Stochastic Finance

Nicolas Privault

Notes on Stochastic Finance

This version: April 25, 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 [90]. 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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This version: April 25, 2013http://www.ntu.edu.sg/home/nprivault/indext.html

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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 and embeddedvideos in Chapters 8, 9, 11 and 14, that may require using Acrobat Readerfor viewing on the complete pdf file. Clicking on an exercise number insidethe solution section will send to the original problem text inside the file.Conversely, clicking on the problem number sends the reader to the corre-sponding solution, however this feature should not be misused.

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Contents

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

1 Assets, Portfolios and Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . 111.1 Definitions and Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2 Portfolio Allocation and Short-Selling . . . . . . . . . . . . . . . . . . . . . 121.3 Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.4 Risk-Neutral Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.5 Hedging of Contingent Claims . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.6 Market Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.7 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2 Discrete-Time Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.1 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.2 Portfolio Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.3 Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.4 Contingent Claims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.5 Martingales and Conditional Expectation . . . . . . . . . . . . . . . . . . 352.6 Market Completeness and Risk-Neutral Measures . . . . . . . . . . . 402.7 The Cox-Ross-Rubinstein (CRR) Market Model . . . . . . . . . . . . 42Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3 Pricing and hedging in discrete time . . . . . . . . . . . . . . . . . . . . . . 473.1 Pricing of Contingent Claims . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.2 Hedging of Contingent Claims - Backward Induction . . . . . . . . 513.3 Pricing of Vanilla Options in the CRR Model . . . . . . . . . . . . . . 533.4 Hedging of Vanilla Options in the CRR model . . . . . . . . . . . . . 553.5 Hedging of Exotic Options in the CRR Model . . . . . . . . . . . . . . 593.6 Convergence of the CRR Model . . . . . . . . . . . . . . . . . . . . . . . . . . 66Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

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4 Brownian Motion and Stochastic Calculus . . . . . . . . . . . . . . . . 734.1 Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.2 Wiener Stochastic Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.3 Ito Stochastic Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.4 Stochastic Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.5 Geometric Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.6 Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . 96Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5 The Black-Scholes PDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.1 Continuous-Time Market Model . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.2 Self-Financing Portfolio Strategies . . . . . . . . . . . . . . . . . . . . . . . . 1055.3 Arbitrage and Risk-Neutral Measures . . . . . . . . . . . . . . . . . . . . . 1095.4 Market Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115.5 The Black-Scholes PDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125.6 The Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205.7 Solution of the Black-Scholes PDE . . . . . . . . . . . . . . . . . . . . . . . . 123Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

6 Martingale Approach to Pricing and Hedging . . . . . . . . . . . . . 1316.1 Martingale Property of the Ito Integral . . . . . . . . . . . . . . . . . . . . 1316.2 Risk-neutral Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1346.3 Girsanov Theorem and Change of Measure . . . . . . . . . . . . . . . . 1376.4 Pricing by the Martingale Method . . . . . . . . . . . . . . . . . . . . . . . . 1396.5 Hedging Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

7 Estimation of Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1597.1 Historical Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1597.2 Implied Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1607.3 The Black-Scholes Formula vs Market Data . . . . . . . . . . . . . . . . 1627.4 Local Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1667.5 Stochastic Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1707.6 Volatility Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

8 Exotic Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1858.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1858.2 The Reflexion Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1908.3 Barrier Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1998.4 Lookback Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2188.5 Asian Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

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9 American Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2699.1 Filtrations and Information Flow . . . . . . . . . . . . . . . . . . . . . . . . . 2699.2 Martingales, Submartingales, and Supermartingales . . . . . . . . . 2709.3 Stopping Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2729.4 Perpetual American Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2829.5 Finite Expiration American Options . . . . . . . . . . . . . . . . . . . . . . 294Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

10 Change of Numeraire and Forward Measures . . . . . . . . . . . . . 31310.1 Notion of Numeraire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31310.2 Change of Numeraire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31510.3 Foreign Exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32310.4 Pricing of Exchange Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32810.5 Hedging by Change of Numeraire . . . . . . . . . . . . . . . . . . . . . . . . . 330Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

11 Forward Rate Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33911.1 Short Term Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33911.2 Zero-Coupon Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34211.3 Forward Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35011.4 The HJM Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35611.5 Forward Vasicek Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36011.6 Modeling Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36411.7 The BGM Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

12 Pricing of Interest Rate Derivatives . . . . . . . . . . . . . . . . . . . . . . 38112.1 Forward Measures and Tenor Structure . . . . . . . . . . . . . . . . . . . . 38112.2 Bond Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38412.3 Caplet Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38512.4 Forward Swap Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38812.5 Swaption Pricing on the LIBOR . . . . . . . . . . . . . . . . . . . . . . . . . . 390Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394

13 Credit Risk, CDSs and CDOs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40713.1 Stochastic Default . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40713.2 Defaultable Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41213.3 Credit Default Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41313.4 Correlated default times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41513.5 Merton model of credit risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42613.6 Modeling the default times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42813.7 Collateralized debt obligations (CDOs) . . . . . . . . . . . . . . . . . . . . 433Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439

