Faculty of Business Department of Finance, Operations, and

Brock University Faculty of Business

Department of Finance, Operations, and Information Systems FNCE 4P17

Derivatives, Part II Winter 2009

Instructor: Hatem Ben Ameur Office: TA 318 Email: [email protected] Phone: (905) 688–5550 x 5874 Class time: Wednesdays 15:30–17:00 and Fridays 14:00–15:30 (TA 307) Description: A derivative is a contract whose return depends on the price movements of some underlying assets. There are three main families of derivative contracts: options, futures, and swaps. They all have the ability to reduce risk; thus, are widely used for hedging purposes. This course covers advanced topics related to options, futures, and swaps. We focus on their structure, evaluation, and hedging properties. Complex products, such as exotic options, callable bonds, and swaptions are considered. Materials: The following textbook is required, and will be used frequently during the term. 1. Robert W. Kolb and James A. Overdahl, 2007, Futures, Options, and Swaps, 5th

Edition, Blackwell. The following textbooks offer useful alternate explanations, and are listed in increasing order of difficulty. 2. John C. Hull, 2005, Fundamentals of Futures and Options, 6th Edition, Prentice Hall. 3. Don M. Chance and Robert Brooks, 2007, An Introduction to Derivatives and Risk

Management, 7th Edition, Thomson. 4. John C. Hull, 2008, Options, Futures, and Other Derivative Securities, 7th Edition,

Prentice Hall. Monitoring the news about financial derivatives is recommended as an important complement to classroom work. Selected articles from business newspapers such us The Financial Times, The Globe and Mail, The National Post, and The Wall Street Journal will be discussed in class.

mailto:[email protected]

Grading: The grading policy is based on: • four assignments each worth 5%; • two midterm exams each worth 30%; • and a quiz worth 20%. The midterm exams are cumulative. The assignments will improve your skills, and help you prepare for the exams. For each assignment, you will be given selected problems to solve, articles to comment on, and empirical experiments to implement. A quiz is planned at the last week. Office hours will be fixed during the first class. In every aspect of the course, students must comply with the Brock University Honour Code. Outline:

Week Chapter 1–2 Risk-Neutral Pricing 3 Option Sensitivities 4 Pricing American Options

5–6 Options on Stock Indexes, Foreign Currencies, and Futures – First Midterm Exam

7 Pricing Corporate Securities 8 Exotic Options 9 Interest-Rate Options – Second Midterm Exam 10 Long-Term T-Bond Futures Contract

11–12 Swap Contracts and Swaptions – Quiz

Hatem Ben Ameur Derivatives, Part II Brock University, FNCE 4P17

Derivatives, Part II

Hatem Ben Ameur

Brock University �Faculty of Business

Finance, Operations, and InformationSystems

FNCE 4P17

Winter 2010

1


1 Risk-Neutral Pricing

Topics Covered:

1� The Present-Value Principle

2� Pricing in the Binomial Model

3� Risk-Neutral Pricing of �nancial assets

4� Pricing in the Black and Scholes Model

2


1.1 The Present-Value Principle

Let t be the present time and T an investment horizon.The discount factor from time T to time t, indicated by�t;T , is the present value at time t of one dollar to bereceived with certainty at time T .

The compound factor from time t to time T is the futurevalue to be received with certainty at time T of one dollarinvested at time t. In a perfect market, the compoundfactor is the inverse of the discount factor:

ct;T =1

�t;T.

In a perfect market, an asset whose future cash �ows andtheir occurrence dates are known in advance with cer-tainty can be evaluated using the present-value principle:

PVt =Xtn>t

�t;tnCFtn,

3


where �t;tn, for tn > t, is the discount factor fromtime tn to time t, CFtn the future cash �ow (in dol-lars) promised at time tn, and PVt the asset�s presentvalue at time t (in dollars).

For this equation to hold, the asset�s cash �ows and theiroccurrence dates have to be known in advance with cer-tainty.

Exercise: Can a Treasury bond be evaluated using thePV principle? Give the assumptions under which a stockcan be evaluated using the PV principle. Does the PVprinciple apply for a corporate bond, a European call op-tion, or a forward/futures contract?

Exercise: The nominal interest rate is �xed at rnom =4% (per year), and interest is compounded semi-annually.Give the semi-annual interest rate r:5 (in % per six months),which is relevant to compound interest, the compoundfactor c0;1, and the discount factor �0;1. Compute theequivalent annual interest rate r1 (in % per year), which

4


would apply if interest were compounded annually. This isthe e¤ective interest rate. Compute again the compoundfactor c0;1 and the discount factor �0;1. Compute theequivalent monthly interest rate r1=12 (in % per month)and quarterly interest rate r0:25 (in % per quarter). Com-pute again the compound factor c0;1 and discount factor�0;1. Recognize that r:5, r1, r1=12 and r:25 are all equiv-alent rates. The continuously compounded interest raterc (in % per year) is a nominal rate, which would ap-ply if interest were compounded at each second (or evena fraction of a second). Compute again the compoundfactor c0;1 and the discount factor �0;1. Recall that�

1 +a

n

�n! ea, when n!1,

where a 2 R and n 2 N.

An arbitrage opportunity is an investment strategy thatguarantees a riskless pro�t.

Suppose the PV principle applies for a �nancial asset. Ifthe asset is not traded at its (fair) present value, arbitrage

5


would be possible. Borrow at the risk-free rate, and buythe asset if it is undervalued. Short sell the asset, andsave at the risk-free rate if the asset is overvalued.

1.2 Pricing in the Binomial Model

We consider a market for a saving account (the risklessasset) and a stock (the risky asset). Trading activitiestake place only at the current time t0 = 0 and at horizont1 = T . No trading is allowed in between. All positionsare then closed at the horizon.

In addition, the stock price is assumed to move from itscurrent level S0 according to a one-period binomial tree:

p Sup1 = uS0

%S0

&1� p Sdown1 = dS0

t0 = 0 t1 = T

.

6


The stock price can rise by a factor of u or drop by afactor of d, where u > d.

The probabilities p and 1�p de�ne the physical probabil-ity measure P , under which investors evaluate likelihoodsand make decisions in the real world.

The parameters u and d can be seen as volatility para-meters. The greater is u� d, the higher the volatility ofthe stock return.

Example: Assume that the stock price is currently quotedat $100, and can either increase by 25% or decrease by20%. The factors u and d are

u = 1:25, and d = 0:8.

7


In this case, the one-period binomial tree is

Sup1 = uS0 = 1:25� 100 = 125

%S0

&Sdown1 = dS0 = 0:8� 100 = 80

t0 = 0 t1 = T

.

The price can move upward from $100 to $125, or down-ward from $100 to $80.

The (periodic) risk-free rate is indicated by r, and is ex-pressed in % over the time period [t0, t1].

The binomial tree is arbitrage free if, and only if, thefollowing property holds:

d < 1 + r < u.

Indeed, in the case of an upward movement Sup1 = uS0,the rate of return on the stock u � 1 (in % per period)

8


must exceed the risk-free rate r (in % per period). Other-wise, the stock would be overpriced, and an arbitrage op-portunity would arise. Similarly, the risk-free rate r mustexceed the rate of return on the stock under a downwardmovement, that is, d� 1.

Example (continued): The risk-free rate is r = 7%

(per period). Is the model arbitrage free?

The binomial tree is simple, but viable. It is used here togo through the fundamentals of options pricing.

The goal now is to characterize the (present) value of aEuropean call option C0 in the one-period binomial tree,as a function of the stock price S0, the option strike priceX, the volatility parameters u and d, and the risk-freerate r.

Consider a European call option on a stock with a matu-rity date T = t1 and a strike price X.

9


The hedge portfolio consists of holding � shares of thestock and a single signed call option on that stock withan initial value of

H0 = �S0 � C0,

and a terminal value of

H1 = �S1 � C1.

For a speci�c level of �, the hedge portfolio is riskless,and, by the no-arbitrage principle, must earn the risk-freerate. A formula for the (present) value of the call optioncan therefore be derived.

The value of the hedge portfolio moves along the binomialtree as follows:

Hup1 = �S

up1 � Cup1

%H0

&Hdown1 = �Sdown � Cdown1

.

10


The hedge portfolio is riskless if, and only if,

Hup1 = Hdown1 .

Solving for � gives

� =Cup1 � Cdown1

Sup1 � Sdown1

=�C

�S,

which depends on the known parameters S0, X, u, andd.

Given the hedge parameter �, the hedge portfolio, as ariskless investment, should earn the risk-free rate:

H0 = �S0 � C0

=H11 + r

=Hup1

1 + r=Hdown1

1 + r.

Solving for C0 gives

C0 =p�Cup1 + (1� p�)Cdown1

1 + r,

11


where

p� =1 + r � du� d

2 (0; 1) .

The probabilities p� and 1� p� de�ne the so-called risk-neutral probability measure P �, which is not related inany way to the physical probability measure P .

In sum, the value of the European call option can beexpressed as a weighted average (an expectation) of itspromised cash �ow, which is discounted at the risk-freerate:

C0 = E��C11 + r

j S0�,

where E� [:] is the expectation sign under the risk-neutralprobability measure P �, C0 the call-option value at timet0, and C1 the call-option value at time t1.

The pricing formula discounts the risky cash �ow of thecall option by the risk-free rate as if investors were riskneutral, while they are not. This is done via a major

12


correction. We move from the real world, seen under thephysical probability measure P , to a risk-neutral world,seen under the risk-neutral probability measure P �.

Example (continued): Consider a European call optionon the previously mentioned stock with a strike price ofX = $100.

1. Compute the risk-neutral probabilities for upward anddownward movements.

2. Draw the one-period binomial tree for the call option.

3. Compute the hedge ratio.

4. Use the formula to compute the value of the calloption.

13


5. Check that the hedge portfolio earns the risk-freerate.

6. Compute in two di¤erent ways the value of its asso-ciated European put option.

The risk-neutral probabilities are:

p� =1 + r � du� d

=1 + 7%� 0:81:25� 0:8

= 0:6

and

1� p� = 0:4.

The one-period binomial tree for the stock, the call op-

14


tion, and the hedge portfolio are:

Sup1 = 125

% Cup1 = 25

S0 = 100 Hup1 = 0:56� 125� 25 = 45

C0 = 14:02 &� = 0:556 Sdown1 = dS0 = 0:8� 100 = 80H0 = 42:02 Cdown1 = 0

Hdown1 = 0:56� 80� 0 = 45

t0 = 0 t1 = T

,

where

C0 =0:6� 25 + 0:4� 0

1 + 7%,

� =25� 0125� 80

,

H0 = 0:56� 100� 14:02.

The call-option value veri�es:

C0 = �S0 �H0,and

C1 = �S1 �H1,15


where C1 is the call-option value at time t1, that is, thecall-option payo¤ in this example.

In other words, one can fully replicate the call optionusing a strategy that consists of holding � shares of theunderlying stock and borrowing at the risk-free rate.

