, 12, 298abstract Bayes formula, 272adapted process, 75, 361admissible portfolio strategy, 97affine model, 297American forward contract, 262American option

finite expiration, 252perpetual, 239

annuity numeraire, 344annuity numeraire, 365arbitrag

opportunity, 11arbitrage, 11

absence of, 12continuous time, 97discrete time, 29

arbitrage price, 41, 124Asian option, 155, 205attainable, 15, 20, 41, 99

backward induction, 46, 49Barrier forward contract, 223

down-and-in, 224, 475down-and-out, 224, 476up-and-in, 224, 471up-and-out, 224, 473

Barrier options, 163down-and-in, 166down-and-out, 166, 173, 175, 181up-and-in, 166up-and-out, 166, 171, 173, 176

barrier options, 153Bernoulli distribution, 433BGM model, 327

binomial distribution, 433Black caplet formula, 341Black-Scholes

formula, 102, 111, 125, 282, 327PDE, 99, 101, 109, 114, 178, 181, 413

with jumps, 400Black-Scholes calibration, 143bond

defaultable, 363option, 339pricing PDE, 299, 324, 526zero-coupon, 297

Brownian motion, 67

call option, 4call/put duality, 283cap, 343cap pricing, 343caplet, 341

pricing, 341Cauchy distribution, 431CEV model, 297change of measure, 122change of numeraire, 271, 283characteristic function, 443Chasles relation, 79CIR model, 296Clark-Ocone formula, 54, 203complete market, 16, 20, 124completeness

continuous time, 98discrete time, 37

Compound Poisson martingale, 394compound Poisson process, 375, 397conditional

expectation, 436, 441probability, 426

541

N. Privault

conditional expectation, 31conditional survival probability, 359conditioning, 426contingent claim, 15, 29, 37, 41

attainable, 15, 20, 99continuous-time limit, 61continuous-time model, 93correlation problem, 323coupon

bond, 298rate, 303

coupon bond, 364Courtadon model, 296credit default swap, 364CRR model, 38

default rate, 361Delta, 101, 104, 113, 131

hedging, 129, 287, 288density

function, 429marginal, 432

discounted asset prices, 28discrete distribution, 433distribution

Bernoulli, 433binomial, 433Cauchy, 431discrete, 433exponential, 430gamma, 431Gaussian, 430geometric, 433lognormal, 210, 431marginal, 437negative binomial, 434Pascal, 434Poisson, 434uniform, 430

Doob-Meyer decomposition, 260Dothan model, 297, 304Dupire PDE, 150

enlargement of filtration, 363entitlement ratio, 6, 104, 106, 145, 146equivalent probability measure, 14, 18,

122Euler discretization, 417event, 422exchange options, 284exotic option, 29, 47exotic options, 151


expectation, 434conditional, 436, 441

exponential distribution, 360, 430exponential model, 402exponential Vasicek model, 296

failure rate, 360Fatou’s lemma, 234filtration, 68, 227

enlargement, 363finite differences

explicit method, 411, 414implicit method, 412, 415

floorlet, 343foreign exchange, 278foreign exchange option, 281forward

contract, 62, 101, 114, 460measure, 338rate, 306swap rate, 308

forward contract, 62, 101, 114, 460American, 262, 487

forward rate, 305spot, 307, 319, 341swap, 308

gamma distribution, 431gamma process, 383Garman-Kohlagen formula, 281Gaussian

cumulative distribution function, 62distribution, 102, 430random variable, 444

generating function, 90, 443geometric

distribution, 433geometric Brownian motion, 84Girsanov theorem, 122, 123, 276

jump processes, 389, 397

heat equation, 108, 411hedging, 16, 45, 49, 52, 127hedging by change of numeraire, 286hedging strategy, 128hedging with jumps, 404hitting probability, 236hitting time, 231HJM

condition, 313model, 311, 363

Ho-Lee model, 297Hull-White model, 297, 312

542


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independence, 426, 429, 430, 433, 436,441, 443–445

independent increments, 118, 392indicator function, 428instantaneous forward rate, 307Interest rate model

affine, 297Constant Elasticity of Variance, 297Courtadon, 296Cox Ingersoll Ross, 296Dothan, 297, 304exponential Vasicek, 296Ho-Lee, 297Hull-White, 297Vasicek, 295

inverse Gaussian process, 383, 384Ito

isometry, 77process, 81, 100stochastic integral, 73, 76, 117

Ito formula, 81, 133with jumps, 381

Ito table, 83with jumps, 382

Levy process, 383Laplace transform, 443law of total expectation, 437law of total probability, 426, 437LIBOR

model, 309rate, 309swap rate, 310, 346, 348

Lipschitz function, 284lognormal approximation, 210lognormal distribution, 431lookback option, 182

call, 194put, 154, 182, 187

marginaldensity, 432distribution, 437

Margrabe formula, 284market completeness, 16, 20, 37Markov property, 284, 287martingale, 31, 117, 228

compound Poisson, 394continuous time, 98discrete time, 34method, 123Poisson, 392submartingale, 228supermartingale, 228

transform, 34, 117maximum of Brownian motion, 156mean hitting time, 239mean reversion, 295Merton model, 403Milshtein discretization, 418moment generating function, 443Musiela notation, 311

negative binomial distribution, 434negative inverse Gaussian process, 384Nelson-Siegel, 320numeraire, 97, 269numeraire invariance, 286

optimal stopping, 252option

on average, 154on extrema, 152writer, 16

optionalsampling, 232stopping, 232

Partial integro-differential equation, 401partition, 426, 437Pascal distribution, 434path dependent options, 52path integral, 49payoff function, 5, 151PDE

