Damien Lamberton, Bernard Lapeyre, Nicolas Rabeau, Francois Mantion Introduction to Stochastic...

~.

r

I.I

Introduction to

Stochastic CalculusApplied to Finance

Damien LoolbertonL'Universite ffSfarne la Vallee

France

and

Bernard LapeyreL'Ecole Nationale des Ponts et Chaussees

France

Translated by

Nicolas RabeauCentre for Quantitative Finance

Imperial College, London. and

Merrill Lynch Int. Ltd., London

and '

Francois MantionCentrefor Quantitative Finance

Imperial CollegeLondon

CHAPMAN & HALUCRCBoca Raton London New York Washington, D.C.

L-G~jj(5~3

L3(;/3

'9-96

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IntroductionOptionsArbitrage and put/call parityBlack-Scholes model and its extensionsContents of the bookAcknowledgements

1 Discrete-time models1.1 Discrete-time formalism1.2 Martingales and arbitrage opportunities1.3 Complete markets and option pricing1.4 Problem: Cox, Ross and Rubinstein model

2 Optimal stopping problem and American options2.1 Stopping time2.2 The Snell envelope2.3 Decomposition of supermartingales2.4 Snell envelope and Markov chains2.5 Application io American options2.6 Exercises

viiVll

Vlll

IX

X

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1148

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17171821222325

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First edition 1996First CRC reprint 2000

© 1996 by Chapman & Hall

No claim to original U.S. Government worksInternational Standard Book Number 0-412-71800-6

Printed in the United States of America 2 3 4 5 6 7 8 9 0Printed on acid-free paper

3 Brownian motion and stochastic differential equations3.1 General comments on continuous-time processes3.2 Brownian motion3.3 Continuous-time martingales3.4 Stochastic integral and Itocalculus3.5 Stochastic differential equations3.6 Exercises

29293132354956

vi

4 The Black-Scholes model4.1 Description of the model4.2 Change of probability. Representation of martingales4.3 Pricing and hedging options in the Black-Scholes model4.4 American options in the Black-Scholes model4.5 Exercises

5 Option pricing and partial differential equations5.1 European option pricing and diffusions5.2 Solving parabolic equations numerically5.3 American options5.4 Exercises

6 Interest rate models6.1 ModeIling principles6.2 Some classical models6.3 Exercises

7 Asset models with jumps7.1 Poisson process7.2 Dynamics of the risky asset7.3 Pricing and hedging options7.4 Exercises

8 Simulation and algorithms for financial models8.1 Simulation and financial models8.2 Some useful algorithms8.3 Exercises

AppendixAl Normal random variablesA2 Conditional expectationA3 Separation of convex sets

References

Index

Contents

636365677277

9595

103110118

121121127136

141141143150159

161161 ,168170

173'173,174178

179

183

Introduction

The objective of this book is to give an introduction to the probabilistic techniquesrequired to understand the most widely used financial models. In the last fewyears, financial quantitative analysts have used more sophisticated mathematicalconcepts, such as martingales or stochastic integration, in order to describe thebehaviour of markets or to derive computing methods.

In fact, the appearance of probability theory in financial modeIling is not recent.At the beginning of this century, Bachelier (1900), in trying to build up a "Theoryof Speculation' , discovered what is now called Brownian motion. From 1973, thepublications by Black and Scholes (1973) and Merton (1973) on option pricing andhedging gave a new dimension to the use of probability theory in finance. Sincethen, as the option markets have evolved, Black-Scholes and Merton results havedeveloped to become clearer, more general and mathematicaIly more rigorous. Thetheory seems to be advanced enough to attempt to make it accessible to students.

Options

Our presentation concentrates on options, because they have been the main motivation in the construction of the theory and stilI are the most spectacular example ofthe relevance of applying stochastic calculus to finance. An option gives its holderthe right, but not the obligation, to buy or seIl a certain amount of a financial asset,by a certain date, for a certain strike price.

The writer of the option needs to specify:

• the type of option: the option to buy is caIled a call while the option to seIl is aput;

• the underlying asset: typicaIly, it can be a stock, a bond, a currency and so on.

viii Introduction Introduction ix

• the amount of an underlying asset to be purchased or sold;

• the expiration date: if the option can be exercised at any time before maturity,it is called an American option but, if it can only be exercised at maturity, it iscalled a European option;

• the exercise price which is the price at which the transaction is done if theoption is exercised.

The price of the option is the premium. When the option is traded on an organisedmarket, the premium is quoted by the market. Otherwise, the problem is to pricethe option. Also, even if the option is traded on an-organised market, it can beinteresting to detect some possible abnormalities in the market.

Let us examine the case of a European call option on a stock, whose price attime t is denoted by St. Let us call T the expiration date and K the exerciseprice. Obviously, if K is greater than ST, the holder of the option has no interestwhatsoever in exercising the option. But, if ST > K, the holder makes a profit ofST - K by exercising the option, i.e. buying the stock for K and selling it backon the market at ST. Therefore, the value of the call at maturity is given by

(ST - K)+ = max (ST - K,O).

If the option is exercised, the writer must be able to deliver a stock at price K.It means that he or she must generate an amount (ST - K)+ at maturity. At thetime of writing the option, which will be considered as the origin of time, Sr isunknown and therefore two questions have to be asked: .

1. How much should the buyer pay for the option? In other words, how should weprice at time t = 0 an asset worth (ST - K)+ at time T? That is the problem,of pricing the option.

2. How should the writer, who earns the premium initially, generate an amount(ST - K)+ at time T? That is the problem of hedging the option.

Arbitrage and put/call parity

We can only answer the two previous questions if we make a few necessaryassumptions. The basic one, which is commonly accepted in every model, is theabsence of arbitrage opportunity in liquid financial markets, i.e. there is no risklessprofit available in the market. We will translate thatinto mathematical.terms in thefirst chapter. At this point, we will only show how we can derive formulae relatingEuropean put and call prices. Both the put and the call which have maturity T andexercise price K are contingent on the same underlying asset which is worth St attime t. We shall assume that it is possible to borrow or invest money at a constantrate r.

Let us denote by Ct and P; respectively the prices of the call and the put at timet. Because of the absence of arbitrage opportunity, the following equation called

put/call parity is true for all t < T

Ct - Pt = St - K e-r(T-t).

To understand the notion of arbitrage, let us show how we could make a risklessprofit if, for instance,

c, .; Pt > S, - K e-r(T-t).

At time t, we purchase a share of stock and a put, and sell a call. The net value ofthe operation is

Ct - Pt - St.

If this amount is positive, we invest it at rate r until time T, whereas if it is negativewe borrow it at the same rate. At time T, two outcomes are possible:

• ST > K: the call is exercised, we deliver the stock, receive 'the amount K andclear the cash account to end up with a wealth K + er(T -t) (Ct - P, - St) .> O.

• ST ::; K: we exercise the put and clear our bank account as before to finishwith the wealth K + er(T-t)(ct - Pt - St) > O.

In both cases, we locked in a positive profit without making any initial endowment:this is an example of an arbitrage strategy.

There are many similar examples in the book by Cox and Rubinstein (1985).We will not review all these formulae, but we shall characterise mathematicallythe notion of a financial market without arbitrage opportunity.

Black-Scholes model and its extensions

Even though no-arbitrage arguments lead to many interesting equations, they arenot sufficient in themselves for deriving pricing formulae. To achieve this, weneed to model stock prices more precisely. Black and Scholes were the first tosuggest a model whereby we can derive an explicit price for a European call on a'stock that pays no dividend. According to their model, the writer of the option canhedge himself perfectly, and actually the call premium is the amount of moneyneeded at time 0 to replicate exactly the payoff (ST - K)+ by following theirdynamic hedging strategy until maturity. Moreover, the formula depends on onlyone non-directly observable parameter, the so-called volatility.

It is by expressing the profit and loss resulting from a certain trading strategyas a stochastic integral that we can use stochastic calculus and, particularly, Itoformula, to obtain closed form results. In the last few years, many extensions ofthe Black-Scholes methods have been considered. From a thorough study of theBlack-Scholesmodel, we will attempt to give to the reader the means to understandthose extensions. r

x Introduction Introduction Xl

Contents of the book

The first two chapters are devoted to the study of discrete time models. Thelink between the mathematical concept of martingale and the economic notionof arbitrage is brought to light. Also, the definition of complete markets andthe pricing of options in these markets are given. We have decided to adopt theformalism of Harrison and Pliska (1981) and most of their results are stated in thefirst chapter, taking the Cox, Ross and Rubinstein model as an example.

The second chapter deals with American options. Thanks to the theory of.optimal stopping in a discrete time set-up, which uses quite elementary methods,we introduce the reader to all the ideas that will be developed in continuous timein subsequent chapters.

Chapter 3 is an introduction to the main results in stochastic calculus that we willuse in Chapter 4 to study the Black-Scholes model. As far as European options areconcerned, this model leads to explicit formulae. But, in order to analyse Americanoptions or to perform computations within more sophisticated models, we neednumerical methods based on the connection between option pricing and partialdifferential equations. These questions are addressed in Chapter 5.

Chapter 6 is a relatively quick introduction to the main interest rate models andChapter 7 looks at the problems of option pricing and hedging when the price ofthe underlying asset follows a simple jump process.

In these latter cases,' perfect hedging is no longer possible and we must definea criterion to achieve optimal hedging. These models are rather less optimisticthan the Black-Scholes model and seem to be closer to reality. However, theirmathematical treatment is still a matter of research, in the framework of so-calledincomplete markets.

Finally, in order to help the student to gain a practical understanding, we haveincluded a chapter dealing with the simulation of financial models and the use ofcomputers in the pricing and hedging of options. Also, a few exercises and longerquestions are listed at the end of each chapter.

This book is only an introduction -to a field that has already benefited fromconsiderable research. Bibliographical notes are given in some chapters to helpthe reader to find complementary information. We would also like to warn thereader that some important questions in financial mathematics are not tackled.Amongst them are the problems of optimisation and the questions of equilibriumfor which the reader might like to consult the book by D. Duffie (1988).·

A good level in probability theory is assumed to read this book: The reader isreferred to Dudley (1989) and Williams (1991) for prerequisites. Ho~ever, somebasic results are also proved in the Appendix.

Acknowledgements

This book is based on the lecture notes taught at l'Ecole Nationale des Pontset Chaussees since 1988. The-organisation of this lecture series would not have

",i

t

been possible without the encouragement ofN. Bouleau. Thanks to his dynamism,CERMA (Applied Mathematics.Institute of ENPC) started working on financialmodelling as early as 1987, sponsored by Banque Indosuez and subsequently byBanque Intemationale de Placement.

Since then, we have benefited from many stimulating discussions with G. Pagesand other academics at CERMA, particularly O. Chateau and G. Caplain. A fewpeople kindly jead the earlier draft of our book and helped us with their remarks.Amongst them are S. Cohen, O. Faure, C. Philoche, M. Picque and X. Zhang.Finally, we thank our colleagues at the university and at INRIA for their adviceand their motivating comments: N. El Karoui, T. Jeulin, J.E Le Gall and D. Talay.

1

Discrete-time models

The objective of this chapter is to present the main ideas related to option theorywithin the very simple mathematical framework of discrete-time models. Essentially, we are exposing the first part of the paper by Harrison and Pliska (1981).Cox, Ross and Rubinstein's model is detailed at the end of the chapter in the formof a problem with its solution.

1.1 Discrete-time formalism

1.1.1 Assets

A discrete-time financial model is built on a finite probability space (0, F, P)equipped with a filtration, i.e. an increasing sequence of o-algebras included inF: Fo, F1 , •.. , F N. Fn can be seen as the information available at time nandis sometimes called the a-algebra of events up to time n. The horizon N willoften correspond to the maturity of the options. From now on, we will assumethat Fo = {0,O}, FN = F = P(n) and Vw E 0, P ({w}) > O. The marketconsists in (d + 1) fiflanci;l assets, whose prices at time n are given by thenon- negative random variables S~, S~, ... ,S~, measurable with respect to_:fn(investors know past arid present prices but obviously not the future ones). nievector Sn = (S~, S~, .... , S~) is the vector of prices at time n. The asset indexedby 0 is the riskless asset and we have sg = 1. If the return of the riskless assetover one period is constant and equal to r, we will obtain S~ ~ (1 + rt.'Thecoefficient /3n = 1/ S~ is interpreted as the discount factor (from time n to time0): if an amountAn is invested.in.the riskless a~et at time 0, then one dollarwillbe available at time n. The assets indexed by i = 1 ... d are called risky assets.----- ..

1.1.2 Strategies

Atrading strategy is defined as a stochastic Rrocess (i.e. a§e.q~e in the discrete- .(:(. 0 1 d)) . d+1 i

c!!.~e)rP = rPn' ~n"'" ~n O~n~N In lR where rPn denotes the number of

;=0

Its discounted value is

Remark 1.1.1 The equality if>n,Sn = if>n+l.Sn is obviously equivalent to

if>n+l,(Sn+l'- Sn) = if>n+l.Sn+l - if>n,Sn,

The value ofthe portfolio at time n is the scalar product

d

Vn(~) = »;s: = Lif>~S~.

3

n

Vn(if» = Vo(if» + L if>j . !::J.Sj,j=1

where !::J.Sj is the vector Sj - Sj-l = {JjSj - {Jj- 1Sj-l.

Proof. The equivalence between (i) and (ii) results from Remark 1.1.1. Theequivalence between (i) and (iii) follows from the fact that if>n,Sn = if>n+l,Sn ifand only if «s; = if>n+l.Sn. 0

This proposition shows that, if an investor follows a self-financing strategy, thediscounted value of his portfolio, hence its value, is completely defined by the~itial wealth and the strategy (if>~, ... , if>~) O:::;n:::;N (this is only justified because

!::J.SJ = 0). More precisely, we can prove the following proposition.

Proposition 1.1.3 For any predictable process (( if>~, . . . , if>~))O<n<N and for

any Fo-measurable variable Yo, there exists a unique predictablepr~ce~s (if>~) O<n'<N

such that the strategy ¢) =' (if>o, if>1, ... , if>d) is self-financing and its initial value-is Yo. " ,

Proof. The self-financing condition implies

Vn(if» if>~+if>~S~+"'+if>~S~~

Discrete-time formalism

(iii) For any n E {l, ... , N},~

~ (1 -1 . d -d)Vo + L.J if>j!::J.Sj + ... + if>j!::J.Sj .j=1

which defines if>~. We just have to check that if>0 is predictable, but this is obviousITw~"~Ifeequation

Discrete-time models2

Vn(if» = {In (if>n,Sn) =«:s:with'{Jn = 1/S~ and s: = (1, (JnS;, ... , (JnS~) is the v~tor of disco~ntedprices.

A strategy is called self-financing if the following equation is satisfied for allnE {O,I, ... ,N-I}

if>n,Sn = if>n+l' S;".

The interpretation is the following: at time ~, once the new prices S~~, arequoted, the investor readjusts his positions from if>n to if>n+l without bringing Q!

consuming any wealth.~~..:..

shares of asset i held in the portfolio at time n. if> is predictable, i.e.

{

if>b is Fo-measurableViE{O,I, ... ,d}

and, for n ~ 1: if>~ is F n_ 1-measurable.

This assumption means that the positions in the portfolio at time n (if>~, if>~, ... , if>~)

,are decided with respect to the information available at time (n -1) and kept untiltime n when new quotations are available.

or toVn+l(if» - Vn(if» = if>n+dSn+l -Sn).

At time n +'1, the portfolio is worth if>n+l,Sn+l a~d «: ,Sn+l - if>n+l,Sn isthe net gain caused by the price changes between times nand n + I-:--Hence;-tI1eprofit or loss realised by following a self-financing strategy is only due to the pricemoves.

The following proposition makes this clear in tenns of discounted prices.

Proposition 1.1.2 The following are equivalent

(i)' The strategy if> is self-financing.

(ii): For any n E {l, ... , N},n

Vn(if» = Vo(if» + L if>j . !::J.Sj,j=1

where 6.Sj is the vector Sj - Sj-l.

o

II' '

1.1.3 Admissible strategies and arbitrage

We did not make any assumption on the sign of the quantities if>~. If if>~ -: 0, wehave borrowed the amount 1if>~1 in the riskless asset. If if>~ < °for i ~ 1, we saythat we are short a number if>~ of asset i. Short-selling and borrowing is allowedbut the value of our portfolio must be' positive at all times.

Definition 1.1.4 A'~trategy if> is admissible if it is self-financing and !jVn( if» ~ °foranyn E {O,I, ... ,N}.

4 Discrete-time models Martingales and arbitrage opportunities 5

The following proposition is a very useful characterisation of martingales.

Proposition 1.2.4 An ~daptedsequence ofreal random variables (M~) i~ a ~ar-tingale ifand only iffor any predictable sequence (Hn), we have .-,

E (t. n; 6.u;) = O.

Proof. If (Mn) is a ~artingale, the sequence (~n) defined by Xo = 0 ~~d,for n ;::: 1, X n = bn=1 Hn6.Mn for any predictable process (Hn) is also amartingale, by Proposition 1.2.3. Hence, E(XN) = E(Xo) = O. Conversely, wenotice that if j E {I, ... , N}, we can associate the sequence (H n ) defined byH n =0 for n # j + 1 and Hj+l = lA, for any Frmeasurable A. Clearly, (Hn )

is predictable and E (2::=1 Hn6.Mn) = 0 becomes

E(iA (Mj+l - M j)) ~O.

Therefore E (!'v!j+ 11 F j ) = u; 0

Definition 1.2.2 An adapted sequence (Hn)05,n5,N of random variables is predictable if, for all n ~ 1, n; is Fn~1 measurable.

Proposition 1.2;3 Let (Mn)05,n5,N be a martingale and (Hn)O<n<N a predictable sequence with respect to the filtration (Fn)O<n<N' Den-ote 6.Mn =M n - Mn-1. The sequence (Xn)05,n5,N defined by --

X o HoMo

X n = HoMo + H 1.6M1 + ... + Hn6.Mn for n ~ 1

is a martingale with respect to (Fn)05,n5,N'

(Xn) is sometimes called the martingale transform of (Mn) by (Hn). A consequence of this proposition and Proposition 1.1.2 is that if the discounted prices ofthe assets are martingales, the expected value of the wealth generated by followinga self-financing strategy is equal to the initial wealth.Proof. Clearly, (Xn ) is an adapted sequence. Moreover, for n > 0

E (Xn+l - XnlFn)

, E (Hn+lUv/n+l - Mn)IFn)

= Hn+lE (Mn+1 - MnlFn) since Hn+l is Fn-measurable.;.'

= O.

The investor must be able to pay back his debts (in riskless or risky asset) at anytime..

The notion of arbitrage (possibility of riskless profit) can be formalised asfollows:

Definition 1.1.5 An arbitrage strategy is an admissible strategy with zero initialvalue and non-zero final value.

Most models exclude any arbitrage opportunity and the objective of the nextsection is to characterise these models with the notion of martingale.

1.2 Martingales and arbitrage opportunities

In order to analyse the connections between martingales and arbitrage, we mustfirst define a martingale on a fj.nite probability space. The conditional expectationplays a central role in this definition and the reader can refer to the Appendix fora quick review of its properties.

1.2.1 Martingales and martingale transforms

In this section, we consider a finite probability space (D, F, P), with F = P(D)and Vw E D, P ({w}) > 0, equipped with a filtration (Fnh~n::;N (withoutnecessarily assuming that F N = F, nor Fo = {0, D}). A sequence '(Xn)O::;n::;Nofrandom variables is adapted to the filtration!f for any n, X!, is Fn-measurable.

Definition 1.2.1 An adapted sequence (Mn)O::;n~N of real random variables is:

o a martingale ifE (Mn+1IFn) = Mnfor all n :S N - 1;

o asupermartingale ifE (Mn+lIFn) :S Mnforallri:S N -1;

o asubmartingale ifE (Mn+lIFn) ~ Mnforalln:S N-1.

These definitions can be extended to the multidimensional case: for instance, asequence (Mn)O<n<N of JRd-valued random variables is a martingale if each

.component is.a real-valued.martingale, .In a financial context, saying that the price (S~)O::;n::;N of the asset i ~a

martingale implies that, at each time n, the best estimate (in the least-squaresense) of S~+1 is given by S~.

The following properties are easily derived from the previous definition and standas a good exercise to get used to the concept of conditional expectation.

1. (Mn)O::;n5,N is a martingale if and only if

E (Mn+jIFn) == u; Vj ~ 0

2. If (Mnk:~o is a martingale, thus for any n: E (Mn) = E (MQ) .

3. The sum of two martingales is a martingale.

4. Obviously, similar properties can be shown for supennartingales and submartingales.

Hence

'A E (Xn+1IFn) = E (XnIFn) = X n.That shows that (Xn ) is amartingale .. o

where A is the event {Gn(¢) < o}. Because ¢ is predictable and A is Fn

measurable, 'ljJ is also predictable. Moreover

1.2.2 Viable financial markets

Let us get back to the discrete-time models introduced in the first section.

Definition 1.2.5 The market is viable if there is no arbitrage opp0'!!!:Eity.

Lemma'1.2.6 jf the market is viable, any~ble process (¢i , ... ,¢d) satis-fies -- -" -

ti.!!J1!l{I·Proof. Let us assume thatGN(¢) E r. First, ifGn(¢) 2: 0 for all n E {O,... , N}the market is obviously not viable. Second, if the Gn (¢) are not all non-negative,

we define n = sup {kiP (G k (¢) < 0) > o}. It follows from the definition ofn

that

n ~ N - 1, P (Gn(¢) < 0) > 0 and 'v'm > n Gm(¢) 2: O.

We can now introduce a new process 'ljJ

7Martingales and arbitrage opportunities

(bI) To any admissible process (¢;" ... , ¢~) we associate the process defined by

If.the.strategy is admissi~le and its initial value is zero, then E* (VN (¢)) = 0,

with VN (¢) 2:0. Hence VN (¢) = 0 since P* ({w}) > 0, for all wEn.(b) The proof of the converse implication is more tricky. Let us call r the convexcone of strictly positive random variables. The market is viable if and only if forany admissible strategy ¢: Vo (¢) = 0 => VN (¢) ¢ r.

That is the cumulative discounted gain realised by following the self-financingstrategy¢;', ... , ¢~. According to Proposition 1.1.3, there exists a (unique) process(¢~) su~h that the strategy ((¢~, ¢;" ... , ¢~)) is self-financing with zero initialvalue. C!n (.¢) is the discounted value of this strategy at time n and because themarket IS vIabl~, th~ fact tha! this value is positive at any time, i.e Gn (¢ ) 2: 0 forn = 1, ... , N, implies ~hat G N (¢) = O. The following lemma shows that even ifwe do not assume that Gn(¢) are no~-negative, we still have GN(¢) ¢ r. '(b2) The set V of random varia~es GN(¢), with ¢ predictable process in IRd

; isclearly a vector subspace of IR(where IRo is the set of real random variablesdefined on n). According to Lemma 1.2.6, the subspace V does not intersect rThere~ore.it?oesnoti~tersecttheconvexcompacts·etK = {X E fI Ew X(w):::I} WhICh IS included m r. As a result of the convex sets separation theorem (seethe Appendix), there exists (oX (w)tEo such that:

1. vx E K, L oX(w)X(w) > O.

I,


if j ~ n

if j > n

if j ~ nif j > nu

{

0,

Gj('ljJ)= lA(Gj(¢)-G1;l(¢))

thus, Gj

('ljJ) 2: 0 for all j E {O,... , N} and GN ('ljJ) > 0 on A. That contradictsthe assumption of market viability and completes the proof of the lemma. 0

6

Theorem 1.2.7 The market is viable if and onl)' if there exists-a-probabil.ity .. ....------.----"- - -----_/ -

~:_YYiY.q.le~~he:-diSt;ounteJ£pric.:s._(fl!S~e..~~_P* -

martingales.~ (a) Let us assume that there exists a probability P* equivalent to P underwhich discounted prices are martingales. Then, for any self-financing strategy

(¢n), (1.1.2) impliesn

Vn(¢) = Vo(¢) + L ¢j.f:::.Sj.j=l

Thus by Proposition 1.2.3, (Vn (¢)) is a P" - martingale. Therefore VN (¢) and

Vo (¢) have the same expectation under P*: ,

E* (VN (¢)) =E* (Vo (¢)).

t Recall that two probability measures P I and P2 are equivalent if and only if for any eventA. PI (A) = °¢} P2 (A) = 0, Here, P" equivalent to P means that. for any wEn.

p·({w}»o,

w

2. For any predictable ¢

w

From Property 1: we deduce that oX(w) > 0 for all wEn, so that the probabilityP* defined by '. !, .'

P* ({w}) = oX(w)Ew' EO oX(w')

is equivalent to P. ,Moreover, if we denote by E* the expectation under measure P*, Property 2.

means that, for any predictable process (¢n) in IRd,

It follows that for all i E {I, ... ,d} and any predictable sequence (¢~) in JR, wehave

E* (t¢;6.8;) = O.J=I

Therefore, according to Proposition 1.2.4, we conclude that the discounted prices(8~), ... , (8~) are P* martingales. 0

8 Discrete-time models Complete markets and option pricing 9

Theore.~ 1.3.4 A viable m~rket is complete if and only if there exists a uniqueprobability measure P equivalent to P under which discounted prices are mar-tingales. .

The probability P* will appear to be the computing tool whereby we can deriveclosed-form pricing formulae and hedging strategies.

Proof. (a) Let us assume that the market is viable and complete. Then, anynon-neg~tlve, FN-~~asurable random variable h can be written as h = VN (¢),where ¢ IS an admissible strategy that replicates the contingent claim h. Since ¢is self-financing, we know that

the last equality coming from the fact that ·Fo = {0, O}. Therefore

P** ({w}) == (1 + X(w)) P*({w})211Xlloo .

and, since h is arbitrary, PI = P 2 on the whole a-algebra F N assumed to beequal to F.

(b) Let us assume that the market is viable and incomplete. Then, there existsa random variable h ~ 0 which is not attainable. We call V the set of randomvariables of the form

(1.1)

N

Uo+ L ¢n.6.8n,n=I

)

h _ N

SO = VN (¢) = Vo (¢) + L ¢j.6.8j.N j=I .

Thus, if PI and P: are two probability measures under which discounted prices

are martingales, (Vn (¢)) is a martingale under both PI and P2

It followsO<n<N .that, for i = 1 or i = 2 --

":here Uo is Fo-measlJrable and ((¢~'''''¢~))o<n<N is an JRd-valued pre-dictable process, - -

It follows fr,?m Proposition 1.1.3 and Remark 1.3.2 that the variable hiSf}y doesnot belong to V. Hence, V is a strict subset of the set of all random variables on(0, F). Therefore, if P* is a probability equivalent to P under which discountedprices are riJ.~ngales and if we define the following scalar product on the setof random ~ariables (X, Y) t-+ E: (XY), we notice that there exists a non-zerorandom vanable X orthogonal to V. We also write

':

* Or more generally a contingent claim.

1.3 Complete markets and option pricing

1.3.1 Complete markets

We shall define a European option" of inaturity N by giving its payoff h 2: 0,FN-measurable. For instance, a call on the underlying SI with strike price Kwill be defined by setting: h = (S}y - K) +. A put on the same underlying asset

with the same strike price K will be defined by h = (K - S}y)+. In those twoexamples, which are actually the two most important in practice, h is a function ofS N only. There are some options dependent on the whole path of the underlyingasset, i.e. h is a function of SO, S1>' .. , SN. That is the case of the so-called Asianoptions where the strike price is equal to the average of the stock prices observedduring a certain period of time before maturity..

Definition 1.3.1 The contingent claim defined by h is attainable if there exists anadmissible strategy worthli at time N.

Remark 1.3.2 In a viable financial market, we just need to find a self-financingstrategy worth h at maturity to say that h is attainable. Indeed, if ¢ is a selffinancing strategy and if P* is a probability measure equivalent to P under which

discounted prices are martingales, then (Vn (¢)) is also a P*-martingale, being

a martingale transform. Hence, for n E {O,... , N} Vn(¢) = E* (VN(¢)IFn).

Clearly, if VN (¢) ~ 0 (in particular if VN (¢) = h), the strategy ¢ is admissible.

, Definition 1.3.3 The market is complete if every contingent claim is attainable.

To assume that a financial market is complete is a rather restrictive assumption thatdoes not have such a clear economic justification as the no-arbitrage assumption.The interest of complete markets is that it allows us to derive a simple theoryof contingent claim pricing and hedging. The Cox-Ross-Rubinstein model, thatwe shall study in the next section, is a very simple example" of complete marketmodelling. The following theorem gives a precise characterisation of complete,viable financial markets.

The sequence (Vn ) is a P* -rnartingale, and consequentlyO::;n::;N

for any predictable process ((¢~, ... ,¢~))O::;n::;N· It follows from Proposition

1.2.4 that (Sn)O::;n::;N is a P" -m.artingale. 0

J.3.2 Pricing and hedging contingent claims in complete markets

The market is assumed to be viable and complete and we denote by P* the uniqueprobability measure under which the discounted ~rices of finan~ial assets aremartingales. Let h be an .1'N-measurable, non-negative random variable and ¢ bean admissible strategy replicating the contingent claim hence defined, i.e.

11Complete markets and option pricing

model will show how we can compute the option price and the hedging strategyin practice.

and

J.3.3 Introduction to American options

Since an American option can be exercised at any time between 0 and N, we shalldefine it as a positive sequence (Zn) adapted to (.1'n), where Zn is the immediateprofit made by exercising the option at time n. In the case of an American optionon the stock SI with strike price K, Zn = (S~ - K) +; in the case of the put,

Zn = (K - S~) -t-' In order to define the price of the option associated with(Zn)O::;n::;N, we shall think in terms of a backward induction starting at time N.Indeed, the value of the option at maturity is obviously equal to UN = ZN. Atwhat price should we sell the option at time N ., I? If the holder exercises straightaway he will earn Z N -1, or he might exercise at time N in which case the writermust be ready to pay the amount ZN. Therefore, at time N - 1, the writer hasto earn the maximum between Z N -1' and the amount necessary at time N - 1 togenerate ZN at time N. In other words, the writer wants the maximum betweenZ N -1 and the value at time N - 1 of an admissible strategy paying off Z N at time

N, i.e. S~_IE* (ZNI.1'N- I)' with ZN = ZN/S~. As we see, it makes sense to

price the option at time N - 1 as

UN- I = max (ZN-I, S~_IE* ( ZNI ~N-l)) .

By induction, we defin~ the American option price for n :: 1, ... , N by

_ Un- I = ~ax ( Z~-I' S~_IE* ( ~~ l.1'n- 1) ) .

If we assume that the interest rate over one period is constant and equal to r,

S~ = (1 + r)"

Un- I = max ( Zn-I, 1~ r E* (Un l.1'n-l )) ,

let ii; = Un / S~ be the discounted price of the American option.

Proposition 1.3.6 The "sequence (Un) is a P* -supermartingale. It is the. O<n<N

smallest P* -supermartingale that domin~te~ the sequence (Zn) '.:. O<n<N

We should note that, as opposed to the European case, the discounted price ofthe American option is generally not a martingale underP".Proof. From the equality .,

Un- I = max ( Zn-I, E* (Un l.1'n-l ) ) ,

it follows that (Un)O::;n::;N is a supermartingale dominating (Zn)O::;n::;N. Let us


'-'that is Vo(¢) = E* (h/S~) and more generally

Vn (¢ ) = S~E* (; l.1'n) , n = 0,1, ... ,N., . N

At any time, the value of an admissible strategy replicating h is completely determined by h. It seems quite natura} to call Vn (¢) the price of the option: that is thewealth needed at time n to replicate h at time N by following the strategy ¢. If, at

time 0, an investor sells the option for'

E.* (:~),he can follow a replicating strategy ¢ in order to generate an amount h at time N.In other words, the investor is perfectly hedged.

Remark 1.3.5 It is important to notice that the computation of the option priceonly requires the knowledge of P* and not P. We could have just considered ameasurable space (fl,.1') equipped with the filtration (.1'n). In other words, wewould only define the set of all possible states and the evolution of the informationover time. As soon as the probability space and the filtration are specified, wedo not need to find the true probability of the possible events (say, by statisticalmeans) in order to price the option. The analysis of the Cox-Ross-Rubinstein

10

with II XII"" = sUPwEn IX(w)l. Because E* (X) = 0, that defines a new probability measure equivalent to P and different from P*. Moreover

E** (t, ¢n.~Sn) = 0

now consider a supermartingale (Tn)O~n~N that dominates (Zn)O~n~N. ThenTN 2: lJN and if i; 2: ii; we have

Tn- l 2: E* (Tn l.1'n-l) 2: E* (Un l.1'n-1 )

1.4 Problem: Cox, Ross and Rubinstein model

The Cox-Ross-Rubinstein model is a discrete-time version of the Black-Scholesmodel. It considers only one risky asset whose price is Sn at time n, 0 ~ n ~ N, 'and a riskless asset whose return is r over one period of time. To be consistentwith the previous sections, we denote S~ = (1 + r)".