If the signer decides to replicate the option, he insuresthe promised payment to the option holder at maturity.

A market in which one can fully replicate all derivativecontracts is called a complete market. The binomial treeand the Black and Scholes model are two examples.

Exercise: Compute the present value of the stock in thebinomial tree.

The stock price, discounted at the risk-free rate, is saidto verify the martingale property.

16


The multi-period binomial tree is a simple extension ofthe one-period binomial tree. Both are arbitrage freeunder the same condition, that is,

d < 1 + r < u.

In a multi-period binomial tree, the stock price movesaccording to aMarkov process, and veri�es the martingaleproperty.

The call-option value at time t can then be expressed asfollows:

Ct = E��Ct+11 + r

j Ft�

= E��Ct+11 + r

j St�,

where Ft represents the information available to investorsup to time t.

This formula holds for both European call and put op-tions. Pricing European options in a multi-period bino-mial tree is done as follows:

17


1. Start at maturity, and evaluate the option;

2. Use the risk-neutral pricing formula, and evaluate theoption backward in time;

3. Stop at the origin.

Exercise (continued): Extend the one-period binomialtree to a three-period binomial tree. Evaluate the Euro-pean call option, and its associated put option. Checkthat the call-put parity holds.

1.3 Risk-Neutral Pricing

In a risky environment, it is possible to extend the present-value (PV) evaluation principle, as long as the marketmodel is arbitrage free. The martingale property is a

18


su¢ cient condition for the market model to preclude ar-bitrage.

The present value of an asset can be expressed as follows:

PVt = E�24Xtn>t

�t;tnCFtn j Ft

35 ,where E� [�] is the expectation sign under the risk-neutralprobability measure, �t;tn, for tn > t, the discount factorfrom time tn to time t, CFtn the cash �ow promised bythe asset at time tn, and Ft the information availableto investors up to time t. This formula supports marketrisk, credit risk, and random payment dates.

This formula discounts the risky cash �ows of a �nancialasset using their associated risk-free rates, while higherrates should be used. An adjustment is therefore made:the expectation is computed under the risk-neutral prob-ability measure, rather than the physical probability mea-sure.

This is the risk-neutral pricing formula, which holds aslong as the market model is arbitrage free.

19


1.4 Pricing in the Black and Scholes Model

Black and Scholes considered a frictionless market for astock (the risky asset) and a saving account (the risklessasset) in which trading takes place continuously.

The stock price process fStg is assumed to follow a geo-metric Brownian motion characterized by

ST = SteR,

where R is the continuously compounded rate of returnon the stock over [t, T ]. The annualized rate of returnon the stock is R= (T � t) (in % per year). The rates ofreturn on the stock over successive and non-overlappingtime intervals are independent.

The Black and Scholes model turns out to be arbitragefree. Under the risk-neutral probability measure P �, therate of return on the stock R is random, and follows anormal distribution:

R =�r � �2=2

�(T � t) + �

pT � tZ,

20


where r is the continuously compounded risk-free rate (in% per year), � the volatility of stock returns (per year),T � t the option�s remaining time to maturity (in years),and Z a random variable that follows the standard normaldistribution N (0, 1).

The stock price process is said to be lognormal, sincethe natural logarithm of a future stock price, given thecurrent stock price, follows a normal distribution.

Black and Scholes used a hedge portfolio, and derived aformula for a European call option on a stock paying nodividend:

Ct = E� he�r(T�t)max (0, ST �X) j Sti

= N (d1)St �Xe�r(T�t)N (d2) ,

where

d1 =ln (St=X) +

�r + �2=2

�(T � t)

�pT � t

, and

d2 = d1 � �pT � t.

21


Here, St is the current stock price, X the option�s strikeprice, � the volatility of stock returns, T � t the option�sremaining time to maturity, r the risk-free rate (in % peryear), and N (�) the cumulative function of the standardnormal distribution."

Please establish the second term ofthe Black and Scholes formula.

#

The hedge portfolio used in the Black and Scholes modelconsists of a short position on the call option and a longposition on � shares of the underlying stock. This port-folio is similar to the one used in the binomial modelwith the di¤erence that it has to be continuously ad-justed. The Black and Scholes model is not realistic, butremains very useful in practice.

1.5 Assignment 1

Exercise 1: This exercise shows how to replicate a Eu-ropean call option in the Black and Scholes model using

22


the � hedging strategy, based on the underlying stockand the saving account.

Consider 1; 000 European call options on a stock payingno dividend. The size of each option contract is of 100.The initial stock price is St = $49, the strike price X =

$50, the risk-free rate r = 5% (per year), the volatility� = 20% (per year), and the time to maturity T � t =20=52 = 0:3846 (in years).

1� Check that the call-option value isCt = $240; 052:73.

2� Compute the � coe¢ cient at the start.

The stock price at maturity is ST = $5714; thus, the call

option expires in the money.

3� Compute the call-option payo¤ paid by the seller tothe buyer at maturity.

23


The path of the stock price, observed once a week, was asfollows: week0 49 (in dollars), week1 4818, week2 47

38,

week3 5014, week4 5134, week5 53

18, week6 53, week7

5178, week8 5138, week9 53, week10 49

78, week11 48

12,

week12 4978, week13 5038, week14 52

18, week15 51

78,

week16 5278, week17 5478, week18 54

58, week19 55

78,

week20 5714.

4� Use the � hedging strategy, rebalanced weekly, toreplicate the European call option. Use Excel, and �ll inthe following hedging table. Explain.

Week Stock Delta Shares Cost Cum. Interest# Price � Purch. Cost ($000)0 49

1 4818...19 557820 5714

5� Repeat this exercise for the following path of stockprices: week0 49 (in dollars), week1 4934, week2 52,

24


week3 50, week4 4838, week5 4814, week6 48

34, week7

4958, week8 4814, week9 48

14, week10 51

18, week11 51

12,

week12 4978, week13 4978, week14 48

34, week15 47

12,

week16 48, week17 4614, week18 4818, week19 46

58, week20

4818.

6�� Simulate a random path for the stock price anddo again question 4�. De�ne your code parameters asfollows: s for St, X for the option strike price, sigma for�, Tau for the remaining time to maturity T � t, r forthe risk-free rate, a and b for the lower and upper boundof the interval [a, b], respectively, and delta for the hedgeratio �.

7�� Set the number of observation dates N as an addi-tional parameter. Do again question 6� with increasingN 2 f20, 40, 80, 160g. De�ne the hedge error e (ep-silon) and show that e ! 0 when N ! 1. This ques-tion is not mandatory, but very instructive. It is worthan additional 10 points in the �rst mid-term exam (thedeadline for question 7�).

25


Exercise 2: In the Black and Scholes model, the stockprice follows a lognormal process, which is a Markovprocess. The following transition parameter keeps trackof the behaviour of the stock price.

1. Show that the following transition parameter can beexpressed as

P � (ST 2 [a, b] j St)

= N

0@ln (b=St)��r � �2=2

�(T � t)

�pT � t

1A�N

0@ln (a=St)��r � �2=2

�(T � t)

�pT � t

1A ,where N (:) is the cumulative distrubution functionof N (0; 1).

2. Comment on the case where a = X and b = 1.Give an example, and support your argument by se-lected numbers.

26


3. Set St = $100 and r = 4%. Fill in the followingTables for

P � (ST 2 [a; b] j St) .

Set T � t = 1 (year) when � changes and � = 20%(par year) when T�t changes. Intrepret your results.

� T � t[a; b] 10% 20% 30% :5 1 2[70; 80][80; 90][90; 100][100; 110][110; 120]

.

27


2 Sensitivity Analysis

Topics Covered:

1. Single-Option Sensitivity Analysis

2. Portfolio Sensitivity Analysis

28


2.1 Single-Option Sensitivity Coe¢ cients

The value of an option contract, v, depends on several in-puts, including the underlying asset price S, the volatilityof asset returns �, the option�s remaining time to matu-rity T � t, and the risk-free rate r.

The sensitivity of an option with respect to its underlyingasset price is

�v

�S,

where �S is a small change in the underlying asset price,and �v is the associated change in the option value. Allother inputs are assumed to be �xed.

A sensitivity coe¢ cient cannot directly be computed fromreal-life observations. Indeed, the inputs that a¤ect theoption value change together over time. A sensitivity co-e¢ cient can, however, be computed via a market model,especially when a closed-form solution is available for the

29


option value. The Black and Scholes model is an exam-ple.

The delta coe¢ cient, indicated by �, is the �rst partialderivative of v with respect to S, that is,

� =@v

@S

= lim�S!0

�v

�S,

when the right-hand quantity exists.

Exercise: The call-option parameters are S = $100,X = $100, � = 0:3 (per year), T � t = 180 (days), andr = 8% (per year). Use Table 14.1, and approximate �at S = $100. Approximate the change in the call-optionvalue if the stock price drops by $0:5.

The gamma coe¢ cient, indicated by �, is the second

30


partial derivative of v with respect to S, that is,

� =@2v

@S2

=@

@S

�@v

@S

�=@�

@S,

when the right-hand quantity exists.

Table 14.4 and Table 14.5 provide � and �, as functionsof some relevant parameters, for European call and putoptions in the Black and Scholes model.

[Please explain Figure 14.1 on page 478.]

Exercise: Comment on Figure 14.4 on page 484, andFigure 14.13 on page 491 in conjunction with Figure 14.1on page 478. Please adjust the shape of the curve of �.

31


The �rst-order Taylor expansion is

�v � ��S,

where �S is a small change in the stock price. All otherparameters are kept �xed.

Exercise: Use a Taylor expansion of order one, and ap-proximate the change in the call-option value if the stockprice drops by $0:5. Compare with the true value.

The second-order Taylor expansion, which is more accu-rate than the �rst-order one, is

�v � ��S + 12� (�S)2 .

Again, all the parameters that a¤ect the option value arekept �xed, except the stock price.

Exercise: Use a second-prder Taylor expansion, and ap-proximate the change in the call-option value if the stockprice drops by $0:5. Compare with the true value. UseTable 14.6 on page 480.

32


The vega coe¢ cient, indicated by V, is the �rst partialderivative of v with respect to �, that is,

V = @v

@�

= lim��!0

�v

��,

where �� is a small change in the volatility of stockreturns. Here, all the parameters that impact the optionvalue are kept �xed, except the volaltility of stock returns.

Exercise: How does a call-option value behave as a func-tion of the volatility of stock returns? Answer the samequestion for a put-option value. Use Table 14.4, Ta-ble 14.5, and Figure 14.8. Check if Figure 14.8 is correct.

Exercise: Approximate the change in the put-option valueif the volatility rises by 1%. Compare with the true value.Use Table 14.6 on page 480.

33


The theta coe¢ cient, indicated by �, is the �rst partialderivative of v with respect to t, that is,

� =@v

@t= � @v

@ (T � t)

= lim�t!0

�v

�t,

where �t is a small change in time, for example, one dayfrom the present time. Here, all parameters that impactthe option value are kept �xed, except the time index.