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14 Stochastic Calculus for Jump Processes . . . . . . . . . . . . . . . . . . . 44314.1 The Poisson Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44314.2 Compound Poisson Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44914.3 Stochastic Integrals with Jumps . . . . . . . . . . . . . . . . . . . . . . . . . . 45214.4 Ito Formula with Jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45414.5 Stochastic Differential Equations with Jumps . . . . . . . . . . . . . . 45914.6 Girsanov Theorem for Jump Processes . . . . . . . . . . . . . . . . . . . . 464Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470

15 Pricing and Hedging in Jump Models . . . . . . . . . . . . . . . . . . . . . 47315.1 Risk-Neutral Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47315.2 Pricing in Jump Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47415.3 Black-Scholes PDE with Jumps . . . . . . . . . . . . . . . . . . . . . . . . . . 47615.4 Exponential Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47815.5 Self-Financing Hedging with Jumps . . . . . . . . . . . . . . . . . . . . . . . 481Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484

16 Basic Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48716.1 Discretized Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48716.2 Discretized Black-Scholes PDE . . . . . . . . . . . . . . . . . . . . . . . . . . . 48916.3 Euler Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49316.4 Milshtein Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494

Appendix: Background on Probability Theory . . . . . . . . . . . . . . . . 497Probability Spaces and Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497Probability Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501Conditional Probabilities and Independence . . . . . . . . . . . . . . . . . . . . 502Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503Probability Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505Expectation of a Random Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510Conditional Expectation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517Moment Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522

Exercise Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597

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Chapter 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603Chapter 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612Chapter 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629Chapter 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635Chapter 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641Background on Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647

Author index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655

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List of Figures

0.1 As if a whole new world was laid out before me. . . . . . . . . . . . . . 30.2 Graph of the Hang Seng index - holding a put option might be

useful here. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50.3 Sample price processes simulated by a geometric Brownian motion. . . 60.4 Infogrames stock price curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.1 Arbitrage - 2006 retail prices around the world for the Xbox 360. . . . 141.2 Absence of arbitrage - the Mark Six Investment Table. . . . . . . . . . 151.3 Separation of convex sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.1 Illustration of the self-financing condition (2.3). . . . . . . . . . . . . . . . . . 31

4.1 Sample paths of a one-dimensional Brownian motion. . . . . . . . . . . . . 764.2 Two sample paths of a two-dimensional Brownian motion. . . . . . . . . 774.3 Sample paths of a three-dimensional Brownian motion. . . . . . . . . . . . 774.4 Step function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.5 Geometric Brownian motion started at 1. . . . . . . . . . . . . . . . . . . . . . 96

5.1 Illustration of the self-financing condition (5.2). . . . . . . . . . . . . . . . . . 1065.2 Graph of the Black-Scholes call price function with strike K = 100. 1165.3 Graph of the stock price of HSBC Holdings. . . . . . . . . . . . . . . . . . . . 1165.4 Path of the Black-Scholes price for a call option on HSBC. . . . . . . . . 1175.5 Time evolution of the hedging portfolio for a call option on HSBC. . 1185.6 Graph of the Black-Scholes put price function with strike K = 100. . 1195.7 Path of the Black-Scholes price for a put option on HSBC. . . . . . . . . 1195.8 Time evolution of the hedging portfolio for a put option on HSBC. . 1205.9 Time-dependent solution of the heat equation. . . . . . . . . . . . . . . . . . 1215.10 Option price as a function of the volatility . . . . . . . . . . . . . . . . . . . . 127

6.1 Drifted Brownian path. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1356.2 Delta of a European option. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

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6.3 Gamma of a European option. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1486.4 Option price as a function of the underlying and of time to maturity 1536.5 Delta as a function of the underlying and of time to maturity . . . . . . 1546.6 Gamma as a function of the underlying and of time to maturity . . . . 1556.7 Option price as a function of the underlying and of time to maturity 1566.8 Delta as a function of the underlying and of time to maturity . . . . . . 1576.9 Gamma as a function of the underlying and of time to maturity . . . . 157

7.1 The fugazi: its a wazy, its a woozie. Its fairy dust. . . . . . . . . . . 1607.2 Implied volatility of Asian options on light sweet crude oil futures. . . 1627.3 Graph of the (market) stock price of Cheung Kong Holdings. . . . . . . 1627.4 Graph of the (market) call option price on Cheung Kong Holdings. . 1637.5 Graph of the Black-Scholes call option price on Cheung Kong Holdings.1637.6 Graph of the (market) stock price of HSBC Holdings. . . . . . . . . . . . . 1647.7 Graph of the (market) call option price on HSBC Holdings. . . . . . . . 1647.8 Graph of the Black-Scholes call option price on HSBC Holdings. . . . 1657.9 Graph of the (market) put option price on HSBC Holdings. . . . . . . . 1657.10 Graph of the Black-Scholes put option price on HSBC Holdings. . . . 1667.11 Call option price vs ALSTOM underlying. . . . . . . . . . . . . . . . . . . . . . 1667.12 Euro / SGD exchange rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