The risky asset is modelled as follows; between two consecutive periods therelative price change is either a or b, with -1 < a < b:

{Sn(l+a)

Sn+l = Sn(1 + b).

The initial stock price So is given. The set of possible states is then n = {I +a, 1 + b}N, Each N -tuple represents the successive values of the ratio Sn+dSn,n = 0,1, , N - 1. We also assume that.1'o = {0, n} and .1' = pen). Forn = 1, , N, the a-algebra .1'n is equal to a(SI"'" Sn) generated by therandom variables SI ,... ,Sn. The assumption that each singleton in n has a strictlypositive probability implies that P is defined uniquely up to equivalence. We nowintroduce the variables Tn = Sn/Sn-I, for n = 1, ,N. If (XI, ... ,XN) isone element of n, P{(XI, ... , XN)} = P(TI = Xl, ,TN = XN). As a result,knowing P is equivalent to knowing the law of the N -tuple (TI, T2 ;' ... , TN). We'also remark that for n 2: 1,.1'n = a(TI, ... ,Tn).

1. Show that the discounted price (Sn) is a martingale under P if and only ifE(Tn+II.1'n) = 1"+ r, '<In E {O,1, ... , N - I}.

The equality E(Sn+I'IFn) = s: is equivalent to E(Sn+dSnIFn) == 1, since Sn is,Fn-measurable and this last equality is actually equivalentto E(Tn+dFn) = 1 + r .

2. Deduce 'that r must belong to[c, b[ for the market to be arbitrage-free.

If the market is viable, there exists a probability P* equivalent to P, under which (Sn)is a martingale.Thus, according to Question 1. '

E*(Tn+IIFn) = 1 + r

and therefore E*(Tn+d = 1 + r . Since Tn+l is either equal to 1 + a or 1 + b withnon-zero probability, we necessarilyhave (1 + r) E]1 + a, 1 + b[.

Discrete-time models Problem: Cox, Ross and Rubinstein model13

3. Give examples of arbitrage strategies if the no-arbitrage condition derived inQuestion (2.) is not satisfied. '

Assumefor instance that r :S a. By borrowingan amount So at time0, we can purchaseone share of the risky asset. At time N, we pay the loan back and sell the riskyasset. We realised a profit equal to SN - So(1 + r)N which is always positive, sinceSN 2: So(1 + a)N. Moreover, it is strictly positive with non-zero probability. There isarbitrage opportunity. If r 2: b w,e can make a riskless profit by short-selling the riskyasset.

4. From now o~, w.e assume that r E Ja, b[ and we write p = (b - r)/(b _ a).ShowTthat (~n) IS a P-martingale if and only if the random variables TI, T

2,

. : ., N are mdependent, identically distributed (lID) and their distribution isgrven by: P(TI = 1 + a) = p = 1 - P(TI = 1 + b).Conclude that the marketis arbitrage-free and complete.

IfTi are independent and satisfy P(Ti = 1 + a) =P = 1 - P(Ti = 1 + b), we have

E(Tn+dFn) = E(Tn+d = p(1 + a) + (1 - p)(1 + b) = 1 + r

and thus, (Sn) is a P-martingale, according to Question I. '

CO,nversely, if for n = 0,1, .. : , N - I, E(Tn+ljFn) = 1 + r, we can write

p + a)E (1{Tn+1 = l+a} IF n ) + (1 + b)E (1{Tn+ 1=1+b}IFn)= 1 + r.

Then, the followingequality

E(1{Tn+1=l+a}IFn) + E (1{Tn+1=1+b}IFn) = 1,:

~mplie~ that E (l{Tn+l=l+a}jF~) = P and E (1{Tn+1=1+b}jFn) = 1.- p. Byinduction, we prove that for any xi E {I + a, 1 + b},

P (TI = Xl, ... .t; = x;,) = IIPii=l

wherePi = P if Xi = 1+ a and Pi = 1 - pif Xi = 1+ b.That shows that the variablesT, are lID under measure P and that P(Ti = 1 + a) = p.

We have shown that the very fact that (Sn) is a P-martingale uniquely deterrni~esthe distribution of the N-tuple (TI, T2 , • • • , TN) under P, hence the measure P itself.Therefore, the market is arbitrage-freeand complete.

5. We denote by C; (resp. Pn ) the value at time n, ofa European call (resp. put)on a share of stock, with strike price K and maturity N.

(a) Derive the put/call parity equation

C; - Pn = Sn - K(1 + r)-(N-n),

knowing the put/call prices in their conditional expectation form.If we deEote E* the expectation with respect to the probabilitymeasure P* underwhich (Sri) is a martingale, we have "

len - Pn = (1 + r)-(N-n)E* ((SN - K)+ - (K - SN )+IFn)= (1 +r)-(N-n)E* (SN - KIFn)= Sn-K(1+r)-(N-n),

o

12

Tn- l 2: max ( Zn-l, E* (Un l.1'n-1 )) = Un-I'

A backward induction proves the assertion that (Tn) dominates (Un).

whence

the last equality comes from the fact that (Sn) is a P" -martingale.

(b), Show that we can write en = c(n, Sn) where c is a function of K, a; b, randp.

When we write SN = S« n::n+l Ti, we get

c; = (1 + r)-(N-n)E" ((Sn ,IT t: - K) : Fn)..=n+l +

1514 Discrete-time models Problem: Cox, Ross and Rubinstein model

where, for each N, the random variables XI' areIll), belong to

{-ajVN, ajVN},

and their mean is equal to J.LN, with limN400(NJ.LN) = J.L. Show that thesequence (YN ) converges in law towards a Gaussian variable with mean J.Land variance a 2 •

Wejust need to study the con,:ergenceof the characteristic function tPYN of YN. Weobtain

Since under the probability P", the random variable n::n+l T, is independent ofF« and since S« is Fn-measurable, Proposition A.2.5 in the Appendix allows us towrite: C« = c(n, Sn), where c is the function defined by

c(n, x)(l +r)-(N-n)

= E" (x IT t: - K)i=n+l +

N-n""' (N - n).! . rl (1 _ p)N-n-i (x(I + a)i (1 + b)N-n-i - K) .L.J (N-n-J)!J! +i=O

6. Show that the replicating strategy of a call is characterised by' a quantity H n =b.(n, Sn-l) at time n, where b. wi11 be expressed in terms offunction c.

We denote H~ the number of riskless assets in the replicating portfolio. We have

H~(I + r)" + HnSn = c(n, Sn).

'Since H~ and H; are Fn_1-measurable, they are functions of Sv, . . . ,Sn-l only and,since s: is equal to Sn-l (l + a) or Sn-l (1 + b), the previous equality implies

H~(l + r)n + HnSn..:.1(I + a) = c(n, Sn-l(I +a»

andH~(I +rt + HnSn-1(I + b) = c(n, Sn--:l(I + b».

Subtracting one from the othtr, it turns out that

( )c(n,x(I+b»-c(n,x(I+a»

D. n, x = x(b _ a) :

, 7. We can now use the model to price a call or a put with maturity T on a singlestock. In order to do that, we study the asymptotic case when N convergesto 'infinity, and r = RT/N, log((l + a)j(l + r)) = -ajVN and log((l +b)j(l + r)) = ajVN. The real number R is interpreted as the instantaneousrate at al1 times between 0 and T, because eRT = limN4oo(1 + r)N. a 2 canbe seen as the limit variance, under measure P", of the variable log(SN), whenN converges to infinity. -, '

(a) Let (YN )N~l be a sequence ofrandom variables equal to

YN=xi'+xf+'''+x~

r,"

N

tPYN(U) = E(exp(iuYN» = IIE (exp(iuXf"»)i=l

= (E(exP(iuXf»)t

(1 + iUJ1.N - a2u2 / 2N + a(I/N») N.

Hence, limj, ..... oo tPYN (u) = exp (iuJ1. - a 2 u2/2), which proves the convergencein law.

(b) Give explicitly the asymptotic prices of the put and the call at time O.For a certain N, the put price at time 0 is given by

pJN) = (1 + RT/N)-N'E" (K':'" ~o IT Tn)

n=l +

= E",((l + RT/N)-NK - So exp(YN») +

with YN ' = L::=I log(Tn/(I + r». According to the assumptions, ~he variables

Xf" = log(Ti/(l + r» are valued in {-a/VN,a/VN}, and are liD underprobability P", Moreover

, 2 trlVN -trlVNE"(Xi

N) = (1 _ 2p)~ = - e - e ~.

VN etrl VN - e-trl VN VNTherefore, the sequence (YN) satisfies the conditions of Question 7.(a), with J1. =_a

2/2. If we write 'Ij;(y) = (Ke- R T- SoeY)+, we are able to write

IPJN) E" ('Ij;(YN» I '

= IE" (((1 + RT/N)-N K - So exp(YN») +

- (Ke-RT

-Soexp(yN»)+)1

< K 1(1 + RT/N)-N _ e-RTI.

Since 'Ij; is a bounded t, continuous function and because the sequence (YN) converges in law, we conclude that

I'm p'(N) = lim E" ('Ij;(YN»N~oo 0 N .....oo

t It is precisely to be able to work with a bounded function that we studied the put first.

(

2.1 Stopping time

The buyer of an American option can exercise its right at any time until maturity.The decision to exercise or not at time n will be made according to the information available at time n. In a discrete-time model built on a finite filtered space(S1, F, (Fn)O:::;n:::;N ,P), the exercise date is described by a random variable calledstopping time.

Definition 2.1.1 A random variable IJ taking values in {O,1,2, ... , N} i~'a stopping time if, for any nE {O,1", . ,N},

{IJ = n} E F~.

Remark 2.1.2 As in the previous chapter, we assume that F = P(S1) andP( {w}) > 0, Vw E S1.~ This hypothesis is nonetheless not essential: if it doesnot hold, the results presented in this chapter remain true almost surely. However,we will not assume Fo = {0, S1} and FN = F, except in Section 2.5, dedicatedto finance. .

~ ,

Remark 2.'1.3 The reader can verify, as an exercise; that IJ is a stopping time ifand only if, for any n E {O,1, ... ,Nt

{IJ :::; n} E Fn .

The purpose of this chapter is to address the pricing and hedging of Americanoptions and to establish the link between these questions and the optimal stoppingproblem. To do so, we will need to define the notion of optimal stopping time,which will enable us to model exercise strategies for American options. We willalso define the Snell envelope", which is the fundamental concept used to solve. theoptimal stopping problem. The application of these concepts to American optionswill be described in Section 2.5.

Optimal stopping problemand American options

2Discrete-time models

= _1_1+00

(Ke-RT _ Soe-u2/2+UY)+e-y2/2dy .

.j21i -00

"The integral can be expressed easily i~ termsof the cumulative normaldistribution

F, so thatlim pJN) = Ke- RTF(-d

2) - SoF(-d1) ,N-+oo

wheredl = (log(x/K) + RT + a 2/2)/a,d2 = d1 - a and

F(d) = _1_1d

e-,,2/ 2dx ..j21i -00

Thepriceofthecallfollows easilyfromput/call paritylimN-+oo C~N) = SoF(d 1 )

K e- RT F(d2 ) .

Remark 1.4.1 We note that the only non-directly observable parameter is .a ..Itsinterpretation as a variance suggests that it should be estimated by statisticalmethods. However, we shall tackle this question in Chapter 4.

Notes: We have assumed throughout this chapter that the risky assets were notoffering any dividend. Actually, Huang and Litzenbe~ger (19~8) a~p.ly the sameideas to answer the same questions when the stock IS carrying dIvId.en~s. 1?etheorem of characterisation of complete markets can also be proved WIth infiniteprobability spaces (cf. Da1ang, Morton and Willinger (1990) and ~orton (1989)).In continuous time, the problem is much more tricky (cf. Hamson and Kreps(1979), Stricker (1990) and Delbaen and Schachermayer (~994)). Th~ theory ofcomplete markets in continuous-time was developed by H~rns~n and Ph~ka .( 198.1,1983). An elementary presentation of the Cox-Ross-Rubmstem model IS given mthe book by I.e. Cox and M. Rubinstein (1985).

16

We will use this equivalent definition to generalise the concept of stopping timeto the continuous-time setting. .

Let us introduce now the concept of a 'sequence stopped at a stopping time'. Let(Xn)O<n<N be a sequence adapted to the filtration (Fn)O<n<N and let v be astopping time. The sequence stopped at time v is defined as- -

X~ (w) = Xv(w)/\n (w),

i.e., on the set {v = j} we have

1918 Optimal stopping problem and American options The Snell envelope

Proposition 2.2.1 The random variable defined by

Vo = inf {n ~ 0IUn = Zn} (2.1)

is a stopping time and the stopped sequence (U) . . l. n/\vo O<n<N IS a martmga e.Proof. Since UN = ZN, Vo is a well-defined element of {O 1 N} andhave ' , ... , . we

n

2.2 The Snell-envelope .

In this section, we consider an adapted sequence (Zn)05.n5.N, and define thesequence (Un)05.n5.N as follows:

Note that XN(w) = Xv(w)(w) (= X j on {v = j}).

Proposition 2.1.4 Let (Xn ) be an adapted sequence and v be a stopping time.The stopped sequence (X~)O<n<N is adapted. Moreover; if(Xn) is amartingale(resp. a supermartingale), then rX~) is a martingale (resp. a supermartingale).

Proof. We see that, for n ~ 1, we have

{VO = ~} = {Uo = Zo} E Fo,and for k ~ 1

{vo = k} = {Uo > Zo} n··· n {Uk-I> Zk-d n {Uk = Zd E r;To demonstrate that (U~o) is a martingale, we write as in the proof of Proposition

2.1.4:n

U:;o = Un/\vo == ti; + L ¢>j 6.Uj,j=1

where ¢>j = I{vo~j}. So that, for n E {OJ 1, ... ,N - I},

u» u» ( .n+l - n = ...¢>n+l Un+1 - Un)

= l{n+l5.vo} (Un+l - Un).

By definition, U'; = max (Zn, E (Un+lIFn)) and on the set {n + 1 ~ VO}, U; >Zn. Consequently Un = ;E (Un+lIFn) and we deduce .

Uvo uvo 1 .n+l - n = {n+l5.vo} (Un+l - E (Un+lIFn))

and taking the conditional expectation on both sides of the equality

E ((U::+ 1 --: U~o) IFn) = l{n+l5. vo}E ((Un+l - E(Un+I!Fn))IFn)because {n + 1 ~ vol E Fn (since the complement of {n + 1 < vol is {v <n}). - 0 -

Hence

E ((U:;+1 - U:;o) \Fn) = 0,which proves that U'» is a martingale. . oIn the remainder, we s?all note Tn,N the set of stopping times taking values in

{n, n + 1.'... , N}. Nonce that Tn,N is a finite set since f! is assumed to be finite.The martingale property of the sequence U'» gives the following result whichrelates the concept of Snell envelope to the optimal stopping problem.Corollary 2.2.2 The stopping time Vo satisfies

Uo = E (ZvoIFo) = sup) E (ZvIFo).. . vETo.N

If we. think ~f Zn as the total winnings of a gambler after n games, we see thatstopping ~t tlmevvo. maximises the expected gain given Fo.Proof. Since U ° IS a martingale, we have

"In ~ N:"-1.

if j ~ nif j > n.

XJ

XnX~ = {

= ZN= max (Zn, E'(Un+lIFn))

Xv/\n =x, + L ¢>j ix, - X j-1),j=1

where ¢>j = I {j~v}' Since {j ~ v} is the complement of the set {v < j} ={v ~ j - I}, the process (¢>n)O<n<N is predictable.

It is clear then that (Xv/\n)~<::<N is adapted to the filtration (Fn)O~n~N.

Furthermore, if (Xn) is a martingale, (Xv/\n) is also a martingale with respect to(Fn ) , since it is the martingale transform of (Xn ) . Similarly; we can show thatif the sequence (Xn ) is a supermartinga1e (resp. a submartingale), the stoppedsequence is stilla supermartingale (resp. a submartinga1e) using the predictabilityand the non-negativity of (¢>j )05.j 5. N. 0

The study of this sequence is motivated by our first approachof American options(Section 1.3.3 of Chapter 1). We already know, by Proposition 1.3.6 of Chapter1, that (Un)O<n<N is the smallest supermartingale that dominates the sequence(Zn)O<n<N. W;call it the Snell envelope of the sequence (Zn)O<n<N.

By definition, Ui; is greater than Zn (with equality for n =,N) a;din the case ofa strict inequality, Un = E(Un+lIFn). It suggests that, by stopping adequately thesequence (Un), it is possible to obtain a martingale, as the following propositionshows.

Therefore

But, since U" is a supermartingale,

On the other hand, if 1/ E Io,N, the stopped sequence U" is a supermartingale. So

that

21Decomposition ofsupe rmartingales

2.3 Decomposition of supermartingales

The following decomposition (commonly called 'Doob decomposition') is usedin viable complete market models to associate any supermartingale with a tradingstrategy for which consumption is allowed (see Exercise 5 for that matter).

Proposition 2.3.1 Every supermartingale (Un)05,n5,N has the unique followingdecomposition:

Un = M n - An,

where (Mn) is a martingale and (An) is a non-decreasing, predictable process,null at O.

andMn+1 - Mn = Un+1 - E (Un+11.rn) .

(Mn) and (An) are entirely determined using the previous equations and we seethat (Mn) is a martingale and that (An) is predictable and non-decreasing (because(Un) is a supermartingale). 0

Suppose then that (Un) is the Snell envelope of an adapted sequence (Zn)' Wecan then give a characterisation of the largest optimal stopping time for (Zn) usingthe non-decreasing process (An) ~f the Doob decomposition of (U~):

Proposition 2.3.2 The largest optimal stopping time for (Zn) is given by

_{N . if AN = 01/_ - inf in, An+1 =I- O} if AN =I- O.

Proof. It is straightforward to see that I/m", is a stopping time using the fact that\An)o5,n5,N is predictable. From Un = M n - An and because Aj = 0, forJ S ~, we deduce that U"fn", = M"fnu and conclude that U"fn", is a martingale. Toshow the optimality of !{n'"" it is sufficient to prove

U"fn'" =Z"fn""

N-l )

v.: - "L: 1{~",=j}Uj + 1{"fn",";N}UNj=O

N-l

L: 1{"fnu=j} max (Zj, E (Uj+d.rj)) + 1{"fnu=N}ZN,j=O

We note that

Un+1 - Un = M n+1 - M n - (An+1 - An) .

So that, conditioning both sides with respect to F.n and using the properties of M~A.. .

- (An+1 - An) = E (Un+ll.rn) - Un

Proof. It is clearly seen that the only solution for n = 0 is M o = Uo and Ao = O.Then we must have

Optimal stopping problem and American options

= E (ZlIn l.rn) ,where I/n = inf {j ~ nlUj = Zj}.Definition 2.2.4 A stopping time 1/ is called optimalfor the sequence (Zn)o<n <Nif - -

E (ZIII.ro) = sup E (ZII!.rO) ~Io,N

We can see that I/o is optimal. The following result gives a characterisation ofoptimal stopping times that shows that I/o is the smallest optimal stopping time.

Theorem 2.2.5 A stopping time v is optimal ifand only if

{ZII = UII (2.2)and (UlIl\n)05,n5,N is a martingale.

Proof. If the stopped sequence U" is a martingale, Uo = E(UIII.ro) and consequently, if (2.2) holds, Uo = E(ZIII.ro). Optimality of 1/ is then ensured byCorollary 2.2.2.

Conversely, if 1/ is optimal, we have

Uo = E (ZIII.ro) S E (UIII.ro) .

. ,(based on the supermartingale property of (U~) we get

E (UlIl\nl.ro) = E (UIII.ro) = E (E (UIII.rn)l.ro).

But we have UlIl\n ~ E (UIII.rn), therefore u-; =' E (VIII.rn), which provesthat (U::) is a martingale. 0

E (UIII.ro) = E (ZIII.ro)

and since UII ~ ZII' UII = ZII'Since E (UIII.ro) = Uo and from the following inequalities

Uo ~ E (UlIl\nl.fo) ~ E (U1I1l"0)

Remark 2.2.3 An immediate generalisation of Corollary 2.2.2 gives

u; sup E (ZIII.rn)liE/noN .

Uo > E (UNI.ro) =E (UIII.ro)

> E (ZIII.ro),

which yields the result. 0

20

22 Optimalstoppingproblem and American options Application to American options 23

We have E (Uj+IIFj) = M, - Aj+l and, on the set {vrnox = j}, A j = 0 andAj+l > 0, so Uj = M j and E (Uj+IIFj) = M j - Aj+l < Uj. It follows thatU, = max (Zj, E (Uj+IIFj)) = Zj. So that finally

Ulfuv. = Z lfuv. •

It remains to show that it is the greatest optimal stopping time. If v is a stoppingtime such that v 2: Vrnax and P (v > vrnax) > 0, then

E(Uv) = E(Mv) - E(Av) = E(Uo) - E(Av) < E(Uo)

2.4 Snell envelope and Markov chains

The aim of this section is to compute Snell envelopes in a Markovian setting. Asequence (Xn)n~O of random variables taking their values in a finite set E iscalled a Markov chain if, for any integer n 2: 1 and any elements xo, Xl,' .. ,Xn-l, X, Y of E, we have

P(Xn+l = y!Xo = Xo, ... ,Xn- l = Xn-l, X n = x) =P(Xn+l = ylXn = x) .

The chain is said to be homogeneous if the value P(x, y) = P (Xn+l = ylXn = x)does not depend on n. The matrix P = (P(x, y))(X,Y)EEXE' Indexed byE x .E,is then called the transition matrix of the chain. The matrix P has non-negative

, entries and satisfies: LYEE P(x, y) = 1 for all x E' E; itis said to be a stochastic

matrix. On a filtered probability space ( n,F, (F";)O:::;n~N,P), we can define tlie

notion of a Markov chain with respect to the filtration:

Definition 2.4.1 A sequence (Xn)O:::;n:::;N of random variables taking values inE is a homogeneousMarkov chain with respectto thefiltration (Fn)O<n<N' withtransition matrixP, if (Xn) is adapted and iffor any real-valuedfunctIon f onE, we have " ,

E (f (Xn~d IFn) = P f (Xn) ,

whereP f represents thefunction whichmapsx E E toP f(x) = LYEE P(x; y)f(y).Note that, if one interprets real-valued functions on E as matrices with a singlecolumn indexed by E, then P f is indeed the product of the two matrices P andf. It can also be easily seen that a Markov chain, as defined at the beginningof the section, is a Markov chain with respect to.its natural filtration, definedbyF n = a(Xo, ... ,Xn ) . ' . C

The following proposition is an immediate consequence of the latter definitionand the definition of a Snell envelope. '

Proposition 2.4.2 Let (Zn) be an adapted sequence defined by Zn = 'ljJ(n, X n),where (Xn) is a homogeneous Markov chain with transition matrix P, takingvalues in E, and ib is afunctionfrom N x E to JR. Then, the Snell envelope (Un)

and consequently

'<In:S N-1.= ZN= max (Zn, S~E* (Un+l/ S~+lIFn))

In Section 1.3.3 of Chapter 1, we defined the value process (Un) of an Americanoption described by the sequence (Zn), by the system

{, UN

Un

where (Mn) is a Pt-martingale and (An) is an increasing predictable process,null at O. Since the market is complete, there is a self-financing strategy ¢suchthat

Thus, the sequence CUn) 'defined by ii; = Un/S~ (discounted price of the option)is,the Snell envelope, under P* , of the sequence (Zn). We deduce from the aboveSection 2.2 that . '

ii; = sup' E* ( ZvlFn), vETn.N

2.5.1 Hedging American options

2.5 Application to American options

From now on, we will work in a viable complete market. The modelling will be

based on the filtered space (fl, F, (Fn)O:::;n:::;N ,P) and, as in Sections 1.3.1 and

1.3.3 of Chapter 1, we will denote by P* the unique probability under which thediscounted asset prices are martingales.

u(n,') = max ('ljJ(n, .), Pu(n + 1, .)) .

u(N, x) = 'ljJ(N, x) '<Ix E E

of the sequence (Zn) is given by Un = u(n, X n), where the function u is definedby

and, for n :s N - I,

u; = S~ sup E* (SZ~ IFn) .vETn.N v

From Section 2.3, we can write .

ii; = Mn - An,

o VN (¢) = SNMN,

i.e., VN (¢) = MN. For the seque?ce (Vn (¢)) is a P* -martingale, w~ have

Vn(¢) E* (VN(¢)!Fn )

E* (MNIFn )

II,

oand UV cannot be a martingale, which establishes the claim.

24 Optimal stopping problem and American options Exercises 25

and consequently

Therefore

If Cn ~ Zn for any n then the sequence (cn), which is a martingale under P",appears to be a supermartingale (under P") and an upper bound of the sequence(Zn) and consequently

Un = Vn(cP) - An,

where An = S~An. From the previous equality, it is obvious that the writer of theoption can hedge himself perfectly: once he receives the premium Uo = Vo(cP),he can generate a wealth equal to Vn(cP) at time n which is bigger than Un and afortiori Zn.

What is the optimal date to exercise the option? The date of exercise is to bechosen among all the stopping times. For the buyer of the option, there is no pointin exercising at time n when U« > Zn, because he would trade an asset worth Un(the option) for an amount Zn, (by exercising the option). Thus an optimal date T

of exercise is such that UT ~ ZT' On the other hand, there is no point in exercisingafter the time

/I"", = inf {j, A j+! i- O}

(which is equal to inf {j, Aj +! i- 0}) because, at that time, selling the optia,n

provides the holder with a wealth UJ.iruu = VJ.iruu (cP) and, following the strategycP from that time, he creates a portfolio whose value is strictly bigger than the

.option's at times /1m.. + I, /1max + 2, ... , N. Therefore we set; as a second condition,T :::; /I""" which allows us to say that eF is a martingale. As..a result, optimal datesof exercise are optimal stopping times for the sequence (Zn), under probabilityp ". To make this point clear, let us considerthe writer's point of view.If he hedgeshimself using the strategy cP as defined above and if the buyer exercises at time T

which is not optimal, then UT > ZT or AT > O. In both cases, the writer makes aprofit VT(cP) - ZT = UT + AT - Zro which is positive.

2.5.2 American options and Europeanoptions '

Proposition 2.5.1 Let Cn be the value at time n ofan American option describedby an adapted sequence (Zn)O<n<N and let Cn be the value at time n of the European option defined by the FN--measurable random variable h = Z N. Then,we have Cn ~ Cn.

Moreover, if Cn ~ Zn for any n, then

Cn =Cn \In E {O,I, ... ,N}.-

The inequality Cn ~ Cn makes sense since the American option entitles the holderto more rights than its European counterpart. c.. '.

Proof. For the discounted value (Cn ) is a supermartingale ~nder p. , we have

Cn ~ E· (CNIFn) = E· (cNIFn) = cn.

Hence c, ~ Cn.

Cn~C;' \lnE{O,I, ... ,N}.

o

Remark 2.5.2 One checks readily that if the relationships of Proposition 2.5.1did not hold, there would be some arbitrage opportunities by trading the options.

To illustrate the last proposition, let us consider the case of a market with asingle risky asset, with price Sn at time n and a constant riskless interest rate,equal to r ~ 0 on each period, so that S~ = (I + r)". Then, with notations ofProposition 2.5.1, if we take Zn = (Sn - K)+, Cn is the price at time n of aEuropean call with maturity N and strike price K on one unit of the risky assetand Cn is the price of the corresponding American call. We have

cn (I + r)-NE· ((SN - K)+IFn)

> E· (SN - K(I + r)-NIFn) ,

- -N= Sn-K(I+r) ,

using the martingale property of (Sn). Hence: ~n ~ s; - K(I + r)-(N-n) ~Sn - K, for r ~ O. As Cn .~ 0, we also have c., 2: (Sn - K)+ and by Proposition2.5.1,Cn = Cn. There is equality between the price of the European call and theprice of the corresponding American call. ,

This property does not hold for the put, nor in the case of calls on currencies ordividend paying stocks.Notes: For further discussions .on the Snell envelope and optimal stopping, onemay consult Neveu (1972), Chapter VI and Dacunha-Castelle and Duflo (1986),Chapter 5, Section 1. For the theory of optimal stopping in the continuous case,see EI Karoui (1981) and Shiryayev (1978).

2.6 Exercises

Exercise 1 Let /I be a stopping time with respect to a filtration (Fn)O<n<N.We denote by F; the set of events A such that A n {/I = n}. E Fn f'(;r anynE{O, ... ,N}. .

I. Show that F; is a sub-a-algebra of FN. F; is often called 'a-algebra of eventsdetermined prior to the stopping time /I'.

2. Show that the random variable /I is Fv-measurable.

3. Let X be a real-valued random variable. Prove the equality

N

E(XIFv) = L l{v=j}E(XIFj).j=O

26 Optimal stopping problem and American options Exercises 27

Vn(¢ ) = Vo(¢) + L ¢j.~Sj - L 1'j.j=l j=l

Vn(¢ ) = Vo(¢ ) + L¢j·~Sj - L1'j/SJ-1'j=l j=l

2. In the remainder, we assume that the market is viable and complete and wedenote by P' the unique probability under which the assets discounted pricesare martingales. Show that if the pair (¢, 1') defines a trading strategy withconsumption, then (V~(¢)) is a supermartingale under P".

3. Let (Un) be an adapted sequence such that (Un) is a supermartingale underP". Using the Doob decomposition, show that there is a trading strategy withconsumption (¢,1') such that Vn (¢ ) = Un for any n E {O,... , N}.

4. Let (Zn) be an adapted sequence. We say that a trading strategy with consumption (¢, 1') hedges the American option defined by (Zn) if Vn(¢) 2: Zn forany n E {O,1, .. '. , N}. Show that there is at least one trading strategy withconsumption that hedges (Zn), whose value is precisely the value (Un) of theAmerican option. Also, prove that any trading strategy with consumption (¢, 1')hedging (Zn) satisfies Vn(¢) 2: Un, for any n E {O,1, ... , N}. '

6. Show that the hedging strategy of the American put is determined by a quantityHn = ~(n, Sn-d of the risky asset to be held at time n, where ~ can bewritten as a function of Pam.

Exercise 5 Consumption strategies. The self-financing strategies defined inChapter 1 ruled out any consumption. Consumption strategies can be introducedin the following way: at time n, once the new prices S~, . . . ,S~ are quoted, theinvestor readjusts his positions from ¢n to ¢n+1 and selects the wealth 1'n+1 to beconsumed at time n + 1. Any endowment being excluded and the new positionsbeing decided given prices at time n, we deduce

¢n+1,Sn = ¢n,Sn - 1'n+l. (2.3)

So a trading strategy with consumption will be defined as a pair (¢,1'), where¢ is a predictable process taking values in IRd+1, representing the numbers ofassets held in the portfolio and l' = bnh<n<N is a predictable process takingvalues in IR+, representing the wealth consumed at any time. Equation (2.3) givesthe relationship between the processes ¢ and l' and replaces the self-financingcondition of Chapter 1.

1. Let ¢ be a predictable process taking values in IRd+1 and let l' be a predictable

process taking values in IR+. We set Vn(¢) '= ¢n,Sn and Vn(¢ ) = s;»:Show the equivalence between the following conditions:

(a) The pair (¢, 1') defines a trading strategy with consumption.(b) For any n E {l, .. :,N},

n

n

n

n

(c) For any n E {I, ... ,N},

E (Un) = sup E (Zv).vETn,N

Exercise 3 Show that v is optimal according to Definition 2.2.4 if and only if

E(Zv)= sup E(ZT)'TETo,N

Exercise 4 The purpose of this exercise is to study the American put in the modelof Cox-Ross-Rubinstein. Notations are those of Chapter 1.