Exercise: How does the call-option value behave as afunction of the time index? Use Table 14.4.

The call-option value decreases over time. This is thetime-decay phenomenon for a call option.

[Please explain the time-decay phonomenon from Figure 14.1.]

34


Exercise: A call-option holder knows with certainty thathe will lose value over time. Is it pro�table to hold a calloption? Explain.

All the parameters that impact the option value are keptconstant, except the stock price and the time index. Then,the �rst-order Taylor expansion for a multivariate functioncan be written as

�v � ��S +��t.

Exercise: Approximate the change in the call-option valueif the stock price rises by $0.5 over one trading day. Thereare 252 trading days per year. Compare with the truevalue. Use Table 14.6 on page 480.

Finally, the rho coe¢ cient, indicated by �, is the �rstpartial derivative of v with respect to r, that is,

� =@v

@r

= lim�r!0

�v

�r,

35


where �r is a small change in the risk-free rate. Here, allparameters that impact the option value are kept �xed,except the risk-free rate.

Exercise: Approximate the change in the call-option valueif the stock price rises by $0.5 over one trading day, andthe risk-free rate drops by 1%. Compare with the truevalue. Use Table 14.6 on page 480.

The single-option sensitivity coe¢ cients, as de�ned above,can be extended to all derivative contracts.

2.2 Portfolio Sensitivity Coe¢ cients

Consider a potfolio of derivative contracts and their un-derlying assets. The value of this portfolio is a linearcombination of its basic components. For each compo-nent, compute the sensitivity coe¢ cient. Then, apply

36


the same linear combination, and obtain the associatedportfolio-sensitivity coe¢ cient.

Consider the hedge portfolio considered in the previouschapter:

P = �S � C,

where � 2 R is the delta coe¢ cient computed at S.The delta coe¢ cient of this portfolio is

@P

@S= �

@S

@S� @C@S

= �� 1��= 0.

The hedge portfolio is therefore insensitive to small changesin the stock price. This portfolio is called �� neutral.

A��neutral portfolio is not necessarily ��neutral, V�neutral,��neutral, or ��neutral ; however, ��neutral port-folios and the like can be obtained. For example, a port-folio combining two call options and their common un-derlying asset can be made �� neutral. Indeed, the

37


value of the portfolio is

P = nsS + n1C1 + n2C2,

where nS, n1, and n2, are the number of shares, �rstcall option, and second call option, respectively. Solvethe following equation for nS, n1, and n2:(

ns + n1�1 + n2�2 = 00 + n1�1 + n2�2 = 0

,

and determine the associated �� neutral portfolio.

Example: Set ns = 1, �1 = 0:6151, �2 = 0:4365,�1 = 0:0181, and �2 = 0:0187. Show that n1 =�5:1917 and n2 = 5:0251. Identify the associated�� neutral portfolio.

Following the same rules, a sensitivity analysis can bedone for simple and complex strategies that combine deriv-ative contracts and their underlying assets.

A long straddle is a portfolio of a long position on a calloption and a long position on a put option, both with

38


the same parameters. The pro�t function of the straddle,under the assumption that it is held till maturity, is

� = CT � C0 + PT � P0= max (0, ST �X)� C0 +max (0, X � ST )� P0,

where CT and C0 are the call-option values at maturityand at the start, respectively; and PT and P0 are the put-option values at maturity and at the start, respectively.Similarly, the pro�t function of the straddle, under theassumption that it is held up to time t, is

� = Ct � C0 + Pt � P0,

where Ct and Pt are functions of the stock price at timet, among other parameters.


Exercise: Consider the straddle associated to the exer-cise priceX = $100, described in Table 14.7 on page 495.Consider the straddle value at the start, that is, C0+P0,as a function of the stock price, the volatility of stock

39


returns, the remaining time to maturity, and the risk-freerate. Compute its sensitivity coe¢ cients, and interpretyour results. How will the straddle behave if the stockprice drops by $1 and the volatility rises by 2% over thenext three trading days?

A strangle is similar to a straddle except that its associ-ated call and put options have di¤erent strike prices.


Exercise: Consider the strangle de�ned in Figure 14.16on page 497. Answer the same questions as for the strad-dle.

A butter�y spread is a portfolio that employs three calloptions associated to the same parameters except fortheir strike prices.

Consider the butter�y spread taken from Table 14.7 onpage 495, which consists of a long position on the call

40


option with a strike price of X1 = $90, a long positionon the call option with a strike price of X3 = $110, anda short position on two call options with a strike price ofX2 = $100.


Exercise: Consider the butter�y spread in Figure 14.17on page 499. Answer the same questions as for the strad-dle.

Consider two call options associated to the same para-meters except for their strike prices. A bull spread is aportfolio of a long position on the call option with thelowest strike price and a short position on the call optionwith the highest strike price.

Exercise: Consider the bull spread de�ned in Figure 14.18on page 500. Answer the same questions asked for thestraddle.

41


2.3 Assignment 1 Part 2

Do exercises no 1, 2, 3, 4, 9, and 16 on pages 505�506.

Exercise: Consider the call option in Table 14.6.

1� Use the formulas in Table 14.1�Table 14.4 and com-pute the call-option sensitivity coe¢ cients in Table 14.6.

2� Use Excel and draw Figure 14.1, Figure 14.9, andFigure 14.13. Comment your results.

42


3 Pricing American Options

Topics Covered:

1. Exercise Value and Holding Value

2. Pricing American Options in the Black and ScholesModel

(a) The American Call Option

(b) Transition Parameters for Pricing American Op-tions

3. Pricing American Options in the Binomial Model

43


3.1 Exercise Value and Holding Value

The American version of a European option gives itsholder the additional right to exercise the option early,before maturity.

Thus, an American option is worth more than its asso-ciated European option. The di¤erence in value, calledthe early exercise premium, stems from the early-exercefeature.

Let Vt (s) be the value of an American option, and vt (s)be the value of its associated European option at time twhen St = s. Uppercase letters are used for Americanoptions, and lowercase letters are used for their Europeancounterparts.

Let V et (s) and Vht (s) be the exercise value and the hold-

ing value of an American option at time t when St = s,respectively. The exercise value is also called the intrinsicvalue.

44


Example: The exercise value of a vanilla call option is

Cet (s) = max (0, s�X) , for t � T ,

where St = s is the current underlying asset price, andX is the exercise price. Similarly, the exercise value of avanilla put option is

P et (s) = max (0, X � s) , for t � T .

The holding value of an American option is

V ht (s) = E� h�t;uVu j St = si ,

where E� [�] is the expectation sign under the risk-neutralprobability measure, u is the �rst decision date after t,�u=�t is the discount factor from time u to time t, andSt = s is the stock price at time t. The holding valueveri�es

V ht (s) � 0, for all s.

with the convention that

V hT (s) = 0, for all s.45


The holding value represents the future potentialities ofthe contract, discounted back to the current date. Ex-cept for a few exceptions, the holding value cannot becomputed in a closed form, and must be approximatedin some way. The reason is that Vu itself depends on anexpectation, which itself depends on an expectation, andso on. In sum, the holding value is a high-dimensionalintegral. Computing the holding value is the most chal-lenging issue in pricing American options.

The value of an American option is

Vt (s) = max�V et (s) , V

ht (s)

�, for all s,

and the optimal exercise policy consists of exercising theoption at time t when the underlying asset is at St = s

if, and only if,

V et (s) > Vht (s) ,

and holding the contract for at least another moment if,and only if,

V et (s) � V ht (s) .46


This is to say that the option holder acts in an optimalway.

A European option can be seen as a particular case of anAmerican option, where the exercise values veri�es

V et (s) = 0, for t < T and all s,

and

V eT (s) = VT (s) , for all s.

The following general results hold:

Vt (s) � V ht (s) � vt (s) , for all s,

and

Vt (s) � V et (s) , for all s.

The exercise premium at time t when St = s is de�nedas Vt (s)� vt (s) � 0.


47


The time value of an American option at time t whenSt = s is de�ned as

Vt (s)� V et (s) � 0.

The larger the time value of an American option, thelonger is its early exercise.

3.2 Pricing American Options in the Black

and Scholes Model

3.2.1 The American Call Option

Call-option values admit the following bounds:

ct � St �Xe�r(T�t),

and

Ct � St �X.48


An American call option on a stock that pays no dividendcannot be exercised optimally before maturity.

Indeed, one has

Cet = St �X � St �Xe�r(T�t) � ct � Cht .

Exercising the call option before maturity is clearly sub-optimal. When the underlying stock pays dividends, how-ever, it may be optimal to exercise the call option early,just before an ex-dividend date.

"Please draw two paths for the stock price,

with and without dividend paying.

#

Consider a call option on a dividend-paying stock. LetD1; : : : ; DN be the dividends, and t1; : : : ; tN their as-sociated ex-dividend dates over the option�s life [t, T ].Fisher Black approximated the American call-option valueas follows.

49


1. Adjust downward the current stock price by the PVof all dividends:

S�t = St �NXn=1

Dne�r(tn�t).

This is somewhat equivalent to adjust Stn downwardby Dn at tn, for n = 1; : : : ; N .

2. Compute the European call-option value using theBlack and Scholes formula:

c = BS (S�t , X, �, T � t, r) .

3. For each ex-dividend date tm, adjust the call-option�sstrike price downward by the PV of all remainingdividends, including the one that is about to go ex-dividend, that is,

X�m = X �Xn�m

Dne�r(tn�tm).

This is equivalent to adjust upward the stock priceby the same amount.

50


4. Compute the American call-option value using theBlack and Scholes formula, under the assumption ofan early exercise just before tm, that is,

cm = BS (S�t , X�m, �, tm � t, r) .

5. The approximate call-option value and its optimalexercise policy are given by

bc = max fc; c1; : : : ; cNg .The intuition behind this approximation is that the Amer-ican call-option can be exercised only prior to an ex-dividend date or at maturity.

Example: The parameters of the American call optionare St = $60, X = $60, � = 20% (per year), T �t = 180 (days), r = 9% (per year), D1 = D2 = $2,t1 = 60 (days), t2 = 150 (days). Approximate its valueand identify its optimal exercise policy. Check that the(approximate) optimal exercise policy, as given by Black�s

51


approximation, consists of exercising the call option early,at time t2.

The optimal exercise policy, provided by Black�s approx-imation, is deterministic while it must be expressed as afunction of time and the stock price as follows: exercisethe call option early at the ex-dividend date tn if, andonly if, the stock price is su¢ ciently high at that time.

Merton extended the Black and Scholes model to a �-nancial asset that pays dividends continuously at a �xedrate � (in % per year). Think of the underlying asset asa stock index. The extension is easily done by adjustingdownward the current stock price as follows:

S�t = Ste��(T�t).

Similarly, in the Black-Scholes-Merton model, an Ameri-can call option can be exercised early, at any time beforematurity if, and only if, the stock price is su¢ ciently high.