8.1 Brownian motion Bt and its supremum Xt. . . . . . . . . . . . . . . . . . . . 1878.2 A function with no last point of increase before t = 1. . . . . . . . . . . . . 1878.3 Brownian motion Bt and its moving average. . . . . . . . . . . . . . . . . . . 1898.4 Reflected Brownian motion with a = 1. . . . . . . . . . . . . . . . . . . . . . . 1918.5 Probability density of the maximum of Brownian motion. . . . . . . . . . 1928.6 Density of the supremum of geometric Brownian motion. . . . . . . . . . . 1938.7 Joint probability density of B1 and its maximum over [0,1]. . . . . . . . . 1958.8 Heat map of the joint probability density of B1 and its maximum

over [0,1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1968.9 Probability density of the maximum of drifted Brownian motion. . . . . 1988.10 Graph of the up-and-out call option price with B > K. . . . . . . . . . . 2038.11 Graph of the up-and-out put option price (8.12) with B = 80 > K = 60.2088.12 Graph of the up-and-out put option price (8.13) with

K = 100 > B = 80. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2098.13 Graph of the down-and-out call option price with B < K. . . . . . . . . . 2108.14 Graph of the down-and-out call option price with K > B. . . . . . . . . . 2118.15 Graph of the down-and-out put option price with K > B. . . . . . . . . . 2128.16 Delta for the up-and-out option. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2168.17 Graph of the lookback put option price. . . . . . . . . . . . . . . . . . . . . . . 2248.18 Graph of the normalized lookback put option price. . . . . . . . . . . . . . . 2298.19 Black-Scholes put price in the decomposition (8.32). . . . . . . . . . . . . . . 2308.20 Function hp(, z) in the decomposition (8.32). . . . . . . . . . . . . . . . . . . 2318.21 Graph of the lookback call option price. . . . . . . . . . . . . . . . . . . . . . . 2348.22 Normalized lookback call option price. . . . . . . . . . . . . . . . . . . . . . . . . . 238

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8.23 Graph of the underlying asset price. . . . . . . . . . . . . . . . . . . . . . . . . . . 2388.24 Graph of the lookback call option price. . . . . . . . . . . . . . . . . . . . . . . . 2398.25 Running minimum of the underlying asset price. . . . . . . . . . . . . . . . . . 2398.26 Black-Scholes call price in the normalized lookback call price. . . . . . . 2408.27 Function hc(, z) in the normalized lookback call option price. . . . . . . 2408.28 Delta of the lookback call option. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2438.29 Rescaled portfolio strategy for the lookback call option. . . . . . . . . . . . 2438.30 Graph of the Asian option price. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2508.31 Lognormal approximation to the Asian option price. . . . . . . . . . . . . . 251

9.1 Drifted Brownian path. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2719.2 Evolution of the fortune of a poker player vs number of games played. 2719.3 Stopped process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2749.4 Graphs of the option price by exercise at L for several values of L. . 2859.5 Animated graph of the option price depending on the values of L. . 2869.6 Option price as a function of L and of the underlying asset price. . . . 2879.7 Path of the American put option price on the HSBC stock. . . . . . . . 2879.8 Graphs of the option price by exercising at L for several values of L. 2929.9 Graphs of the option prices parametrized by different values of L. . . . 2939.10 Expected Black-Scholes European call price vs (x, t) 7 (xK)+. . . 2959.11 Black-Scholes put price function vs (x, t) 7 (K x)+. . . . . . . . . . . . 2969.12 Optimal frontier for the exercise of a put option. . . . . . . . . . . . . . . . . 2969.13 Numerical values of the finite expiration American put price. . . . . . . . 2989.14 Longstaff-Schwartz algorithm for the American put price. . . . . . . . . . 2999.15 Comparison between Longstaff-Schwartz and finite differences. . . . . . 299