1. Show that the price Pn, at time n, of an American put on a share with maturityN and strike price K can be written as

Pn= Pam(n, Sn)

where Pam(n, x) is defined by Pam(N, x) = (K - x)+ and, for n :::; N - 1,

(f(n+1,x))

Pam(n,x) = max (K - x)+, 1 + r , '

with f(n + 1,x) = pPam(n + 1,.:r(1 + a)) + (1 ., p)P~;.,,(n+ 1,x(1 + b))andp = (b- r)/(b - a).

2. Show that the function Pam(O,.) can be expressed as

Pam(O,x) = sup E' ((1 + r)-V(K - xVv)+) ,vETo,N

where the sequence of random 'variables (Vn)O::;n::;N is defined by: Vo 1and, for n 2: 1, Vn = TI7=1 Ui, where the U/s aresome random variables. 'Give their joint law under P".

3. From the last formula, show that the function x !-t' Pam(O,x) is convex andnon-increasing. "

4. We assume a < 0. Show that there is' a real number x' E [0, K] such that, forx:::; x', Pam (0, x) = (K - x)+ and; for x E ]x', K/(l + a)N[, Pam(O, x) >(K - x)+.

5. An agent holds the American put at time 0. For which values of the spot Sowould he rather exercise his option immediately?

and more generally

(Hint: first consider the case T = N.)

Exercise 2 Let (Un) be the Snell envelope of an adapted sequence (Zn). Withoutassuming that :Fo is trivial, show that

E (Uo) = sup E (Zv) ,vETo,N

4. Let T be a stopping time such that T 2: v, Show that:Fv C Fr.

5. Under the same hypothesis, show that if (M n ) is a martingale, we have

5. Let x be a non-negative number representing the investor's endowment and let'Y = bnh<n<N be a predictable strategy taking values in IR.+. The consumption process (1-n) is said to be budget-feasible from endowment x if there is apredictable process ¢ taking values in IR.d+ 1, such that the pair (¢, 'Y) definesa trading strategy with consumption satisfying: Vo(¢ ) = x and Vn (¢ ) 20, forany n E {O, ... , N}. Show that bn) is budget-feasible from endowment x if

and only if E$ (L::=l 'Yj / SJ-l) ~ x.

28 Optimal stopping problem and American options

3

Brownian motion andstochastic differential

•equations

The first two chapters of this book were dealing with discrete-time models. Wehad the opportunity to see the importance of the concepts of martingales, selffinancing strategy and Snell envelope. We are going to elaborate on these ideas ina continuous-time framework. In particular, we shall introduce the mathematicaltools needed to model financial assets and to price options. In continuous-time,the technical aspects are more advanced and more difficult to handle than indiscrete-time, but the main ideas are fundamentally the same.

Why do we consider continuous-time models? The primary motivation comesfrom the nature of the processes that we want to model. In practice, the pricechanges in the market are actually so frequent that a discrete-time model canbarely follow the moves. On the other hand, continuous-time models lead to moreexplicit computations, even if numerical methods are sometimes required. Indeed,the most widely used model is the continuous-time Black-Scholes model whichleads to an extremely simple formula. As we mentioned in the Introduction, theconnections between stochastic processes and finance are not recent. Bachelier(1900), in his dissertation called Theorie de la speculation, is not only amongthe first to look at the properties of Brownian motion, but he also derived optjonpricing formulae.

We will be giving a few mathematical definitions in order to understandcontinuous-time models. In particular, we shall define the Brownian motion sinceit is the core concept orthe Black-Scholes model and appears in most financialasset models. Then we shall state the concept of martingale in a continuous-timeset-up and, finally, we shall construct the stochastic integral and introduce thedifferential calculus associated with it, namely the Ito calculus. -

It is advisable that, upon first reading, the reader passes over the proofs in smallprint. as they-are very technical.

3.1 General comments on continuous-time processes

What do we exactly mean by continuous-time processes?

From now on, ,a Brownian motion is assumed to be standard if nothing else ismentione~. In that case, the distribution of X, is the following:

1 (X2)

--exp -- dxV2ii 2t'

where dx is the Lebesgue measure on IR. '

The r7-algebra associated with r is defined as

FT = {A E A, for any t ~ 0 , A n {r ~ t} E Ftl .

This r7-algebrarepresents the information available before the random time r. Onecan prove that (refer to Exercises 8, 9, 10, 11 and 14):

Proposition 3.1.6

• If S is a stopping time, S is F s measurable.

• If S is a stopping time, finite almost surely, and (Xt)t~O is a continuous,adapted process, then X s is F s measurable.

• IfSand T are two stopping times such that S ~ T P a.s., then F s eFT.

• If Sand T are two stopping times, then S 1\ T = inf(S, T) is a stopping time.In particular; if S is a stopping time and t is a deterministic time S 1\ t is astopping time.

3.2 Brownian motion

A particularly important example of stochastic process is the Brownian motion. Itwill be the core of most financial models, whether we consider stocks, currenciesor interest rates.

Definition 3.2.1 A Brownian motion is a real-valued, continuous stochastic process (Xdt~o, with independent and stationary increments. In other words:

• continuity: P a.s. the map S I--t X, (w) is continuous.

• independent increments: If S ~ t, X, - X s is independent ofF s = r7(Xu , U ~

s).• stationary increments: if'S ~ t, X, - X; and X t - s - Xo have the same

probability law.

This definition induces the distribution of the process X t , but the result is difficultto prove and the reader ought to consult the book by Gihman and Skorohod (1980)for a proof of the following theorem. '

Theorem 3.2.2 If (Xt)t>o is a Brownian motion, then X; - Xo is a normalrandom variable with mean rt and variance r7 2 t , where rand o are constant realnumbers.

Remark 3;2.3 A Brownian motion is standard if

31

E (Xi) = t.E(Xd = 0,X o = 0 P a.s.

Brownian motion

• We will only work with processes that are indexed on a finite time interval[O,T].

30 Brownian motion and stochastic differential equations

Definition 3.1.1 A continuous-time stochastic process in a space E endowed witha a- algebra E is afamily (Xt)tEIR+ of random variables defined on a probabilityspace (n, A, P) with values in a measurable space (E, £).

Remark 3.1.2

• In practice, the index t stands for the time.

• A process can also be considered as a random map: for each w in n we associatethe map from IR+ to E: t -t Xt(w), called a path of the process.

• A process can be considered as a map from IR+ x n into E. We shall alwaysconsider that this map is measurable when we endow the product set IR+ x nwith the product e-algebra B(IR+) x A and when the set E is endowed withE.

.As in discrete-time, we introduce the concept offiltrat(on.

Definition 3.1.3 Consider the probability space (n, A, P), a filtration (Ft)t>o isan increasing family ofa-algebras included in A -

The o-algebra F t represents the information available at time t. We say that aprocess (Xtk:~o is adapted to (Ftk~o, if for any t, X, is Ft-measurable.

Remark 3.1.4 From now on, we will be working with filtrations which have thefollowing property

If A E A and if P(A) =0, 'then for any t, A EFt ..

In other words F t contains all the P-null sets of A. The importance of this technicalassumption is that if X = Y P a.s. and Y is Ft-measurable then we can showthat X is also Ft-measurable.

We can build a filtration generated by a process (Xt)t>o and we write Ft =r7(Xs , S ~ t). In general, this filtration does not satisfy -the previous condition.However, if we replace Ft by :Ft which is the o-algebra generated by both Ftand N (the o-algebra generated by all the P-null sets of A), we obtain a properfiltration satisfying the desired condition. We call it the natural filtration of theprocess (Xth~o. When we talk about a filtration without mentioning anything, itis assumed that we are dealing with the natural filtration of the process that we areconsidering. Obviously, a process is adapted to its natural filtration.

As in discrete-time, the concept of stopping time will be useful. A stopping time. is a random time that depends on the underlying process in a non- anticipative

way. In other words, at a 'given time t, we know if the stopping time is smaller thant. Formally, the definition is the following: .

Definition 3.1.5 r is a stopping time with respect to the filtration (Fdt>o if r isa random variable in IR+ U {+oo}, such that for any t 2: 0 -

{r~t}EFt.

The following theorem emphasises the Gaussian property of the Brownian motion.We have just seen that for any t, X, is a normal random variable. A stronger resultis the following:

Theorem 3.2.4 If (Xdt?o is a Brownian motion and if 0 ~ tt < ... < t« then(Xt1, ... , X tn ) is a Gaussian vector.

The reader ought to consult the Appendix, page 173, to recall some properties ofGaussian vectors.Proof. Consider 0 ~ tl < ... < t«. then the random vector (Xt1, X t2 X t1, ... , X t n - X tn_1) is composed of normal, independent random variables(by Theorem 3.2.2 and by definition of the Brownian motion). Therefore, thisvector is Gaussian and so is (Xt1, ... ,Xt n ) . · 0

33

E ((Xt - X s)2 + 2Xs(Xt - Xs)IFs)

= E ((Xt - Xs)2IFs) + 2XsE (Xt - XsIFs) ,

and since (Xdt?o is a martingale E (Xt - XsIFs) = 0, whence

Continuous-time martingales

3. exp (aXt - (a 2/2)t) is an Frmartingale.

Proof. If s ~ t then X, - X, is independent of the a-algebra F s . Thus E(Xt

XsIFs) = E(Xt - X s)' Since a standard Brownian motion has an expectationequal to zero, we have E(Xt - X s) = O. Hence the first assertion is proved. Toshow the second one, we remark that

Brownian motion and stochastic differential equations32

o

E (eO'(X'-X')IFs) = E (eO'(X'-X'))

E (eO'x,-,)

E (e0'9vr=s)

exp (~a2(t - S))

That completes the proof.

E (e-X9) =/+00 e-x:r. e-:r.2/2 dx =e-X

2/2.

-00 ~On the other hand, if s < t

E (eO'X.-0'2t/2IFs) ="eO'X.-0'2t/2E (i(X'-X')IFs)

because X; is Fs-measurable. Since X, - X, is independent of Fs ' it turns out

that

If (Mt)t>o is a martingale, the property E (M; IFs) = M s, is also true if t and sare bounded .stoppingtimes. This result is actually an adaptation of Exercise 1 inChapter 2 to the continuous case and it is called the optional sampling theorem. Wewill not prove this theorem, but the reader ought to refer to Karatzas and Shreve(1988), page 19.

E ((Xt - X s)2JFs ) = E (X't-s)t - s.

The last equality is due to the fact that X, has a normal distribution with meanzero and variance t. That yields E (Xl - tlFs ) = X; - s, if s < t.

Finally, let us recall that if 9 is a standard normal random variable, we know

that

Because the Brownian motion has independent and stationary increments, it fol

lows that

We shall also need a definition of a Brownian motion with respect to a filtration(Ft ) .

Definition 3.2.5 A real-valued continuous stochastic process is an (Ft)-Brownianmotion if it satisfies: '

• For any t 2: 0, X, is Ft-measurable.

• Ifs ~ t, X t - Xs.isindependimtofthea-algebraFs.

• If s ~ t, X t - X; and X t- s - X o have the same law.

Remark 3.2.6 The first point of this definition shows that a(Xu , u ~ t) eFt.Moreover, it is easy to check that an Ft-Brownianmotion is also a Brownianmotion with respect to its natural filtration.·

3.3 Continuous-time martingales

As in discrete-time models, the concept of martingale is a crucial tool to explainthe notion of arbitrage. The following definition is an extension of the one indiscrete-time.

Definition 3.3.1 Let us consider a probability space (n, A, P) and a filtration(Fdt?o on this space. An adaptedfamily (Mt)t?o ofintegrable random variables,i.e. E(IMtI) < +00 for any tis:

• a martingale if, for any s ~ t, E (MtIFs) = M s;

• asupermartingaleif,foranys ~ t, E(MtIFs) ~ u;• a submartingale if, for any s ~ t, E (MtIFs) 2: Ms.

Remark 3.3.2 It follows from this definition that, if (Mt)t>o is a martingale, thenE(Md = E(Mo) for any t. .-

Here are some examples of martingales.

Proposition 3.3.3 If(Xt}t?o is a standard FrBrownian motion:

I. X, is an Ft-martingale.

2. Xl - t is an Fi-martingale. (

Remark 3.3.5

E ( sup IMtl2) ~ 4E(IMT I2).

°9:S:TThe proof of this theorem is the purpose of Exercise 13.

The optional sampling theorem is also very useful to compute expectationsinvolving the running maximum of a martingale. If M, is a square integrablemartingale, we can show that the second-order moment of sUPo9:S:T IMti can bebounded. This is known as the Doob inequality.

Theorem 3.3.7 (Doob inequality) If(Mt)o9:S:T is a continuous martingale, we

have

The case a < 0 is easily solved if we notice that

Ta = inf.{s 2: 0, -X. = -a},

where (-Xdt~o is an Ft-Brownian motion because it is a continuous stochasticprocess with zero mean and variance t and with stationary, independent increments.

o

Stochastic integral and Ito calculus 35

By letting a converge to 0, we show that P(Ta < +00) = 1 (which means thatthe Brownian motion reaches the level a almost surely). Also


• This result implies that if 7 is a bounded stopping time then E(MT ) = E(Mo)(apply the theorem with 71 = 0,72 = 7 and take the expectation on both sides).

• If M, is a submartingale, the same theorem is true if we replace the previousequality by .

Proposition 3.3.6 Once again, we consider (Xt)t>o an Ft-Brownian motion. Ifa is a real number, we call Ta = inf {s 2: 0, X, =~} or +00 if that set is empty.

Then, Ta is a stopping time, finite almost surely, and its distribution ischaracterised by its Laplace transform

Theorem 3.3.4 (optional sampling theorem) If (Mt)r2.o is a continuous martingale with respect to the filtration (Ft)t>o, and if 71 and 72 are two stoppingtimes such that 71 ~ 72 ~ K, where K- is a finite real number, then M T 2 isintegrable and

We shall now apply that result to study the properties of the hitting time of a pointby a Brownian motion.

Proof. We will assume that a 2: O. First, we show that Ta is a stopping time.Indeed, since X, is continuous ,.

That last set belongs to F t , and therefore the result is'proved. In the following, wewrite x 1\ y = inf(x, y).

Let us apply the sampling theorem to the martingale M; = exp (aXt - (a 2 / 2)t ).We cannot apply the theorem to Ta which is not necessarily bounded; however, ifn is a positive integer, Ta 1\ n is a bounded stopping time (see Proposition 3.1.6),and from the optional sampling theorem

E (MTa/\n) = 1.

On the one hand MTa/\n = exp (aXTa/\n - a 2 (Ta 1\ n)/2) ~ exp (aa). Onthe other hand, if Ta < +00, limn--Hoo MTa/\n = MTa and if Ta = +00,X, ~ a at any t, therefore limn-Hoo MTai\n = O. Lebesgue theorem implies thatE(I{Ta<+oo}MTJ = 1, i.e. since X Ta = a when T~< +00

E (I{Ta<+oo} exp ( - ~2 Ta ) ) =e-:u a.

,.'l3.4 Stochastic integral and Ito calculus

In a discrete-time model, if we follow a self-financing strategy ¢ = (Hn)o<n<N'the discounted value of the portfolio with initial wealth Vo is - -

n

s>YO + L Hj (5j - 5j - 1 ) .

j=l

That wealth appears to be a martingale transform under a certain probabilitymeasure such that the discounted price of the stock is a martingale. As far ascontinuous-tim~ models are concerned, integr~ls of the form JH.dS. will helpus to describe the same idea.

However, the processes modelling stock prices are normally functions of one orseveral Brownian motions. But one of the most important properties of a Brownianmotion is that, almost surely, its paths are not differentiable at any point. In otherwords, if (X t ) is a Brownian motion, it can be proved that for almost every wEn,there is not any time t in lR+ such that dXt / dt exists. As a result, we are not ableto define the integral above as

rt rtdX

io f(s)dX. = io f(s) ds' ds.

Nevertheless, we are able to define this type of integral with respect to a Brownian

36 Brownianmotion and stochasticdifferential equations Stochastic integraland Ito calculus 37

motion, and we shall call them stochastic integrals. That is the whole purpose ofthis section.

If we include sand t to the subdivision to = 0 < t1 < ... < tp = T, and if we. t

call Mn = fon HsdWs and 9n =Ftn for 0 :::; n :::; p, we want to show that Mn

is a 9n-martingale. To prove it, we notice that

as a result

E(r/>:(Xi-Xi-iY) = E(E(r/>:(Xi-:-Xi-d219i-I))

E (r/>:E((X i - Xi.-d219i~I)) 1

Since X, is a martingale, E(Xj - Xj-119i-d = O. Therefore, if i < i.E (r/>ir/>j (Xi - Xi-d (Xj - X j-1 )) = o. If j > i we get the same thing. Finally, if i = i.

(3.1)n n

L L E (r/>ir/>j(Xi - Xi-d(Xj - Xj-d)·i=l j=l

with r/>i 9i-1 -measurable. Moreover, X n = Wtn is a 9n-martingale (since (Wt)t>ois a Brownian motion). (Mn)nE[O,pj turns out to be a martingale transformof (Xn)nE[O,pj' The Proposition 1.2.3 of Chapter 1 allows us to conclude that(Mn)nE[O,pj is a martingale. The second assertion comesfrom thefact that

Also, if io< i. we have

E (r/>ir/>j(Xi - Xi-d(Xj ~ Xi-d)

E (E (r/>ir/>j (Xi - Xi-I) (Xi - X j-1 ) 19j-1))

~ = E (r/>ir/>j(Xi - Xi..:.dE (Xi - Xj-119j-dfNote that I(H)t can be written as

I(H)t = L r/>i(Wt.f\t - Wt._1f\t),l~i~p

3.4.1 Constructionofthe stochastic integral

Suppose that (Wt}t~O is a standard Ft-Brownian motion defined on a filteredprobability space (fl, A, (Ft)t>o, P). We are about to give a meaning to the

expression f~ f (s,w)dWs for;certain class of processes f (s,w) adapted to thefiltration (Ft}t~o, To start with, we shall construct this stochastic integral for aset of processes called simple processes.Throughout the text, T will be a strictlypositive, finite real number.

Definition 3.4.1 (Ht}O<t<T is called a simple process if it can be written as- - ,p

Ht(w) = ~::>t>i(w)lJt'_l,t.j(t)i=l

where 0 = to < t1 < ... < tp = T and r/>i is F t._ 1 -measurableand bounded.

Then, by definition, the stochastic integral of a simple process H is the continuousprocess (I(H)t)O~t~T defined for any t E '~k, tk+d as c

I(H)t = L r/>i(Wt• - Wt._J + r/>k+dWt - Wtk)·1~i9

which proves the continuity oft t--+ I(Hk We shall write f~ HsdWs for I(HkThe following proposition is fundamental.

Proposition 3.4.2 If(Ht)o~t~T is a simple process:

• (f~ HsdWs) is a continuousFt-martingale.09~T

• E ( ([ H~dW.)') ~ E (1,' H;d'). .

• E (i~?I[ H.dW-I') < 4E ([H;d')Proof. In order to prove this proposition, we are going to use discrete-time

processes. Indeed, to show that (J~ HsdWs) is a 'martingale, w~ just need to

check that, for any t > s,

: E (I tHudWulFs) =1;HudWu.

E ( (Xi ~ X i- i)219i-I) = E ((Wt• - Wti_l )2) = t; - ti-1. (3.2)

From (3.1) and (3.2) we conclude that

The continuity, of t -t f~HsdWs is clear if we look at the definition. The thirdassertion is just a consequence ofDoob inequality (3.3.7) applied to the continuous

martingale (f~ Hs_dWs) '. 0t~O

r r IJ\

38 Brownian motion and stochastic differential equationsrI Stochastic integral and Ito calculus 39

. (3.7)

That implies that (J(H)t}O<t<T does not depend on the approximating sequence.On the other hand, (J(H)SO~t5,T is a martingale, indeed .

A proof of this result can be found in Karatzas and Shreve (1988) (page 134,problem 2.5).

If H E Hand (Hn)n?o is a sequence of simple processes converging to H inthe previous sense, we have

Therefore, there exists a subsequence HcP(n) such that

thus, the series whose general term is I(HcP(n+l)) - I(HcP(n)) is uniformly convergent, almost surely. Consequently I(HcP(n))t converges towards a continuousfunction which will be, by definition, the map t t-+ J(H}t. Taking the limit in(3.6), we obtain

Moreover, for any t, lim n -+ + oo I(Hn)t = J(H}t in L2(HiP) norm and, becausethe conditional expectation is continuous in L 2 (n,P), we can conclude.

From (3.7) and from the fact that E(I(Hnm = E (JoT IH';I.2 dS} it fol-

lows that E(J(H);) = E (JoT IHs l2 dS). In the same way, from (3.7) arid since

E(SUPt5,T I(Hnm ~ 4E (JOT IH';1 2dS), we prove (3.4).

The uniqueness of the extension results from the fact that the set of simpleprocesses isdense in H. :

We now prove (3.5). First of all, we notice that (3.3) is still true if H E H.This is justified by the fact that the simple processes are dense in H and by(3.7). We first consider stopping times of the form T = L:1<i<n ti1A" whereo < tl < ... < tn = T and the Ai'S are disjoint and F t ; measurable, and weprove (3.5) in that case. First

2. 1fT is an Fi-stopping time

r-] HsdWs = ITl{s5,T}HsdWs' (3.5)

Proof. We shall use the fact that if (Hs)s5,T is in,H, there. exists a sequence(H';) s5,T of simple processes such that ' ,

lim E (iT IHs - H';1 2dS) = O.

n-++oo 0

Remark 3.4.3 We write by definition

iT H.dW. =ITHsdWs -It

HsdWs-

If t ~ T, and if A E Ft. then s -t 1Al {t<.} Hs is still a simple process and it iseasy to check from the definition of the integral that

ITlAHs1{t<s}dWs = lA iT n.sw.. (3.3)

Now that we have defined the stochastic integral for simple processes and statedsome of its properties, we are going to extend the concept to a larger class ofadapted processes H

H = { (Ht)05,t5,T, (Ft}~?o - adapted process, E (ITH;dS) < +00} .

Proposition 3.4.4 Co~sider (Wt}t?o an Ft-Browriian motion. There exists aunique linear mapping J from H to the space of continuous Ft-martingalesdefined on [0, T], suc~ that:

1. If (Ht}t5,T is a simple process, P a.s. for any a ~ t ~ T, J(H)t = I(Hk

2. If t s T, E (J(H);) = E (I tH;dS).

This linear mappingisunique in the following sense, if both J and J' satisfy theprevious properties then

P a.s. '10 ~ t ~ T, J(H)t = J'(H}t.

We denote,for HE H, I tH.dW. = J(H}t.

On top of that, the stochastic integral satisfies the following properties:

Proposition 3.4.5 If (Ht}O<t<T belongs to H then

1. We have

Proof. It is easy to deduce from the extension property and from the continuityproperty that if H E 1t then P a.s. Vt ~ T, J(H)t = J(Hk

Let H E ic. and define t; = inf {O ~ s ~ T, J; H~du 2: n} (+00 if that setis empty), and H;' = H, l{s:STn}'

Firstly, we show that Tn is a stopping time. Since {Tn ~ t} = U6 H~du 2: n},we just need to prove that J6 H~du is Frmeasurable. This result is true if H is a

simple process and, by density, it is true if H E 11.. Finally, if H E il, J6 H~du

is also Ft-measurable because it is the limit of J6(Hu A K)2du almost surely asK tends to infinity. Then, we see immediately that the processes H;' are adaptedand bounded, hence they belong to 1t. Moreover .

it H;'dWs =it l{s:STn}H;,+ldWs,

and relation (3.5) implies that


Remark 3.4.7 It is crucial to notice that in this case (J6 HsdWs) is notO<t<T

necessarily a martingale. - -

Thus, on the set U; H~du < n}, for any t ~ T, J(Hn)t = J(Hn+lk Since

Un~oUoT H~du < n} = UoTH~du < +oo}, we can define almost surely a

- ·Tprocess J(H)t by: on the set Uo H~du < n},

Vt ~ T J(H)t = J(Hnk

The process t ~ J(H)t is almost surely continuous, by definition. The extensionproperty is satisfied by construction. Wejust need to prove the continuity propertyof J. To do so, we first notice that

\ .

P (SUPt:STIJ(H)tl2:~) < P (I; H;ds 2: liN)

+P (l{foT H;'du<l/N} SUPt:ST IJ(H)tl2: €).If we call TN = inf {s ~ T, Jos H~du 2: liN} (+00 if this set is empty), then

on the set {J; H~du < liN}, it follows from (3.5) that, for any t ~ r.r-,


Tn converges almost surely to T. By continuity of the map t ~ J6 HsdWs wecan affirm that, almost surely, J;n HsdWs converges to JOT HsdWs. On the otherhand

(

T ' . T 2) (T )E 11{s:ST}HSdW. -11{s:STn}HsdWs = E 11{T<s:STn}H;dS .

This last term converges to 0 by dominated co~vergence, therefore l: 1{S:STn}HsdWsT '

converges to Jo l{s:ST}HsdWs in L 2(0, P) (a subsequence converges almostsurely). That completes the proof of (3.5) for an arbitrary stopping time. 0

and then JoT l{s:ST}HsdWs = J; HsdWs.In order to prove this result for an arbitrary stopping time T, we must notice that

T can be approximated by a decreasing sequence of stopping times of the previousform. If

also, each l{s>qlA.Hs is adapted because this process is zero if s ~ t, and isequal to lA; H, otherwise, therefore it belongs to 1t. It follows that

T

JoT l{s>T}HsdWs = LilA. l{s>qHsdWsl<i<n a

- - T T

L lA;1 u.sw, = 1 HsdWs,l:Si:Sn t, T

2. Continuity property: If (Hn)n>o is a sequence of processes in il such that

JoT (H,;)2 ds converges to 0 in ;robability then SUPt:s'T IJ(Hnhl converges too in probability.

Consistently, we write J6 HsdWs for J(H)t.

In the modelling, we shall need processes which only satisfy a weaker integrabilitycondition than the processes in 1t, that is why we define '

il = {(Hs)o:SS:ST (Fth~o - adapted process, iT H;ds < +00 P a.s. } .

The following proposition defines an extension of the stochastic integral from 1tto ii. ' ,Proposition 3.4.6 There exists a unique linear mapping I from il into the vectorspace ofcontinuous processes defined on [0,TJ,such that:

1. Extension property: 1f(Ht)09:ST is a simple process then

Pa.s.\fO ~ t ~ T, J(H)t 7' I(Hit.

whence, by applying (3.4) to the process s t-t H. I{.5,TN} we get

P (~~~ IJ(H)tl ~ E) < P (iT H;ds ~ ~ )

+4/E2E (1: H; I {.5:. T N }ds)

< P (iT H~ds ~ ~ ) + N4E2'

As a result, if f: (H;l ds converges to 0 in probability, then SUPt<T IJ(Hn)tlconverges to 0 in probability. -_ In order to prove the linearity of J, let us consider two processes belonging to

1i, called Hand K, and the two sequences H[' and K[' defined at the beginningT T·'

of the proof such that fo (H"; - H.)2ds and fo (K"; - K.)2ds converge to 0

in probability. By continuity of J we can take the limit in the equality J(>..Hn +J.LKn)t = >"J(Hn)t +J.LJ(Kn)t, to prove the continuity of 1.

Finally, the fact that if HE it then f: (Ht-H[,)2dt converge to 0 in probabilityand the continuity property yields the uniqueness ofthe extension. 0

We are about to summarise the conditions needed to define the stochastic integralwith respect to a Brownian motion and we want to specify the assumptions thatmake it a martingale.Summary:Let 'us consider an Ft-Brownian motion (Wt)t>o and an Ft-adapted process

(Hd095,T. We are able to define the stochastic-integral (J~ H.dW. )05,t5,T as

soonasf: H;ds < +00 P a.s. By construction.the process Ij'[ H.dW.)o5,t5,T

is a martingale ifE (foT

H;ds) < +00. This condition is not necessary. Indeed,

the inequality E (1: II.;ds) < +00is satisfied if and only if

E (SUp (t H.dW.) 2) < +00.05,t5:.T Jo

This is proved in Exercise IS.

3.4.2 Ito calculus

It is now time to introduce a differential calculus based on this stochastic integral.It will be called the Ito calculus and the main ingredient is the famous Ito formula.

In particular, the Ito formula allows us to differentiate such a function as t t-tf (Wd if f is twice continuously differentiable. The following example will simplyshow that a naive extension of the classical differential calculus is bound to fail.Let us try to differentiate the function t --+ W? in terms of 'dWt'. Typically, for a


differentiable function f(t) null at the origin, we have f( t)2 = 2 f~ f (s )j(s)ds =

2 f~ f(s)df(s). In the Brownian case, it is impossible to have a similar formula

W? = 2 f~ W. dWs- Indeed, from the previous section we know that f~ W. dW.

is a martingale (because E (f~ W,;ds) < +00), null at zero. If it were equal to

W? /2 it would be non-negative, and a non-negative martingale vanishing at zerocan only be identically zero.

We shall define precisely the class of processes for which the Ito formula isapplicable.

Definition 3.4.8 Let (0, .1', (Fdt~o, P) be ajilteredprobability space and (Wt)t>oan FtcBrownian motion. (Xt )05, t5,T is an IR-valuedIto process ifit can be writtenas

where

• X o is Fo-measurable.

• (Kt)05,t5,T and (Ht)095,T are Fe-adapted processes.

• f: IK.lds < +00 P a.s.T •

• fo IH.12ds < +00 P a.s.

We can prove the following proposition (see Exercise 16) which underlines theuniqueness of the previous decomposition.;J ,

Proposition 3.4.9 If (Mt )05, t5,T is a continuous martingale such that

M t = it K..ds, with P a.s.iT

IK.lds < +00,

thenP a.s. 'lit ~ T, u, = O.

This implies that:

- An Ito process decomposition is unique. That means that if

x, = X o + it K.ds + rH.dW. =' Xb + rK~ds + rH~dW.0: Jo Jo Jo

then '

Xo = x; dP a.s. H. = ti; ds x dP a.e. K. = K; ds x dPa.e.

- If (Xt )05, t5,T is a martingale of the form X o + f~ K.ds + f~ H.dW., thenK, = 0 dt x dP a.e.

We shall state Ito formula for continuous martingales. The interested reader shouldrefer to Bouleau (1988) for an elementary proof in the Brownian case, i.e. when(Wd is a standard Brownian motion, or to Karatzas and Shreve (1988) for acomplete proof.

It turns out that

and

i t dS lit ( 1)log(St) = log(So) + _s + - -2 a 2S;ds .a Ss 2 a Ss

Using (3.9), we get

}Ii = Yo + it (J.L - a 2/2) dt + it adWt,


J; Ssds and J~ SsdWs exist and at any time t

P a.s. s, = Xo + it J.LSsds + it aSsdWs.

To put it in a simple way, let us do a formal calculation. We write }Ii = log(St)where St is a solution of (3.8). St is an Ito process with K; = J.LSs and H; = aSs.Assuming that St is non-negative, we apply Ito formula to f(x) =log(x) (at leastformally, because f(x) is not a C 2function!), and we obtain

Yt = log(St) = log(So) + (J.L - a 2/2) t + aWt.

Taking that into account, it seems that

St = Xo exp ((J.L - a 2/2) t + aWt)

is a solution of equation (3.8). We must check that conjecture rigorously. We haveSt = f(t, Wd with

roJ f(t,x)=xoexp((J.L-a2/2)t+ax).

Ito formula is now applicable and yields

St = f(t, Wd

= f(O, W o) + it f:(s, Ws)ds

+ it f~(s, Ws)dWs +.~ it'f~/x(S, Ws)d(W, W)S'

Furthermore, since (W, W)t = t, we can write

and finally

. it. itSt = Xo + a SsJ.Lds + a SsadWs"

Remark 3.4.11 We could have obtained the same result (exercise) by applyingIto formula to St = ¢(Zd, with Zt = (J.L-a2 /2)t+aWt (which is an Ito process)and ¢(x) =nxo exp(x).

We have just proved the existence of a solution to equation (3.8). We are about toprove its uniqueness. To do that, we shall use the integration by parts formula.

,

I

It[,~I

tf

II

Ii

(3.9)

(3.8)

Brownian motion and stochastic differential equations

s, ~ x'o + it s. (J.Lds + adW:).

This is often written in the symbolic form

as, ~ St (J.Ldt + adWt), So = xo·

f(Xt) = f(Xo) + it f'(Xs)dX s + ~ it j"(Xs)d(X,X)s

whe re, by definition

W t2

..:.. t = 2itWsdWs'

Since E (J; Ws2 ds ) < +00, it confirms the fact that W? - t is a martingale.