52


Along the same lines, an American put option on a stockcan be exercised early at any time before maturity, whetheror not the stock pays dividends. The optimal exercise pol-icy is as follows: exercise the put early if, and only if, thestock price is su¢ ciently low.

[Please explain Figure 15.3 and Figure 15.4 on page 524.]

Several approximations have been proposed for pricingAmerican options. Some of them are available in thetextbook. All of them su¤er from two major disadvan-tages. They are neither accurate nor general.

3.2.2 Transition Probabilities for Pricing AmericanOptions

Transition probabilities under the risk-neutral probabil-ity measure completely characterize the dynamics of thestock price. The following numerical procedure can be

53


used for pricing American options as long as the transi-tion probabilities are known in a closed form.

Let G =na0 = 0; : : : ; ap; ap+1 =1

obe a grid points

for the stock price. Any choice for the ai, for i =1; : : : ; p, will work as long as they cover a large domainand verify

�a! 0, when p!1.

Consider the transition probabilities

P ��Stn+1 2 [ai, ai] j Stn = ak

�, for all i and k,

which are known in a closed form in the Black-Scholes-Merton model, among other models.

Consider an American option characterized by its exercisevalue V et , which can be exercised early at the decisiondates t0; : : : ; tN = T . Assume that the option-valuefunction is known at a decision date tn+1 and the gridpoints G. This is not a strong assumption since the optionvalue is known everywhere at maturity. The numericalprocedure acts backward in time from tn+1 to time tn,as follows.

54


1. Interpolate the value function at time tn+1 by apiecewise constant approximation:

bVtn+1 �Stn+1� = pXi=0

�n+1i � I�Stn+1 2 [ai, ai]

�,

where I (�) is the indicator function, and �n+1i =bVtn+1 (ai), for all i.2. Move backward in time from time tn+1 to time tn,and consider the holding value

V htn (ak) = E� he�r(tn+1�tn)Vtn+1 �Stn+1� j Stn = aki ,

which is not known in closed form.

3. Approximate the holding value function as follows:

eV htn (ak)= E�

he�r(tn+1�tn) bVtn+1 �Stn+1� j Stn = aki

= e�r(tn+1�tn)Xi

�n+1i �

P ��Stn+1 2 [ai, ai] j Stn = ak

�.

55


4. Approximate the option value at tn and at the gridpoints G usingeVtn (ak) = max �V etn (ak) , eV htn (ak)� , for all ak in G,and identify the optimal exercise policy.

5. Interpolate eVtn (ak), for all ak in G, to bVtn (s), forall s > 0.

6. Follow the same procedure, and go from time tn totime tn�1, backward in time to time t0.

7. Reach any level of desired precision by increasing thegrid size p sincebVt0 �St0�! Vt0

�St0

�, when p!1.

This numerical procedure has been proposed by Ben Ameuret al for American call and put options in European Jour-nal of Operational research, and for American Asian op-tions in Management Science. It is used by several �nan-cial institutions all over the world.

56


3.3 Pricing American Options in the Bi-

nomial Model

The binomial model can be adjusted to be consistent withthe Black-Scholes-Merton model:

V BMt ! V BSMt , when �t! 0.

The convergence speed is believed to be 0:5, meaningthat the error

e =j V BMt � V BSMt j ,

is cut by half if the time increment �t is divided by 4.The �rst part of this result is hard to prove, and thesecond is an open question.

For the binomial model to be consistent with the Black-Scholes-Merton model, set

u = e�p�t, d =

1

u, and p� =

e(r��)�t � du� d

.

57


Example: Consider an American call option on a stockthat pays dividends continuously at a �xed rate � =

13:75% (per year). The option parameters are as follows:St0 = $60, X = $60, � = 20% (per year), T �t = 180(days), r = 9% (per year), and �t = 90 (days). Com-pute its early exercise premium, and identify its opti-mal exercise policy. Please check that u = 1:104412,d = 0:905460, p� = 0:416663, and that the two-periodbinomial tree for the stock price is

Suu2 = 73:18%

Su1 = 66:26% &

S0 = 60 Sud2 = Sdu2 = 60& %

Sd1 = 54:32&

Sdd2 = 49:19

.

58


If the stock pays discrete dollar dividends over the op-tion�s life, Black�s intuition applies to adjust the binomialtree, though with a minor modi�cation.

1. Adjust downward the stock price at the present datet0 by the PV of all dividends over the option�s life:

S�t0 = St0 � PV all dividends.

This is somewhat equivalent to adjust St downwardby Dt, for all ex-dividend dates.

2. Create the binomial tree as usual.

3. At each decision tn, adjust upward the stock priceby the PV of all remaining dividends (including theone that is to be paid).

4. Go backward through the adjusted binomial tree, andompute options values as usual.

59



Read the chapter, do the examples, and exercises no 1�7on page 537. This part is to be prepared for the midtermexam, but should not be handed in as part of Assign-ment 2.

Do exercises no 8, 17, 19.

Exercise (Bonus � worth 10 points in the secondmid-term exam): This exercise is to be handed in af-ter the reading week. Use VB-Excel. In all cases, yourprogram must be fully automated.

1� Implement Black�s procedure for pricing an Americancall option on a stock that pays discrete dollar dividends.Let the number of ex-dividend dates �exible. Solve theexample shown in the textbook.

2� Implement the analytical approximation of Ameri-can option prices, and draw Figure 15.3 and Figure 15.4.Solve the examples shown in the textbook.

60


3� Implement the binomial tree on a stock that paysdollar dividends. Let the number of ex-dividend dates�exible. Solve the examples in the textbook. Report thebinomial trees reported in the textbook.

61


4 Exotic Options

Topic Covered:

1. Forward-Start options

2. Instalment Options

3. Compound Options

4. Chooser Options

5. Barrier Options

6. Binary Options

7. Asset-or-Nothing Options62


8. Lookback Options

9. Asian Options

63


4.1 Forward-Start Options

Standard call and put options are known as vanilla op-tions. Other option contracts, known as exotic options,are traded on exchanges and in over-the-counter markets.They are supposed to match hedgers�needs and require-ments. Forward-start options are examples.

Under a forward-start option, the holder pays the pre-mium before the option starts at parity. There are threekey dates: the date the contract is signed t, the date theoption starts T1, and the date the option matures T2.Date T1 is called the grant date.

Forward-start options are often used in compensation pack-ages for executives.

Proposal: The presentation on executive options willcover forward-start options, as well as other option con-tracts used in compensation packages.

64


Forward-start options can be evaluated using the risk-neutral evaluation principle. Consider a European forward-start option in the Black and Scholes model. The valueof the European forward-start option at the start is

vt

= E�he�r(T1�t)vBST1

�ST1; X = ST1; �; T2 � T1; r

�j St

i,

since it starts at parity. On the other hand, one has

vBST1

�ST1; X = ST1; �; T2 � T1; r

�= ST1f (�; T2 � T1; r) ,

where f is a function of the volatility of the stock log-returns �, the option�s life T2 � T1, and the risk-freerate, but not the stock price ST1. Using the martingaleproperty, the value of the European forward-start optionis

vt = E� he�r(T1�t)ST1 j Sti� f (�; T2 � T1; r)

= St � fBS (�; T2 � T1; r)= vBSt (St; St; �; T2 � T1; r) ,

whether it is a call or a put option.65


If the stock underlying the option pays continuous divi-dends at a constant dividend rate � (in % per year), themartingale property is adjusted as follows:

E�he�r(T1�t)ST1 j St

i= e��(T1�t)St,

and the value of the European forward-start option isconsequently adjusted as follows:

vt = E� he�r(T1�t)ST1 j Sti� f (�; T2 � T1; r; �)

= e��(T1�t)St � fBSM (�; T2 � T1; r; �)= e��(T1�t)vBSMt (St; St; �; T2 � T1; r) .

All in all, compute the value of a European forward-startoption at the grant date as if the stock price were equalto the current stock price, and then discount this valueback from the grant date to the start date at the dividendrate.

Valuing American forward-start options can be done back-ward in time through a stochastic dynamic program, asit can be done for American vanilla options.

66


4.2 Instalment Options

The holder of a European instalment option must payinstalments at certain decision dates to keep the optionalive. Therefore, at each decision date, the holder mustchoose between the following:

1. to pay the instalment, which keeps the option alivetill the next decision date;

2. not to pay the instalment, which puts an end to thecontract.

The main bene�ts for the holder of an instalment op-tion are twofold. First, risk management with instalmentoptions is �exible. Like American options, instalment op-tions are well suited for hedging cash �ows whose timingis uncertain. Second, the hedging cost structure with in-stalment options is �exible too. Instead of paying lump

67


sums for hedging instruments, corporations may enter aninstalment option at a low initial cost, and adjust the in-stalment schedule according to their cash forecasts andliquidity constraints. This feature is particularly attractivefor corporate treasurers who massively hedge interest-rateand currency risks with forwards, futures, or swaps, be-cause option contracts imply an entry cost that may beincompatible with a temporary cash shortage.

Consider an instalment option with positive instalments�0; : : : ; �N , to be paid at the decision dates t0 = 0; : : : ;tN = T . The initial instalment �0 plays the role of theoption�s premium. By the risk-neutral evaluation princi-ple, the net holding value of this instalment option atdecision date tn is

vnhtn (s)

= E�[e�r(tn+1�tn)vtn+1(Stn+1) j Stn = s]� �n,

and the overall value is

vtn (s) = max�0, vnhtn (s)

�, for n = 0; : : : ; N � 1.

68


The value of the instalment option at maturity is

vtN (s) =

(max (0; s�X) , for an instalment call optionmax (0; X � s) , for an instalment put option

.

The exit strategy at time tn is as follows: put an end tothe option�s life if, and only if,(

Stn < a�n, for an instalment call option

Stn > b�n, for an instalment put option

,

where a�n and b�n are two thresholds that are functions of

the option parameters. The threshold is determined bysolving for s = Stn the following equation:

vnhtn (s) = 0.

One way of pricing the instalment option is via backwarditeration: from the known function vtN , compute vtN�1,then from vtN�1 compute vtN�2, and so on, down to vt0.Neither the value function vtn, for n = 0; : : : ; N�1, northe threshold v�n (a

�n or b

�n) is known in a closed form.

They must be approximated in some way. Stochastic69


dynamic programming can be used (Ben Ameur et al.,2006, European Journal of Operational Research).

Example: Consider an instalment call option with onlyone intermediate instalment �1 to be paid at t1 < t2 =

T . The net holding value at the decision date t1 is

cnht1 (s) = E�[e�r(T�t1)cT (ST ) j St1 = s]� �1

= E�[e�r(T�t1)max (0; ST �X) j St1 = s]� �1= cBSMt1

�St1; X; �; T � t1; r; �

�� 1,

where X is the option�s strike price, and the overall valueis

ct1 (s) = max�0; cBSMt1

�St1; X; �; T � t1; r; �

�� 1

�.