11.1 Graph of t 7 rt in the Vasicek model. . . . . . . . . . . . . . . . . . . . . . . . 34011.2 Graphs of t 7 P (t, T ) and t 7 er0(Tt). . . . . . . . . . . . . . . . . . . 34711.3 Graph of t 7 P (t, T ) for a bond with a 2.3% coupon. . . . . . . . . . . . 34811.4 Bond price graph with coupon rate 6.25%. . . . . . . . . . . . . . . . . . . . . . 34811.5 Graph of T 7 f(t, T, T + ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35211.6 Stochastic process of forward curves. . . . . . . . . . . . . . . . . . . . . . . . . . 35711.7 Forward rate process t 7 f(t, T, S). . . . . . . . . . . . . . . . . . . . . . . . . . 36011.8 Instantaneous forward rate process t 7 f(t, T ). . . . . . . . . . . . . . . . . 36111.9 Forward instantaneous curve in the Vasicek model. . . . . . . . . . . . . 36211.10 Forward instantaneous curve x 7 f(0, x) in the Vasicek model. . . 36211.11 Short term interest rate curve t 7 rt in the Vasicek model. . . . . . . 36311.12 Market example of yield curves (11.23). . . . . . . . . . . . . . . . . . . . . . . 36311.13 Graph of x 7 g(x) in the Nelson-Siegel model. . . . . . . . . . . . . . . . . 36411.14 Graph of x 7 g(x) in the Svensson model. . . . . . . . . . . . . . . . . . . . 36511.15 Comparison of market data vs a Svensson curve. . . . . . . . . . . . . . . . 36511.16 Graphs of forward rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36611.17 Forward instantaneous curve in the Vasicek model. . . . . . . . . . . . . . . 36611.18 Graph of t 7 P (t, T1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36711.19 Graph of forward rates in a two-factor model. . . . . . . . . . . . . . . . . . 369

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11.20 Random evolution of forward rates in a two-factor model. . . . . . . . . 37011.21 Graph of stochastic interest rate modeling. . . . . . . . . . . . . . . . . . . . . 372

12.1 Forward rates arranged according to a tenor structure. . . . . . . . . . . . 381

13.1 Different Gaussian copula graphs for = 0, = 0.85 and = 1. . . . . 41813.2 Different Gaussian copula density graphs for = 0, = 0.35 and

= 0.999. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41913.3 Function x 7 (1(x) + ( r)T t/) for > r, = r, and

< r. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42713.4 A representation of CDO tranches. . . . . . . . . . . . . . . . . . . . . . . . . . . . 43413.5 A Titanic-style representation of cumulative tranche losses. . . . . . . . . 43513.6 Function fk(x) = min((xNk1)+, Npk). . . . . . . . . . . . . . . . . . . . 43713.7 Cumulative historic default rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43813.8 Internal Ratings-Based formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439

14.1 Sample path of a Poisson process (Nt)tR+ . . . . . . . . . . . . . . . . . . . . . 44414.2 Sample path of a compound Poisson process (Yt)tR+ . . . . . . . . . . . . 45014.3 Sample trajectories of a gamma process. . . . . . . . . . . . . . . . . . . . . . . 45714.4 Sample trajectories of a stable process. . . . . . . . . . . . . . . . . . . . . . . . 45814.5 Sample trajectories of a variance gamma process. . . . . . . . . . . . . . . . 45814.6 Sample trajectories of an inverse Gaussian process. . . . . . . . . . . . . . . 45914.7 Sample trajectories of a negative inverse Gaussian process. . . . . . . . . 45914.8 Geometric Poisson process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46114.9 Ranking data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46114.10 Geometric compound Poisson process. . . . . . . . . . . . . . . . . . . . . . . 46214.11 Geometric Brownian motion with compound Poisson jumps. . . . . . 46314.12 SMRT Stock price. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463

16.1 Divergence of the explicit finite difference method. . . . . . . . . . . . . . . . 49116.2 Stability of the implicit finite difference method. . . . . . . . . . . . . . . . . . 493

S.1 Market data for the warrant #01897 on the MTR Corporation. . . . . 547S.2 Price of a digital call option. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558S.3 Risky hedging portfolio value for a digital call option. . . . . . . . . . . . . 559S.4 Riskless hedging portfolio value for a digital call option. . . . . . . . . . . 559S.5 Average return by selling at the maximum vs selling at maturity T . . 562S.6 Average return by selling at the minimum vs selling at maturity T

as a function of T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563S.7 Graph of the up-and-in long forward contract price with

K = 60 < B = 80. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567S.8 Delta of the down-and-in long forward contract with K = 60 < B = 80.568S.9 Graph of the up-and-out long forward contract price with

K = 60 < B = 80. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569

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S.10 Delta of the up-and-out long forward contract price withK = 60 < B = 80. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570

S.11 Graph of the down-and-in long forward contract price withK = 60 < B = 80. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571

S.12 Delta of the down-and-in long forward contract with K = 60 < B = 80.571S.13 Graph of the down-and-out long forward contract price with

K = 60 < B = 80. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572S.14 Delta of the down-and-out long forward contract with

K = 60 < B = 80. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573S.15 Lookback call option as a function of maturity time T . . . . . . . . . . . . . 574S.16 Cashflow data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635S.17 CDS price data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636

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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 paperA 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 lEcole 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,Kiyoshi, 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 dubbedthe 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 Bacheliers 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 [Bacheliers]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.

Fig. 0.1: [14] As if a whole new world was laid out before me.

In recognition of Bacheliers contribution, the Bachelier Finance Society wasstarted in 1996 and now holds the World Bachelier Finance Congress every

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2 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.

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 paperBond 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 and

This has to be put in relation with the modern development of risk societies; soci-eties increasingly preoccupied with the future (and also with safety), which generatesthe notion of risk.