We now want to.tackle the problem of finding the solutions (Sdt~o ofr

. (X,Xk = it H;ds,a .

We are actually looking for an adapted process (St)t~O such that the integrals

3.4.3 Examples: Ito formula in practice

Let us start by giving an elementary example. If f(x) = x 2 and Xt = W t, weidentify K, = °and H, = 1, thus

i t '1 rW t

2 = 2 a WsdWs + '2 io 2ds.

Theorem 3.4.10 Let (Xt )09 $ T be an Ito process

x, = Xo + it Ksds + it HsdWs,

and f be a twice continuously differentiable function, then

it 'it ita f'(Xs)dX s = a f'(Xs)Ksds + a f'(Xs)HsdWs'

Likewise, if(t, x) -+ f(t, x) is a function which is twice differentiable with respectto x and once with respect to t, and ifthese partial derivatives are continuous withrespectto (t,x) (i.e. f is a junction ofclass C1,2), ltd formulabecomes

f(O, Xo) + it f:(s, Xs)ds

+ it f~(s,Xs)dXs+ ~ it f~lx(S:Xs)d(X.,X)s.

44

o

Proof. By Ito formula

(Xt + Yi)2 = (Xo + YO)2

By subtracting equalities 2 and 3 from the first one, it turns out that

XtYi = XoYo +I tX.dY. +I t

Y"dX. +I tH.H~ds.

47Stochastic integral and Ito calculus

In this case, we have

(X, Z)t = (1" X.adW., - 1" Z.adW.)t = -l,t a2x.z.u.

Therefore

This process is given by'0

Remark 3.4.14

d(XtZt) = X tZt{(-J.L+a2)dt-adWt)

+ XtZt (J.Ldt + adWd - X tZta2dt =o.Hence, XtZt is equal to XoZo, which implies that

"It ~ 0, P a.s. x, = XOZ;I = St.

The processes X, and Zt being continuous, this proves that

, P a.s. "It ~ 0, X, = XOZ;I = St.

We have just proved the following theorem:

Theorem 3.4.13 If we consider two real numbers a, J.L and a Brownian motion(Wth>o and a strictly positive constant T, there exists a unique Ito process(SdO;:::;T which satisfies,for any t ~ T,

s, = Xo +I tS. (J.Lds + adW.).


Proposition 3.4.12 (integration by parts formula) Let X, and Yi be two Ito processes, x, = X o+ J; K.ds + J; H.dW. and Yi = Yo + J; K~ds + J; H~dW•.Then

XtYi = XoYo +I t

X.dY. +I t

Y.dX. + (X, Y)t

with the following convention

(X, Y)t =I t

H.H~ds.

+2J;(X. + Y.)d(X. + Y.)

+ J;(H. + H~)2ds'

xg + 2 J; X.dX. + J; H;ds

2 rt rt ,2= Yo + 2 Jo Y.dY. + Jo H. ds.

" .

We now have the tools to show that equation (3.8) has a unique solution. Recallth~ ,

Sc = Xo exp ( (J.L - a2/2) t + aWt)

is a solution of (3.8) and assume that (Xt)t>o is another one. We attempt tocompute the stochastic differential of the quantity XtS;I. Define . '

, Zt = ~~ = exp (( -J.L +a2/2) t -q-Wt) ,

J.L' =' -J.L + a2 and a' = -a, Then Zt = exp((J.L' - a,2/2)t + a'Wt) and theverification that we have just Gone shows that

Zt = 1 + r Z.(J,L'ds + a'dW.) = 1 + r z, ((-J.L + ( 2) ds - adW.).~ ~ ,

From the integration by parts formula, we can compute the differential of X;Zt

• The process St that we just studied will model the evolution of a stock price inthe Black-Scholes model.

• When J.L = 0, St is actually a martingale (see Proposition 3.3.3) called theexponential martingale ofBrownian motion.

Remark 3.4.15 Let e be an open set in IR and (Xt)O:::;t:::;T an Ito process whichstays in e at all times. If we consider a function f from e to lR which is twicecontinuously differentiable, we can derive an extension of Ito formula in that case

f(Xt).~ f(;o) +I t!'(X.)dX. + ~ I t

!"(X.)H;ds.

This result allows us to apply Ito formula to log(Xd for instance, if X t is a strictlypositive process.

3.4.4 Multidimensional Ito formula

We apply a multidimensional version of Ito formula when f is a function of'severalIto processes which are themselves functions of several Brownian motions. This

it P it ,X i = Xi + Kids + '" Hi,idWit 0 s L..J s s

o j:=l 0

then, if 1 is twice differentiable with respect to x and once differentiable withrespect to t, with continuous partial derivatives in (t, x)

b (t, X t) dt + a (t, X t) dWtZ

Remark 3.5.2 Formally, we often write equation (3.10) as

.{ ex.Xo

Stochastic differential equations 49

• (J~ HsdWti

, J~ H~dW/)t = 0 if i ;i j.

• (J~ tt.aw], J~ H~dWDt = J~ HsH~ds.This definition leads to the cross-variation stated in the previous proposition.

3.5 Stochastic differential equations

In Section 3.4.2, we studied in detail the solutions to the equation

x, = x + it Xs(/Lds + adWs)'

We can now consider some more general equations of the type

x, = Z + it b(s,Xs)ds + it a(s,Xs)dWs. (3.10)

These equations are called stochastic differential equations and a solution of (3.10)is called a diffusion. These equations are useful to model most financial assets,whether we are speaking about stocks or interest rate processes. Let us first studysome properties of the solutions to these equations.

The following theorem gives sufficient conditions on b and a to guarantee theexistence and uniqueness of a solution of equation (3.10).

Theorem·3.5.3 Ifb and a are continuous functions and if there exists a constantK < +00 such that

3.5.I Ito theorem

Whatdowe mean by a solution of(3.1O)?

Definition 3.5:1 We consider a probability space (n, A,.P) equipped with a filtratioh (Ft)t~o, We also have functions b : lR+ x lR -+ IR., a : lR+ x lR -+ IR., aFo-measurable randomvariable Z and finally an Ft-Brownian motion (Wdt~o.

-1 solution to equation (3./0) is an Fi-adapted stochastic process (Xt)t~O thatsatisfies:

• For any t ~ 0, the integrals J~ b(s, Xs)ds and J~ a(s, Xs)dWs existt . t .'

llb(s,Xs),ds < +00andila(s,XsWds < +00 P a.s.

• (Xt)t~O satisfies (3./0), i.e._. "'

t t

'It ~ 0 P a.s. .x,t = Z +1b(s,Xs) ds +1a (s, X s ) dWs'


version will prove to be very useful when we model complex interest rate structuresfor instance.

Definition 3.4.16 We call standard p-dimensional FrBrownian motion an lRP

valued process (Wt = (Wl, . . . , Wi) k:~o adapted to Ft, where all the (Wnt~Oare independent standard FrBrownian motions.

Along these lines, we can define a multidimensional Ito process.

Definition 3.4.17 (Xt)09~T is an Ito process if

with:

• ax: = Kids + ,",P Hi,idWjS 5 L...,,-J==l 5 5'

• d(Xi Xi) = ,",P Hi,m Hi,mds, s L...,,-m=l 5 s .

Remark3.4.19 If (Xs)O<t<T and (Y.)O<t<T are two Ito processes, we can define formally the cross-va-ri~tion of X and Y (denoted by (X, Y)s) through thefollowing properties:

• (X, Y)t is bilinear and symmetric.

• (J~ Ksds, x')t = 0 if (Xt)09~T is an Ito process.

where:

• K t and all the processes (Hi) are adapted to (Ft).

• JoT IKslds < +00 ~ a.s.

• JoT (H;) 2ds < +00 P a.s.

Ito formula becomes:

Proposition 3.4.18 Let (Xl, ... , X;') be n Ito processes

1(0,XJ, ... ,Xfn + it ~~ (s,X~,; .. ,X:)ds'

+tit ::.(s,x~,... ,X:)dX;i=l 0 ,

+~ t it 8821 (s,X~, ... ,X:)d(Xi.,Xi)s

2 .. 0 xixi',J=l .

We deduce that ~ is a mapping from E to E with a Lipschitz norm bounded from

. above by k(T) = (2(K2T2+ 4K2T)) 1/2. If we assume that T is small enoughso that k(T) < 1, it turns out that ~ is a contraction from E into E, Thus ithas a fixed point in E, Moreover, if X is a fixed point of ~, it is a solution of(3.10). That completes the proof of the existence forT small enough. On the otherhand, a solution of (3.10) which belongs to E is a fixed point of~. That provesthe uniqueness of a solution of equation (3.10) in the class E, In order to provethe uniqueness in the whole class of Ito processes, we just need to show that asolution of (3.10) always belongs to E, Let X be a solution of (3.10), and defineT; = inf{s 2: 0, IXsl > n} andr(t) = E (suPO~S<tIlTn IXsI2). It is easy tocheck that fn(t) is finite and continuous. Using the same comparison argumentsas before, we can say that .

-c)

E (suPO~u~tIlTn IXuI2

) < 3 ( E(Z2) + E (J~IITn K(1 + IXsl)ds) 2

+4E (J~IITn K2(1 + IXsI)2ds))

< 3 (E(Z2) + 2(K2T + 4K2)

x f~ (1 + E (sUPO~U~SIlTn IXuI2

) ) dS) .

This yields the following inequality

r(t) ::; a + bit r(s)ds.

In order to complete the proof, let us recall the Gronwall lemma.

Lemma 3.5.4 (Gronwalll~mma) Iff is a continuous function such that for any0::; t ::; T, f(t) ::; a + b fo f(s)ds, then f(T) ::; a(1 + ebT).

Proof. Let us write u(t) =e-bt f; f(s)ds. Then,

i u'(t) = e-bt(f(s) - bit f.(s)ds) ::; ae-bt.o .

By first-order integration we obtain u(T) ::; alb and f(T) ::; a(1 + ebT). 0


J. Ib(t,x) - b(t, y)1 + loo(t, x) - oo(t,Y)1 ::; Klx - yl2. Ib(t,x)1 + loo(t,x)l::; K(1 + Ixl)3. 'E(Z2) < +00then, for any T 2: 0, (3.10) admits a unique solution in the interval [0,T].Moreover, this solution (Xs)O~s~T satisfies

E ( sup IXsI2) < +00O~s~T

The uniqueness of the solution means that if (Xt)09~T and (Yi)09~T are twosolutions oj(3.1O), then P a.s. '10 ::; t ::; T, X, = Yi,

Proof. We define the set

E = {(XS)O~S~T' Ft-adapted continuous processes,

sU~h that E (~~~IXsI2) < +oo} .Together with the norm IIXII = (E (sUPO<s<T IXsJ2)) 1/2 E is a complete normedvector space. In order to show the existence of a solution, we are going to usethe theorem of existence of a fixed point for a contracting mapping. Let ~ be thefunction that maps a process (Xs)O~s~T into a process (~(X)s)O~s~T definedby .

~(X)t = Z + it b(s, Xs)d~ + it oo(s, Xs)dWs'

If X belongs to E, ~(X) is well defined and furthermore if X and Y are both inE we can use the fact that (a + b)2 ::; 2(a2+ b2) to write that

1~(X)t - ~(Y)tI2 <' 2 (suPO~t~T'lf~(b(S,Xs) - b(s, Ys))ds I2

+sUPO~t~T If~(oo(s,~s) - oo(s, Ys))dWsI2)

and therefore by inequality (3.4)

E (sup 1~(X)t - ~(Y)tI2)-sr

< 2E (sup (it Ib(s, X s) .: b(s, Ys)ldS) 2)09~T 0 ,

+8E (iT (oo(s, X s) - oo(s, Ys))2dS)

< 2(K2T2+ 4K2T)E ,( sup IXt - Yi1 2)

09~T

Stochastic differential equations 51

In our case, we have t" (T) < K < +00,where K is a function ofT independentof n. It follows from Fatou lemma that, for any T,

E ( sup IXsI2) < K < +00.O$;s$;T

Therefore X belongs to [; andthat completes the prooffor small T. For an arbitraryT, we consider a large enough integer n and think successively on the intervals[0, Tin), [Tin, 2TIn), ...,[(n - l)TIn, T). 0

52 Brownian motion and stochastic differential equations Stochastic differential equations 53

that if AI, , An are real numbers and if 0 ::; t l < ... < t-: the random variableAlXtl + + AnXtn is normal. To convince ourselves, we just notice that

x; = xe-eti + O'e-et•1+00

l{s$;t;}eCSdWs = mi + it Ji(s)dWs.

Then AIXtt + '" + AnXtn = E7=I Aimi + J~ (E7=1 Adi(S)) dWs which isindeed a normal random variable (since it is a stochastic integral of a deterministicfunction of time).

It can be' written explicitly. Indeed, if we consider yt = x,» and integrate byparts, it yields

3.5.2 The Omstein-Ulhenbeck process

Omstein-Ulhenbeck process is the unique solution of the following equation:

dyt = dXtect + Xtd(eet) + d(X, eC'}t.

Furthermore (X,eC')t = 0 because d(e ct) = ceetdt. It follows that dytO'ectdWt and thus .

-ct -ct it cSdWX; = xe + ae e s-

o "This enables us to compute the mean and variance of Xt:

E(Xd= xe-et + O'e-ctE (it eCSdws) = xe-et

(3.11)

In other words, we are looking for a process (Xt)O<t<T with values in lRn,adapted to the filtration '(Ftk~o and such that P a.s. ,fo~any t and for any i ::; n

• z = (ZI, ... , Z") an Fo-measurable random variable in lRn.

We are also interested in the following stochastic differential equation:-v

3.5.3 Multidimensional stochastic differential equations

The analysis of stochastic differential equations can be extended to the case when. processes evolve in lRn. This generalisation proves to be useful in finance when

we want to model baskets of stocks or currencies. We consider

• W = (WI, ... , W1') an lR1' -valued Ft-Brownian motion.

• b : lR+ x lRn ---+ lRn. b(s , x) = (b1( s, x), .. : , b"(s , x) ).

• a: lR+ X lRn ---+ lRnx1' (which is the set of n x p matrices),

O'(s,x) = (O'i,j(s,x)h$;i$;n ,I$;j$;1"

= -cXtdt + O'dWt= x{

ax,Xo

(since E (J;(e~S)2ds) < +00, J~ ecsdWs is a martingale null at time 0 and

therefore its expectation is zero). Similarly , .

E ((Xt- E(Xt))2)

.'E (e-'" (I,' e"dW')'),.O'2~-2ctE (lte2csdi) '

1 - e-2ct= 0'2 _

2c

We can also prove that X, is a normal random variable, since X; can be written as

J~ j(s)dWs where j(.) is a deterministic function oftim~ and J~ !2(s)d~ < +00(see Exercise 12). More precisely, the process (Xtk~o IS GaUSSIan. ThIS means

The theorem of existence and uniqueness of a solution of (3.11) can be stated as:

Theorem 3.5~5 If x E lRn, we denote by [z] the Euclidean norm of x and if

a E lRnx1', 10'1 2 = EI<i<n 1< '< O'L· We assume that__ , _1_1' .

I. Ib(t,x) -b(t,y)1 + IO'(t,x) - O'(t,y)1 ::; Klx - yl2.. lb(t,x)1 + IO'(t,x)l::; K(1 + Ixl)

3. E(IZI2) < +00then, there exists a unique solution to (3.11). Moreover. this solution satisfies foranyT

E ( sup IXsI2) < +00.O$;s$;T

The proof is very ,similar to the one in the scalar case.

3.5.4 The Markov property of the solution ofa stochastic differential equation

The intuitive meaning of the Markov property is that the future behaviour of theprocess (Xtk~o after t depends only on the value X; and is not influenced by thehistory of the process before t. This is a crucial property of the Markovian modeland it will have great consequences in the pricing of options. For instance, it willallow us to show that the price of an option on an underlying asset whose price isMarkovian depends only on the price of this underlying asset at time t.

Mathematically speaking, an Fradapted process (Xt)t>o satisfies the Markovproperty if, for any bounded Borel function f and for any ~ and t such that s s t,we have

E (f (Xt) IFs) =E (f (Xt) IXs) .

We are going to state this property for a solution of equation (3.10). We shalldenote by (X;'X, s ;:::- t) the solution of equation (3.10) starting from x at time tand by X" = Xo,x the solution starting from x at time O. For s ;::: t, X;'x satisfies

Xt,x = x ~ 1sb (u Xt,X) du + _1 s

a (u Xt,X) dWs " u 'u u·t t

A priori, X.t,x is defined for any (t, x) almost surely. However, under the assumptions of Theorem 3.5.3, we can build a process depending on (t, x, s) which isalmost surely continuous with respect to these variables and is a solution of theprevious equation. This result is difficult to prove and the interested reader shouldrefer to Rogers and Williams (1987) for the proof.

The Markov property is a consequence of the flow property of a solution of astochastic differential equation which is itself an extension of the flow property ofsolutions of ordinary differential equations.

Lemma 3.5.6 Under the assumptions ofTheorem 3.5.3, if s ;::: t

Remark 3.5.8 The previous equality is often written as

P a.s. E (f (Xt) IFs) = ¢(Xs),

with ¢(x) = E (f(X;·X)).

55

Proof. Yet again, we shall only sketch the proof of.this theorem. For a full proof,the reader ought to refer to Friedman (1975).

x'The-flow property shows that, if s ::; t, Xf = X:' '. On the other hand, wecan prove that X;,x is a measurable function of x and the Brownian increments(Ws+u - W s , u ;::: 0) (this result is natural but it is quite tricky to justify (seeFriedman (1975)). If we use this result for fixed sand t we obtain X;·x(x, W s+u ~ w., u ;::: 0) and thus

Stochastic differential equations

Indeed, if t ::; s

X~·x x + J; b (u, X~) du + J; a (u, X~) dWu

x: + J/ b (u, X~) du + J/ a (u, X~) dWu'

The uniqueness of the solution to this equation implies that X~,x = X;'x, fort ::; s. 0

In this case, the Markov property can be stated as follows:

Theorem 3.5.7 Let (Xtk~o be a solution of (3.10). It is a Markov process withrespect to the Brownian filtration (Ftk~o. Furthermore, for any bounded Borelfunction f we have


Proof. We are only going to sketch the proof of this lemma. For any x, we have

P a.s. X;'x = x + 1sb (u, X~,X) du +1s

a (u, X~·X) dWu.

It follows that, P a.s. for any y E JR,

X;·y = y + ~s b.(u,X~'Y) du +ls

a (u,X~;Y) dWu,

and also

X;·x; =X:+1sb(u,X~'X;)dU~1s

a(u,X~·X;)dWu.

These results are intuitive, but they can be proved rigorously by using the continuityofyH x», We can also notice that X: is also a solution of the previous equation.

x: = (X:, W s+u - w., u;::: 0),

where X: is Fs-measurable and (Ws+u, - Ws)u~o is independent of F s.If we apply the result of Proposition A.2.5 in the Appendix to X s , (Ws+u

Ws)u~o, .and F" it turns out that

E (f ( (X: ,W s+u - w., u;::: 0))1 F s )

E (f ((x, W s+u - W s; u;::: O)))lx=x;

= E (f (X:·X))lx=x~ .

-0

The previous result can be extended to the case when we consider a function ofthe whole path of a diffusionafter time s. In particular, the following theorem isuseful when we do computations involving interest rate models.

Theorem 3.5.9 Let (Xt)t>o be a solution of(3.1O) and r(s, x) be a non-negative

56 Brownian motion and stochastic differential equations Exercises 57

We define

Ie(f) = L ai(Xt .+1 - XL;).0:Si:SN-1

Prove that I, (f) is a normal random variable and compute its mean and variance. In particular, show that

2. ~rom ~his, show that there exists a unique linear mapping I from L 2(lR+, dx)into Z (fl,F,P),suchthatI(f) = Ie(f),when 1belongs to 11. andE(I(f)2) =11f1l£2, for any 1 in L2(lR+).

3. Prove that if (Xn)n>O is a sequence of normal random variables 'with zeromean which converge to X in L2(fl, .1',P), then X is also a normal random

(lR,B(lR))Xs(w)

2. Prove that the mapping

([0, t] x n,B([O, t]) x Ft ) --+(s,w) f-----+

Exercise 9 Let S be a stopping time, prove that Sis Fs-measurable.

Exercise 10 Let Sand T be two stopping times such that S ~ T P a.s. ProvethatFs eFT.

Exercise 11 Let S be a stopping time almost surely finite, and (Xt)t>o be anadapted process almost surely continuous. -

l. Prove that, P a.s., for any s

x, = lim '" l[k/n (k+1)/n[(S)Xk/n(w).n-t+oo L..J '

k~O

is measurable.

3. Conclude that if S ~ t, X s is Fcmeasurable, and thus that X s is Fs-measurable.

Exercise 12 This exerciseis an introduction to the copcept of stochastic integra

tion. We want to build an integral of the form s: l(s)dXs, where (Xt)t>o is anFt-Brownian'motion and I(s) is a measurable function from (lR+, B(lR+)) into

(lR, B(lR)) such that s: j2 (s )ds < +00. This type of integral is called Wienerintegral and it is a particular case of Ito integral which is studied in Section 3.4.

We~recall that the set 11. of functionsof the form LO<i<N-1 ai1jt.,ti+d, with

ai E lR, and to = 0 ~ t1 ~ ... ~ tN is dense in the space[,2(lR+, dx) endowed

with the norm II/ilL2 = (Jo+oo j2(s)ds) 1/2.

l. Consider a;E lR, and 0 = to ~ t1 ~ ... ~ i», and call

1 = L ai 1 jti,t.+d·0:Si:SN-1

¢(x) = E (e-f.' r(u,X:"~)du I(Xt'X)) .

It is also written as

E (e- f.' r(u,X,,)du1 (Xt) IFs) = E (e-f.' r(u,X:"~)du I(X;'X)) Ix=x.

Remark 3.5.10 Actually, one can prove a more general result. Without gettinginto the technicalities, let us just mention that if ¢ is a function of the whole pathof X, after time s, the following stronger result is still true:

Pa.s. E(¢.(X:, t~s)IFs)= E(¢(Xt'X, t~s))lx=x.'

Remark 3.5.11 When' b and a are independent of x (the diffusion is said to behomogeneous), we can show that the law of X:;t is the same as the one of X~,x,which implies that if 1 is a bounded measurable function, then

E (J(X:;t)) = E (J(X~'X)).

We can extend this result and show that if r is a function of x only then

E (e-I." r(X:"~)du1(X:;t) ) = E (e-f; r(X~'~)du I(X~'X)) .

In that case, the Theorem 3.5.9 becomes

E (e-f.' r(X,,)du1 (Xt) IFs) = E (e-fo'-' r(X~'~)du I(X~~~)) Ix=x.

with

measurable function. For t > s

P a.s. E (e- f.' r(u,X,,)du1 (Xt) IFs) = ¢(Xs)

3.6 Exercises

Exercise 6 Let (Mtk:~.o be a martingale such that for any t, E(Ml) < +00.Prove that if s ~ t .

E ((Mt - Ms)2IFs) = E (Mt2 -'M;IFs) .

Exercise 7 Let X, be a process with independent stationary increments and zeroinitial value such that for any t, E (Xl) < +00. We shall also assume that themap t f-t E (Xl) is continuous. Prove thatE (Xt ) = ct and that Var(Xt) = c't,where c and c' are two constants.

Exercise 8 Prove that, if T is a stopping time,

F r = {A E A, for all t ~ 0 , A n {T ~ t} E Ft }

is a a-algebra.

58 Brownian motion and stochastic differential equations Exercises 59

variable with zero-mean. Deduce that if f E L2(IR+, dx) then 1(1) is a normal

random variable with zero mean and a variance equal to s: f2 (s )ds.

4. We consider f E L2(IR+, dx), and we define

z, = it f(s)dXs = JI]O,tJ(s)f(s)dX s,

prove that Zt is adapted to Ft, and that Zt - Zs is independent of F, (hint:

begin with the case f E H).

5. Prove that the processes Zt, Z; - f; P(s)ds, exp(Zt - ~ f; f2(s)ds) are

Frmartingales.

Exercise 13 Let T be a positive real number and (Mt)095,T be a continuousFt-martingale. We assume that E(Mj.) is finite.

1. Prove that (IMtI)095,T is a submartingale.

2. Show that, if M* ., sUP05,t5,T IMtl,

AP (M* ~ A) ~ E (IMTll{M·~.q).

(Hint: apply the optional sampling theorem to the submartingale IMtl betweenT /\ T and T where T = inf{t ~ T, IMti ~ A} (if this set is empty T is equal

to +00).)

3. From the previous result, deduce that for positive A

E((M* /\ A)2) ~ 2E((M* /\ A)IMTI).

M·I\A(Use the fact that (M* /\ A)P = fo pxp-1dx for p = 1,2.)

4. Prove that E(M*) is finite and

E ( sup IMtI2

) ~ 4E(IMT I2

) .°95,T

Exercise 14

1. Prove that if 5 and 5' are two Frstopping times then 5 /\ 5~ = inf(5, 5') and5 V 5' = sup(5, 5') are also two Ft-stopping times.

2. By applying the sampling theorem to the stopping time 5 V s prove that

~ (Ms l{s>s}IFs) =u, l{s>s}'

3. Deduce that for s ~ t

E (Msl\t1{s>s}IFs) = Msl{s>s}'

4. Remembering that MSl\s is Fs-measurable, show that t -t MSl\t is an Ftmartingale.

Exercise 15

1. Let (Ht)05,t5,T be an adapted measurable process such that foT H'fdt < 00,

a.s. Let u, = f; HsdWs. Show that if E (sUPO<t<T Ml) < 00, then

E (foT H; dt) < 00. Hint: introduce the sequence ~f~topping times Tn =

inf{t ~ a If; H;ds = n} andsho~thatE(Mj.I\TJ=E (foTl\Tn H;ds).

2. Letp(t,x) = 1/vT=texp(-x2/2(1- t)), for a ~ t < 1 and x E IR, andp(l, x) = O. Define M; = p(t, Wt), where (Wt)O<t<1 is standard Brownianmotion. - -

(a) Prove that

(b) Letop

H, = ox (t, Wd·

Prove that fol

Hldt < 00, a.s. and E (fol Hldt) =+00.

Exercise 16 Let (Mt)05,t5,T be a continuous Ft-martingale equal to f; tc,e«where (Kt)O.::;t5,T is an Ft-adapted process such that foT IKslds < +00 P a.s,

'. T1. Moreover, we assume that P a.s. fo IKslds ~ C < +00. Prove that if we

write tf = Ti/n for a ~ i ~ n, then

lim E (~' (Mtn - Mt~ ) 2) = O.n---t+oc> L....J 1. 1-1

i=1

2. Under the same assumptions, prove that

~

Conclud~that M T = a P a.s. ,and thus P a.s. 'Vt~ T, M, = O.

3. foT

IKslds is now assumid to be finite'almost surely as opposed to bounded.

We shall accept the fact that the random variable f; IKslds is Ft-m~asurable.Show that Tn defined by

t; = inf{O ~ s ~ T, it IKslds ~ n}

,(or T if this set is empty) is astopping time. Prove that P a.s. limn-Hex> Tn =T. Considering the sequence of martingales (MtI\TJt~O' prove that

-P a.s. 'Vt~ T, M t = O.

6. If A 2: fJ. and A 2: 0, prove that

P(Wt :::; u; wt 2: A) = P(Wt 2: 2A - fJ., wt 2: A) = P(Wt 2: 2A - fJ.),

and if A :::; fJ. and A 2: 0

P(Wt :::; u; wt 2: A) = 2P(Wt 2: A) - P(Wt 2: fJ.).


4. Let M t be a martingale of the form J~ n.sw, + J~ Ksds with J~ H'[ds <. +00 P a.s. and J~ IKslds < +00 P a.s. Define the sequence of stopping

times Tn = inf{t :::; T,J~ H;ds 2: n}, in order to prove that K; = 0 dt xP a.s.

Exercise 17 Let us call X, the solution of the following stochastic differentialequation

{ax, = (p.xt + fJ.')dt + (oXt + a')dWtXo = O.

We write St = exp ((fJ. - a2 /2)t + aWt).1. Derive the stochastic differential equation satisfied by St- 1

.

2. Prove thatd(XtS;l) = St- 1 ((fJ.' - aa')dt + a'dWd·

3. Obtain the explicit representation of X t .

Exercise 18 Let (Wt k:~o be an Ft -Brownian motion. The purpose of this exerciseis to compute the law of (Wt , sUPs9 Ws ) .

1. Consider 5 a bounded stopping time. Apply the optional sampling theorem tothe martingale Mt ' = exp(izWt + z2t / 2), where z is a real number to provethat if 0 s u :::; v then

E (exp (iz(Wv+s - Wu+s)) IFu+s) =exp (-Z2(V - u)/2) .

2. Deduce that w,f = Wu+s - Ws is an 'Fs+u-Brownian motion independentof the a-algebra Fs-

3. Let (Yi)t>o be a continuous stochastic process independent of the a-algebra Bsuch that-E(suPO<s<K I~ I) < +00. Let T be a non-negative B-measurablerandom variable bounded from above by K: Show that

E (YTIB) = E (Yi)lt=T'

We shall start by assuming that T can be written as I:l<i<n ti1A" whereo< tl < ... < t« = K, and the Ai are disjoint B-measurablesets.

4. We denote by r>' the inf{s 2: 0, W s > A}, prove that if f is a bounded Borelfunction we have '

E (i(Wt)I{T>'~t}) = E (I{T>'~t}¢(t - r>.)) ,

where¢(u) = E(f(Wu+ A)). NoticethatE(f(Wu +A)) = E(f(-Wu+A))and prove that

E (f(Wt)l{ T>'91} ) = E (i(2A - Wt) l{T>'9}) .

5. Show that if we write Wt = sUPs9 Ws and if A 2: 0

P(Wt :::; A, Wt 2: A) = P(Wt 2: A, wt 2: A) ::;:; P(Wt 2: A).

Conclude that wt and IWtl have the same probability law.

Exercises

7. Finally, check that the law of (Wt , Wt) is given by

2(2y-x) ((2Y- X)2)l{o~y} l{x~y} ...,fi;i3 exp - 2t dxdy.

61

II·I

4

The Black-Scholes model

Black and Scholes (1973) tackled the problem of pricing and hedging a Europeanoption (call or put) on a non-dividend paying stock. Their method, which is basedon similar ideas to those developed in discrete-time in Chapter 1 of this book,leads to some formulae frequently used by practitioners, despite the simplifyingcharacter of the model. In this chapter, we give an up-to-date presentation of theirwork. The case of the American option is investigated and some extensions of themodel are exposed.

4.1 Description of the model

4.1.1lhe behaviour ofprices

The model suggested by Black and Scholes to describe the behaviour of prices isa continuous-time model with one risky asset (a share with price St at time t) anda riskless asset (with price Sp at time t). We suppose the behaviour of Sp to beencapsulated by the following (ordinary) differential equation:

dSP = rSpdt, (4.1)

where r is a non-negative constant. Note that r is an instantaneous interest rateand should not be confused with the one-period rate in discrete-time models. Weset sg = 1, so that Sp = eft for t ~ O. .

We assume that the behaviour of the stock price is determined by the followingstochastic differential equation:

,dSt = St (J.tdt + adBt) , (4.2)

where J.t and a are two constants and (B t) is a standardBrownian motion.The model is valid on the interval [0, T] where T is the maturity of the option.

As we saw (Chapter 3, Section 3.4.3), equation (4.2) has a closed-form solution~ .

s, = So exp (J.tt - ~2 t + aBt) ,

4.2 Change of probability. Representation of martingales

4.2.1 Equivalent probabilities

Let (fl, A, P) be a probability sp.ace. A probability measure Q on (fl, A) IS

absolutely continuous relative to P if '.

Change ofprobability. Representation ofmartingales 65

1. iT IHfldt + l TH;dt < +00 a.s.

2. Hfsf + u.s, = HgSg + tt-s; + I t

H~dS~ + I t

n.as; a.s.

for all t E [0, T].We denote by St =' e-rt S, the discounted price of the risky asset. The followingproposition is the counterpart of Proposition 1.1.2 of Chapter 1.