Black-Scholes, 99, 101, 109integro-differential, 401variational, 255

PIDE, 401Poisson

compound martingale, 375, 397distribution, 434process, 369

Poisson process, 361portfolio, 10, 26portfolio strategy, 94

admissible, 97, 99predictable process, 34, 44, 379predictable representation, 127, 130pricing, 41, 47

with jumps, 398probability

conditional, 426density function, 429distribution, 429measure, 425

equivalent, 14, 18space, 421

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processgamma, 383inverse Gaussian, 383predictable, 34, 44, 379stable, 383stopped, 231variance gamma, 383

put option, 4

randomvariable, 427

ratedefault, 361forward, 305forward swap, 308instantaneous forward, 307LIBOR, 309LIBOR swap, 310, 346, 348

recovery rate, 363reflexion principle, 155replication, 16risk-neutral measure, 14, 397

continuous time, 119risk-neutral measures


riskless asset, 62, 93

self-financing portfolio, 286, 288continuous time, 93, 95, 405discrete time, 27

short-selling, 10, 19, 105spot forward rate, 307, 319, 341stable process, 383stochastic

calculus, 80default, 361differential equations, 87integral, 43, 71, 75

with jumps, 378

process, 25stopped process, 231stopping time, 230, 361

theorem, 232strike price, 15submartingale, 228super-hedging, 16supermartingale, 228survival probability, 359Svensson parametrization, 320swap, 308

measure, 271, 344, 354swaption, 345

tenor structure, 337, 364tower property, 34, 35, 44, 45, 49, 78,

118, 130, 299, 437, 442, 451, 525two-factor model, 324

uniform distribution, 430

vanilla option, 29, 47variance, 439variance gamma process, 383, 384variational PDE, 255Vasicek model, 295volatility

historical, 141implied, 142local, 148smile, 143surface, 142

warrant, 6, 104

yield, 307, 341curve, 319

zero-coupon bond, 297

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Author index

Achdou, Y. 150

Barrieu, P. 210Bermin, H. 203Bjork, T. 23, 321Bosq, D. 371Bosq, S. 371Brace, A. 327Brigo, D. 301, 325

Carr, P. 209, 210Chan, C.M. 31Chen, R.R. 364Cheng, X. 364Cont, R. 378, 383, 389, 393Cox, J.C. 38, 296

Dana, R.A. 194de Chavez J., Ruiz 52Devore, J.L. 421Di Nunno, G. 129Doob, J.L. 232, 260Dothan, L.U. 297, 304Dudley, R.M. 75Duffie, D. 363Dufresne, D. 210Dupire, B. 150

El Karoui, N. 271, 287El Khatib, Y. 203, 205Elliott, R.J. 363Eriksson, J. 31, 178

Fabozzi, F. 364Folland, G.B. 68Follmer, H 9, 14, 16, 37, 61Fouque, J.P. 141

Garman, M.B. 281Gatarek, D. 327

Geman, H. 271, 287Gerber, H.U. 267Guo, X. 362, 363

Heath, D. 313Huang, J.Z. 364Hull, J. 312

Ikeda, N. 78Ingersoll, J.E. 296

J. Persson, J. 31, 178Jacod, J. 363, 421Jamshidian, F. 286, 287, 340Jarrow, R. 313, 362, 363Jeanblanc, M. 194, 363, 407Jeulin, Th. 363

Kallenberg, O. 443Kohlhagen, S.W. 281

Lamberton, D. 52, 214Lando, D. 361, 363Lapeyre, B. 214Levy, E. 210Liu, B. 364Longstaff, F.A. 257, 259

Margrabe, W. 284Matsumoto, H. 207Menn, C. 362, 363Mercurio, F. 301, 325Merton, R. 285Meyer, P.A. 260Morton, A. 313Musiela, M. 327

Nguyen, H.T. 371

Øksendal, B. 129

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Papanicolaou, G. 141Pintoux, C. 305Pironneau, O. 150Pitman, J. 421Privault, N. 52, 54, 75, 78, 118, 129,

203, 286, 305, 312, 322, 323, 325,340, 342, 349, 407

Proske, F. 129Protter, P. 82, 88, 123, 129, 130,

276, 299, 300, 421

Revuz, D. 68Rochet, J.-C. 271, 287Rogers, C. 216Ross, S.A. 38, 296Rouault, A. 210Rubinstein, M. 38

Schied, A. 9, 14, 16, 37, 52, 61Schoenmakers, J. 349Schroder, M. 209, 210Schwartz, E.S. 257, 259Shi, Z. 216

Shiryaev, A.N. 98, 99Shiu, E.S.W. 267Shreve, S. 162, 171, 186, 220, 241,

243, 264, 292, 470Singleton, K. 363Sircar, K.R. 141Steele, J.M. 253

Tankov, P. 378, 383, 389, 393Teng, T.-R. 286, 342, 349Turnbull, S.M. 210

Uy, W.I. 305

Vasicek, O. 295, 301

Wakeman, L. 210Watanabe, S. 78White, A. 312Widder, D.V. 108Williams, D. 52Wong, H.Y. 31

Yor, M. 68, 207, 210, 363

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This book is an introduction to the stochastic calculus and PDE ap-proaches to the pricing and hedging of financial derivatives, including vanillaoptions and exotic options. The presentation is done both in discrete andcontinuous-time financial models, with an emphasis on the complementar-ity between algebraic and probabilistic methods. It also covers the pricingof some interest rate derivatives, American options, exotic options such asbarrier, lookback and Asian options, and stochastic models with compoundPoisson jumps. The text is accompanied with a number of figures and simu-lations, and includes 20 examples based on actual market data.

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Documents

Notes on Stochastic Finance-Nicolas Privaut