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swaptions 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 .

Fig. 0.2: 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 price (or exercise price) and at a predefined date Tcalled 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.



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

(ST ) = (K ST )+ =K ST , ST K,

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

in Figure 0.3.

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


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 (KST )+(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.

Fig. 0.4: Infogrames stock price curve.

An illustration - pricing and hedging in a binary model

We close this introduction with a simplified illustration of the pricing andhedging technique in a binary model. Consider a risky stock price S valuedS0 = $4 at time t = 0, and taking only two possible values

S1 =

$5$2

at time t = 1. In addition, consider an option that yields a payoff P whosevalues are contingent to the data of S:

P =

$3 if S1 = $5$0 if S1 = $2.



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At time t = 0 we choose to invest units in the risky asset S, while keeping$ on our bank account, meaning that we invest a total amount

S0 + $ at t = 0.

The following issues can be addressed:

a) Hedging: how to choose the portfolio allocation {, $} so that the value

S1 + $

of the portfolio matches the future payoff P at time t = 1 ?

b) Pricing: how to determine the amount S0 + $ to be invested in such aportfolio at time t = 0 ?

Hedging means that at time t = 1 the portfolio value matches the futurepayoff P , i.e.

S1 + $ = P.

This condition can be rewritten as

P =

$3 = $5 + $ if S1 = $5,$0 = $2 + $ if S1 = $2,

i.e. 5+ = 32+ = 0,

which yields

= 1$ = $2.

In other words, we buy 1 unit of the stock S at the price S0 = $4, and weborrow $2 from the bank. The price of the option contract is given by theportfolio value

S0 + $ = 1 $4 $2 = $2.at time t = 0.

Conclusion: in order to deliver the random payoff P =

$3 if S1 = $5$0 if S1 = $2.

at time t = 1, one has to:

1. receive $2 (the option price) at time t = 0,

2. borrow $ = $2 from the bank,

3. invest those $2 + $2 = $4 into the purchase of = 1 unit of stock valuedat S0 = $4 at time t = 0,

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4. wait until time t = 1 to find that the portfolio value evolved into

P =

$5 + $ = 1 $5 $2 = $3 if S1 = $5, $2 + $ = 1 $2 $2 = 0 if S1 = $2.

so that the option contract is fulfilled whatever the evolution of S.

We note that the initial amount of $2 can be turned to P = $3 (%50 profit)... or into P = $0 (total ruin).

Thinking further

1) The expected gain of our portfolio is

IE[P ] = $3 P(P = $3) + $0 P(P = $0)= $3 P(S1 = $5)= $3 P(S1 = $5).

In absence of arbitrage opportunities (fair market) this expected gain IE[P ]should equal the initial amount $2 invested in the option. In that case weshould have IE[P ] = $3 P(S1 = $5) = $2P(S1 = $5) + P(S1 = $2) = 1.from which we can infer the probabilities

P(S1 = $5) =2

3

P(S1 = $2) =1

3.

(0.1)

We see that the stock S has twice more chances to go up than to go down ina fair market.

2) Based on the probabilities (0.1) we can also compute the expected valueIE[S1] of the stock at time t = 1. We find

IE[S1] = $5 P(S1 = $5) + $2 P(S1 = $2)= $5 2

3+ $2 1

3= $4

= S0.



N. Privault

This means that, on average, no profit can be made from an investment onthe risky stock. In a more realistic model we can assume that the riskles bankaccount yields an interest rate equal to r, in which case the above analysis ismodified by letting $ become $(1 + r) at time t = 1, nevertheless the mainconclusions remain unchanged.

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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 [34]. 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+1is defined by

x y = x0y0 + x1y1 + + xdyd.The vector

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

)denotes the prices pi(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 (,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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N. Privault

S(0) = (1 + r)pi(0).

1.2 Portfolio Allocation and Short-Selling

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

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

) Rd+1,

in which (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

pi =di=0

(i)pi(i)

at time t = 0.

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

S =di=0

(i)S(i).

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

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

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

4. Return the asset to its owner, with possibly a (small) fee p > 0.

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

(1 + r)pi(i) S(i) p > 0,

which is positive provided S(i) < (1 + r)pi(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 (0)pi(0) > 0 on the riskless asset no 0.

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

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

4. Refund the amount (1 + r)(0)pi(0) > 0 with interest rate r > 0.

At the end of the operation the profit made is

(i)S(i) ((1 + r)(0)pi(0)) = (i)S(i) + (1 + r)(0)pi(0)

= (0)pi(0)

pi(i)S(i) + (1 + r)(0)pi(0)

= (0)pi(0)

pi(i)

(S(i) (1 + r)pi(i)

)= (i)

(S(i) (1 + r)pi(i)

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



N. Privault

> 0,

or S(i) (1 + r)pi(i) per unit of stock invested, which is positive providedS(i) > pi(i) and r is sufficiently small.