Proposition 4.1.2 Let ¢ = ((H?, H t)')09::;T be anadapted process with vdlues

inIR2, satisfying JoT IH?ldHJoT Hldt < +ooa.s. Weset: Vi(¢) = HfS?+HtSt

and frt(¢) = e"':rtVi(¢). Then, ¢ defines a self-financing strategy ifand only ,if

frt(¢) = Vo(¢) +it HudSu a.s. (4.3)

for all t E [0, T].Proof. Let us consider the self-financing strategy ¢. From equality :

dVt(¢) = -rfrt(¢)dt + e-rtdVi(¢)

which results from the differentiation of the product of the processes (e~rt) arid(Vi (¢)) (the cross-variation term d{e- r

. , v: (¢)kis null), we deduce

dfrt(¢) . - _re-rt (Hfert + HtSt) dt + e-rtHfd(ert) + «<n.as,H, (_re- rtSt dt + e-rtdSt)

HtdSt,

which yields equality (4.3). The converse is justified similarly. 0

Remark 4.1.3 We have not imposed any condition of predictability on strategiesunlike in Chapter 1. Actually, it is still possible to define a predictable process incontinuous-time but, in the case of the filtration of a Brownian motion, it-does notrestrict the class of adapted processes significantly (because of the continuity ofsample paths of Brownian motion).In our study of complete discrete models, we had to consider at some stage a probability measure equivalent to the initial probability and under which discountedprices of assets are martingales. We were then able to design self-financing strategies replicating the option. The following section provides the tools which allowus to apply these methods in continuous time.

VA E A P (A) = 0 ~ Q (A) =0.


T . iT1HfdSf = 0 Hfrertdt

is well-defined, as is the stochastic integralT "T . T1HtdSt = 1 (H~St/-L) dt +1O"HtStr/Bt,

since the map t f-t St is continuous, th~s bounded on [0, T] almost surely.

Definition 4.1.1 A self-financing strategy is defined by a pair ¢ ofadapted pro-s

cesses (Hf)o::;t::;T and (Ht)09::;T satisfying:

dVi (¢) = HfdSf + HtdSt·

To give a meaning to this equality, we set the condition

iT IHfldt < +00 a.s. and iT H;dt < +00 a.s.

Then the integral

4.1.2 Self-financing strategies

A strategy will be defined as it process ¢ = (¢t)09::;T=( (H?, Ht)) with values in

IR?, adapted to the natural filtration (Ft ) of the Brownian motion; the componentsH? and H, are the quantities of riskless asset and risky asset respectively, held inthe portfolio at time t. The value of the portfolio at time t is then given by

Vi (¢) = Hf S?+tt.s;In the discrete-time models, we have characterised self-financing strategies by theequality: Vn+l (¢) - Vn(¢) = ¢n+dSn+l - Sn) (see Chapter 1, Remark 1.1.1).This equality is extended to give the self-financing condition in the continuous-

time case

where So is the spot price observed at time O. One particular result from this modelis that the law of St is lognormal (i.e. its logarithm follows a normal law).

More precisely, we see that the process (St) is a solution of an equation of type(4.2) if and only if the process (log(St)) is a Brownian motion (not necessarilystandard). According to Definition 3.2.1 of Chapter 3, the process (Sd has the

following properties:

• continuity of the sample paths;' .

• independence of the relative increments: if u :s t, StlS« or (equivalently), therelative increment (St - Su) / Su is independent of the O"-algebra O"(Sv, v :s u);

• stationarity of the relative increments: if u :s t, the law of, (St - Su)/Su isidentical to the law of (St-u - So)/So·

These three properties express in concrete terms the hypotheses of Black and

Scholes on the behaviour of the share price.

64

Theorem 4.2.1 Q is absolutely continuous relative to P ifand only if there existsa non-negative random variable Z on (n, A) such that

VA E A Q(A) = i Z(w)dP(w).

Z is called density ofQ relative to P and sometimes denoted dQ/dP.

The sufficiency of the proposition is obvious, the converse is a version of theRadon-Nikodym theorem (cf. for example Dacunha-Castelle and Duflo (1986),Volume 1, or Williams (1991) Section 5.14).

The probabilities P and Q are equivalent if each one is absolutely continuousrelative to the other. Note that if Q is absolutely continuous relative to P, withdensity Z, then P and Q are equivalent if and only if P (Z > 0) = 1.

4.2.2 The Girsanov theorem

Let (n, F, (Ft)o~t~; , p) be a probability space equipped with the natural fil

tration of a standard Brownian" motion (Bt)O<t<T' indexed on the time interval[0, T]. The following theorem, which we admit, is knownas the Girsanov theorem(cf. Karatzas and Shreve (1988), Dacunha-Castelle and Duflo (1986), Chapter 8).

Theorem 4.2.2 Let ({1t)09~T be an adapted process satisfying l{ O;ds < 00

a.s. and such that the process (Lt)O<t<T defined by

t; = exp (-it e.s», - ~ it O;dS)is a martingale. Then, under the probability p(L) with density LT relative to P,the process (Wt)o~t~T"defined by Wt = B; + f~ Osds, is a standard Brownian

motion.

Remark 4.2.3 A sufficient condition for (Lt)09~T to be a martingale is:

(4.4)

66 The Black-Scholes model Pricing and hedging options in the Black-Scholes model 67

Theorem 4.2.4 Let (Mt)09~T be a square-integrable martingale, with respectto the filtration (Ft)09~T. There exists an adapted process (Ht)O<t<T such that

E (foT H;ds) < +00 and - -

Vt E [0,T] u, = Mo + it n.s», a.s.

Note that this representation only applies to martingales relative to the naturalfiltration of the Brownian motion (cf. Exercise 26).

From this theorem, it follows that if U is an FT-measurable, square-integrablerandom variable, it can be written as

U = E (U) + iT u.s», a.s.,

where (Ht) is an adapted process such that E (foT

Hlds) < +00. To prove it,

consider the martingale M; = E (UIFt ) . It can also be shown (see, for example,Karatzas and Shreve (1988» that if (Mt)O~t~T is a martingale (not necessarilysquare-integrable) there is a representation similar to (4.4) with a process satisfying

only foT

Hlds < 00, a.s. We will use this result in Chapter 6.

4.3 Pricing and hedging options in the Black-Scholes model

4.3.1 A probability under which (St) is a martingale

We now consider the model introduced in Section 4.1. We will prove that thereexists a probability equivalent to P, under which the discounted share price St =e-rtSt is a martingale. From the stochastic differential equation satisfied by (St),we have

From Theorem 4.2.2, with Ot = (J.L - r) / a, there exists a probability P" equivalentto P under which (Wt)O<:t"<T is a standard Brownian-motion. We will admitthat the definition of the stochastic integral is invariant by change of equivalentprobability (cf. Exercise 25). Then, under the probability P., we deduce fromequality (4.5) that (St) is a martingale and that

St = So exp(aWt - a 2t/2).

(see Karatzas and Shreve (1988), Dac~nha-Castelle and Duflo (1986». The proofof Girsanov theorem when (Ot) is constant is the purpose of Exercise 19:

4.2.3 Representation ofBrownian martingales

Let (Bt)O<t<T be a standard Brownian motion built on a probability space(n, F, P) -;~d let (Ft)O~t~T be its natural filtration. Let us recall (see Chapter 3, Proposition 3.4.4) that if (Ht)O<t<T is an adapted process such that

E (JoT Hldt) <" 00, the process (J~ lIs-dBs) is a square-integrable martin

gale, null at o. The following theorem shows that any Brownian martingale can be

represented in terms of a stochastic" integral.

dSt = -re-rt St dt +«r'as,= St ((J.L - r)dt + adBt).

Consequently, if we set W t = 13t + (J.L - r)t/a,

dSt = StadWt. (4.5)

"Ct Vo +I tHudSu

= Vo +I tHuaSudWu.

Under the probability p:;', SUPtE[O,Tj Vt is square-integrable, by definition of admissible strategies. Furthermore, the preceding equality shows that the pro.c~ss

(lit) is a stochastic integral r~la~ve to (Wt ): It follows (cf..Chapter 3, ProyoSItlOn3.4.4 and Exercise 15) that (Vt ) ISa square-mtegrablemartmgale under P . Hence

Vt = E* (VTIFt) ,

69

"

Remark4.3.3 When the random variable hces: be written as h = f(ST)' we canexpress the option value Vt at time t as a function oft and St. We have indeed

',\It E* (e-r(T~t) f(ST )IFt)

= E* (e':"r(T-t) f ( Ster(T-t)e"(WT-W, )~(,,2/2)(T-t))IFt) .

b

The random variable S, is .r.:t-measurable and, under P*, W T - Wt is independentof Ft. Therefore, from Proposition A.2.5 of the Appendix, we deduce

Vt =F(t, St),

\It(</JY = ertMt = E* (e-r(T-t)hIFt).

This expression clearly shows that \It (</J) is a non-negative random variable, withsUPO<t<T \It(</J) square-integrable under P* and that VT(</J) = h. We have foundan admissible strategy replicating h. 0

where

F(t, x) = E* (e-r(T-t) f (xe r(T-t)e,,(WT-W,j_(,,2/2)(T-t)) ) . (4.7)

Since, under P*, WT - W t is a zero-mean normal variable with variance T - t, 1+~ -~~dF(t, x) = e-r(T-:t) f (xe(r-,,2/2)(T-t)+"Yv'T-t) e y.-~ ~

Pricing and hedging options in the Black-Scholes model

and consequently

\It = E* (e-r(T-:-t) hlFt) . (4.6)

So we have proved that if a portfolio (HO, H) replicates the option defined byh, its value is given by equality (4.6). To complete the proof of the theorem, itremains to show that the option is indeed replicable, i.e. to find some processes(H2) and (Ht) defining an admissible strategy, such that

H~S~ + n.s, = E* (e-r(T-t) hlFt) .

Under the probability P*, the process defined by M; = E*(e-rThIFt} is asquare-integrable martingale. The filtration (Ft ) , which is the natural filtration of(B t ) , is also the natural filtration of (Wt ) and, from the theorem ofrepresentationof Brownian martingales, there exists an adapted process (Kt)O<t<T such that

E* (JoT K~ds) < +00 and - -

\It E [0, T] u, = Mo +I tKsdWs a.s.

o· - 0 - .The strategy </J = (H ,H), WIth H, = Kt/(aSt) and H'; = M; - HtSt, IS then,from Proposition 4.1.2 and equality (4.5), a self-financing strategy; its value attime t is given by


Vt = E* (e-r(T-t) hlFt) . "

Thus, the option value at time t can be. naturally defined by the expression

E* (e-r(T-t) hJFt}. ..' . o. .Proof. First, let us assume that there IS an admissible strategy (H ,H), replicatingthe option. The value at time t of the portfolio (H2, H t) is given by

\It == H~S~ + u.s;and, by hypothesis, we haveVr = h. Let "Ct =Vte- rt be the discounted value

- 0 -Vt = n; + u.s;

Since the strategy is self-financing, we get from, Proposition 4.1.2 and equality

(4.5)

4.3.2 Pricing

In this section, we will focus on European options. A European option will bedefined by a non-negative, FT-measurable, random variable h. Quite often, h canbe written as f(ST)' (f(x) = (x-K)+ inthecaseofacall,f(x) = (K-:z)+in the case of a put). As in the discrete-time setting, we will define the optionvalue by a replication argument. FOf technical reasons, we will limit our study to

the following admissible strategies:

Definition 4.3.1 A strategy </J = (m, H t) 09~T is admissible if it is self-financing

and if the discounted value "Ct(</J) = H2 + Ht!;t_ofthe corresponding portfolio is,for all t; non-negative and such that SUPtE[O,Tj \It is sguare-integrable under P*.

An option is said to be replicable if its payoff at maturity is equal to the finalvalue of an admissible strategy. It is clear that, for the 'option defined by h to bereplicable, it is necessary that h should be square-integrable under P* . In the caseof a call (h = (ST - K)+), this property indeed holds since E*(S}) < 00; note

that in the case of a put, h is even bounded.Theorem 4.3.2 In the Black-Scholes model, any option defined Hy a non-negative,:F -measurable random variable h, which is square-integrable under the probabi~ty P*, is replicable and the value at time t ofany replicating portfolio is given

by

68


2. The 'implied' method: some options are quoted on organised markets; the priceof options (calls and puts) being an increasing function of a (cf. Exercise 21),we can associate an 'implied' volatility to each quoted option, by inversionof the Black-Scholes formula. Once the model is identified, it can be used toelaborate hedging schemes.

In those problems concerning volatility, one is soon confronted with the imperfections of the Black-Scholes model. Important differences between historicalvolatility and implied volatility are observed, the latter seeming to depend uponthe strike -price and the maturity. In spite of these incoherences, the model isconsidered as a reference by practitioners.

70

F can be calculated explicitly for calls and puts. If we choose the case of the call,where f(x) = (x - K)+, we have, from equality (4.7)

F(t,x) = E· (e-r(T-t) (xe(r-0'2/2)(T-t)~0'(WT-W.)- K)+)

= E (xeO'Y6g- 0'26/2 _ K e-r6) +

where 9 is a standard Gaussian variable and 0 = T - t.Let us set

log (xl K) + (r + (2/2) 0 Ind - and' d2 = d1 - avO.1- aVO

Pricing and hedging options in the Black-Scholes model 71

Using these notations, we have

, I F'(~, x) =e-rtF (t, xert) ,

we have Vi = fi'(t, St) and, for tcT; from the Itoformula

so that P-(t, St) ca~ be written as

'-, P (t, St) = P (0, So) + it ~~' (u, Su) dSu

t - t 2 -

18F ( -) 11 8 F ( -) --+ 0; at w.S« du+ 0 28x2 U,Su d(S,S)u, ,

From equality dSt = aStdWt, we deduce

- - 2 -2d(S, S)u = a ~udu,

t - , / t

P(t,St)'=p(o,so}+1a~~ (u,~u)SudWu+ 1,Kudu .

Since p(t, St) is a martingale under P", the process K, is necessarily null (cf.

4.3.3 Hedging calls and puts

In the proof of Theorem 4.3.2, we referred to the theorem of representation ofBrownianmartingales to show the existence of a rep,licating portfolio. In practice,a theorem of existence is not satisfactory and it is essential to be able to build areal replicating portfolio to hedge an option.

When the option is defined by a random variable h = f (ST ), we show that it ispossible to find an explicit hedging portfolio. A replicating portfolio must have,at any time t, a discounted value equal to

Vi =e-rtF(t, St),

where F is the function defined by equality (4.7). Under large hypothesis on f(and, in particular, in the case of calls and puts where we have the closed-formsolutions of Remark 4.3.3), we see that the function F is ofclass Coo on [0, T[xIR..If we set

(4.9)F(t, x) = tee:"N(-d2) - xN(-dd·

The reader will find efficient methods to compute N(d) in Chapter 8.

Remark 4.3.4 One of the main features of the Black-Scholes model (and oneof the reasons for its success) is the fact that the pricing formulae, as well asthe hedging formulae we will give later, depend on only one non-o?servableparameter: a, called 'volatility' by practitioners (the drift parameter J.L disappearsby change of probability). In practice, two methods are used to evaluate a:

I. The historical method: in the present model, a2T is the variance oflog(ST) andthe variables 10g(STISo), log(S2TIST), ... , 10g(SNTIS(N-l)T) are independent and identically distributed. Therefore, a can be estimated by statisti~almeans using the asset prices observed in the past (for example ~y calculatingempirical variances; cf. Dacunha-Castelle and Duflo (1986), Chapter 5).

1 jd 2N(d) == -,-, e- X /2dx.~ -00

Using identical notations and through similar calculations, the price of the put is

equal to

where

F(t,x) _ E [(xeO'Y6g- 0'26/2 _Ke-r6) l{g+d22:0 } ]

j +oo ) e-y2

/2= ( xeO'Y6y-0'2 6/2 _ Ke- r6 ---dy

-d2 ~

I. )e-y2

/2

(xe- 0'Y6y- 0'26/2 _ Ke- r6 ---'-dy.

. ~-00

Writing this expression as the difference of two integrals and in the first one using

the change of variable z = y + aVO, we obtain ,

F(t, x) = xN(d1 ) - Ke-r6N(d2), (4.8)

and in the case of the put

P (t, St) = P (0, SO) + it a ~~ (u, S~) SudWu

- ( -) t aP ( -) -= F O,So + io

ax «,s; es;The natural candidate for the hedging process H, is then

4.4 American options in the Black-Scholes model

4.4.1 Pricing American options

We have seen in Chapter 2 how the pricing of American options and the optimalstopping problem are related in discrete-time setting. The theory of optimal stopping in continuous-time is based on the same ideas as in discrete-time but is farmore complex technically speaking. The approach we proposed in Section 1.3.3of Chapter 1, based on an induction argument, cannot be used directly to priceAmerican options. Exercise 5 in Chapter 2 shows that, in a discrete model, it ispossible to associate any American option to a hedging scheme with consumption.

American options in the Black-Scholes model 73

Definition 4.4.1 A trading strategy with consumption is defined as an adaptedpro.cess if> = ((Hf, H t))095,T, with values in IR?, satisfying thefollowing properties:

1. iT IHfldt + I THtdt < +00 a.s.

. rt rt

2. H?sf + u.s; = HgSg + HoSo + ioH~dS~ + in HudSu - c, for all

I t E [0, Tj, where (Ct )05,t5,Tis an adapted, continuous, ~on-decreasingprocessnull at t = 0; Ct corresponds to the cumulative consumption up to time t.

An American option is naturally defined by an adapted non-negative process(ht )09 5, -r: ' To sim?lify, we wil.l only study processes of the form ht = 'lj;(St) ,where w IS a .;:ontmuous function from IR+ to IR+, satisfying: 'lj;(x) ~ A +B«, 'Vx E IR ,for some non-negative constants A and B. For a call, we have:'lj;(x) = (x - K)+ and for a put: 'lj;(x) = (K - x)+.

The tra~ing str~tegy with consumption if> = ((Hf, H t) )05,t5,T is said to hedgethe Amencan option defined by ht = 'lj;(St) if, setting Vt (if» = Hf Sf + HtSt,we have ..

'Vt E [O,Tj Vt(if» 2': 'lj;(St} a.s.

De~ote by <1>'" the set of tradi~g strategies with consumption hedging the Americanoption defined by ht = 'lj;(St). If the writer of the option follows a strategy if> E <1>"',he or she p~ssesses a~ an~ time t, a wealth at least equal to 'lj;(St), which is preciselythe payoff If the opuon IS exercised at time t. The following theorem introducesthe minimal value of a hedging scheme for an American option:

Theorem 4.4.2 Let u be the map from [0, Tj x IR+ to IR defined by

u(t, x) = sup E··[e-r(r-t)'lj;(xexp((r_ (a 2/2))(T-t)+a(Wr -Wd))]

rE7't,T

wh~reTi,T represents the set ofstopping times taking values in [t, Tj. There existsa strategy if> E <1>'" such that Vt (¢) = u(t, St),for all t E [0, Tj. Moreover; forany strategy if> E <1>"', we have: Vt(if» 2': u(t,St),forallt E [O,Tj.

To overcome technical difficulties, we give only the outlines of the proof (see~aratzas and Shreve (1988)Jor details). First, we show that the process (e-rtu( t, St))IS t~e Snell env~lope of the process (e-rt'lj;(St)), i.e. the smallest supermartingalewhich bounds It from above under p ". As it can be proved that the discountedvalue of a ~ading ~trategy 'with consumption is a supermartingale under P", wede~uce the inequality: ~(if» 2': u(t, St), for any strategy if> E <1>"'. To show theexistence of a strategy- if> such that Vt(¢) = u(t, St), we have to use a theoremof decomposition of supermartingales similar to Proposition 2.3.1 of Chapter 2 aswell as a theorem of representation of Brownian martingales.

. It i~ ~atural t~ ~onsider u(t; St) as a price for the American option at time t,smce.lt IS the minimal value of a strategy hedging the option.

The Black-Scholes model72

Chapter 3, Exercise 16). Hence

aP ( -) aFH, = s: t, St = ax (t, St) .

If we set Hf = P (t, St) - tt.s; the portfolio (Hf, H t) is self-financing and its

discounted value is indeed Vi = P (t, St).

Remark 4.3.5 The preceding argument shows that it is not absolutely necessaryto use the theorem of representation of Brownian martingales to deal with optionsof the fomi f (ST). . .

Remark 4.3.6 In the' case of the call, we have, using the same notations as inRemark 4.3.3,

aF-a (t,x) = -N(-d1 ) .x .

This is left as an exercise (the easiest way is to differentiate under the expectation).This quantity is often called the 'delta' of the option by practitioners. Moregenerally, when the value at time t of a portfolio can be expressed as ll1(t, St),the quantity (alII/ ax) (t, St), which measures the sensitivity of the portfolio withrespect to the variations of the asset price at time t, is called the 'delta' of theportfolio. 'gamma' refers to the second-order derivative (a 2 1l1 / ax2 ) (t, St), 'theta'to the derivative with respect to time and 'vega' to the derivative of III with respectto the volatility a.

4.4.2 Perpetual puts, critical price

In the case of the put, the American option price is not equal to the European oneand there is no closed-form solution for the function u. One has to use numericalmethods; we present some of-them in Chapter 5. In this section we will only usethe formula .

u(t, x) = sup E* (K~-r(r-t) - xexp (:--a2(7 - t)/2 + a(Wr - Wd)) +rET.,T .

. (4.10)

to deduce some properties of the function u. To make our point clearer, we assumet = O. In fact, it is always possible to come down to this case by replacing Twith

On the other hand, we have

E* ((ST - e-rTK)+!Fr) ~ E* ((ST - e-rTK)IFr) ~ s, - e-rTK'.,

since (St) is a martingale under P*. Hence,

E* ((ST - e-rTK)+ IFr) .~ s; ~ e-rrK

since r ~ 0 and by non-negativity of the left-hand term,

E* ((ST - e-rTK)+IFr) ~ (Sr - e-rrK) + .

We obtain the desired inequality by computing the expectation of both sides. , 0

Remark 4.4.3 Let 7 be a stopping time taking values in [0,T]. The value at time 0of an admissible strategy in the sense of Definition 4.3.1 with value 'l/J(Sr) at time 7

is given by E* (e- rr'l/J(Sr)), since the discounted value of any admissible strategyis a martingale under P*. Thus the quantity u(O, So) = sUPrE70,T E* (e- rr'l/J(Sr))is the initial wealth that hedges the whole range of possible exercises.

As in discrete models, we notice that the American call price (on a non-dividendpaying stock) is equal to the European call price:

Proposition 4.4.4 If in Theorem 4.4.2, 'l/J 'is given by 'l/J(x) = (z - K)+, for allreal x, then we have

u(t,x) = F(t,x)

where F is the function defined by equation.iaB) corresponding to the Europeancall price.

Proof. We assume here that t = 0 (the proof is the same for t > 0). Then it issufficient to show that, for any stopping time 7,

E*(e-rr(Sr - K)+) :::; E*(e-rT(ST - K)+) = E*(ST - e-rTK)+.

is given by the formulae

75

(4.11)

American options in the Black-Scholes model

T - t. Equation (4.10) becomes

u(O,x)= sup E*(Ke-rr-xexp(aWr-(a27/2)))+.

rE70,T .

Let us consider a probability space (0, F, P), and let (Bt)o<t<oo be a standardBrownian motion defined on this space and IR+. Then, we get

u(O, x) sup E(Ke-rr-xexp(aBr-(a27/2)))+rE70,T



(4.12)

noting To,oo the set of all stopping times of the filtration of (Bt)t~o and To,T theset of all elements of To,oo with values in [0,T]. The right-hand term in inequality(4.12) can be interpreted naturally as the value of a 'perpetual' put (i.e. it can beexercised at any time). The following proposition gives an explicit expression forthe upper bound in (4.12).

Proposition 4.4.5 The function

Uoo(x) = sup E.[(Ke-rr - xexp (aBr - (a27/2)))+ l{r<oo}] (4.13)rE70.<X>

Uoo(x) = K - x for x:::; z"

( X) -"'IUoo(x) = (K - z") x* for x > x"

with z" =K,/(l +,) and, = 2r/a2 .'.Proof. From formula (4.13) we deduce that the function uoo is convex, decreasingon [O,oo[ and satisfies: Uoo(x) ~ (K - x)+ and, for any T > 0, Uoo(x) ~

E(Ke-rT - xexp (aBT - (a2T/2)))+, which implies: Uoo(x) > 0, for all

x ~ O. Now we note x" = sup{x ~ Oluoo(x) = K - x}. From the properties ofuoo we have just stated, it follows that

"Ix:::; z" Uoo(x) = K - x and "Ix >x* Uoo(x) > (K - x)+. (4.14)

On the other hand, the Spell envelope theory in continuous time (cf. E1 Karoui(1981)(Kushner (1977), aswell as Chapter 5)enables us to show

Uoo(x) = E [(Ke,,"':'rrz- xexp (aBr% - (a27

x/2)))+ 1h<00}]

where 7x is the stopping time defined by 7x = inf{t ~ 01 e-rtuoo(Xt)«<;« - Xt)+} (with inf 0 = 00), the process (Xf) being defined by: Xf =x exp ((r - a 2 / 2)t + a Bt ) . The stopping time 7 x is therefore an optimal stoppingtime (note the analogy with the results in Chapter 2).

It follows from (4.14) that

7x = inf{t ~ 0IX: :::; z"} = inf {t ~ Ol(r - a2/2)t + aBt :::; log(x* /x)}.

The Black-Scholes model74

where, = 2r1a2 . The derivative of this function is given by

z'Y- 1

¢'(z) = - (K, - b + l)z).. x'Y

It results that ifx ~ K,I(,+l),maxz ¢(z) = ¢(x) = K -xandifx > K,/b+I), max, ¢(z) = ¢ (K,Ib + 1)), and we recognise the required expressions. 0

Remark 4.4.6 Let us come back to the American put with finite maturity T.Following the same arguments as in the beginning of the proof of Proposition4.4.5, we see that, for any t E [0, T[, there exists a real s(t) satisfying

"Ix ~ s(t) u(t,x) = K - x and "Ix> s(t) u(t,x) > (K - x)+. (4.15)

Taking inequality (4.12) into account, we obtain s(t) ~ z", for all t E [0, T[. Thereal number s(t) is interpreted as the 'critical price' at time t: if the price of theunderlying asset at time t is less than s(t), the buyer of the option should exercisehis or her option immediately; in the opposite case, he should keep it.

76Introduce, for any z E IR+, the stopping time Tx,z defined by

Tx,z = inf{t ~ 0IX: ~ z}.

. .., iven by T = T "We fix xWith these notations, the optimal stoppmg ume IS gi x x,x'and we note ¢ the function of z defined by

..1.( ) . E ( -rTZ '1 }(K - x: ) ).'I' Z = e " {Tz,.<OO T z •• +

S. I'S' optimal the function ¢ attains its maximum at point z = x": Wemce T x x"' . . . d . * and

are going to calculate ¢ explicitly, then we will maximise It to etermme x

UOO(x) = ¢(x*). ( ) _ (K _ ) If z < x we haveIf z > x, it is obvious that Tx,z = aand ¢ z - x +. -' ,

by the continuity of the paths of (Xnt"~o'

Tx,z = inf{t ~OIX: = z}

Exercises 77

and consequently

we get

it can be seen that

Vz E [O,x] n [O,K] ¢(z) = (K- z) (~)'Y,

4.5 Exercises

Notes: The presentation we have used, based on Girsanov theorem, is inspired byHarrison and Pliska (1981) (also refer to Bensoussan (1984) and Section 5.8 inKaratzas and Shreve (1988». The initial approach of Black-Scholes (1973) andMerton (1973) consisted in deriving a partial differential equation satisfied bythe the call price as a function of time and spot price. It is based on an arbitrageargument and the Ito formula. For more information on statistical estimation of themodels' parameters, the reader should refer to Dacunha-Castelle and Duflo (1986)and Dacunha-Castelle and Duflo (1986) and to the references inthese books.

Exercise 19 The objective of this exercise is to prove the Girsanov theorem 4.2.2in the special case where the process (Ot)is constant. Let (Bt)o9~Tbe a standardBrownian motion with respect to the filtration (Ft)O<t<T and let JL be a real-valuednumber, We set, for a~ t,~ T, i; =exp (- JLBt .: (i.t2 12)t).

1. Sh6w that (Lt)O<t<T ~s a martingale ~elative to the filtration (Ft) and thatE(Lt ) = 1, for all t E [0, T]. .

2. We note p(L.) the density probability Lt with respect to the initial probabilityP. Show that the probabilities p(LT ) and p(L.) coincide on the a-algebra Ft.

3. Let Z be an FT-measurable, bounded random.variable, Show that theconditional expectation of Z, under the probability p(LT), given F«, is

E'(LT)(ZIFt) = E (ZLTIFt).Lt

if z > x .if Z E [0, x] n [0, K]if z E [O,x] n [K,+oo[.

(K - x)+(K - z)E (exp (-rT1og(z/x)/u))

a~(z) = {

The maximum of ¢ is attained on the interval [0, x] n [0, K]. Using the following

formula (proved in Exercise 24)

. E (e-aT&) = exp (~b - IbJv!JL2 + 2a) ,

¢(z) = (K - z)+E (e-rTz,· l h ,. <OO})

(K - z)+E (e- r T z, . ) ·

. . -roo _ a Using the expression of Xt in terms of B t , weWIth, by convention, e -. ,see that, for z ~ x,

Tx,z = inf{t~OI(r-a;)t+aBt=IOg(ZIX)}

inf {t ~ 0l/.tt + e, = ~ IOg(ZIX)} ,

with JL = ria - a12. Thus, if we note, for any real b,

Tb =inf{t ~ OIJLt + B; = b},

79Exercises

Exercise 25

1. Let P and Q be two equivalent probabilities on a measurable space (n, A).Show that if a sequence (Xn ) of random variables converges in probabilityunder P, it converges in probability under Q to the same limit.

2. Notations and hypothesis are those of Theorem 4.2.2. Let (Ht)O<t<T be an

adapted process such that J; H;ds < 00 P-a.s. The stochastic-i.rtegral of(Ht ) relative to B, is well-defined under the probability P. We set

x, = I tn.e», +I t

Hs(}sds.

Since p(L) and P are equivalent, we have J; H;ds < 00 p(L)-a.s. and we

can define; under p(L), the process

lit = I tHsdWs.

The question is to prove the equality of the two processes X and Y. To doso, it is advised to consider first the case of simple processes; and to use the

fact that if (Ht)o9~T is an adapted process satisfying J; H;ds < 00 a.s.,

there is a sequence (Hn) of elementary processes such that JoT (H 5 - H;')2 dsconverges to 0 in probability.

Exercise 26 Let (Bt)O<t<l be a standard Brownian motion defined on the timeinterval [0,1]. We note Crd099 its natural filtration and we consider T anexponentially distributed random variable with parameter A, independent of .rl.For t E [0,1], we note 9t the a-algebra generated by.rt and the random variableT t\ t. '

1. Show that (9t)O<t<l is a filtration and that (Bt)O<t<l is a Brownian motionwith respect to (OS. - -

2. For t E [0,1], we set M, = E (I{T>1}19t). Show that M, is equal to,.. e->'(l-t) l{T>t} a.s. The following property can be used: if 8 1 and 8 2 aretwo sub-a-algebras and X a non-negative random variable such that the aalgebra generated by 8 2 and X are independent of the a-algebra 8 1, thenE ('~18l V 82) = E (XI82 ) , where 8 1 V 82 represents the o-algebragener-ated by 8 1 and 8 2. .

3. Showthat there exists-no path-continuous process (Xt ) such that for all t E[0,1]' P (Mt = X t ) =,1 (remark that we would necessarily have

P ('Vt E [0, IJ M, = X t ) = 1).

Deduce that the martingale (Mt ) cannot be represented as a stochastic integralwith respect to (Bd. .

Exercise 27 The reader may use the results of Exercise '18 of Chapter 3. Let(Wtk~o be an .rt-Brownian motion.

Tt = inf{t 2: 0 I J-Lt + B; = b}

with the convention: inf 0 = 00.

1. Use the Girsanov theorem to show the following equality:

'Va, t > 0 E (e-a(Tt: I\t)) = E (e-a{T~I\t) exp (J-LBT~1\t - J-L; T~ t\ t) ) .

The Black-Scholes model784. We set Wt = J-Lt + Bi, for all t E [0,T]. Show that for all real-valued u and for

all sand t in E [0,T], with s ~ t, we have

E(LT) (eiU(w.-w.) \.rs) = e- u 2(t-s)/2.

Conclude using Proposition A.2.2 of the Appendix.

Exercise 20 Show that the portfolio replicating a European option in the Black

Scholes model is unique (in a sense to be specified).

Exercise 21 We consider an option described by ~ = f.(ST) and w~ note F thefunction of time and spot corresponding to the option pnce (cf. equation (4.7)).