This is the payo¤ from a European call option with matu-rity t1 and strike price �1 on a call option with maturityT and exercise price X, the underlying call option beingwritten on a dividend-paying stock. Options on optionsare also known as compound options. There are mainlyfour types of compound options: calls on calls, calls onputs, puts on calls, and puts on puts.

70


4.3 Chooser Options

The holder of a chooser option has the right to deter-mine whether the chooser will become a vanilla call orput option by a speci�ed choice date. Chooser optionsare also known as as-you-like-it options. For simplicity,assume that the call and put options have the same ex-ercise price and maturity date.

Chooser options may be useful for hedging market risk,which strongly depends on the occurrence of a futureevent.

Example: In 1993, the North American Free Trade Agree-ment (NAFTA) was being discussed, but had not yet beenagreed upon. NAFTA was known to be bene�cial to theMexican peso. Before the NAFTA agreement, chooseroptions were e¤ective for hedging the Mexican peso risk.If NAFTA went through, the chooser option would thenbecome a call option; otherwise, the chooser option wouldbecome a put option on the Mexican peso.

71


There are mainly three key dates to consider: the eval-uation date t, the choice date T1, and the expiry dateT2.

If T1 = t, one has

vt = max�cBSMt , pBSMt

�,

and if T1 = T2, one has

vT = max (cT , pT )

= (ST �X) I (ST > X) + (X � ST ) I (ST � X) ,and, by the risk-neutral evaluation principle,

vt = cBSMt + pBSMt ,

which is the value of the associated straddle.

Now, if T1 2 (t, T2), one has

vT1

= max�cBSMT1

, pBSMT1

�= max

�cBSMT1

, cBSMT1+Xe�r(T2�T1) � ST1e

��(T2�T1)�

= cBSMT1+ e��(T2�T1)max

�0, Xe�(r��)(T2�T1) � ST1

�,

72


where the right-hand side consists of a portfolio of a calloption with a strike price of X and a maturity date ofT2 and e��(T2�T1) put options with a strike price ofXe�(r��)(T2�T1) and a maturity date of T1. The risk-neutral evaluation principle gives

vt = cBSMt (St; X; �; T2 � t; r; �)+

e��(T2�T1)pBSMt

�St; Xe

�(r��)(T2�T1); �; T1 � t; r; ��.

4.4 Barrier Options

A barrier option acts as a vanilla option under the as-sumption that the underlying stock price reaches a barrieralong its path. There are four families of barrier options:down-and-in, down-and-out, up-and-in, and up-and-outbarrier options. They are also known as knock-in andknock-out options.

[Please discuss the payo¤ from a down-and-in call option.]

73


A barrier option is a path-dependent option since its �-nal payo¤ does not only depend on the stock price atmaturity, but also on its behaviour over the option�s life.Barrier options are not traded; however, they are usefulat modeling knock-in and knock-out events.

Example: Several corporate securities can be interpretedas derivatives on a �rm�s value. For example, Merton�smodel of a corporate discount bond assumes that the�rm can only go bankrupt at maturity, which is a majorlimitation. One way to improve Merton�s model is tode�ne bankruptcy as

Vt < bt, for t 2 [0, T ] ,

where Vt is the �rm�s value at time t, and bt is a bar-rier related to the debt�s amortization. Instalment andcompound options are also useful at modeling corporatecoupon bonds, since bondholders can put an end to thecorporate bond if certain instalments are not paid.

74


4.5 Binary Options

A binary option pays a certain amount if an event hap-pens, but nothing otherwise. This event is usually relatedto the performance of an underlying �nancial asset.

A cash-or-nothing call option pays its holder a �xed amountif the underlying �nancial asset exceeds a given strikeprice X at maturity T . The value of a binary option is

cBinaryt = E�

he�r(T�t)AI (ST > X)

i= e�r(T�t)AP � (ST > X)

= e�r(T�t)AN�dBSM2

�,

where A is the amount to be paid if the binary optionexpires in the money, and N

�dBSM2

�is the normal dis-

tribution function evaluated at dBSM2 . Along the samelines, the value of a cash-or-nothing put option is

pBinaryt = Ae�r(T�t)N

��dBSM2

�.

75


An asset-or-nothing call option promises its holder theunderlying asset under the condition that the underlyingasset exceeds a given strike price at maturity. This is aplain vanilla call option with X = 0 whose value is

cBinaryt = Ste

��(T�t)N�dBSM1

�.

The value of an asset-or-nothing put option is

pBinaryt = Ste

��(T�t)N��dBSM1

�.

4.6 Lookback Options

The payo¤ function from a lookback option is based onextreme values.

The payo¤ from a European option to buy an underlyingasset at the lowest price along its path is

ST � min[0, T ]

St,

76


and the payo¤ from a European option to sell at thehighest price is

max[0, T ]

St � ST .

Under Black, Scholes, and Merton�s model, lookback op-tions can be evaluated in a closed form, since the proper-ties of lognormal processes and their extreme values areknown.

4.7 Asian Options

The payo¤ from an Asian option is based on averageprices, rather than terminal prices. Asian options are use-ful in markets where there are major players that can ma-nipulate market prices, at least locally in time. Examplesinclude oil and oil-product markets.

77


For example, a European Asian call option is character-ized by the following �nal payo¤:

max�0, ST �X

�= max

0,St0 + � � �+ StN

N + 1�X

!.

Other exotic options do exist, such as exchange options,basket options, rainbow options, and gap options, amongothers. They may be helpful to hedge for speci�c risks,or better analyse fundamental �nancial securities.


Read the chapter, and do exercises no 1, 5, and 6 on page616.

78


5 Options on Stock Indices, Cur-

rencies, and Futures

Topics Covered:

1. Adjusting Basic Models for Continuous Dividends

2. Options on Stock Indices

3. Options on Foreign Currencies

4. Options on Futures

79


5.1 Options on Stock Indices

Merton extends the Black and Scholes model for a stockthat pays dividends continuously at a �xed rate � (in %per year). From now on, the Black and Scholes modelwill be indicated by BS, and the Black-Scholes-Mertonmodel by BSM.

In BSM, e��(T�t) shares of stock at time t is equivalentto one share of stock at time T , the dividends beingreinvested in additional shares over [t, T ]. Thus, BSMcall- and put-option values are obtained by exchangingSt in BS by

Ste��(T�t).

For example, the BSM formula for a European call optionis

cBSMt = N (d1)Ste��(T�t) �Xe�r(T�t)N (d2) ,

80


where

d1 =ln�Ste

��(T�t)=X�+�r + �2=2

�(T � t)

�pT � t

=ln (St=X) +

�r � � + �2=2

�(T � t)

�pT � t

, and

d2 = d1 � �pT � t.

The binomial model can also be adjusted for a stockthat pays continuous dividends. The adjustment is donethrough risk-neutral probabilities:

p� =e(r��) � du� d

.

Options in BSM admit the following bounds:

cBSMt � Ste��(T�t) �Xe�r(T�t),

and

pBSMt � Xe�r(T�t) � Ste��(T�t).

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Exercise: Use the no-arbitrage property, and establishthese lower bounds.

The put-call parity for European options in the BSM is

cBSMt +Xe�r(T�t) = pBSMt + Ste��(T�t),

and for American options is

Ste��(T�t)�X � CBSMt �PBSMt � St�Xe�r(T�t).

Exercise: Use the no-arbitrage property, and establishthe put-call parity for European call and put options.

5.2 Options on Stock Indices

Options on stock indices are traded in exchanges andover-the-counter markets. Some indices track the move-ment of the overall stock marker, and others track themovement of particular industries.

82


Example: The S&P 100 and S&P 500 stock indices arebased on selected 100 and 500 US stocks, respectively.Options on S&P 100 are of American style, while optionson S&P 500 are of European style.

Since the delivery of a stock index involves high transac-tion costs, stock-index options are settled in cash usingthe following payo¤ function:

Cet = m�max (0; It �X) ,

for a call option, and

P et = m�max (0; X � It) ,

for a put option. Here, It is the stock-index level at timet, m is the multiplier, and X is the option�s strike price.The scalar m plays the role of the option�s size, and isusually set at 100.

Other exchange-traded options related to stock indicesare the so-called LEAPS, CAPs, and FLEX options.

83


A stock index is a �nancial asset that provides its holderwith several discrete dividends throughout all over theyear. The assumption that dividends are distributed con-tinuously at a �xed rate is acceptable in this context. Thedividend rate can easily be infered from futures prices, orestimated from historical observations. Thus, the BSMformula can be used for pricing.

Exercise: A European call option on a stock index hasthe following parameters: It = 350, X = 340, � =

20%, T � t = 150 days, r = 8% (per year), � = 4%

(per year). Set m = 1. Check that cBSMt = $25:92.Compute pBSMt .

A call option on a stock index can be used to hedgeagainst upward movements of the stock market, and aput option on a stock index, to hedge against downwardmovements of the stock market.

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5.3 Foreign Currency Options

Options on foreign currencies are traded both in exchangesand in over-the-counter markets.

For example, options on the Canadian dollar are tradedon the Philadelphia Stock Exchange.

A foreign currency continuously pays its holder interestat the foreign risk-free rate. Thus, BMS applies with� = rf , where rf is the foreign risk-free rate (in % peryear).

For example, the European call-option value is

cBSMt = N (d1)Ste�rf(T�t) �Xe�r(T�t)N (d2) ,

where

d1 =ln�Ste

�rf(T�t)=X�+�r + �2=2

�(T � t)

�pT � t

=ln (St=X) +

�r � rf + �2=2

�(T � t)

�pT � t

, and

d2 = d1 � �pT � t.

85


Example: Consider a European call on the British poundwith the following parameters: St = 1:4 (US$ per Britishpound), X = 1:5 (US$ per British pound), � = 50%

(per year), T � t = 200 days, r = 8% (per year), andrf = 12% (per year). Use a �ve-period binomial tree,and price the European call option. Compute the valueof the associated put option.

The parameters of the binomial tree are

u = e��t = 1:1180, d =1

u= 0:847452,

and

p� =e(r�rf)�t � d

u� d= 0:445579.

Figure 16.1 gives the �ve-period binomial tree. Risk-neutral evaluation is then used to value the call option.The result is cBSMt = 0:1519 (US$ per British pound).Given the call-option value, compute the put-option value.

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5.4 Options on Futures Contracts

Unlike options on stocks, options on futures contracts aresettled in cash.

Let t, T1, and T2 � T1 be the current date, the option�smaturity date, and the futures�delivery date, respectively.The futures�delivery date needs not be equal to the op-tion�s maturity date. Suppose that the futures contractis active at the current date t.

If a call option on a futures contract is exercised at t, thecall-option holder acquires a long position in the under-lying futures contract plus a cash amount equal to

m��ft� �X

�,

where m is the futures�size, t the current date, ft thefutures price at t, ft� the last settlement price, and Xthe option�s strike price.