City Currency US$

Tokyo 38,800 yen $346Hong Kong HK$2,956.67 $381Seoul 378,533 won $400Taipei NT$12,980 $404New York $433Sydney A$633.28 $483Frankfurt e399 $513Paris e399 $513Rome e399 $513Brussels e399.66 $514London 279.99 $527Manila 29,500 pesos $563Jakarta 5,754,1676 rupiah $627

Fig. 1.1: Arbitrage - 2006 retail prices around the world for the Xbox 360.

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

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

i) pi 0, [start from 0 or even with a debt]

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

iii) P( 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.

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Fig. 1.2: Absence of arbitrage - the Mark Six Investment Table.

In the table of Figure 1.2 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.

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 purchase 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. Totalcost: $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 stock



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at 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.

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 is called a risk-neutral mea-sure if

IE[S(i)] = (1 + r)pi(i), i = 1, 2, . . . , d. (1.2)

Here, IE denotes the expectation under the probability measure P. Notethat for i = 0, the condition IE[S(0)] = (1 + r)pi(0) is always satisfied by

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definition.

In other words, P is called risk neutral because taking risks under Pby 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)pi(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 all A F . (1.3)

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

pi =di=0

(i)pi(i) =1

1 + r

di=0

(i) IE[S(i)] =1

1 + rIE[ S] > 0,

because P( S > 0) > 0 as P( S > 0) > 0 and P is equivalent to P, andthe condition pi > 0 contradicts Definition 1.1-(i). The proof of necessityrelies on the theorem of separation of convex sets by hyperplanes Theorem 1.2below, cf. Theorem 1.6 of [34]. It can be briefly sketched as follows. Giventwo financial assets with net discounted gains X,Y and a portfolio made ofone unit of X and c unit(s) of Y , the absence of arbitrage opportunities canbe reformulated by saying that for any portfolio choice determined by c R,we have

X + cY 0 = X + cY = 0, P a.s., (1.4)i.e. a riskless (no loss) portfolio can not entail a stricly positive gain. In otherwords, if one wishes to make a strictly positive gain on the market, one hasto accept the possibility of a loss. In order to show the absence of arbitrageopportunities implies the existence of a risk-neutral probability measure Punder which all risky investments have zero discounted return, i.e.

IEP [X] = IEP [Y ] = 0, (1.5)



N. Privault

we apply the convex separation Theorem 1.2 to the convex set

C = { (IEQ[X], IEQ[Y ]) : Q P}in R2, where P is the family of probability measures Q on equivalent to P.If (1.5) does not hold under any probability measure Q P then 0 / C andthe convex separation Theorem 1.1 shows the existence of c R such that

IEQ[X] + c IEQ[Y ] 0, Q P, (1.6)

and IEP [X] + c IEP [Y ] > 0 for some P P. This shows that X + cY 0a.s. while P(X + cY > 0) 6= 0, which contradicts the absence of arbitrage.

Next is a version of the separation theorem for convex sets, cf. e.g. Theo-rem 4.14 of [51].

Theorem 1.2. Let C1 and C2 be two disjoint convex sets in R2. Then thereexists a, b R such that we have

y1 a+ bx1 and a+ bx2 y2,

for all (x1, y1) C1 and (x2, y2) C2 (up to exchange of C1 and C2).

Fig. 1.3: Separation of convex sets.

1.5 Hedging of Contingent Claims

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

Inequality (1.6) might be reversed, in this case we choose (1,c) as portfolioallocation.

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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) con-tract 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 such that

C = S.

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

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

2. invest the amount

pi =di=0

(i)pi(i)

in this portfolio at time t = 0,

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

The above shows that in order to attain the claim, an initial investment piis needed at time t = 0. This amount, to be paid by the buyer to the issuer



N. Privault

of the option (the option writer), is also called the arbitrage price of thecontingent claim C, and denoted by

pi(C) := pi. (1.7)

The action of allocating a portfolio such that

C = S (1.8)

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 (1.8) would be maintained.

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

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 finance.

Theorem 1.3. A market model without arbitrage is complete if and only ifit admits only one risk-neutral measure.

Proof. cf. Theorem 1.40 of [34]. Theorem 1.3 will give us a concrete way to verify market completeness bysearching for a unique solution P to Equation (1.2).

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1.7 Example

In this section we work out a simple example that allows us to apply Theo-rem 1.1 and Theorem 1.3.

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

= {, +},

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

P({}) > 0 and P({+}) > 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 expect-ing an important binary decision of yes/no type, which can lead to twodistinct scenarios called and +.

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

S(1) : R

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

Precisely, we set

S(1)() = a, and S(1)(+) = b,

i.e. the value of S(1) becomes equal a under the scenario , and equal to bunder the scenario +, 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 on = {, +} is characterized by the data of two numbers P({}) [0, 1]and P({+}) [0, 1], such that

P() = P({}) + P({+}) = 1. (1.9) The case a = b leads to a trivial, constant market.