1. Show that if f is non-decreasing (resp. non-increasing), F(t, x) is a non

decreasing function (resp. ,non-increasing) of x.2 We assume that f is convex. Show that F(t,x) is a convex ~unction ~f z, a

. d . function of t if r - 0 and a non-decreasing function of a manyecreasmg - f J ,. ality:

case. (Hint: first consider equation (4.7) and make us~ 0 ensen ~ mequ .cI>(E(X)) ~ E (cI>(X)), where cI> is a convex function and X IS a random

variable such that X and cI>(X) are integrable.)

We note Fe (resp. Fp) the function F obtained when f(x) = (x ~ K)+ (resp.3. f(x) = (K _ x)+). Prove that Fe(t,.) and Fp(t,.) are non-negatlvefor t < T.

Study the functions Fe(t,.) and Fp(t, .) in the neighbourhood of 0 and +00.

Exercise 22 Calculate under the initial probability P, the probability that a Eu-

ropean call is exercised. . .

Exercise 23 Justify formulae (4.8) and (4.9) and calculate for a call and a put the

delta, the gamma, the theta and the vega (cf. Remark 4.3.6).

Exercise 24 Let (Btk?o be a standard Brownian motion. For any real-valued J-L

.and b, we set

2. Prove the inequality'

'Va,t>O E(e-a(T~t\t)exP(J-LBT~I\t-~2T~t\t)l{t<Tn) ~e-at.

3. Deduce from above and Proposition 3.3.6 that .

'Va> 0 E (e-aTt: 1{Tt: <oo}) = exp (J-Lb-lbIJ 2a+ J-L2) .

4. Calculate P (Tt < 00).

The Black-Scholes model 81Exercises

dSt

.

S = JLdt + udWt,. t.

where (Wt}tE[O,Tj is a standard Brownian motion on a probability space (O,\F, P),JL and o are real-valued, with o > 0. We note (Ft)tE[O,Tj the filtration generatedby (WdtE[O,Tj and assume that Ft represents the accumulated information up totime t

J'( . . ..'

3. Prove that there exists a probability P" equivalent to P, under which thediscounted stock price is a martingale. Give its density with respect to P.

4. In the remainder, we will tackle the problem of pricing and hedging a call withmaturity T and strike price K.

(a) Let (H?, Hl) be a self-financing strategy, with value \It at time t. Show thatif (\It/SP) is a martingale under P" and if VT = (ST - K)+, then

'TIt E [0, T) Vi = F(t, St),

where F is the function defined by

F(!, x) = E" (x exp (J.T u(, )dW. - ~ J.T U'U)d,) -: K e- J.T .(.)d.) +

and (Wd is a standard Brownian motion under p ".

(b) Give an expression for the function F and compare it to the Black-Scholesformula.

(c) Construct a hedging strategy for the call (find H?and Ht ; check the selffinancing condition).l

I

Problem 2 Garman-Kohlhagen modelThe Garman-Kohlhagen model (1983) is the most commonly used model to priceand hedge foreign-exchange options. It derives directly from the Black-Scholesmodel. To clarify, we shall concentrate on 'dollar-franc' options. For example, aEuropean call on the dollar, with maturity T and strike price K, is the right to buy,at time T, one dollar for K francs.

We will note S; the price of the dollar at time t, i.e. the number of francs-perdollar. The behaviour of S, through time is modelled by the following stochasticdifferential equation:

1. Express S, as a function of So, t and Wt. Calculate the expectation of St.

2. Show that if JL > 0, the process (St)tE[O,Tj is asubmartingale.

3. Let U; = 1/St be the exchange rate of the franc' against the dollar. Show thatUc satisfies the following stochastic differential equation

dUt (2 .- = u - JL)dt - udWt.u,

!I

Problem 1 Black-Scholes model with time~dependent parameters ~e consider once again the Black-Scholes model, assuming that ~e ass~t p~ces aredescribed by the following equations (we keep the same notations as In this ch~~-

J .

ter)

{

dSP = r(t)Spdt

dS t = St(JL(t)dt+ u(t)dBd

. where ret), JL(t), u(t) are detenninistic functions of time, continuous on [0, T).

Furthermore, we assume that inftE[o,T] u(t) > 0: .

2.(a) .Let (X

n) be a sequence of real-valued, zero-mean normal random vari.ables

converging to X in mean-square. Show that X is a normal random variable.

(b) By approximating o by simple .functi?ns, show that J~ u(s )dBs is anormalrandom variable and calculate ItS varIance. 0 • . .

1. Prove that

s, = So exp (~t JL(~)ds + ~t u(s)dBs _ ~ ~t u2(S)dS)

.

You may consider the process

z, = s, exp --:- .: JL(s)ds + ~t u(s)dBs :- ~ ~t u2(s)dS)

.

Deduce that if x~ JL

. (a2T ) (2), - JL + aT)E (

aWT1 .}) -exp -- + 2a), N 1m .e {WT~~,inf.::;TW.~A -. . 2 vT

2. Let H ~ K; we are looking for an analytic formula for

C = E (e-rT(XT - K)+1{inf.::;TX.~H}) ,

where X, = x exp ((r - 'u2 /2) t+ uWt) . Give a finan~iaUnterpretati.?n to

this value and give an expression for the probability p that makes Wt -

(r / o - o /2) t + W t ~ standard Brownian motion. _

3. Write C as the expectation under Pof a random variable function only ofWT

and sUP~~s~TWs . . ,.

4. Deduce an analytic formula for C.

82 The Black-Scholes model Exercises

Deduce that if 0 < J.L < a2, both processes (St)tE[O,T] and (Ut)tE[O,TJ aresubmartingales. In what sense does it seem to be paradoxical?

IIWe would like to price and hedge a European call on one dollar, with maturity Tand strike price K, using a Black-Scholes-type method. From his premium, whichrepresents his initial wealth, the writer of the option elaborates a strategy, definingat any time t a portfolio made of HP francs and H, dollars, in order to create, attime T, a wealth equal to (ST - K)+ (in francs).

At time t, the value in francs of a portfolio made of Hp francs and H, dollarsis obviously

Vt = H~ + n.s; (4.16)

We suppose that French francs are invested or borrowed at the domestic rate TOand US dollars are invested or borrowed at the foreign rate TI' A self-financingstrategy will thus be defined by an adaptedprocess ((HP, Ht»tE[O,Tj, such that

dVt = ToHpdt + TIHtStdt + HtdSt (4.17)

where Vt is defined by equation (4.16).

1. Which integrability' conditions must be imposed on the processes (HP) and(Hd so that the differential equality (4.17) makes sense?

2., Let Vi = e-rotVt be the discounted value of the (self-financing) portfolio(HP, Ht}. Prove the equality

,dVt = Hte-rotSt(J.L + TI - To)dt+Hte-rotStadWt.

3.

(a) Show that there exists a probability P, equivalent to P, under which theprocess

TiT _ J.L + TI - TO t UTt't't - + t't'ta

is a standard Brownian motion.(b) A self-financing strategy is.said to be admissible if its discounted value Vi

is non-negative for all t and if SUPtE[O,Tj (Vi) is square-integrable under

P. Show that the discounted value of an admissible strategy is a martingaleunder P.

4. Show that if an admissible strategy replicates the call, in other words it is worthVT = (ST - K)+ at time T, then for any t ~ T the value of the strategy attime t is given by

where

F(t,x) = E(xexP(-(TI.+(a2j2»(T-t)+a(WT - Wt»)

- K e-~o(T-t)). +

1I'

I

83(The symbol E stands for the expectation under the probability P.)

5. Show (through detailed calcUlation) that

F(t,x) = e-r,(T-t)xN(dd - Ke-ro(T-t)N(d2)

where N is the distribution function of the standard norrnallaw d, an

dl

10g(xjK) + (TO - TI + (a2j2»(T - t)a..;T=t

d2 10g(xjK) + (TO - TI - (a2j2»(T - t)a..;r-=-t

6. The next step is to show that the option is effectively replicable.

(a) We set St = eh -ro)tSt. Derive the equality

es, = aStdWt.

(b) Let ~ be the function defined by F(t, x) = e-rotF(t, xe(ro-r,jt) (F is thef~nctl_on defined in Question 4). We set C, = F(t S) and G - -rotcF(t, St}. Derive the equality , t t - e t =

- aFsc, = -(t St)ae-rotS dW';ax ' t t-

(c) renlicati that the c.all is replicable and give an explicit expression for therep icatmg portfolio ((Hp , Ht».

7. fors« down a put-call parity relationship, similar to the relations~iP we gave

dor stocksh' alndd grve an example of arbitrage opportunity when this relationshipoes not 0 .

Proble~ 3 Option to exchange one asset for another .

W~ tO~~der a fin2anci~1 market in which there are two risky assets with respectivepnces t a~d St at tlI~e t and a riskless asset with price So = ert at time t

~~::S~~~~~e~:nt~;1~~~:~~!s and Slover time are modelled by the followin~

{

dS! S! (J.Lldt + aldB!)

/ dS~ Sl (J.L2dt + a2aB l)where (BI) [ . and (B2) .'. d t tE O,T] t tE[O,Tj are two Independent standard Brownian mo

nons efined on a probability space (0 F P). with :> 0 . d ' , ,J.LI,J.L2,al anda2arerealnumbersabiesa11 an ~2 > O. We note!t the a-algebra generated by the random vari~

• and B. for s ~ t. Then the processes (B I) d (B2):rre (Fd-Brownian motions and, for t ~ s, the vecto/ (~IO~];~B2 ~ ~JO)?JIndependent of F.. - t·. , t . • IS

.J'

I

We study the pricing and hedging of an option giving the right to exchange one ofthe risky assets for the other at time T.

1. We set (h = (ILl - r) /0"1 and (h = (IL2 - r) /0"2, Show that the process definedby

u, = exp ( -e.e; - B2B; - ~(B~ + B~)t) ,

is a martingale with respect to the filtration (Ft)tE[o,T]'

2. Let P be the probability with density MT with respect to P. We introduce theprocesses WI and W 2defined by Wl = Bf +B1t and Wl = B; +B2t. Derive,under the probability P, the joint characteristic function of (Wl ,Wl). Deducethat, for any t E [0, T], the random variables Wl and wl ar~ independentnormal random variables with zero-mean and variance t under P.

In the remainder, we' will admit that, under the probability P, the processes

(Wl )o:9~T"and(Wl)O:9~Tare (Ft)-indepe~dent st;n?~d Brownian motionsand that, for t 2: s, the vector (Wl - W s

1,Wt - W s ) IS independent of Fs •

3. Write Sland Sl as functions of SJ, S5,!Vl and Wl and show that, under P,the discounted prices Sl = e:"Sl and Sl == e:"S; are martingales.

We want to price and hedge a European option, with maturity T, giving to theholder the right to exchange one unit of the asset 2 for cine unit of the as~et.1: :0do so we use the same method as in the Black-Scholes model. From hIS initialwealth, the premium, the writer ?f the opt~on builds a strategi' defini~g a~ anytime t a portfolio made of HP Units of the riskless asset and H; and H; Units ofthe assets 1 and 2 respectively, in order to generate, at time T, a wealth equal to(St - Sf)+· A trading strategy will be defined by the three adapted processesHO, HI and H 2.

·11

1. Define precisely th~ self- financing strategies'and prove that, if Vt = e-rtVt isthe discounted value of a self-financing strategy, we have

d1-r - HI - rtSl dW l + H2e-rtS20" dW 2Yt - t e t 0"1 t t t 2 t .

2. Show that if the processes (HI )o~t~~ and (Hl)o:9~T of a self-financingstrategy are uniformly bounded (which means that: 30 > 0, Vet,w) E [O,~] xfl, IHt(w)1 ::; 0, for i = 1,2), then the. discounted value of the strategy IS amartirigale under P.

3. Prove that if a self-financing strategy satisfies the hypothesis of the previousquestion and has a terminal value equal to VT = (St .:... Sf)+ then its value atany time t < T is given by

(4.18)

85where the function F is defined by

- ( (WI I) ~ ,,2 )F(t, Xl, X2) = E x1eO'I r-W• "'-T(T-t) _ x2e0'2(Wf-W,2)-=f(T-t) +'

(4.19)the ~ymb~l E repres~nting the expectation under P.The existence of a strategyhaving this value will be proved later on. We will consider in the remainderthat the value of the option (St - Sj)+ at time t is given by F(t, Sf, Sl).

4. Find a parity relationship between the value of the option with payoff (Sl _sj)+ and the symmetrical option with payoff (Sf - St)+, similar to :theput-call parity relationship previously seen and give an example of arbitrageopportunity when this relationship does not hold.

ill

The objective of this section is to find an explicit expression for the function Fdefined by (4.19) and to establish a strategy replicating the option.

1. Let 91 and 92 be two independent standard normal random variables and let Abe a real number.

(a) Show that under the probability p(A), with density with respect to P givenby ._

dP(A) 2__ = eAgI-A /2

dP 'the random Gaussian variables 91- A arid 92 are independent standard

variables.

(b) Deduce that for all real-valued Y1, Y2, Al and A2' we have

E (exp(Yl + A19d' - exp(Y2 + A292))+ .

=eYI+A~/2N-(Yl - Y2 + A?) _ eY2+A~/2N (Y1 - Y2 -A~)JA~+A~ JA~+A~ ,

where N is the standard normal distribution function.

2. Deduce from the previous question an expression for F using the function N.3. We set c, = «:"F(t, Sf, S;). Noticing that

/ :

. c, = F(t, sf, S;) = E (e- rT (S} - Sf)+ 1F t) ,

prove the equality

- of - - of _ _ac, = !lx (t,SI,S;)O"l e-rtSf dwl + ~(t,SI,S;)0"2e-rtSt2dWr

u 1 uX2

Hint: use the fact that if (Xt ) is an Ito process which can be written as X =t t t

X o + fo J1dW; .: f0 7 ; dw ; + f~ Ksds and if it is a martingale under P,then K, = 0, dtdP-almost everywhere.

(ii) For all t E [0, Tj,

HOSo+HtSt = HgSg+HoSo+ r H~dS~+ r HudSu- r c(u)du, a.s.t t 10 . 10 10.

(iii) For all t E [0, Tj, c(t) ~ 0 a.s.

(iv) For all t E [0, Tj. th~ rand.om variable H2S~ + HtSt is non-negative and

sup (Itiffi.+HtSt+ tC(S)dS)tE[O,T] 10

is square-integrable under the probability p ",

1. Let (H2)09~T. (Ht)O~t~T and (c(t))09~T be three adapt~dpro~~~sessatis-fying condition (i) above. We set Vi = H2S~ + HtSt and Vt = e Vi, Then

4. Build a hedging scheme for the option.

Problem 4 A study of strategies with consumptionWe consider a financial market in which there is one riskless asset, with price S~ =ert at time t (with r ~ 0) and one risky asset, with price St at time t: The modelis studied on the time interval [0,Tj (0 ~ T < 00). In the following, (St)O~t~Tis a stochastic process defined on a probability space (0, F, P), equipped witha filtration (Ft)o9~T' We assume that (Ft)O~t~T is the natural filtr~tion of astandard Brownian motion (Bt)o9~T and that the process (St)09~T IS adapted

to this filtration.

We want to study strategies in :ovhich consu~ption is allowed. The dynamic of

(St)09~T is given by the Black-Scholes-model

dSt = St(JLdt + adBt) ,

with JL E IR and a > O. yve note P" the probability with density

exp (-(JBT - (J2T /2)

with respect to P, with (J = (JL -r)/a. Under p •. the process (Wt)09~T.definedby Wt = (JL - r)t/a + Be, is a standard Brownian motion.

A strategy with consumption is defined by three stochastic processes: (H~)09~T,(Ht)O<t<T and (c(t))09~T' H~ and H, represent respectively the quantities. ofriskless asset and risky asset held at time t, and c(t) represents th~ consum?~onrate at time t. We say that such a strategy is admissible if the followmg conditions

hold:(i) The processes (H2)09~T. (Ht)O~t~T and (c(t))09~T are adapted and satisfy

iT (IH~I + H~ + Ic(t)1) dt < oo,a.s.

, 87

II/

We now suppose that the volatility is stochastic. i.e. that the process (St)O<t<T isthe solution of a stochastic differential equation of the.following form: --

Exercises

E' ({C(t)dt)' < 00 and E' ({ e:"'C(t)dt) '" x..

Prove that (c(t))O~t::;T is a budget-feasible consumption process with aninitial endowmentx. Hint: introduce themartingale (Mt)O::;t::;T defined by

Mt = E· (x + foT e-rsc(s)dsIFt) and apply the theore~ of martingales

representation.

(c) An investor with initial endowment x wants to consume a wealth corresponding to the sale of p risky assets by unit of time whenever.S, crosses

~. some. ?arrier K upward (that corresponds to c(t) = pSt1{S,>K}). Whatconditions on p and x are necessary for this consumption process to bebudget-feasible?

show that condition (ii) is satisfied if and only if we have, for all t E [0, Tj.

t:t = Vo + it HudBu -it c(u)du, a.s.

with s. = «r:s; and c(u) = e-ruc(u).

2. We suppose that conditions (i) to (iv) are satisfied and we still note t:t =e-rtVi = e:" (H2S2 + HtSt). Prove that the process (t:t)O<t<T is a super-martingale under probability p. . - -

3. Let (c(t))09~T be an adapted process with non-negative values such that

E· (JOTc(t)dtf ~ 00 and let x > O. We say that (c(t))09~T is a budget

feasible consumption process from the initial endowment x if there exist someprocesses (H2)o~t~T and (Ht)O~t~T such that conditions (i) to (iv) are satis-fied, and furthermore Vo = HgSg + HoSo = x. .

(a) Show that if the process (c(t))O<t<T is budget-feasible from the initial

endowment x then E· (JoT e-rtc(t)dt) .~ x. '

(b) Let (c(t))09~T be an adapted process. with non-negative values and suchthat

dSt = St(JLdt + a(t)dBt), (4.20)

where JL E IRand (a(t) )O::;t~T is 'an adapted process. satisfyingI

Vt E [O,Tj al ~ a(t) ~ (,2,

for so~e constants.o, and az such that 0 < al < az- We consider a Europeancall With maturity T and strike price K on one unit of the risky asset. We know


I. ,

86

(see Chapter 5) that if the process (a(t) )O<t'<T is constant (with a(t) = a for anyt) the price of the call at time tis C(t, St), where the function C(t, x) satisfies

ec a2x2 a2c ec-(t,x) + ---;:;-z(t,x) + rx-;:)(t,x) - rC(t,x) =aat 2 ox ox

on [0, T[x10, oo[

'C(T, x) = (x - K)+.

We note C1 the function C corresponding to the case a = a1 and C2 the functionC corresponding to the case a = a2. We want to show that the price of the call attime 0 in the model with stochastic volatility belongs to [C1(0, So), C2(0, So)l.

Recall that if (OdO<t<T is a bounded adapted process, the process (Ldoi;t~T

d~fined by i, = exp ~.r; e.se, - ~ I; O;ds) i~ a martingale.

1. Prove (using the price formulae written as expectations) that the, functionsx I-t C1(t,x) and x I-t C2(t,x) are convex.

2. Show that the solution of equation (4.20) is given by

s, = S~ exp (Ilt + it a(s)dBs - ~ it a2(S)ds) ..

3. Determine a probability P* equivalent to P under which the process definedby Wt = B, + I;(Il- r)/a(s)ds is a standard Brownian motion.

4. Explain why the price of the call at time ais given by

. Co = E* (e:-rT(ST - K)+) .

5, We set S~ = e-rtSt. Show that E* (Sn s S6elT~t.I

6. Prove that the process defined by

. l't .ec .M, = 0 e-ru a} (u, Su)a(u)SudWu

is a martingale under probability P* . .

7. Using Ito formula and Questions 1 and 6, show that «rc, (t, Sd is a sub-martingale under probability .P*. Deduce that C1(0, So) ~ Co. . ,

8. Derive the inequality Co s C2(0, So).

Problem 5 Compound option )We consider a financial market offering two investment opportunities.. The firsttraded security is a riskless asset whose price is equal to Sp = ert at time t (withr ~ 0) and the second security is risky and its price is denoted by S, at timet E [0, T1. Let (SdO~t~T be a stochastic process defined on a probability space(n,:F, P), equipped with a filtration (:Ft)O~t~T, We assume that (:Fd09~T is

89the natural filtration generated by a standard Brownian motion (Bdo<t<T andthat (St)O~t~T follows a Black-Scholes model - -

with Il E IR and a > O.

.We wan~ to study an example of compound option. We consider a call optionWI.th matunty T I E10, T[ and strike price K 1 on a call of maturity T and strikepnce K. The.value of this option at time T1 is equal to

h = (C(T1, STl) - Kd+,

where C(t, x) is the price of the underlying call, given by the Black-Scholesformula.

1.

(a) Graph function x I-t C(T1, x). Show that the line y = x - Ke-r(T-Tt} isan asymptote (hint: usethe put/call parity).

(b) Show that the equation C(T1 , x) = K 1 has a unique solution Xl.

2. Show that at time t < T1 , the compound option is 'worth G(T _ t S) hG · dfi db 1 , t,wereIS e ne y ,

G(O, xl~ E [e-'~ (C ( T" xe(.-"n';-,"",) - K,n'with 9 bei~g a standard normal random variable.

3.

(a) ~how that x I-t G(O, x) is an increasing convex function .

(b) We now want to compute G explicitly. Let us denote by· N the standardcumulative normal distribution. Prove that ' ,

G(O,x) = E [e-rIlC (n,xe(r-lT2/2)lI+lTV69) 1{9>~d}] _ K1e-rIlN(d),

i. where

d = 10g(X/!1) + (r- a2 /2) 0a../O

(c) Show that if 91 is a st~ndard normal variable independent of 9, we can writeO~/= T - T1 and characterise G by, ,"

. G(O, x) + K 1e- r llN(d)

E [(xelT(V69+V919t) - "22 (1I+IIt) _ K e-r(II+II l») lA] ,where the event A is defined by

(log(X/Kr) + (r _~2) (0 + (1))

and 9 > -d}.

90 The Black-Scholes model Exercises 91

(d) From this, derive a formula for G(B,x) in terms of Nand N2 the twodimensional cumulative normal distribution defined by

N2(Y,Y1,P) = P(g < y,g+ pg1 < yd for y,Y1,P E JR.

4. Show that we can replicate the compound option payoff by trading the under-

lying call and the riskless bond.

Problem 6 Behaviour of the critical price close to maturityWe consider an American put maturing at T with strike price K on a share of riskyasset S. In the Black-Scholes model, its value at time t < T is equal to P(t, St),

when P is defined by

.( -rT I1W~_.,.2T)+P(t,x) = sup' E Ke - xe 2 ,

TE70.T-t

To T-t is the set of stopping times with values in [0, T - t] and (Wt)O$;t$;T is astandard P·-Brownian' motion. We also assume thatr > O. For t E [0,T[, we

denote by s( t) the critical ~rice defined as,

s(t) = inf{x > 0 I P(t,x) > K - z}.

we recall that limt-+T s(t) = K.1. Let P, be the function pricing the European put with maturity T and strike

price K

Pe(t, x) =E(e-r(T-t) K .. xeI1VT-tg- .,.; (T-t)) + ,

where 9 is a standard normal variable. Show that if t E [0, T[, the equationP; (t, x) = K - x has a unique solution in ]0,K[. Let us call it se(t).

2. Show that s(t) ~ se(t), for any t E [0, T[.

3. Show that

liminf KA) > E (liminfK~) - aKg) +t-+T . T - t - t-+T T - t

We shall need Fatou l~mma: for any sequence (Xn)nEN of non-negative ran

dom variables, E(lim infn -+oo X n ) ~ lim infn -+oo E(Xn ) .

4.

(a) Show that for any real number 1/,

E(1/ - Kag)+ > 1/.

(b) Deduce that

. K-se(t) . K-s(t)lim = hm = +00.t-+T VT - t t-+T VT - t

Problem 7 Asian option

We conside~ a ?nan~ial market offering two investment opportunities. The firsttraded secunty 1S a nskless asset whose price is equal to Sp = ert at time t (withr ~ 0) and the second security is risky and its price is denoted by St at timet E [0, T]. Let .(St)O$;t?T be a stochastic process defined on a probability space(f!, F, P), equipped with a filtration (Ft)O<t<T. We assume that (:F.) .h 1 fil

. - - t O<t<T 1St e natura tration generated by a standard Brownian motion (Bdo<t~;: andthat (SdO$;t$;T follows a Black-Scholes model - -

dSt = St(/Ldt + adBd,

with./L E JR and a > ~. We shall denote by p. the probability measure withde.ns1ty exp (-BBT - B T/2) with respect to P, where B = (/L - r)/a. UnderP , th~ proce~s (Wt)O$;t$;T' defined by Wt = (/L - r)t/a + Bt is a standardBrownian motion.

We are going to study the option whose payoff is equal to

h ~ (~ { S,dt - Krwhere K is a positive constant. '

I

1. Explain briefly why the Asian option price at time t (t ~ T) is given by

V, ~ E' [,-dT- ' ) (~ { S.da -KrrJ

2. Sh(j.wthat on the event { J;, Sudu ~ KT}, we have

e-r(T-t) it 1 ~ e-r(T-t)n. Vi = T' Sudu + S, - K e-r(T-t).

o rT

3. We define s, = e:"s.. for t E [0,T].

(a) Derive the inequality (

E·(St:Ke-rT)+ ~E·[e-rT(ST-K)+].

(Use conditional expectations given F t ) .

(b) Deduce that

Vo s E· [e- rT (ST - K)+] ,

i.e, the Asian option price.is smaller than its European counterpart.

(c) For t ~ u, we denote by Ct,u the value at time t of a European call maturing

93

(b) Deduce that

, (erT

- 1 )Vo -va ~ Soe-rT rT - exp ((rT/2) - (12T /12) .

Exercises

Vo = e-rTE (So exp ((r - (12 /2)(T /2) + (1VT/39) - K) + ,

where 9 is a standard normal variable. Give a closed-form formulator Vt interms of the normal distribution function. 0

3. Prove the inequality

n


at time u with strike price K. Prove the following inequality

e-r(T-t)t (lit )+ 1 iTVt < - S du - K + - e-r(T-u)Ct duo- T t 0 U T t ,u

We denote by (~t)09~T the process defined by

~t = ~t (~I t

Sudu - K) .1. Show that (~t)O~t~T is the solution of the following stochastic differential

equation:

92

2.

(a) Show that

v. ~ ;-'(T-'IS,E' [((.+ ~t S~dU) \1"] ,with S~ = exp ((r - (12 /2)(u - t) + (1(Wu - Wt}) .

(b) Conclude that Vi = e-r(T-t) StF(t, ~t), with

F(t,O ~E' (u ~t '>:'duf

3. Find a replicating strategy to hedge the Asian option. We shall assume that thefunction F introduced earlier is of class C 2 on [0,T[ x IR and we shall use Itoformula.

ill

The purpose of this section is to suggest an approximation of Vo obtained byconsidering the geometric average as opposed to the arithmetic one. We define

Vo ~ e-,TE' (exp (~ [tn(S,)dt) -K) +

1. Show that Vo ~ Vo.

2.

(a) Show that under measure P", the random variable JoT Wtdt is normal withzero mean and a variance equal to T 3/3.

5

Option pricing and partialdifferential equations

In the previous chapter, we saw how we could derive a closed-form formula forthe price of a European 'option in the Black-Scholes environment. But, if we areworking with more complex models or even if we want to price American options,we are not able to find such explicit expressions. That is why we will often requirenumerical methods. The purpose of this chapter is to give an introduction to someconcepts useful for computations.

Firstly, we shall show how the problem of European option pricing is related toa parabolic partial differential equation (PDE). This link is basedon the conceptof the infinitesimal generator of a diffusion. We shall also address the problem ofsolving the PDE numerically. '

The pricing of American options is rather difficult and we will not attemptto address it in its whole generality. We shall concentrate on the Black-Scholesmodel and, in particular, we shall underline the natural duality between the Snellenvelope and a parabolic system of partial differential inequalities. We shall alsoexplain how we can solve this kind of system numerically.

We shall only use classical numerical methods and therefore we will just recallthe few results that we need. However, an introduction to numerical methods tosolve parabolic PDEs can be found in Ciarlet and Lions(1990) or Raiviart andThomas (1983).

i-:

5.1 European option pricing and diffusions

In a Black-Scholes environment, the European option price is given by

Vi = E (e-r(T-t) I(ST)!.rt)

with I(x) = (x - K)+ (for a call), (K - x)+ (for a put) and

S - x e(r-u2/2)T+uWT

T - 0 .

where

Infact, we should point out that the pricing ofa European option is only a particularcase of the following problem. Let (Xtk~o be a diffusion in ffi., solution of

97Europeanoption pricing and diffusions

Proof. Ito formula yields

Hence

E (J (Xt)) = f(x) +E (It Af (X:) dS) .

Moreover, since the derivatives of f are bounded by a constant K] and sinceIb(x)1 + 1(1(x)1 :::; K(1 + Ixl) we can say that

E(~~~ IAf(X:)I) s Kj (1 + E(~~~ IX:12) ) < +00.

!heref?re, since x .H Af(x) and s H X: are continuous, the Lebesgue theoremIS applicable and yields ,

d" l (1 rdt E (f (Xt))lt=o =l~ E t 1

0Af(X:)dS) = Af(x).

The differential operator A is called the infinitesimal generator of the diffusion(Xt ) . The re~der can refer to Bouleau (1988) or Revuz and Yor (1990) t t dsome properties of the infinitesimal generator of a diffusion 0 s u Y

J .•

The ProPositio,n 5.1.1 can also be extended to the time-dependent case. We assumethat b~nd (1 satisfy th.e assumptions ofTheorem 3.5.3 in Chapter 3 which guaranteethe existence and unIqueness of a solution of equation (5.1).

o

f(Xt) = f(Xo) +it f'(Xs)(1(Xs)dWs

+it [~(12(Xs)JII(Xs)+ b(Xs)f'(Xs)] ds

~nd ther~sult follows from the fact that the stochastic integral J~ f'(Xs)(1(Xs)dW

sIS a martingale, Indeed, ~ccording to Theorem 3.5.3 and since 1(1(x)1 is dominatedby K(1 + Ix!), we obtain .

, :

Remark5.1.2 If we denote by X{ the solution of (5.3) such that Xx = .follows from Proposition 5.1.1 that 0 x, It

(5.1)

(5.2)

Optionpricing and partial differential equations

"Ix E ffi. u(T, x) = f(x)

(au/at + Atu - ru) (t, x) = 0 Vet, x) 'E [0,T] x ffi.{where

(Atf)(x) = (12(t, x) f"(x) + bet,x)f'(x)..2, . .

Before we prove this result, let us explain why the operator At appears naturallywhen we solve stochastic differential equations.

where band (1 are real-valued functions satisfying the assumptions of Theorem3.5.3 in Chapter 3 and ret, x) is a bounded continuous function modelling theriskless interest rate. We generally want to compute

Vt = E (e-J.T r(s,X.)dsf(XT )IFt) .

In the same way as in the Black-Scholes model, Vt can be written as

Vi = F(t, X t )

F(t,:x) =E (e-J.T r(s,X;'%)dsf(X~X)) ,

and X;'x is the solution of (5.1) starting from x at time t. Intuitively

F(t, x) ~ E (e-J.T r(s,X.)dsJ(XT )!Xt '= x) .

Mathematically, this result is a consequence of Theorem 3.5.9 in Chapter 3. Thecomputation of Vt is therefore equivalent to the computation of F(t,x):'Undersome regularity assumptions that we shall specify, this function F( t, x) is theunique solution of the following partial differential equatiori

5.1.1 Infinitesimal generatorofa diffusion

We assume that band (1 are time independent and we denote by (Xtk:~o thesolution of '

dXt = b(Xt) dt + (1 (.Xt ) dWt. (5.3)

Proposition 5.1.1 Let f bea 0 2 function with boundedderivativesand A be thedifferentialoperator that maps a 0 2 function f to Af such that

(AI) (x) = (12 (x) f",(x) + b(x)J'(x)._ 2 '

Then, the process M, = f(Xd - J~ Af(Xs)ds is an Ft-l'TIfrtingale.