87


Example: Consider a futures call option on copper witha strike price X = $0:70 (per pound of copper). Onefutures contract is on 25,000 pounds of copper. Thefutures price for delivery in one month is ft = $0:81

(per pound of copper), and the last settlement price isft� = $0:80 (per pound of copper). If the call-optionholder exercises his right, he is given a long position onthe futures contract and a cash amount of

25; 000� (0:80� 0:70) = $2; 500.If desired, the position in the futures contract can beclosed out immediately, which would leave the investorwith the following cash amount:

m��ft� �X

�+m�

�ft � ft�

�= m� (ft �X)= 25; 000� (0:81� 0:70)= $2; 750.

The futures price for delivery at T is related to its under-lying asset price:

ft = Ste(c�y)(T�t),88


where St is the price of the futures�underlying asset att, c the cost of carry, and y the convenience yield. Thecost of carry measures the storage cost plus the interestrequired to �nance the asset less the income earned on theasset. For example, for 1- a non-dividend paying stock,set c = r and y = 0, 2- a stock index, set c = r� � andy = 0, 3- a foreign currency, set c = r � rf and y = 0,and 4- a commodity held for consumption, the generalformula applies.

For an investment asset underlying the futures contract,one has

ft = Ste(r��)(T�t),

which is equivalent to

St = fte�(r��)(T�t).

Substituting this expression for St in the BSM formulagives Black�s formula, which turns out to be independentof �. Indeed, the holding in�ows and out�ows are alreadyincluded in the futures price.

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The call-option value is

cBt = N (d1) fte�r(T�t) �Xe�r(T�t)N (d2) ,

where

d1 =ln�fte

�r(T�t)=X�+�r + �2=2

�(T � t)

�pT � t

=ln (ft=X) + �

2 (T � t) =2�pT � t

, and

d2 = d1 � �pT � t.

Black�s formula is equivalent to the BSM formula withSt replaced by ft, and � by r, as if a futures option werean option on an underlying asset that continuously paysdividends at the domestic risk-free rate.

Example: Discuss the option on the stock index futureson page 546.

European as well as American futures options can be eval-uated through a binomial tree, adjusted as follows:

p� =1� du� d

.

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Read the chapter, do the examples, and exercises no 1�5on page 564. This part is to be prepared for the midtermexam, but should not be handed in as part of Assign-ment 2.

Do exercises no 6 and 8. The question dealing withBarone-Adesi and Waley�s formula is not mandatory.

Exercise: The sensitivity parameters in the BS modelare known in closed form (see Chapter 2). Give these pa-rameters in closed form in the BSM model, and explainyour approach. Give the associated formulas for optionson stock indices, foreign currencies, and futures. In thiscontext, single- as well as multiple-asset sensitivity analy-sis work, modulo some adjustments.

91


6 Pricing Corporate Securities

Topics Covered:

1. Common Stock as a Call Option on the Firm

2. Senior and Junior Debts

3. Callable Bonds

4. Convertible Bonds

5. Warrants

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6.1 Common Stock as a Call Option

Consider a model for a company with a simple capitalstructure, consisting of one common stock and a pure-discount bond.

The �rm value Vt at the current time t veri�es

Vt = St +Bt,

where St and Bt are the stock price and bond value attime t, respectively. This equation holds under the con-dition that the company is active, that is,

Vt � Bt.

Since the corporate bond bears some credit risk, the PVevaluation principle cannot be used. Risk-neutral pricingis a viable alternative. Here, the stock and the corporatebond are interpreted as option contracts on the �rm.

The following result, established by Merton, is a clue topricing corporate securities. The common stock can be

93


seen as a European call option on the �rm with an exer-cise price and expiry date equal to the bond�s face valueand maturity, respectively. Indeed, if the �rm value atmaturity exceeds the bond face value, that is,

VT � FV,

then

ST = VT � FV � 0.

Conversely, if

VT < FV,

then the �rm will go bankrupt, the bondholder will takecontrol of the �rm, and the stock will end up worthless,that is,

ST = 0, and BT = VT .

In sum, the stock price at maturity can be expressed as

ST = max (0; VT � FV) .94


This results in interpreting the stock as a call option onthe �rm.


The current stock price is then

St = cBSt (Vt; FV; T � t; �V ; r) ,

where cBSt is the Black and Scholes formula for a Euro-pean call option, and �V is the volatility associated tothe �rm, but not to the stock.

Exercise: Single-asset sensitivity analysis applies in thiscontext. Specify the impact on the stock price of a smallchange in the �rm value, time to maturity, �rm volatility,and risk-free rate.

In all cases, the corporate-bond value at maturity is

BT = VT � ST= VT �max (0; VT � FV)= FV�max (0; FV� VT ) .

95


Two useful interpretations result from the above equa-tions. Holding the corporate bond is equivalent to

1. holding the entire company and selling a Europeancall option to the stockholder to buy the companyfor the bond�s face value,

2. holding a riskless bond and selling a put option tothe stockholder to sell the company for the bond�sface value.


The put-call parity gives

Bt = Vt � cBSt (Vt; FV; T � t; �V ; r)= FV� e�r(T�t) � pBSt (Vt; FV; T � t; �V ; r) ,

showing that the corporate bond is worth less than its as-sociated riskless bond. The di¤erence in values representsthe credit worthiness of the issuing company. The higher

96


the put-option value, the lower is the credit worthiness ofthe issuing company.

The common stock and the corporate bond are inter-preted as derivative contracts whose values, St and Bt,depend on the �rm value Vt and its volatility �V . Thereare several methods used to approximate Vt and �V ,some of them from St and �S.

Exercise: Portfolio-sensitivity analysis applies in this con-text. Specify the impact on the corporate-bond value ofa small change in the �rm value, time to maturity, �rmvolatility, and risk-free rate.

Moody�s KMV o¤ers a default-prediction model that isbased on option theory and its applications for corporatesecurities.

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6.2 Senior and Subordinated Debts

Debt contracts di¤er by their securitization levels. For ex-ample, senior debts are reimbursed �rst (if possible), andsubordinated debts are reimbursed next. Subordinateddebts are also called junior debts.

Consider a company with a simple capital structure thatconsists of one common stock, a pure-discount seniorbond with a face value of FVs, and a pure-discount juniorbond with a face value of FVj, both maturing at the sametime T . Let Bst and B

jt be the senior- and junior-bond

values at time t, respectively. Three scenarios can happenat maturity:8>>>>>>>>>><>>>>>>>>>>:

VT < FVs !

ST = 0, BsT = VT , and BjT = 0

FVs � VT < FVs + FVj !ST = 0, BsT = FV

s, and BjT = VT � FVs

VT � FVs + FVj !ST = VT �

�FVs + FVj

�, BsT = FV

s, and BjT = FVj

.

98


Whatever happens at maturity, these equations can besummerized as

ST = max�0, VT �

�FVs + FVj

��,

for the stock,

BsT = FVs �max (0, FVs � VT ) ,

for the senior debt, and

BjT = max (0, VT � FV

s)�max�0, VT �

�FVs + FVj

��,

for the junior debt.

In sum, the common stock can be interpreted as a Euro-pean call option on the �rm with a strike price of FVs+FVj; the senior bond as portfolio of a riskless bond witha face value of FVs and a signed European put optionwith a strike price of FVs; and the junior bond as a port-folio of a European call option with a strike price of FVs,and signed European call option with a strike price ofFVs+FVj. The riskless bond and all option contractsexpire at the corporate bond�s maturity.

[Please explain Figures 17.3 and 17.4 on page 571.]

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6.3 Callable Bonds

A callable bond can be redeemed by the issuer beforematurity for a known call price. A callable can be seen asa portfolio of a straight bond that contains an embeddedcall option at the discretion of the issuer.

Corporate bonds are typically callable, while T-bonds areseldom callable.

Some callable bonds can be redeemed at any time, andothers at speci�ed dates before maturity. However, theinvestor is often protected against the call feature over acertain protection period, for example, the �rst �ve yearsof the bond�s life.

Unlike vanilla call options, call options embedded in bondsare not traded alone, but rather live within their associ-ated bonds.

100


The value of a callable bond is always lower than the valueof its associated straight bond, for it is less desirable.Thus, the value of the embedded call option is

Ct = Bstraightt �Bt,

where Ct is the value at time t of the embedded calloption, Bstraightt and Bt are the values at time t of thestraight bond and the callable bond, respectively. Here,capital letters are used since the bond with its embeddedcall option can be interpreted as an American derivativecontract.

Clearly, the bond issuer is better to call the bond earlywhen interest rates are low, and �nance its activities ata lower cost. The optimal exercise policy for the bondissuer is as follows: redeem the bond before maturity if,and only if, the interest rate is su¢ ciently low.


101


Pricing callable bonds is complex, and can be achievedthrough a dynamic program as for American vanilla op-tions. For simplicity, consider a T-bond that can be re-deemed at each coupon date.

Assume that the callable bond value is known at a givenfuture date tn+1.This is not really a limitation since thebond value is known at maturity, that is, BT = 1 + c,where c is the last coupon. Risk-neutral evaluation gives

Bhn(r) = E�"�n+1�n

Bn+1�rtn+1

�j rtn = r

#,

where frg is the interest-rate process, �n+1=�n is thediscount factor over [tn, tn+1], Bhn the bond holdingvalue at the current decision date tn, and Bn+1 the(overall) bond-value function at the next decision datetn+1. Though highly complex, this expectation can bee¢ ciently computed under several dynamics for the in-terest rate, which are consistent with the no-arbitrageprinciple. See the paper by Ben Ameur et al published inJournal of Economic, Dynamics, and Control.

102


The payo¤ at time tm from the issuer to the investor is(cn + c, if the issuer callsBhn(r) + c otherwise

,

where cn is the call price at time tn, which is known atthe start.

The issuer must redeem the bond at tn if, and only if,the holding value exceeds the exercise bene�t, that is,

Bhn(r) > cn,

which is equivalent to

r < r�n,

where r = rtn is the current interest-rate level at timetn, and r�n is a threshold at time tn that identi�es theoptimal redemption policy.

The overall value function of the callable bond at timetm is

Bn(r) = min�cn, Bhn

�+ c.

103


Finally, the embedded call option can be computed usingthe equation

Ct = Bstraightt �Bt.

Achieving these steps backward in time from maturity tothe start provides the bond value function at each decisiondate, and identi�es its optimal redemption policy.

This methodology can be adjusted a bit to accommodatecorporate callable bonds."

Please discuss the callable bond issued bythe Swiss Confederation.

#

6.4 Convertible Bonds

A convertible bond can be converted into shares of theissuing �rm at the discretion of the bond holder.

104


The conversion ratio is the number of shares received bythe investor upon conversion, indicated here by N .

The conversion ratio N is often �xed at the start, suchthat

N � S0 < B0,

to preclude immediate conversion.

For simplicity, consider a European convertible discountbond with maturity T , face value FV, which can be con-verted into shares at maturity.