N. Privault

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

P({}) > 0 and P({+}) > 0.

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

IE[S(1)] = (1 + r)pi(1). (1.10)

Although we should solve this equation for P, at this stage it is not yet clearhow P appears in (1.10).

In order to make (1.10) more explicit we write the expectation as

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

hence Condition (1.10) for the existence of a risk-neutral measure P reads

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

Using the Condition (1.9) we obtain the system of two equationsaP({}) + bP({+}) = (1 + r)pi(1)

P({}) + P({+}) = 1,(1.11)

with solution

P({}) = b (1 + r)pi(1)

b a and P({+}) = (1 + r)pi

(1) ab a .

In order for a solution P to exist as a probability measure, the numbersP({}) and P({+}) must be non-negative. In addition, for P to beequivalent to P they should be strictly positive from (1.3).

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

a < (1 + r)pi(1) < b, (1.12)

then there exists a risk-neutral (equivalent) probability measure P which isunique, hence by Theorems 1.1 and 1.3 the market is without arbitrage andcomplete.

If a = b = (1 + r)pi(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

S = (1 + r) pi,

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at time t = 1, hence the terminal value 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() and C(+). Therefore,market completeness does not hold when a = b = (1 + r)pi(1).

Note that if a = (1 + r)pi(1), resp. b = (1 + r)pi(1), then P({+}) = 0,resp. P({}) = 0, and P is not equivalent to P.

On the other hand, under the conditions

a < b (1 + r)pi(1) or (1 + r)pi(1) a < b, (1.13)

no (equivalent) risk neutral measure exists, and as a consequence there existarbitrage opportunities in the market.

Let us give a financial interpretation of Conditions (1.13).

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

(0) = (1)pi(1)/pi(0) < 0.In particular, Condition (i) of Definition 1.1 is satisfied, and the investorborrows the amount (0)pi(0) > 0 on the riskless asset and uses it tobuy one unit (1) = 1 of the risky asset. At time t = 1 she sells therisky asset S(1) at a price at least equal to a and refunds the amount(1 + r)(0)pi(0) > 0 she borrowed, with interests. Her profit is

S = (1 + r)(0)pi(0) + (1)S(1) (1 + r)(0)pi(0) + (1)a= (1 + r)(1)pi(1) + (1)a= (1)((1 + r)pi(1) + a) 0, ^

which satisfies Condition (ii) of Definition 1.1. In addition, Condition (iii)of Definition 1.1 is also satisfied because

P( S > 0) = P(S(1) = b) = P({+}) > 0.

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

(1) = (0)pi(0)/pi(1) < 0.This means that the investor borrows a (possibly fractional) quantity(1) > 0 of the risky asset, sells it for the amount (1)pi(1), and in-vests this money on the riskless account for the amount (0)pi(0) > 0. Asmentioned in Section 1.2, in this case one says that the investor shortsells



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the risky asset. At time t = 1 she obtains (1 + r)(0)pi(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

S = (1 + r)(0)pi(0) + (1)S(1) (1 + r)(0)pi(0) + (1)b= (1 + r)(1)pi(1) + (1)b= (1)((1 + r)pi(1) + b) 0, ^

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

(1)b (1)S(1) (1)a

because (1) < 0. We can check as in Part 1 above that Conditions (i)-(iii)of Definition 1.1 are satisfied.

Under Condition (1.12) there is absence of arbitrage and Theorem 1.1 showsthat no portfolio strategy can yield S 0 and P( S > 0) > 0 startingfrom (0)pi(0) + (1)pi(1) 0, although this is less simple to show directly.

Finally if a = b 6= (1+r)pi(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)pi(1) < b,

under which Theorems 1.1 and 1.3 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 = ((0), (1)) suchthat C = S, i.e.

(0)(1 + r)pi(0) + (1)a = C()

(0)(1 + r)pi(0) + (1)b = C(+).(1.14)

These equations can be solved as

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(0) =bC() aC(+)pi(0)(1 + r)(b a) and

(1) =C(+) C()

b a . (1.15)

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

(0)pi(0) =bC() aC(+)

(1 + r)(b a)represents the amount invested on the riskless asset.

Note that if C(+) C() then (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) anondecreasing function, since

(1) =C(+) C()

b a=h(S(1)(+)) h(S(1)())

b a=h(b) h(a)

b a 0,

thus there is no short-selling. This applies in particular to European call op-tions with strike K, for which the function h(x) = (xK)+ is nondecreasing.Similarly we will find that (1) 0, i.e. short-selling always occurs when his a nonincreasing function, which is the case in particular for European putoptions with payoff function h(x) = (K x)+.