96

98 Option pricing and partial differential equations

Proposition 5.1.3 If u( t, x) is a C 1,2 function with bounded derivatives in x and

if X, is a solution of(5./), the process

u, = u(t, X t) - I t(~~ + Asu) (s, Xs)ds

is a martingale. Here, As is the operator defined by

_ 0'2 (s, x) 8 2u 8u(Asu) (x) = 2 8x2 + b(s, x) Bx'

The proof is very similar to that of Proposition 5.1.1: the only difference is thatwe apply the Ito formula for a function of time and an Ito process (see Theorem

3.4.10). .In order to deal with discounted quantities, we state a slightly more general

result in the following proposition.Proposition 5.1.4 Under the assumptions ofProposition 5.1.3, and ifr(t, x) is a

bounded continuous function defined on IR+ x IR, the process

u, = e- I; r(s,X.)dsu(t, Xt)-lt

e- I: r(v,Xu)dv (~~ + Asu - ru) (s, Xs)ds

is a martingale.Proof. This proposition can be proved by using the integration by parts formulato differentiate the product (see Proposition 3.4.12 in Chapter 3)

- f'r(s,X.)ds (t X )e Jo u , t i,

and then applying Ito formula to the process u(t, X t ) .o

European option pricing and diffusions 99

In other words a(t, x) = a(t, x)a* (t, x) where 0'* is the transpose of a(t, x) =(a,j(t, x)).

. Proposition 5.1.5 If(Xt) is a solution ofsystem (5.4) andu(t, x) is a real-valuedfunction of class C 1,2 defined on IR+ x IRn with bounded derivatives in x andalso, r(t, x) is a continuous boundedfunction defined on IR+ x IR, then the proces;

M, = e- Io' r(s,X.)dsu(t, Xd-lte- Io' r(v,Xu)dv (~~ + Asu - ru) (s, Xs)ds

is a martingale.

The proof is based on the multidimensional Ito formula stated page 48.

Remark 5.1.6 The differential operator8/8t+At is sometimes called the Dynkinoperator of the diffusion.

5.1.2 Conditional expectations and partial differential equations

In this se,ction, we want to emphasise the link between pricing a European optionand ~olvmg a parabolic partial differential equation. Let us consider (Xt)t>o asolution of system (5.4), f(x) a function from IRn to IR, and r(t, x) a boundedcontinuous function. We want to compute

Vt =E (e- r r(s,X.)ds f(Xr) 1Ft) .'

In a similar way, as in the scalar case, we can prove that

This result is still true in a multidimensional modeJ. Let us consider the stochastic

differential equation

{

d~:t = bl (t, X t) dt + :E~=I alj (t, X t) dW/(5.4)

dXi' = bn (t, X t) dt + :E~=I anj (t, X t) dW/.

We assume that the assumptions of Theorem 3.5.5 are still satisfied. For any timet we define the following differential operator At which maps a C

2function from

. IRn to IR to a function characterised by

. 1 n . 82 f' n 8f(At!) (x) = -2 L a"j(t,x)8' -.8 (x) + Lbj(t,x) 8x (x),

.. . x, X J . I J',J~I J= 0

where (a,j (t, x)) is the matrix of components p

a,j(t,x) = La'k(t,x)ajk(t,x) 0'

k=1

where

F(t, z) = E (e-I,T r(s,X;,Z)dsf(X~X)) ,

when we denote by xt,x the unique solution of (5.4) starting from x attime t.The following result characterises the function F as a solution of a partial

differential equation. .

Theorem 5.1.7 Let u be a C 1,2 function with a bounded derivative in x defined

on [0, T) x IRn. Ifu satisfies

Vx E IRn u(T, x) = f(x),

and

(~~ + ~tU - ru) (t,x) = 0 V(t,x)E [O,T) x IRn

,

then

V(t, x) E [0, T) xIRn ~(t, x) = F(t, x) =I E (e-r r(s,X;,Z)dsf(X~X)) .


Pr~of. Let us prove the equality u(t, x) = F(t, x) at time t = 0. By Proposition5.1.5, we know that the process

M - - f r(s,X?,Z)ds .(t Xo,X)t-e 0 u , t

101

log(x/K) + (r + (12 /2)(T - t)(1'1/'T - t

1 I d2--. e- X /2dx

V2i -00 '

=

=N(d)

European option pricing and diffusions

The operator At is now time independent and is equal to

A A b. (12 2 a2 at= =-x -+rx-

2 ax2 ax'

It is straightforward to check that the call price given by F(t x) = xN(d ) _K e-r(T-t) N(dl - (1vT _ t) with ' I

o

is a martingale. Therefore the relation E(Mo) = E(MT) yields

u(O,x) '= E (e- JoT r(s,X?'Z)dSU(T,X~'X))

= E (e- JoT r(s,X?,Z)ds j(X~'X))

sinceu(T,x) = j(X). The proof runs similarly fort > 0.

(5.5)

Remark 5.1.8 Obviously, Theorem 5.1.7 suggests the following method to pricethe option. In order to fompute

F(t,x) =E (e- J,T r(s,X:,Z)dsj(X~~))

for agiven j, we just ~eed to find u such tha~

{au + A,u _ ru ~ ° in [0, T] x JR"at _u(T, x) = j(x), "Ix E IRn

.

Problem (5.5) is a parabolic equation with afinal condition (as soon as the functionu(T,.) is given).

For the problem to be well defined, we need to work in a very specific functionspace (see Raviart and Thomas (1983)). Then we can apply some theorems ofexistence and uniqueness, and if the solution u of (5.5) is smooth enough to satisfythe assumptions of Proposition 5.1.4 we can conclude that F = u. Generallyspeaking, we shall impose some regularity.assumptions on the parameters band(1 and the operator At will need to be elliptic, i.e.

3C >0, V(t,x)'E [O,Tj x IRn..

"1(6,.·., ~n) E IRn ~ aij(t, X)~i~j ~ C (t ~?) . (5.6)

Q >=1

5.1.3 Application to the Black-Scholes model

We are working under probability p ". The process (Wt)t::::o is a standard Brownianmotion and the asset price S; satisfies

is solution of the equation

au ' . :'{ at +Ab,u - ru ~ ° in [0, T]x]0, +oo[

, u(T, z) = (z - K)+, "Ix E]O, +00[.The same type of result holds for the put.

Note th~t th~ operato~ Ab. doe~ not satisfy the ellipticity condition (5.6). However, the tnck IS to consider the diffusion X, = log (St), which is solution of

ex, = (r - ~2) dt.+ (1dWt , ,',

since S - S e(r-0'2/2)t+O'w, I . fini .t - ° .' ts m mtesimal generator can be written as

" .-Ab.-log = (12 a2 + (r _(12) ~

2 ax2 2 ax''. .

It is clearly elliptic because (12 > °and, moreover, it has constant coefficients.We write

- '(12 a2( 2) 'a

Ab.-Iog = '"2 aX2' + r ~ ~ ax - r. (5.7)

The ~onnection b~tween the parabolIc problem asso~iatedtoAb8~log and the computation of ~e pnce of an option in the Black-Scholes model can be highlightedas follows:.lf we ~ant to compute the price F(t, x) at time t and for a spot pricex of an option paying off j (ST) at time T, we need to find a regular solution v of

{

:~ (~' x) +Ab.-1ogv(t, x) =° in [0, TJ x IR

(5.8)v(T, x) ,= j(eX

) , "Ix E IR, /

then F(t, z) =v(t, log(x)).

(5.9)

(5.10)

5.2 Solving parabolic equations numerically

{3s E [t,T], X;,x It O} ;

o

103Solving parabolicequationsnumerically

is a bounded stopping time, because r" = T; 1\ T: 1\ T where

tr = inf {O < s < T Xt,x = I}I - - , s

, (

We saw under which conditions the option price coincided with the solution of thepartial differential equation (5.9); We now want to address the problem of solvinga PDE such as (5.9) numerically and we shall see how we can approximate itssolution using the so-called finite difference method. This method is obviouslyuseless in the Black-Scholes model since we are able to derive a closed-formsolution, but it proves to be useful when we are dealing with more general diffusionmodels. We shall only state the most important results, but the reader can referto Glowinsky, Lions and Tremolieres (1976) or Raviart and Thomas (1983) for adetailed analysis.

and indeed T{ is a stopping time according to Proposition 3.3.6. By applying theoptional sampling theorem between 0 and r", we get E(Mo) = E(Mrx), thus bynoticing that if s E [0, "X], Af(X~'X)= 0, it follows that

u(O,x) = E (e- s: ':(S'X~'X)dSu("X,X~~X))

E(1 -J.T r(s,X~,X)ds (T XO,X)){\lsE[t,Tj, X:,xEO}e ° u , T

E(1 -r z

r(s,X~,Z)ds ( x XO,X))+ {3sE[t,Tj, X:,zllO}e ° u r>, r Z•

Furthermore, f(x) = u(T, x) and u(-rx , X~~X) = 0 on the event

Remark 5.1.10 An option on the FT-measurable random variable

, - J.T (X"Z)d . t1{\lsE[t,T], x:,xEo}e ,r • sf(Xi X

)

is called extinguishable. Indeed, as soon as the asset price exits the open set 0, theoption becomes worthless. In the Black-Scholes model, if 0 is of the form ]0, l[or ]1, +oo[weare able to compute explicit formulae for the option price (see Coxand Rubinstein (1985) and exercise 27 for the pricing of Down and Out options).

(- J.T r(s Xo,Z)ds ° ,)

u(O,x) = E 1{\lsE[t,TJ, X:,xEO}e ° '. f(XT'X).

That completes the proof for t = O..

consequently

u(T,x) = f(x) "Ix E O.

As we are about to explain, a regular solution of (5.10) can a~so be expr~ssed interms of the diffusion Xt,x which is the solution of (5.3) starting at x at orne. t.

Theorem 5.1.9 Letu beaC1,2function withboundedderivativeinx that satisfies

equation (5.10). We then have '

(- I,T r(X:,Z)ds f(Xt,X))

V(t,x) E [O,T] x 0, u(t,x) = E 1{\lsE[t,T],.x:,xEo}e T

Proof. We shaltprove the result for t ~ 0 s~nce the argument is simil~ ~ o~h::times There exists an extension of the function u from [0, T] x 0 to [.' ]that is still of class C1,2 . We shall continue to denote by u such an extension. From

Proposition 5.1.4, we know that

- J.'r(XO,X)ds (t Xo,X)M« = eo' u, t

i t - f r(X~,X)dv (au + Au - ru) (s, X~,X)ds'- e ° at

°is a martingale. Moreover

{ T X o x d o} or T if this set is empty"x = inf 0 S s S , s'-'f'

Optionpricing and partial differential equations102

Dnrtial ditterentiai equationson a boundedopen set and computationof5.1.4 r u. su:expectations

. u hout the rest of this section, we shall assume that there is only one asset:°th:t b(x), u(x) and rex) are all time independent. rex) is the riskless rate and

A is the differential operator defined by .

1 a2f(x) af(x)(Af)(x) = '2u(x)2aT + b(x)~,

We denote by A the discount operator such that Af(x) = Af(x) - r(x)f(x).

Equation (5.5) becomes

{

~~(t,~) + Au(t,x) = 0 on [O,T] x nt

u(T,x) = f(x), "Ix E nt., 9) 0 ] b[ as opposed to nt, we need toIf we want to solve problem (5. on = a, . th

consider boundary conditions at a and b. We are going to concentrat~on e ~~s~when the function takes the value zero on the boundaries'.These are e so-ca eDirichlet boundary conditions. The problem to be solved IS then

au (t x) + Au(t, z) = b on [0, T] x 0at '

u(t,a) = u(t,b) = 0 . "It S T

and

"

dXt = (r - O'2/2)dt + O'dWt.

105Solving parabolic equations numerically

Thus

lu(t, x) - UI(t, x)1 < MP (sUPt~s;5T[z + O'(Ws - Wt)1 2: I -lr'TI)

= MP (suPO:5S~T_t Ix + O'Wsl 2: I -lr'T/)

< MP (suPO:S;s:S;T Ix + O'Wsl 2: I -lr'T/) .

and therefore

By Proposition 3.3.6we know that if we define T. = inf {s > 0 W - } thE(exp(-ATa )) = exp(-.J2>:la/). It infers that f~r any a> 0, a~d f~r:n~ ~ en

P (sup w, 2: a) = P (Ta ~ T) s e~TE (e-~To) < e~T -a.,;'2Is:S;T _ e .

Minimising with respect to >. yields

P (sup w, 2: a) ~ exp (_a2) ,

-sr T '

P (SUP(x + O'Ws) 2: a) < exp ( ja -XI2)

s:S;T - O'2T'

Since (-Ws)s~O is also a standard Brownian motion

p (.~~(X +UW.) :5-a)~p(:~~(-X -aW,) ~ a):5 exp (

These two results imply that

P (:~~ Ix + O'Wsl 2: a) s exp ( _laO'~;12) + exp (

and therefore ,

Option pricing and partial differential equations104

5.2.1 Localisation

We want to compute the price of an option whose payoff can be written asf(ST) = f(SoeXT). We write f(x) = !(e). To simplify, we adopt Dirichletboundary conditions. We can prove in that case that the solution u of (5.9) and thesolutions UI of (5.10) are smooth enough to be able to say that

8u(t x) -8; + Au(t, x) = 0 on [0,T] x VI

u(t, I) = u(t, -I) = 0 if t E [0,T]

u(T, x) = f(x) if x EVI.

Problem (5.9) is set on ffi. In order to discretise, we will have to work on abounded open set VI =]- I, l[, where I is a constant to be chosen carefullyin order to optimise the' algorithm. We also need to specify the boundary conditions (i.e. at I and -I). Typically, we shall impose Dirichlet conditions (i.e,u(l) =u(-I) =0 or some more relevant constants) or Neumann conditions (i.e.(8u/8x)(I), (8u/8x)( "':'l)).Ifwe specify Dirichlet boundary conditions, the PDEbecomes

We are going to show how we can estimate the error that we make if we restrictour state space to VI. We shall work in a Black-Scholes environment and, thus, thelogarithm of the asset price solves the following stochastic differential equation

(5.1l)

lu(t, x) - UI(t, x)1 II - Ir'TI- X12)O'2T

+ exp ( _II ~ I~;~+ X12) ) .!his proves that for given t and x, liml--++ UI (t x) = u(t x) ThIS iforrn i 00" • e convergence

even um orrn in t and :: as long as x remains in a compact set of ffi.Remark 5.2.1

UI'(t,X) = E(1 {'v'sE[t,TJ, IX;,zl<l}e-r(T-t) f(X~X))

where X;'x == xexp((r - O'2/2)(s - t) + O'(Ws - Wt )). We assume that thefunction f (hence J) is bounded by a constant M and that r 2: O. Then, it is easyto show that '

If we call r ' = r - 0'2/2

{3s E [t,T], IX;,xl 2: I} C {sUPt:S;s:S;T Ix + r'(s -- t) + O'(Ws - Wt)1 2: I}

C {sUPt:S;s;5T Ix + O'(Ws - Wdl 2: I - Ir'TI} ,

• :t can be proved ~h~t P(suPs:S;T Ws 2: a) = 2P(WT 2: a) (see Exercise 18n Chapter 3). This would lead to a slightly better approximation than the one

above. "

• The ~~damental advantage of the localisation method is that it can be usedfor pncmg American options, and in that case the numerical approximation is

compulsory. The estimate of the error will give us a hint to choose the domain ofintegration of the PDE. It is quite a crucial choice that determines how efficient

. our numerical procedure will be.

5.2.2 Thefinite difference method

Once the problem has been localised, we obtain the following system with Dirichletboundary conditions:

ou(t, x) -ot + Au(t, x) = 0 on [0, T] x 0,

(E) u(t" l) = u(t, -l)= 0 if t E [0, T]

106 Option pricing and partial differential equations Solving parabolic equations numerically

following matrix:

(J 'Y 0 0 0a (J 'Y 0 0

((A h) ij) is.s», iss-:«

0 a (J 'Y 0

00 0 a (J 'Y0 0 0 a (J

where

107

u(T, x) = f(x) if x EO,.

The finite difference method is basically a discretisation in time and space ofequation (E).

We shall start by discretising the differential operator A on 0,. In order to dothis, a function (f(X))XEO, taking values in an infinite space will be associatedto a vector (fi)l<i<N ..We proceed as follows: we denote by (Xi) the sequencedefined by Xi = - -l + 2il/(N + 1), for 0 ~ i ~ N + 1, each fi is somehowan approximation of f(Xi)' We specify boundary conditions on fo, fN+! in theDirichlet case and fo, it, f N, f N +! in the Neumann case.

We consider h = 2l/(N + 1) and Uh = (UUl$i$N a vector in IRN. The

discretised version of the operator A is called Ah and the substitution runs asfollows:

and replace

with

withU i;+-1 _ U i - 1

b(x.) h h, 2h

I

a (J'2 1 ( (J'2)2h2 - 2h r - '2

(J(J'2

= - h2 - r

~

'Y (J'2 1 ( (J'2)= 2h2 + 2h . r - '2 .

If we specify null Neumann conditions, it has the following form:

(J+a 'Y 0 . 0 0a (J 'Y 0 00 a (J 'Y 0

0 (5.12)

0 0 a (J 'Y0 0 0 a (J+'Y

We obtain an operator Ah defined on IRN .

Remark 5.2.2 In the Black-Scholes case (after the usual logarithmic change ofvariables)

A- bs - log (') (J'2 02u (X) ( (J'2) ou(x) ( )u x = - --- + r - ~ -- - ru x.' 2 ox2 2 ox '

is associated with

2 ( 2) .- (J' +1 . . 1 (J' 1 (+1 . 1) .(AhUh)i = 2h2 (u~ - 2ui. + u~- ) + r - '2 2h u~ - u~- - rui..

If we specify null Dirichlet boundary conditions, Ah is then represented by the

This discretisation in space transforms (E) into an ordinary differential equation(Eh ) :

if 0 s t s T

u;.(T) = fh

where!h = (f~h$i$N\ is the vector f~ = f(Xi)' ' .

W~ are now going to discretise this equation using the so-called O-schemes. WeconsIder 0 E [0,1], k a time-step such that T = Mk and, we approximate the

Remark 5.2.3

with e > O. Then:

M N

L L(uh,k)i1jx . - h/ 2,x i+h/ 2j x l](n-I)k,nkj.n=1 i=1

solution Uh of (Eh) at time nk by Uh,k solution of

U;;:k = fh

109

a aao

.'X = " Uh,k

: ..

G (I + (-1 - O)kAh) un~ih,k

T = 1- kOAh.

bi CI aa2'" b2 C2 aa a3 b3 C3T=

Solving parabolic equations numerically

• When 1/2 :::; 0 :::; 1, as h, k tend to a

lim u~ = U in £2 ([0, T] x 01)

lim Ju~ = au/ax in £2 ([0,T] x 01).

• When a:::; 0 < 1/2, as h, k tend to 0, with lim k] h2 = 0, we get

lim u~ = U in £2 ([0,T] x 01)

lim Ju~ = au/ax in £2 ([0, T] x 01)

Remark 5.2.5

• In the case a :::; ~ < 1/2 we say that the scheme is conditionally conver entbecause the algonthm converges only if h, k and k/h2 tend to 0 Th gl.th her tri . . ese a go-

n ms are rat er tricky to Implement numerically and therefore they are rarelyused except when 0 = O.

• In the ca~e 1/2 :::; 0 :::; 1 we say that the scheme is unconditionally convergentbecause It converges as soon as hand k tend to O. '

Finally, we shall examine in detail how we can solve problem (E) . allAt h ti . h,k numenc y.

eac time-step n we are looking for a solution of T X = G where -:

T is a tridiagonal matrix. The following algorithm, known as the Gauss methodsol~es the system with a number of multiplications proportional to N. Denot:X - (xih~i~N, G = (gih'~i~N and

aa a aN-I bN - I CN-Ia \ a a aN bN

The algorithm runs as follows: first, we transform T into a lower triangular matrix


We also call J¢ the approximate derivative defined by

1(J¢)(x) = h (¢(x + h/2) - ¢(x - h/2)).

n decreasing, we solve for each n:

(Eh,k) un+! _ un ;

. h,k -k h,k + OAhUh,k + (1 - O)AhU~:t1 = aif a:::; n:::; M-1.

Theorem 5.2.4 We assumethat band a are Lipschitz and that r is a non-negativecontinuousfunction. Let us recall that Af(x) is equal to 1/2a(x)2(a2f(x)/ax2)+b(x)(af(x)/ax) - r(x)f(x). We assume that the operator A is elliptic

- '. I

(-Au, U)£2(O,) 2: f(lul£2(O,) + Iu 1£2(0,))

• When 0 = a the scheme is explicit because Uh,k is computed directly from

u~tl. But when 0 >' 0, we have to solve at each step a system of the formT~h,k = b, with

{T= (I - OkAh)

. b= (I + (1- '0) kA h ) U~:t1

where T is a tridiagonal matrix. This is obviously more complex and more timeconsuming. However, these schemes are often used in practice because of theirgood convergence properties, as we shall seeshortly,

• When 0 = 1/2, the algorithm is called the Crank and Nicholson scheme. It isoften used to solve systems of type (E) when b = aand a is constant.

• When 0 = 1, the scheme is said to be completely implicit.

We shall now state convergence results of the solution Uh,k of (Eh,k) towardsu(t, x) the solution of (E), assuming the ellipticity condition. The reader ought torefer to Raviart and Thomas (1983) for proofs. We denote by u~(t, x) the function

111

(5.13)(~~ + Atu - ru) (f - u) = 0 in [0, T] x IRn

u(T, x) = I(x) in IRn

American options

Then

u(t,x) = ~(t,x) = sup E (e- J.T r(s,X;'~)ds I (X;'X)) .rETt.T .

Proof. We shall only sketch the proof of this result. For a detailed demonstration,the reader ought to refer to Bensoussan and Lions (1978) (Chapter 3; Section 2)and Jaillet, Lamberton and Lapeyre (1990) (Section 3). We only consider the caset = 0 since the proof is,very similar for arbitrary t. Let us denote by Xt thesolution of (5.4) starting at x at time O. Proposition 5.1.3 shows that the process

M; = e- J;'r(s,X;)dsu(t, Xn- r e- J; r(v,X;)dv (au + A u - ru) (s XX)dsio ' at s. 's

is a martingale. By applying the optional sampling theorem (3.3.4) to this martingale between times 0 and T, we get E(Mr) = E(Mo), and since au/at + Asu-

(on a stock offering no dividend) is equal to the European call price. Nevertheless,there is no explicit formula for the put price and we require numerical methods.

The problem to be solved is a particular case of the following general problem:given a good function I and a diffusion (Xt )t>o in IR". solution of system (5.4),compute the function

~(t, x) = sup E (e - f r(s,X;,z)ds I (X;'X)) .rETt.T

Noticethat ~(t, x) ~ I(x) and for t =T we obtain ~(T, x) = I(x).Remark 5.3.1 It can be proved (see Chapter 2 for the analogy with discrete timemodels and Chapter 4 foqhe Black-Scholes case) that the process

e- Jot r(s,X.)ds~ (t,Xt}

is the smallest martingale that dominates the process I(Xt} at all times.

Wejust stressed the fact that the European option price is the solution ofa parabolicpartial differential equation. As far as American options are concerned, we obtaina similar result in terms of a parabolic system of differential inequalities. Thefollowing theorem, stated in rather loose terms (see Remark 5.3.3), tries to explainthat.

Theorem 5.3.2 Let us assume that u is a regular solution ofthe following systemofpartial differential inequalities:

au. at + Atu - ru ~ 0, u ~ I in [0,T] x IRn

Option pricing and partial differential equations

• ( -r(r-t)1 ( (r_<T2/2)(r-t)+<T(WT-Wtl ))~(t,x) = sup E e xe .

rETt,T .. 'T is th t of

• ( ) is a standard Brownian motion and !t,T IS e se ,and, under P , ~t t~O • [ T] W howed how the American call pncestopping times taking values in t, . e s

where

b'. = b, - Ciai+! /b~+1g~ - g. - c·g' l/b'·+1i-I 11+ 1

. Itt T' X - G' whereWe have obtained an equiva en sys em -,

~ 0 0 ·001

0 0a2 b~ 0O b' 0 0a3 3

T'=. o . ~ . ,o 0 aN-I b'rv_1 ~o 0 0 aN bN

To conclude, we just have to compute X starting frorri the top of the matrix.

Downward:

XI = gUb~ c

For 2 ~ i ~N, i increasingXi = (g~ - aixi_I)/b~,

. e matrix T is not necessarily invertible. However,.we c~n proveRemark5.2.6 Th. I 1+1.\ < Ibl WheneverTis.notmvertlble,theth t it is if for any l we have ai C, - ,. h k

a I I , .' k I the Black-Scholes case, it is easy to c ecprevious algonthm does not wor ..n. I _ 2/21 < a2 /h, i.e. forthat T satisfies the preceding condition as soon as r a -

sufficiently small h. .

5.3 American options

5.3.1 Statement ofthe problem. . notions in continuous time is not straightforward. In

~e;l:~ts~c~~~;:~:l, :e obtained the foll.owing form(ul)a~or(;;~p~\ce)of anAmerican call (f(x) = (x - K)-t) or an American put (f X - +

Vt :;::: ~(t, St)

110

using the Gauss method from.bottom to top.

Upward:

b'rv = bNg'rv = gN . '.For 1 ~ i ~ N - 1, l decreasmg.

Remark 5.3.3 The precise definition of system (5.13) is awkward because, evenfor a regular function f, the solution U is generally not C2

. The proper methodconsists in adopting a variational formulation of the problem (see Bensoussan andLions (1978». The proof that we have just sketched turns out to be tricky becausewe cannot apply the Ito formula to a solution of the previous inequality.

[0, T] x IR

[0, T] X o,

a.e. in

American options

Ifid 113

weconsl er¢(x) = (K -eX) h " "to the price of the American putis t e partial differential inequality corresponding

av -at (t, x) + Abs-1ogv(t, x) $ 0 a.e. in [0, T] xIR

v(t,x)2:¢(x) a.e.in [O,T]xIR

(v~t, x) - ¢(x)) (:~ (t, x) + A:bs-Iogv(t, x)) = 0

veT, x) = ¢(x).

v (t, x) 2: ¢(x) . a.e. in [0, T] x OtJ •

(A) , (V-¢)(:~(t,;r)+A:bS-IOgV(t,x)) =0 a.e. in

veT, x) = ¢~x) ,

av-a(t, ±l) = o.

X r

We can now di ti ,. rscre ise inequality (A) using the fi 't diff .notations are the same as in Section 5 2 2 I ,m e I, eren~es method. The

_ ' . . n particular, M IS the integer such that

The following theorem states the results f . . (5.14)o existence and uniqueness of a solu-

tion to this partial di~ ti I . ., irreren ra mequalttyand establi h th "Amencan put price. IS es e connection With the

Theorem 5.3.4 The inequality (5 14) has a' .vet, x) such that its partial deriv~( . ~mq~e :ont~nuous bounded solutiona2v/ ax2 are locally bounded. More~:s I;h~ e dlls~nbutlO.n sense av / aX,av / at,

er, IS so ution satisfies

v(t,log(x)) = ~(t,x) = sup E* (e-r(r-tlf (xe(r-u2/2)(r-tl+U(WT-Wtl))rET.,T .

The proof of this theorem can be found in Jaillet L b, am erton and Lapeyre (1990),

Numerical solution to this inequality

We are going to show how we can numericall I" .the method is similar to the one us d i th E so ve inequality (5.14). Essentially,problem to work in the interval O~ ~] _ ; l[uropean case, Fir~t, we localise theconditions at ±l. Here is the inequalit ith N' Then, we must Impose boundary

y WI eumann boundary conditions

aVe -b~at t,x)+As-1ogv(t,x)$0 a.e.in [0 ],T X o.

Option pricing and partial differential equations

U(O,x~ = E (e- JoTop, r(s,X;ldsf(X:o~J) .

That proves that u(O,x) $ F(O, z}, and that u(O, x) = F(O,x). We even provedthat Topt is an optimal stopping time (i.e. the supremum is attained for T = Topt).

o

5.3.2 The American put in the Black-Scholes model

We are leaving the general framework to concentrate on the pricing oftheAmerican

put in the Black-Scholes model.We are working under the probability measure P* such that the process (Wt k~:o

is a standard Brownian motion and the stock price St satisfies

dSt = St (rdt + O"dWt) .

We saw in Section 5.1.3 how we can get an elliptic operator by introducing the

process

x, = log (St) = log (So) + (r - 0";) t + O"Wt·

Its infinitesimal generator A is actually time-independent and

2 2 (- 2)A-bs- Iog _ Abs-log _ 0" a .0" a_ - r _ -- + r - - - - r.

. 2 ax2 . 2 ax

U(O, x) 2: sup E (e-JOT r(s,X:lds f(X:)) = F(O, x).rE70,T .

Now, we define Topt = inf{O $ s $ T, u(s,X:) = f(X:)}; we canshow that Topt is a stopping time. Also, for s between 0 and Topt, we have(au/at + Asu - ru) (s, x:) = O.The optional sampling theorem yields /

(_1.TOp' r(s,X:lds X)

U(O,x)=.E e 0 U(Topt,Xrop.) .

Because at time Topt. U(T~Pt,Xrx ) = f(XrX .), we can writeopt opt

U(O, x) 2: E (e- JOT r(s,X;ldSU(T,x:)) .

We recall that U(t,x) 2: f(x), thus U(O,X) 2: E (e- J: r(s,X;ldsf(Xn) .This

proves that

ru s 0

112

I i ,I,() where x - -I + 2il/(N + 1)Mk = T, Ih is the vector given by h = If' Xi • - . IRn we- . . (5 12) If U and v are two vectors m ,

and Ah IS represented by matrix . F' 11 the method is the same as in the. . < "f'vl < t < n Ui < Vi. onna y, . '

write U - V I h -d' - t' ' tionin time leads to the finite dimensional inequalityEuropean case: t e iscre isa I

(Ah,k):

American options

with115

and if 0 ~ n ~ M - 1

Uh,k ~ !h

n+l _ un + k (afhuh k + (1 - O)AhU~~I) ~ 0uh,k h,k , '

. - (- )A- n+l) n I ) - 0(Un+! - Uh k' + k OAhUh,k + (1 - 0 hUh,k ,Uh,k - h - .h,k, . .

. '. N d A- . . by (5 12). If we noteh ( . ) is the scalar product in IR an h IS given .were x,y .

x = Uh.k

G

F = fh,

we have to solve, at each time n, the system of inequalities

\

TX~G

(AD) X ~ F

(T X - G, X - F) = 0,

where T is the tridiagonal matrix

a+b C 0 0 0

a b C -0 0

0 a b C 0T=

00 0 a b C

0 0 0 '.' . a 'b + C

C = _Ok(~+~(r_0-2))2h2 2h 2 '

(AD) is a finite dimensional inequality. We know how to solve this type of

inequality both theoretically and numerically if the matrix T is coercive (i.e.X.TX ~ aX.X, with a > 0). In our case, T will satisfy this assumption ifIr - 0-

2/21 ~ 0-2/h

and if Ir - 0-2/21k/2h < 1. Indeed, this condition impliesthat a and c are negative and, therefore, by using the fact that (a+b)2 ~ 2(a2+b2)we show that

n n n-l

x:Tx = L aXi-lxi + L bx~ + L CXiXi+! + ax~ + cx~i=2 i=I i=1

n

> (a/2) L (xLI + xn~2 .

+ E~=, bx~ + (c/2) E~:/ (x~ + x~+!) + ax~ + cx~

Under the coercitivity assumption, we can prove that there exists a unique solutionto the problem (Ah,k) (see Exercise 28).

The following theorem analyses explicitly the nature of the convergence of asolution of(Ah,k) to the solution of (A). We note

M N

u~(t,x) = L L(uh,k)il]:Z:i-h/2,:Z:i+i/2] x l](n-l)k,~k]'n=I i=1 . .

Theorem 5.3.5 Ifu is a solution of (A),

J. when 0 < 1, the convergence is conditional: if li and k converge to 0 and if" k/h2 converges to 0 then

lim u~ = U in L 2 ([0, T) x (1)

\ .. aulim auk = - in L 2 ([0,T) x 01),

h ax2. when 0 = 1, the convergence is unconditional, i.e. the previous convergence is

true when hand k converge to 0 without restriction.

which is not a solution of (AD).