At maturity, the �rm will default, or the bond will pay itsholder the larger between its face value and its conversionvalue.

In case of default, that is,

VT < FV,

one has

BT = VT , and ST = 0,105


where VT is the �rm value, BT the bond value, and STthe stock price at maturity. Otherwise, the �rm valuever�es

VT � FV,

and the bondholder will decide whether to convert thebond or not. The bond value is

BT = max (FV, N � ST )= FV+max (0, N � ST � FV) .

Please notice that the known amount FV does not repre-sent a riskless straight bond, but a straight bond issuedby the �rm. Indeed, this payment is uncertain, and willonly be received under the condition that

VT � FV.

All in all, the convertible bond can be interpreted as aportfolio of an otherwise identical corporate bond and acall option on N shares of stock with an exercise priceequal to the bond�s face value.

106


Proposal: A possible presentation could focus on couponcorporate bonds, which include both call and conversionfeatures. This presentation would include callable andconvertible bonds with �exible call prices and conversionfactors.

6.5 Warrants

A warrant is similar to a call option except that it iswritten by the underlying company, and typically has along maturity.

If a warrant is exercised, the underlying company issuesa new share for delivery. A dilution e¤ect takes place,and stockholders lose some value. Exercising the warrantmakes sense only when the resulting share price exceedsthe exercise price.

107


Consider a �rm with a simple capital structure, that is,M shares of stock and N European warrants. The �rmvalue and the stock price just before maturity are relatedas follows:

VT� =M � ST�.

At maturity, they are related as follows:

VT =M � ST +N �max (0, ST �X) .

If the warrant expires out of the money, the �rm value andthe stock price move along continuous paths; otherwise,the �rm value remains continuous, while the stock pricejumps downward at maturity. Thus, under exercise atmaturity, one has

VT� =M � ST�= VT

=M � ST +N � wT=M � ST +N � (ST �X) ,

108


which can be summarized as

ST =M � ST� +N �X

M +N.

The warrant value at maturity is

wT = max (0, ST �X)

= max�0,M � ST� +N �X

M +N�X

�= max

�0,

M

M +NST� �

M

N +MX

�=

M

M +N�max

�0, ST� �X

�=

1

1 + NM

�max�0, ST� �X

�.

By the law of one price, the warrant value and its asso-ciated call-option value are related in the same way:

wt =1

1 + NM

� cBSt�St +

N

Mwt, X, �S, T � t, r

�,

where �S is the volatility of the stock log-returns, includ-ing the warrant. This is a formula for the warrant value

109


as a function of the warrant value, which can be solvednumerically.

Proposal: A possible presentation could focus on the re-lationship between option theory and investment projects,known as real options. This presentation would includeoptions to explore, develop, extend, postpone, switch,and shut down a real project.


Read the chapter, and do exercises no 1, 2, 3, 5, 6, 7,and 8 on pages 576 and 577.

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7 The Long-Term T-Bond Futures

Contract (Revisited)

7.1 Futures Contracts

A forward contract commits two parties, one to buy andthe other to sell an underlying asset at a known futuredelivery date (maturity) for an agreed-upon delivery pricealso known as the forward price. No payment is ex-changed up front. Forward contracts are traded over thecounter; therefore, forward-market participants are sub-ject to both market and credit risk.

If prices rise, the long party records a gain since he com-mitted to buying the underlying asset at a lower price,and the short party records a loss. The opposite hap-pens when prices fall. The cumulative loss recorded by

111


one party at maturity can be so great that he can fail toful�ll his obligations.

A futures contract acts as a forward contract except thatit is traded on an exchange, and thus is subject to amargin system and a daily settlement of gains and losses.Unlike a forward contract, which is settled at maturity, afutures contract is settled at the end of each trading day,thereby keeping the loss and counterparty risk at a verylow level.

For each futures contract, the exchange de�nes a set ofterms and conditions related to the contract�s size, quo-tation unit, minimum price �uctuation, grade, tradinghours, delivery terms, daily price limits, and delivery pro-cedures.

First of all, when a futures contract is issued, each partyis invited to open a margin account with his broker, andto deposit an initial margin. The broker is invited to do

112


the same with his clearing �rm, which in turn, does thesame with the clearinghouse.

Next, the trading process continues until the day�s end.The settlement price is then registered, based on the lastfutures price(s). A new settlement price is revealed at theend of each trading day.

Finally, the futures contract is marked to the market atthe end of each trading day. The contract is closed andreopened at the new settlement price, and the value ofboth parties is consequently reset to zero. The di¤erencebetween the new and the last settlement prices, if posi-tive, is subtracted from the seller�s margin account, andadded to the buyer�s margin account. If the di¤erence inprices is negative, then the opposite is done. This is thedaily settlement system.

When a margin account falls below a certain level, knownas the maintenance margin, the investor receives a margin

113


call to bring the margin balance back up to the initialmargin level.

The margin system and daily settlement drastically re-duce counterparty risk, since losses, which are monitoredevery day, remain limited.

To avoid physical delivery, positions on futures contractsare usually closed before the delivery month, and thus set-tled in cash. Liquidity makes it possible to leave futuresmarkets at any time before maturity.

7.2 The T-Bond Futures Contract: The

Margin and the Daily Settlement Sys-

tems

The T-bond futures contract, traded on the CBOT, callsfor the delivery of $100,000 T-bonds with a minimum re-maining life of 15 years at the �rst delivery date, within

114


a delivery month. Thus, at the �rst delivery date, a de-liverable T-bond has at least 15 years to maturity or toits earliest call date. De�ne the reference T-bond un-derlying this futures contract as a hypothetical T-bondwith a maturity of 20 years and a coupon rate of 6%(per year). Delivery months are March, June, Septem-ber, and December. The initial margin is $2,500, andthe maintenance margin is $2,000.

This futures contract is one of the most widely traded inthe world, mainly because its ability to hedge long-terminterest-rate risk.

In the following Table, we consider a short position onthe CBOT T-bond futures contract, which was taken on08/01 at the quoted futures price of 97 � 27. The fu-tures price was thus at $97 + 27=32 per $100 of prin-cipal, which corresponded to $97; 843:75 per $100; 000of principal amount. The seller deposited $2; 500 intohis margin account. The futures price fell to $97; 406:25at the end of the �rst trading day. This was the new

115


settlement price, the old one being $97; 843:75. Theseller recorded a gain, and the buyer recorded a loss. Theabsolute di¤erence between the two settlement prices,that is, $437:50, was subtracted from the buyer�s mar-gin account, and added to the seller�s margin account.The balance of the seller�s margin account then rose to$2; 937:50. The futures price rose to $97; 781:25 atthe end of the second trading day. This was the newsettlement price, the old one having been $97; 406:25.The seller recorded a loss, and the buyer a gain. Theabsolute di¤erence between the two settlement prices,that is, $375:50, was subtracted from the seller�s mar-gin account, and added to the buyer�s margin account.The balance of the seller�s margin account was then at$2; 562:50. . . On 08/08, the balance of the seller�s mar-gin account fell to $1250 just after the contract wasmarked to the market, which is lower than the main-tenance margin of $2000. A margin call of $1250 wasissued the same day, and honoured the next trading day,on 08/11. On 08/18, the last trading day, the settle-ment price fell from $100; 781:25 to $100; 500:00, andthe seller then recorded a gain of $281:25. This traderthen closed his position by inversion, left the market, andwithdrew the remaining balance of $3; 031:25.

116


7.3 Embedded Options

The hypothetical T-bond is typically not traded on themarket, and thereby cannot be delivered. Thus, the shorttrader is given the right to deliver alternative long-termT-bonds. This is the choosing option, which is also calledthe quality option.

To make the delivery fair for both parties, the price re-ceived by the short trader is adjusted according to thequality of the T-bond delivered. This adjustment is madevia a set of conversion factors, which are de�ned by theCBOT as the prices of the eligible T-bonds at the �rst de-livery date under the assumption that interest rates for allmaturities equal 6% per year, and that interest is com-pounded semi-annually. The T-bond, which is actuallyselected for delivery, is known as the cheapest to deliver.

The short trader is given the right to deliver the under-lying T-bond within a delivery month. This is the timing

117


option. The delivery process takes place according to spe-cial features, that is, the delivery sequence and the end-of-month delivery rule. The delivery sequence consistsof three consecutive business days: the position day, thenotice day, and the delivery day. During the position day,the short trader can declare his intention to deliver until8:00 p.m., while the CBOT closes at 2:00 p.m. (Cen-tral Standard Time). This six-hours option is known asthe wild card play. On the notice day, the short traderhas until 5:00 p.m. to state which T-bond will be actu-ally delivered. The delivery then takes place before 10:00a.m. on the delivery day, against a payment based on thesettlement price on the position day. Finally, during thelast seven business days before maturity, trading on theT-bond futures contract stop while delivery, based on thelast settlement price, remains possible according to thedelivery sequence. This is the end-of-month option.

118


7.4 Evaluation

So far, papers in the literature have considered essentiallythe quality option under a �at term-structure of interestrates. The futures contract is then equivalent to a forwardcontract.

Let r be the level of interest rate over the contract�s life[0, T ], and let (c, M) be the deliverable T-bond withmaturity M and coupon rate c. The theoretical futuresprice g�0 and the cheapest to deliver (c

�, M�) are ob-tained by solving the following equation:

max(c;M)

f(c;M)g0 � pT (c:M; r) = 0,

where f(c;M) and pT (c:M; r) are the conversion factorand the fair value of the T-bond (c, M) at maturity,respectively. The conversion factor of the T-bond (c;M),de�ned as

f(c;M) = pT (c;M; 6%) ,

119


veri�es

f(c;M) :

8><>:> 1, if c > 6%= 1, if c = 6%< 1, if c < 6%

.

Thus, the conversion factor f(c;M) adjusts the futuresprice g0 upward if a high-quality T-bond (c, M) is de-livered, and downward if a low-quality T-bond (c, M) isdelivered. No adjustment is made if the reference T-bond(6%, 20) is delivered. The conversion factors make de-livery somewhat fair for both counterparties, but they arenot perfect.

Exercise: Show that the cheapest to deliver bears an ex-treme coupon. For simplicity, assume that the contract�sexpiry date is a coupon date. Identify the cheapest to de-liver as a function of the interest rate r. The last questionis a bonus question, and is not required for the secondmidterm exam.

Ben-Abdallah, Ben-Ameur, and Breton (2009) (GERADreport, and Journal of Banking and Finance) provided a

120


full representation of this futures contract, with all itsembedded options. The most important results are:

1. The quality option is the most valuable embeddedoption (around 2% of par for viable interest-rate lev-els);

2. The timing option is the second most valuable em-bedded option (around 0.2% of par for viable interest-rate levels);

3. The end-of-month delivery rule and the wild cardplay are less important (around 2 and 0.2 basis pointsof par and less, respectively).