The arbitrage price pi(C) of the contingent claim C is defined in (1.7) asthe initial value at t = 0 of the portfolio hedging C, i.e.

pi(C) = pi, (1.16)

where ((0), (1)) are given by (1.15). Note that pi(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 pi(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 pi(C) = pi of the contingent claimC is given by

pi(C) =1

1 + rIE[C]. (1.17)



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Proof. We have

pi(C) = pi= (0)pi(0) + (1)pi(1)

=bC() aC(+)

(1 + r)(b a) + pi(1)C(

+) C()b a

=1

1 + r

(C()

b pi(1)(1 + r)b a + C(

+)(1 + r)pi(1) a

b a)

=1

1 + r

(C()P(S(1) = a) + C(+)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

pi((S(1) K)+) = pi(1) bKb a

(bK)a(1 + r)(b a) .

Here, (pi(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 pi(C) can be obtained by two different methods, i.e. analgebraic method via (1.15) and (1.16) and a probabilistic method from(1.17) 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, (1.15) will be replaced with a partial differ-ential equation (PDE) and (1.17) 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 MonteCarlo team, often trying to determine the same option prices by twodifferent methods.

2. What if we have three possible scenarios, i.e. = {, o, +} 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 (1.11) wouldbe rewritten as

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aP({}) + bP({o}) + cP({+}) = (1 + r)pi(1)

P({}) + P({o}) + P({+}) = 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 has n 2 elements, completenessof 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 pi with price pi0 at time t = 0 and value pi1 = pi0(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), where1 < a < r < b. The return of the risky asset is defined as

R =S1 S0S0

.

a) What are the possible values of R ?b) 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.

c) Does there exist arbitrage opportunities in this model ? Explain why.d) Is this market model complete ? Explain why.e) Consider a contingent claim with payoff C given by

C =

if R = a, if R = b.



N. Privault

Show that the portfolio (, ) defined by

=(1 + b) (1 + a)pi0(1 + r)(b a) and =

S0(b a) ,

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

pi1 + S1 = C.

Hint: distinguish two cases R = a and R = b.f) Compute the arbitrage price pi(C) of the contingent claim C using , pi0,, and S0.

g) Compute IE[C] in terms of a, b, r, , .h) Show that the arbitrage price pi(C) of the contingent claim C satisfies

pi(C) =1

1 + rIE[C]. (1.18)

i) What is the interpretation of Relation (1.18) above ?j) 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.k) Letting pi0 = S0 = 1, r = 5% and a = 8, b = 11, compute the portfolio

(, ) hedging the contingent claim C.l) Compute the arbitrage price pi(C) of the claim C.

Here, is the (possibly fractional) quantity of asset pi and is the quantity held ofasset S.

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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 (,F ,P) is a family (Xt)tT ofrandom variables Xt : 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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pi =(pi(0), pi(1), . . . , pi(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 denotes the values at time t = 1, 2, . . . , N of assets 0, 1, . . . , d, and formsa stochastic process (St)t=0,1,...,N with S0 = pi.

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)

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

2.2 Portfolio Strategies

A portfolio strategy is a stochastic process (t)t=1,...,N Rd+1 where (i)tdenotes the (possibly fractional) quantity of asset i held in the portfolio overthe period (t 1, t], t = 1, 2, . . . , N .

Note that the portfolio allocation

t = ((0)t ,

(1)t , . . . ,

(d)t )

remains constant over the period (t 1, t] while the stock price changes fromSt1 to St over this period.

In other terms,

(i)t S

(i)t1

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

(i)t S

(i)t

represents the value of this investment at the end of (t1, t], t = 1, 2, . . . , N .

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

t St1 =di=0

(i)t S

(i)t1,

when the market opens at time t 1, and becomes

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Discrete-Time Model

t St =di=0

(i)t S

(i)t (2.1)

at the end of (t 1, t], i.e. when the market closes, t = 1, 2, . . . , N .

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

t+1 St =di=0

(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 t.

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

t St = t+1 St, t = 1, 2, . . . , 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+1St1

t+ 1t 1 t tt+1t t t+1

t+1St+1tSt1 tSt t+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 t remains the same throughout the day, howeverthe portfolio value changes from morning to evening due to a change in the



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stock price. Also, 0 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 := t St

the value of the portfolio at time t = 1, 2, . . . , N , with

Vt = t+1 St, t = 0, . . . , N 1,

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

V0 = 1 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, 2, . . . , d, t = 0, 1, 2, . . . , N,

or

Xt :=1

(1 + r)tSt, t = 0, 1, 2, . . . , N.

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

Vt =1

(1 + r)tVt, t = 0, 1, 2, . . . , N.

We have

Vt =1

(1 + r)tt St

=1

(1 + r)t

di=0

(i)t S

(i)t

=di=0

(i)t X

(i)t

= t Xt, t = 1, 2, . . . , N,

andV0 = 1 X0 = 1 S0.

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Discrete-Time Model

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 (t)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 Definition 1.3, a contingent claim is given by the non-negative random payoff C of an option contract at time t = N . For example,in the case of the European call of Definition 0.2, the payoff C is given byC = (SN K)+ where K is called the strike (or exercise) 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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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)

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