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8 Sample of Exams

8.1 First Midterm Exam

Exercise 1 (20%): Consider a three-period binomialtree for a non-dividend-paying stock with the followingparameters: S0 = 100 (in dollars), u = 1:25, d = 0:8,r = 7% (per period), and

p� =1 + r � du� d

.

1. Show that this market model is arbitrage free.

2. Draw the three-period binomial tree for the stock.

3. Evaluate the European put option on this stock witha strike price X = 100 (in dollars) and a maturityT = t3.

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4. Evaluate the associated American put option, andidentify its optimal exercise policy.

5. Evaluate the early-exercise premium associated tothis put option.

6. Consider an Asian put option characterized by thefollowing exercise value:

P etn = max�0, X � Stn

�= max

�0, X �

St0 + � � �+ Stnn+ 1

�,

for n = 0, 1, 2, 3 .

Answer again questions no 3, 4, and 5.

Exercise 2 (20%): Consider two European call optionswith the same parameters except for their strike prices.A bull spread with calls consists of a long position onthe call option c1 with the lower strike price X1, and

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a short position on the call option c2 with the higherexercise price X2. The current time is indicated by t,and the maturity by T . The bull spread�s pro�t functionis de�ned as:

� = vT � vt= c1T � c1t � (c2T � c2t) ,

where vt and vT are the values of the bull spread at timet and T , c1t and c1T are the values of c1 at time t andT , and c2t and c2T are the values of c2 at time t and T ,respectively. The following Table provides the sensitivityparameters of the two call options.

c1 c2Strike price 90 100Value 16:33 10:30Delta 0:7860 0:6151Gamma 0:0138 0:0181Theta �11:2054 �12:2607Vega 20:4619 26:8416Rho 30:7085 25:2515

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The current stock price St is at $100, the volatility � at30% (per year), the remaining time to maturity T � t at180 days, and the risk-free rate r at 8% (per year).

1. Plot the curve of the bull spread�s pro�t, as a func-tion of the stock price at maturity ST .

2. Explain why this strategy is called a bull spread.

3. Suppose that the current stock price rises by $0.1.What is the impact on the bull spread�s value? Whatis the impact on the bull spread�s pro�t? Use Delta,then use Delta and Gamma.

4. Is this bull spread subject to time decay? Explain.

5. Suppose that the current price will rise by $0.1, andthat the volatility will drop by 1% over the next two

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business days. What is the impact on the currentbull spread�s value? What is the impact on the bullspread�s pro�t?

Exercise 3 (20%): Consider an arbitrage-free marketmodel for a non-dividend-paying stock and a savings ac-count that grows at the continuous risk-free rate r (in% per year). Consider a call and a put option on thisstock with an exercise price X and the remaining time tomaturity T � t. Lower-case letters are used to indicateEuropean-option values, and upper-case letters are usedto indicate American-option values. The �rst �ve ques-tions deal with the American call option, and the last �vequestions deal with the American put option.

1. Prove that

ct � St �Xe�r(T�t), for all parameters.

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2. Provide an intuitive explanation for the followingproperty:

Cht � ct, for all parameters,

and give an example where

V ht > vt, for all parameters.

3. Provide an intuitive explanation for the followingproperty:

Ct � max (0, St �X) , for all parameters.

4. Prove that an American call option on a non-dividend-paying stock cannot be exercised optimally beforematurity.

5. Use a graph, and explain why an American call op-tion can be exercised optimally before maturity whenthe underlying stock pays dividends before maturity.

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6. Use the same graph, and plot the exercise- and holding-value functions of an American put option.

7. Use another graph, and plot the value function ofthis American put option.

8. Provide an intuitive explanation why an Americanput option can always be exercised optimally beforematurity.

9. Characterize the optimal exercise policy of an Amer-ican put option at time t 2 [0, T ], as a functionof a threshold a�t , which is itself a function of theoption�s parameters.

10. Provide an intuitive explanation for the behaviour ofa�t over time, from the start to maturity.

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Exercise 4 (20%): Consider a �rm with a very simplecapital structure, that is, a share of stock and a purediscount bond with maturity T and face value FV. The�rm�s value process is indicated by Vt, the bond�s valueprocess by Bt, and the stock price by St, for t 2 [0, T ].The continuously compounded risk-free rate is indicatedby r (in % per year). Examine the �gures, and brie�yexplain each one.

Exercise 5 (20%): Brie�y comment on the businessarticle.

8.2 Second Midterm Exam

Exercise 1 (20%): Consider a three-period binomialtree for a non-dividend-paying stock with the following

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parameters: S0 = $100, u = 1:25, d = 0:8, r = 7%

(per period), and

p� =1 + r � du� d

.

All option contracts considered here are of European style,have an exercise price of X = $100, and expire at timet3 = T .

1. Draw the three-period binomial tree for the stock.

2. Evaluate the asset-or-nothing call option.

3. Evaluate the lookback call option.

4. Evaluate the up-and-in call option with a barrier ofb = $150.

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Exercise 2 (20%): Merton extended the Black and Sc-holes model to a stock that continuously pays dividendsat a �xed rate � (in % per year). The present value ofall dividends paid over a time window is proportional tothe initial stock price, dividend rate, and time period.This model, known as the Black, Scholes, and Mertonmodel, allows one to evaluate European vanilla optionsin a closed form by adjusting downward the stock priceas follows:

S�0 = S0e��T ,

and using the Black and Scholes formula. For example,the European call-option value on a dividend-paying stockis

c0 = S�0N (d1)�Xe�rTN (d2) ,

where

d1 =ln�S�0=X

�+�r + �2=2

�T

�pT

and

d2 = d1 � �pT .

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1. Show that the adjusted stock price S�0 is nothing elsethan the stock price S0 net of all dividends paid overthe option�s life.

2. Explain why a foreign currency can be interpretedas a �nancial asset that continuously pays dividends.Give Black, Scholes, and Merton�s formula for a Eu-ropean call option on a foreign currency.

3. Give the de�nition of a European call option on afutures contract on a �nancial asset.

4. Explain why a call option on a futures contract on a�nancial asset can be evaluated using Black, Scholes,and Merton�s formula. This result was established byBlack.

5. *** The futures price under Black�s model is as-sumed to follow a lognormal process, as it is the case

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for the stock in Black and Scholes�model. Explainwhy Black�s formula is totally unacceptable for a calloption on the CBOT long-term T-bond futures con-tract, though frequently used. Use the convergenceprinciple at the futures contract�s maturity and itsunderlying T-bonds�maturities.

Exercise 3 (20%): Consider an instalment option inthe Black and Scholes model. Let t1; : : : ; tN be the se-quence of decision dates, and let �1; : : : ; �N�1 2 (0, T )be a sequence of instalments. The up-front payment isindicated by �0.

1. De�ne a European instalment option.

2. List the pros of instalment options as hedging tools.

3. Show that an instalment call option with only oneinstalment �1 to be paid at t1 2 (0, T ) can beinterpreted as a compound call option.

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4. Brie�y explain how compound options can be usedto analyse portfolios of corporate bonds.

Exercise 4 (30%): Consider a �rm with a simple capitalstructure, that is, a common stock and a discount bondwith principal amount P and maturity T . Let Vt, Bt,and St be the �rm�s value, the bond�s (present) value,and the stock price at time t, respectively. The �rm playsthe role of the underlying asset in the Black and Scholesmodel, with a known initial value V0 and volatility �V .

1. Interpret the stock and the bond as derivative con-tracts on the �rm.

2. Interpret the expression

P (VT � BT ) ,

where P is the physical probability measure. Whathappens if the risk-neutral probability measure P �

were used.134


3. Prove that

Vt � Bt, for t 2 [0, T ] .

When can the company go bankrupt?

4. Consider now a �rm with a simple capital structure,that is, a common stock, a short-term discount bondwith principal amount P1 and maturity t1, and along-term discount bond with principal amount Pand maturity T . Explain how the stock and the cor-porate debt can be evaluated using compound op-tions, as functions of the �rm�s value V0 and volatil-ity �V . Now, Bt refers to the present value of thebond portfolio at time t. Explain how and when thecompany can go bankrupt, and interpret the expres-sions

�1 = P�Vt1 � Bt1

�and

�T = P�VT � BT j Vt1 > Bt1

�.

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5. This is a numerical investigation that reports thestock price, the present value of the debt, and theterm-structure of default probabilities, as functionsof the �rm�s value V0 and volatility �V . The para-meters are as follows: � = 4% (per year), r = 4%

(per year), t1 = 1 year, P1 = $50, t5 = T = 5

years, and PT = $100. The parameter � is the(instantaneous) rate of return on the �rm under thephysical probability measure, and r is the rate of re-turn on the �rm under the risk-neutral probabilitymeasure.

(a) For each table, comment on the quality of thecorporate debt.

(b) Table 1�Table 3 contain several zeros. Brie�yexplain.

(c) *** How does the default probability �1 behaveas a function of the volatility �V ? Explain.

(d) In Table 1, for �V = 20%, one has �1 = 90:3%and �5 = 13:4%. Interpret this result.

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Table 1: S0, B0, and Default Prob. for V0 = $100

�Vt 0:1 0:2 0:3 0:4t0 S0 0:01599 1:17348 4:63836 9:61668t0 B0 99:98401 98:8265 95:36164 90:38332t1 �1 0:99609 0:90344 0:78278 0:69246t2 �2 0 0 0 0t3 �3 0 0 0 0t4 �4 0 0 0 0t5 �T 0:00956 0:1342 0:26961 0:37824


�Vt 0:1 0:2 0:3 0:4t0 S0 20:57029 24:75861 31:47386 39:17126t0 B0 129:42971 125:24139 118:52614 110:82874t1 �1 0:081431 0:233911 0:284145 0:304564t2 �2 0 0 0 0t3 �3 0 0 0 0t4 �4 0 0 0 0t5 �T 0:002756 0:084375 0:206394 0:316672

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�Vt 0:1 0:2 0:3 0:4t0 S0 70:08769 70:67994 74:29304 80:54610t0 B0 129:91231 129:32016 125:70696 119:45398t1 �1 0:00002 0:01521 0:06292 0:10908t2 �2 0 0 0 0t3 �3 0 0 0 0t4 �4 0 0 0 0t5 �T 0:00005 0:03499 0:14066 0:25391

Exercise 5 (15%): This exercise deals with the CBOTlong-term T-bond futures contract.

1. De�ne the options embedded in this futures contract.

2. Is the cheapest to deliver known in advance? Brie�yexplain.

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3. Ignore the margin system and the timing option, andassume a �at term-structure of interest rates. The(fair) value of the T-bond (c, M) at time t is indi-cated by pt (c;M; r), where r is the level of interestrates.

(a) Is the cheapest to deliver known in advance?Brie�y explain.

(b) Suppose that only the reference T-bond is avail-able for delivery. What is the futures� price atthe start?

(c) *** Suppose that only T-bonds (c1, M1) and(c2, M2) are available for delivery. Give a crite-rion that identi�es the cheapest to deliver, anddetermines the (fair) futures�price at the start.

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