Remark 5.3.9 An implementation of th Boffered in Chapter 8. e rennan and Schwartz algorithm is

Pam(n, x) =max ( (K - x)+,

P(P om n + I, (1+ .)x) + (1 - p)Pom(n + I, (1 +b)X»)l+r

withthefinalconditionP (N x)'- ('K )+. (5.15)in Chapter 1 Section 1 4 that ifth -. - X . On the other hand, we proved

, . , e parameters are chosen as follows: '

r = RT/N

5.3.3 American put p'ricing by a bi ialtnomta method

We shall now explain another numerical method th . .American put in the Black-Scholes d 1 L at IS WIdely used to price the

thmo e. et r a b be thre I b

at -1 < a < r is a fI °- X anP(T

n= 1 + a) = p = (bn ::;:)O/(b _seq)uence 0 ID random variables such that

Ch 2 E a and P(Tn = 1 + b) - 1 p Wi .apter , xercise 4 that the Am . '" - - . e saw Inas '. encan put pnce In this model could be written

Pn =Pam(n, $n), ' ,and that the function P (n x) ld b .equation am , cou e computed by induction according to the

117

(5.16)

l+b

l+a

American options

The computation gives

= exp (-aVT/N)

= exp (+~VT/N)

~ p = (b-r)/(b-a),

then the European option price in this model a . .computed for a riskless rate equal to R d ppr~xl~ates the Black-Scholes pricethat in order to price the Am . an avo atility equal to a. This suggests

Given di .. encan put, we shall proceed as followsome iscrensanon parameter N we fix th I ..'

(5.16) and we compute the price p N ()' eva ues r, a, b,p according toi < n b . d . am n,. at the nodes x(1 + a)n-'(1 + b)i 0 <a - ,y ~n uction of (5,15). It seems quite natural to take pN ( ,-pproxrmation of the American Black S hi' am 0, X) as an, - c 0 es pnce P(O,x). Indeed, we can

'American' downward:Xl = gUb~For 2 ~ i ~ N, increasing i

Xi = (g~ - aXk-d/biXi =SUp(Xi, Ii)'

Jaillet, Lamberton and ~apeyre (1990) prove that, under the previous assumptions,

this algorithm does compute a solution of inequality (AD).

Remark 5.3.7 'The algorithm is exactly the same as in the' European case, apart

from the step Xi = SUp(Xi, Ji). That makes it very effective. ' ,There exist other algorithms to solve inequalities in finite dimensions. Some

exact methods are described in Jaillet, Lamberton and Lapeyre (1990), someiterative methods are exposed in Glowinsky, Lions and Tremolieres (1976),

, .Remark 5.3.8 When we plug in () =' 1 in (Ah,k), and we impose Neumannboundary conditions, the previous algorithm is due to Brennan and Schwartz

(1977).We must emphasise the fact that the previous algorithm only computes the exactsolution of system (AD) if the assumptions stated above are satisfied. In particular,it works specifically for the Americanput. There exist some cases where the resultcomputed by the previous algorithm is not the solution of (AD). The following

example should erase any doubts:

M~(~' T~} F~O)' a=O)

Numerical solution ofa finite dimensional inequality

In the American put case, when the step h is sufficiently small, we can solve thesystem (AD) very efficiently by modifying slightly the algorithm used to solvetridiagonal systems of equations. We shall proceed as follows (we denote by b the

vector (a + b, b, . . . ,b + e))

Upward:,b'tv ~ bNg'tv = s«~For 1 ~ i ~ N - 1, decreasing i

b~ = b, - ca/b~+lg~ = gi - eg~+db~+l


The reader will find the proof of this result in Glowinsky, Lions and Tremolieres

(1976). See also XL Zhang (1994).

Remark 5.3.6 In practice, we nonnally use () = 1 because the convergence is

unconditional.

., (5.17)Write down the equations satisfied by >. and a so that v is continuous withcontinuous derivative at z" . Deduce that if v is continuously differentiable thenz" is a solution of f(x) = x where

1. We denote by u.(t, x) the price of the European put in the Black-Scholesmodel. Derive the system of inequalities satisfied by v = u - u•.

2. We are going to approximate the solution v = u - u. of this inequality bydiscretising it in time, using one time-step only. When we use a totally implicitmethod, show that the approximation v(x) of v(O, x) satisfies

-vex) + T Absv(x) ::; 0 a.e. in ]0, +oo[

119

(5.19)

Exercises

v (t, x) ~ ¢(x) = (K - x)+ - u.(O,x) a.e. in ]0,+oo[

(v(x) - ¢(x)) ( -vex) + T AbSv(x)) = 0 a.e. in ]0, +oof

(5.18)3. Find the unique negative value for a such that vex) = x" is a solution of

-vex) + T Absv(x) = O.

4. We look for a continuous solution of (5.18) with a continuous derivative at x"

{

>'xOt if x ~ x"vex) =

¢(x) otherwise.

K-u.(O,x)f(x) = lalu~(O,x) + 1 + lal

and u~(t, x) = (au.(t, x)jax).

5. Using the closed-form formula for u. (0, x) (see Chapter 4, equation 4.9), prove. that f(O) > 0, that f(K) < K (hint: use the convexity of the function u.) and

that f(x) - x is non-increasing. Conclude that there exists a unique solution tothe equation f(x) = x.

6. Prove that vex) defined by (5.19) where x· is the solution of f(x) = x is asolution of (5.18).

7. Suggest an iterative algorithm (using a dichotomy argument) to compute x"with an arbitrary accuracy.

8. From the previousresults, write an algorithm in Pascal to compute theAmerican put price.

The algorithm that we have just studied is a marginally different version ofthe MacMillan algorithm (see MacMillan (1986) and Barone-Adesi and Whaley (1987».

Prove the uniqueness of a solution of (5.17). . .

~: Show that if M is the identity matrix there exists a unique solution to (5.17).

'. . b S (X) the unique vector Y ~ F such that4. Let p be positive: we denote y p .

VV > F (Y - X + p(MX - G), V - Y) ~ O.

'Show that for sufficiently small p, Sp is a contraction.

5 Derive the existence of a solution to (5.17). .

'. . to a roximate the Black-Scholes American ~ut pnceExercise 29 We arIel :I;~ is a~~lution of the partial differential inequalityu(t,x). Let us reca a .1

aU(t,x)+AbBu(t,x).::;O a.e.in [O,T]x]O,+oo[at . .

u (t, x) ~ (K - x)+ a.e. in [0,T] x ]0, +oo[

(U_(K_X)+).(~~(t,x)+AbSU(t,X)) =0 a.e.in [O,T]x]O,+oo[

u(T, x) = (!< - x)+

5.4 Exercises

. d b (X Y)thescalarproductoftwovectorsX = (xih~i~nExerCISe 28 We eno~e y , . X > Y means that for all i between 1 and .n,and Y = (Yih9~n' The notaltI

l~. ffin M satisfies (X M X) ~ a(X, X) with

Xi ~ Yi. We assume that foram,a > O. We are going to study the system ~

\:::G(Mi -G,X - F) =0.

1. Show that this is equivalent to find X ~ F such that

VV ~ F (M X - G, V ., X) ~ O..

Option pricing and partial differential equations118 . if

. N 0 x) = P(O,x). This result is quite tricky to JUStI Yshow that lim» -++<Xl Pam ( , d P , (1990» and we will not try to(see Kushner (1977) and Lamberton an ages

prove. it herthe. d i th called Cox-Ross-Rubinstein method and it is exposed inThIS me 0 IS e so-details in Cox and Rubinstein (1985).

where'

"

, '

6

Interest rate models

Interest rate models are mainly used to price and hedge bonds and bond options.Hitherto, there has not been any reference model equivalent to the Black-Scholesmodel for stock options. In this chapter, we will present the main features ofinterest rate modelling (following essentially Artzner and Delbaen (1989», studythree particular models and see how they are used in practice.

6.1 Modelling principles

6.1.1 The yield curve

In most of the models that we have already studied, the interest rate was assumedto be constant. In the real world, it is observed that the loan interest rate dependsboth on the date t of the loan emission and on the date T of the end or 'maturity'of the loan.

Someone borrowing one dollar at time t, until maturity T, will have to payback an amount F(t, T) at time T, which is equivalent to an average interest rateR(t, T) given by the equality.

F(t, T) = e(T-t)R(t,T).

If we consider the future as certain, i.e. if we assume that all interest rates(R(t, T))t<T are known, then, in an arbitrage-free world, the function F mustsatisfy -

'TIt < u < s F(t, s) = F(t, u)F(u, s).

Indeed, it is easy to derive arbitrage schemes when this equality does not hold.\

From this relationship and the equality F(t, t) = I, it follows that, if F is smooth,there exists a functio~r(t) such that

'TIt < T F(t, T) =exp (iT r(S)ds)

For an uncertain future, one must think of the instantaneous rate in terms of arandom process: between times t and t + dt, it is possible to borrow at the rater(t) (in practice it corresponds to a short rate, for example the overnight rate).To make the modelling rigorous, we will consider a filtered probability space(n, F, P, (Fdo<t<T) and will assume that the filtration (Ft)O<t<T is the naturalfiltration of a st;ndard Brownian motion (Wt)09ST and that J:T-= F. As in themodels we previously studied, we introduce a so-called 'riskless' asset, whoseprice at time t is given by

1 iTR(t, T) = -T·' r(s)ds.- t t

. The function r(s) is interpreted as the instantaneous interest rate.In an uncertain world, this rationale does not hold any more. At time t, the future

interest rates R(u, T) for T > u > t are not known. Nevertheless, intuitively,it makes sense to believe that there should be some relationships between thedifferent rates; the aim of the modelling is to determine them.

Essentially, the issue is to price bond options. We call 'zero-coupon bond' asecurity paying 1 dollar at a maturity date T and we note P(t, T) the value ofthis security at time t. Obviously we have P(T, T) = 1 and in a world-where thefuture is certain

So - .f r(s)dst - e 0

where (r(t))oStST is an adapted process satisfying JOT Ir(t)ldt < 00, almostsurely. It might seem strange that we should call such an asset riskless since itsprice is random; we will see later why this asset is less 'risky' than the others.The risky assets here are the zero-coupon bonds with maturity less or equal to thehorizon T. For each instant u ~ T, we define an adapted process (P(t, u))o<t<u,satisfying P(u, u) = 1 giving the price of the zero-coupon bond with matu~ity uas a function of time.

In Chapter 1, we have characterised the absence of arbitrage opportunities bythe existence of an equivalent probability under which discounted asset pricesare martingales. The extension of this result to continuous-time models is rathertechnical (cf. Harrison and Kreps (1979), Stricker (1990), Delbaen and Schachermayer (1994), Artzner and Delbaen (1989)), but we were able to check in Chapter4, that such a probability exists in the Black-Scholes model. In the light of theseexamples, the starting point of the modelling will be based upon the followinghypothesis:

(H) There is a probability P* equivalent to P, under which, for all real-valued

is a martingale.

This hypothesis has some interesting conseerty under P* leads to, using the equality p(~~~)e~ I;,deed, the martingale prop-

123

o

Modelling principles

u E [0, T], the process (F(t, u))oStSu defined by

F(t, u) = e- fa' r(s)dsP(t, u)

. . ~(t,U) = E* (F(u,u)/Ft) = E* (e- fa"r(S)ds!Ft)and, eliminating the discounting,

t; = Lo + it HsdWs a.s.S' 0In~e LT is a probability density, we have E(L ) _ _

equivalem to P we have L > 0 d T - 1 - Lo and, forP" ist, To obtain the 'formula (6~~), we :p.sp'tynthm~~efgenera11Y P(Lt > O} = 1 for any

. ( e 0 ormula to the log function To doso, we need to chec.k that P "It E [0 T] L + r t H dW) .f hi ~ ~ , , 0 Jo s s > 0 = 1 The proof

OtIS}act relies in a crucial way on the martin a . . .of Exercise 30 Then the ItA ~ I' g le property and It IS the purpose

. 0 tormu a yields

10g('Lt) = r L1

HsdWs"':' ~ r 2.H 2ds. Jo s 2 Jo £2 s a.s.

which leads to equality (6.3) with q(t) = HtiLt . s

P(t,u) =E* (e- J."r(S)ds!;:,). . t· (6.2)

This equality, which could be compared to foP(t, u) only depend on the behaviour of th rmula (6.1), shows that the pricesprobability P*. The hypothesis w d he process (r(s))OSsST under the

e rna e on t e filtration (;:, ) . 11express the density of the probabilit P* ith t 09ST a ows us tod . Y WI respect to P We denot b L hi

ensity, For any non-negative random . bl X .' e Y T t ISand, if X is Ft-measurable E* (X) _v;(~~) ,w~ have E* (X) = E(XL T )

the random variable i; is th~ density ;rt. P* t.' sedttIng t; ~ E(LTIFt). Thus. . restncte to Ft WIth respect to P.

Proposition 6.1.1 There is an ada d (t E [0,T],' ', . . pte process q(t ))OStST such'that, for all

i; = exp (it q(s)dWs- ~ it q(S)2dS) a.s. (6.3)

Proof. The process (L ) . .filtration of the Brown:a~~~~~~ (;art)I~~a\~relative to (:F.t), which is the naturalthat th . t t- 0 ows (cf. Section 4.2.3 of Chapter 4)

ere exists an adapted process (H ) . fvi rTfor all t E [0, T] t °StST sans ying Jo Hldt. < 00 a.s. and

(6.1)

Interest rate models,

P(t, T) = e- J.T r(s)ds.

and consequently

122

6.1.2 Yield curvefor an uncertainfuture

o

(6.6)

124. > tCorollary 6.1.2 The price at time t of the zero-coupon bond of maturity u -

can be expressedas

P(t u) = E (exp (_jU r(s)ds + jU q(s)dWs - ~ jU q(S~2dS) \Ft) ., t t t (6.4)

P f This follows immediately from Proposition 6.1.1 and fro.m the followingroo . . d able X' 'formula which is easy to derive for any non-negative ran om van.. . '

E (XLTI Ft ) (65)E· (XIFt ) = L

t. .

o

The following proposition gives lm ec~nomic interpretation of the process (q(t))(cf. following Remark 6.1.4). ~ . U

P 't' 613 For'each maturity u there is an adapted process (at )o9~u,roposl Ion . " 'such that, on [0, uj,

dP(t, u) = (r(t) _ afq(t))dt + afdWt.P(t, u) ,

Proof. Since the process (F(t, u))o9~u is a martingale under P~, (F()tLu)L~09~Uis a martingale under P (see Exercise 31). Moreo~er~ we have: P(t, U .t. > 6 ~.si'f IIt E [0 uj Then using the same rationale as In the proof ofPr?po~ltI2on ->: 'or a ,. , U h h t t (OU) dt < 00

we see that there exists an adapted process (Ot )o~t~u SUC t a Jo t

and . _ Ie' O"dW._l Ie'(O~·)2dsF(t, u)Lt = P(O,u)e o' . 2 0 • .

Hence, using the explicit expression of L; and getting rid of the discounting factor

P(t, u) P(O,u) exp (it r(s)ds + fat (O~ - q(s))d~s

_~ it ((O~)2 - q(S)2)dS).

Applying the Ito fo~ula with the exponential fun~tion, we get

dP(t,u) = r(t)dt + (Of - q(t))dWt - ~((Of)2 - ~(t)2)dtP(t,u)

+~(Of - q(t))2dt,2

= (r(t) + q(t)2 - Ofq(t))dt + (Of - q(t))dWt,

which gives the equality (6.6) with ar =Or - q(t).

R k 6 14 The formula (6.6) is to be related with the equality dS? = r(t)S?dt,emar .. . . dW hi h akes the bond

satisfied by the so-called riskless asset. It is the term In t w IC m

Modelling principles 125

riskier. Furthermore, the term r(t) - arq(t) corresponds intuitively to the averageyield (i.e. in expectation) of the bond at time t (because increments of Brownianmotion have zero expectation) and the term -a;:q(t) is the difference betweenthe average yield of the bond and the riskless rate, hence the interpretation of-q(t) as a 'risk premium'. Under probability P*, the process (TVt ) defined by

TVt = Wt - f; q(s)ds is a standard Brownian motion (Girsanov theorem), andwe have

dP(t, u) U -

P(t, u) = r(t)dt + at dWt. (6.7)

For this reason the probability P" is often called the 'risk neutral' probability.

6.1.3 Bond options

To make things clearer, let us first consider a European option with maturity 0on the zero-coupon bond with maturity equal to the horizon T. If it is a call withstrike price K, the value of the option at time 0 is obviously (P(O,T) - K)+ andit seems reasonable to hedge this call with a portfolio of riskless asset and zerocoupon bond with maturity T. A strategy is then defined by an adapted process((HP, Ht))O~t~T with values in rn?, Hp representing the quantity of riskless assetand H, the number of bonds with maturity T held in the portfolio at time t. Thevalue of the portfolio at time t is given by

Vi = HPSp + HtP(t, T) = Hpefo' r(s)ds + HtP(t, T)

and the self-financing condition is written, as in Chapterd, as

dVi = HPdS~ + HtdP(t, T).

Taking into account Proposition 6.1.3, we impose the following integrability con

ditions: foT

IHpr(t)ldt < 00 and foT

(Hta;:)2dt < 00 a.s. As in Chapter 4, wedefine admissible strategies in the following manner:

Definition 6.1.5 A strategy ¢ = ((HP,Ht))o~t~T is admissible if it is self

financing and ifthe discounted value Vi (¢) = Hp +H, F( t, T) ofthe corresponding portfolio is, for all t, non-negative and if SUPtE[O,Tj Vi is square-integrableunderP*. .

The following proposition shows that under some assumptions, it is possible tohedge all European options with maturity 0 < T.

Proposition 6.1.6 We assume sUPO<t<T Ir(t)1 < 00 a.s. and o'[ 1:- 0 a.s. forall t E [0,8]. Let 0 < T and let h be-an Focmeasurable random variable such

thathe - .C r(s),~s is s~uare-integrabl~ under p •. Then there exists an admissiblestrategy whose value at time 0 is equal to h. The value at time t ~ 0 of such astrategy is given by 1

Vi =E* (e- f r(S)dshl Ft) .

Remark 6.1.7 We have not investigated the uniqueness of the probability P* andit is not clear that the risk process (q(t)) is defined without ambiguity. Actually, itcan be shown (cf. Artzner arid Delbaen (1989)) that P* is the unique probabilityequivalent to P under which (p(t, T))O<t<T is a martingale if and only if theprocess (an satisfies aT :f:. 0, dtdP almOsteverywhere. This condition, slightlyweaker than the hypothesis of Proposition 6.1.6, is exactly what is needed to hedgeoptions with bonds of maturity T, which is not surprising when one keeps in mind

the characterisation of complete markets we gave in Chapter 1.

Proof. The method is the same as in Chapter 4. We first observe that if Vi isthe (discounted) value at time t of an admissible strategy ((HP, Ht) )O<t<T, weobtain, using the self-financing condition, the integration by parts forffili"la and

Remark 6.1.4 (cf. equation (6.7))

dVt = HtdP(t, T)

= n.i«, T)aTdWt .

We deduce, bearing in mind that SUPtE[O,TJ Vi is square-integrable under P*, that

(Vi) is a martingale under P*. Thus we have

'It 5, 8 Vi = E* ( Vol Ft )

and, if we impose the condition Vo = h, we get

~ = eI; r(s)ds E* (e - I: r(s)ds hlFt) .

To complete the proof, it is sufficient to find an admissible strategy having thesame value at any time. To do so, one proves that there exists a process (Jt)o<t<oo - -such that Io Jt < 00, a.s, and

he- I: r(s)ds = .E* (he- I: r(S)ds) +10

Jsd~s.

Note that this property is not a trivial consequence of the theorem of representation

of martingales because we do not know whether he-I: r(s)ds is in the a-algebra

generated by the Wt's, t 5, 8 (we only know it is in the a-algebra Fo which canbe bigger (see Exercise 32 for this particular point). Once this property is proved,

it is sufficient to set

H - Jt d HO- E* (h - 1.9

r(S)ds\:F.) - i:-t >: _ T an t - e 0 t Tpet, T)at . at

for t 5, 8. We check easily that ((HP,Ht))o<t<o defines an admissible strategy (the hypothesis sUPO<t<T Ir(t)1 < 00 a~s.-guarantees that the condition

I~ Ir(s)H~lds < 00 holds) ;hose value at time 8 is indeed equal to h. 0

* dr(t~ = a (b* - ret)) dt + adWt (6.9)

where b = b - .\a1a and W - W .\ .according to this model let us -g'i t + t. Before calculating the price of bondsset .' ve some consequences of equation (6.8). If we

127Some classical models

X, = ret) - b,

we see that (Xt ) is a soluti~n of the stochastic differential equation

dXt = -aXtdt + adWt.

which means that (X) . 0 .3.5.2). We deduce th;t rl(t):anr~:t:~~~~l:~beckprocess (cf. Chapter 3, Section

6.2.1 The Vasicek model

In this model, we assume that the process ret) satisfies

dr(t) = a (b - ret)) dt + adWt. . . (6.8)where a, b, a are non-negative t Wi .a const~nt q(t) = _.\, with xEC;.s;::~: e also assume that the process q(t) is

6.2 Some classical models

~~u;t:~n~n~;)e:~~r(~h~ ~how t?at i~ order to calculate the price of bonds, we

pair (r(t), q(t)) under P. ri:a:::~~sm~d;l~t~under P*, or the d.ynamics of thedynamics of ret) under P by a diffusi e are about to examme describe the

q(t~ should have to get a similar e~u~~~~nu~~~:~~ and determ~ne the form thatoptions depend explicitly on 'risk parameters' whichThe~.~e p~lces o~bonds andadvantage of the Heath-Jarrow Mort d 1 . are I cu t to estimate. One

- on mo e which w '11 lai .paragraph 6.2.3, is to provide formulae that oniy dep d e ~~ exp am bnefly indynamics of interest rates under P. en on e parameters of the

ret) = r(O)e-at + b (1- e-at) ~ ae-at it ·asdWe: t (6.10)

°~(~ ~at~i~)) fOl~OWS ~ normal law w~ose mean is given by E(r(t)) = r(O)e-at +

e an variance by Var(r(t)) - a2/2 (1 -2at)P(r(t) < 0) > 0, which is not very satisfacto ; - e r It f~llows that(unless this probability is always very small) Nry r~m a practical point ?f vi.ew

r~t) converges in .law to a Gaussian random' v:i~blea~i;h: t tebnds dto m~mty,

a 12a. ean an variance,. I

. To calculate the price of zero c b dand we use equation (6.9). Fro~ e~:~i~y ~;2~: we proceed under probability P*

P(t,T) = E*(e-J.Tr(S)d~IFt)

Interest rate models126

From equality E* (X:) = xe-as~ we deduce

(

9 ' ) , 1 e-a9E* 1X:d~ = x - a

.t29

(6.15)

(6.16)

dr(t) = (a - br(t))dt +a~dWt

~ dXt,= (a -' bXd dt +O"~dWt on [0,00[.

Some classical models

and in equality (6.14), we get

Var ( r9X"'dS) = 0"2B _ 0"2 (1 _ e-a9) _ z: (1 _ e-a9)2 .Jo s a2 a3 2a3

Going back to equations (6.11) and (6.13), we obtain the following formula

pet, T) = exp [-(T - t)R(T - t, r(t))],

where R(T - t, ret)), which can be seen as the average interest rate on the period[t,T], is given by the formula

R(B,r)=Roo- alB [(Roo - r ) (I - e- a9)_::2 (l_e- a9)2]

with Roo = lim9~oo R(B, r) = b" - 0"2/(2a2). The yield Roo can be interpretedas a long-term rate; note that it does not depend on the 'instantaneous spot rate' r.This last property is considered as an imperfection of the model by practitioners.

Remark 6.2.1 In practice, parameters must be estimated and a value for r mustbe chosen. For r we will choose a short rate (for example, the overnight rate); thenwe will fit the parameters b, a, 0" by statistical methods to the historical data ofthe instantaneous rate. Finally .x will be determined from market data by invertingthe Vasicek formula. What practitioners really do is to determine the parameters,including r, by fitting the Vasicek formula on market data.

Remark 6.2.2 In the Vasicek model, the pricing of bond options is easy becauseof the Gaussian property of the Ornstein-Uhlenbek process (cf. Exercise 33).

6.2.2 The Cox-Ingersoll-Ross model

Cox; Ingersoll and Ross (1985) suggest modelling the behaviour of the instantaneous rate by the following equation:

with 0" and a non-negative, s e JR., and the process (q(t)) being equal to q(t) =-aVT(t), with a E JR.. Note that we cannot apply the theorem of existence anduniqueness that we gave in' Chapter 3 because the square root function is onlydefined on JR.+ and is not Lipschitz. However, from the HOlder property of thesquare root function, one can show the following result,

Theorem 6.~.3 We suppose that (Wt ) is a standard Brownian motion defined on[0,00[. For any real number x 2: 0, there is a unique continuous, adapted process(X t ) , taking values in'JR.+, satisfying Xo = x and

For a proof of this result, the reader is referred to Ikeda and Watanabe (1981), p.221. To be able to study the Cox-Ingersoll-Ross model, we give some properties

(6.12)


we can write

E* (e- ftTX;dS\ Ft) = F(T - t,X;) = F(T - t,r(t) - b*) (6.13)

) E* ( - fos

X:dS) (X"') being thewhere F is the function defi~ed by F(B,x = e , t

. . f U' (6 12) which satisfies Xo= x (cf. Chapter 3, Remarkunique solution 0 equa on .

3.5.11). ) letel We know (cf, Chapter 3) thatIt is possible to calculate F(B, x compe y. 9 '"h (X"') is Gaussian with continuous paths. It follow.s that fo X s ds

~ e pro::~ random variable since the integra! is the limit of Riemann sums fofIS a n~ ts Thus' from the expression of the Laplace transform 0 aGaussian componen. ,

GaUSSian(, -r X~dS) ( E* (19X"'dS) + ~Var (19

X:dS)).E* eo' = exp - s 2 0 ~ .

o

For the calculation of the variance, we write

Var (I,' X:d') = COY (I,' X:d'.[ X:dS)= 1919

Cov(Xf,X~)dudt.. (6.14)

S· ' X'" - xe:" + O"e-at ft easdWs, we have!nce t - )0

= ~2e~a(t+U)E* (it easdWslu easdWs) ,Cov(Xf,X~)

. 1t l\u .= 0"2 e-a(t+u) e2asdso

(e2a(tI\U) - 1)2 -a(t+u) ~= 0" e 2a

= e-b·(T-t)E* (e- r x;dsl Ft) (6.11)

, X* - (t) - b" Since (X*) is a solution of the diffusion equation withwhere t - r . t

coefficients independent of time '

dXt = -aXtdt + O"dWt,

128

.:="

T~ = inf{t ~ 0IX: = O}

Interest rate models130 . x hof this equation. We denote by (Xt) the solution of (6.16) starting at x and TO t e

stopping time defined by

131

'lj;(t).{

-'lj;'(t)

¢'(t) =

Some classical models

is a martingale and the equality E(MT) = Mo leads to (6.17). If F can be writtenas F(t, x) = e-aq,(t)-xt/J(t) , the equations above become ¢(O) = 0, 'lj;(0) = Aand

E (e->'X;) = ( b ) 2a/u2

( Abe-bt )a2/2A(1 - e-bt) + b exp -xa2/2A(1 _ e-bt) + b

= 1 exp ( AL()(2AL + 1)2a/u2 2AL + 1

with L = a2/4b (1 - e-bt

) and ( = 4xb/(a2 (ebt - 1)). With these notations,the Laplace transform of Xi / L is given by the function 94a/u2,(, where 96,( isdefined by

96,((A) = (2A +11)6/2 exp ( - 2AA~ 1) .

This function is the Laplace transform of the non-central chi-square law with 8degrees of freedom and parameter ( (see Exercise 35 for this matter). The densityof this law is given by the function [s.c. defined by .

. )

-(/2I () - e -x/2 6/4-1/2I (r::;() l' 0J6,( x - 2(6/4-1/2 e x 6/2-1 V zt, lor x > ,

where Iv is the first-order modified Bessel function with index u, defined by

When applying Proposition 6.2.5 with JL = 0, we obtain the Laplace transform ofXi

Solving these two differential equations gives the desired expressions for ¢ and'lj;. 0

Iv(x) = (~)v f: (x/2)2n~ 2 n=O n!r(v + n + 1) .

The reader can find many properties of Bessel functions and some approximations of distribution functions of non-central chi-squared laws in Abramowitz andStegun (1970)rChapters 9 and 26.

Let us go back to the Cox-Ingersoll-Ross model. From the hypothesis on theprocesses (r(t)) and (q'(t)), we get

.dr(t) '= (a - (b+ aa)r(t)) dt + av0'ijdWt,

where, under probability P", the process (Wt)O<t<T is a standard Brownianmotion. The price of a zero-coupon bond with m'at~rity T is then given, at time 0,

(6.17)

, ACT + b+ e1'tCT - b)) + 2JL (e1't - 1)'lj;>',I'(t) = a2A (e1't _ 1) + i: b + e1'tCT + b)

with, = Jb2+ 2a

2JL.. . -aq,(t)-xt/J(t) is due to the

Proof The fact that this expectation can be written as e. d th . iti I• .. (XX) I tive to the parameter a an e uu a

additivity property of the process t re a d y, (1990)) Ifcondition x (cf. Ikeda and Watanabe (l?81) , p. 225, ~evuz an or .,for Aand JL fixed, we consider the function F(t, x) defined by

F(t,x):= E (e->.X;e-I'_J: X;dS) ,

it is n~tural t~ look for F as a soiution of the problem

aF 2 a2F . aF .

{

_ ='~x- + (a - bx)- - JLxF. at. Q . ax2 ax

., ->.xv F(O,x)=e .

Indeed, if F satisfies these equations and has bounded derivatives, the Itoformula

shows that, for any T, the process (Mt)o9~T, defined by .r

1.,XT.ds xMt.='e-I' 0 ~ F(T - t, X t )

and

with, as usual, inf 0 = 00.

Proposition 6.2.4

I If a > a2/2, we have P(T~ = 00) = 1, for all x> O.. - h P( x < 00) - 1 for all x > O.2 If 0 < a < a2/2 and b ~ 0, we ave TO - ,

3: If 0 ~ a < a2/2 and b < 0, we have Ph) < 00) E ]0, 1[,jor all x> O.

Thi oposition is proved in Exercise 34. . .' fIS pr. . . hich enables us to characterise the JOint law 0The following proposition, w . d I

(x x ft XXdS) is the key to any pricing within the Cox-Ingersoll-Ross mo e .t 'Jo s '

Proposition 6.2.5 Forany non-negativeAand JL. we have

., E (e->'X; e-I' J: X;dS) = exp (-a¢>',I'(t)) exp (-X1p>-,1' (t))

where thefunctions·¢~,1' and 'lj;>',1' are given by

2 ( 2,e¥ )¢>',I'(t) = - a2 log a2~(e1't _ 1) + "! - b+ e1'tCT-+ b)

Interest rate modeLs Some classicaL modeLs 133

denoting by PI and P 2 the probabilities whose densities relative to P" are givenrespectively by

132

by:

__ E. (e- JOT r(S)dS)P(O,T)

_a<!>(T)-r(O).p(T)e (6.18)

dPl

_ e- J0

9

r(s)ds P(O,T)

dP· - P(O,T)and

dP 2

dP·

- J.9 r(s)dse 0

P(O,O) .

where the functions ¢ and 'ljJ are given by the following formulae

(

th.+ b . ) )2 2'Y· e ........,.-

¢(t) = - a2 log 'Y. _ b. + e"Y·tb· + e-)

and. 2(e"Y· t - l )

.'ljJ(t) ,= 'Y. _ b· + e"Y·tb· + b·)

. b d • - J(b·)2 + 2a2. The price at time t is given bywith b" = + atx an 'Y -

P(t, T) = exp (-a¢(T - t) - r(t)'ljJ(T - t)) .

Let us now price a European call with maturity 0 and exercise pr~ce K, on a ~e~ocoupon bond with maturity ~. We .can sho~ that the hypothesIs of Proposition

6.1.6 holds; the call price at time °ISthus given by 0

Co = E· [e-J09

r(s)ds (P(O,~) - K)+] ,

• [ _J.9 r(s)ds ( _a<!>(T_O)-r(O).p(T-O) _ K) ] 0

=Eeo e . +

= E. (e- J: r(S)dS p(O,T)1{r(O)<r.})

_ KE· (e- J: r(S)dS 1{r(o)<r.})

where r" is defined by

a¢(T - 0) + log(K)r" = - 'ljJ(T _ 0)

. E. ( - t r(s)ds P(O T)) = P(O T) from the martingale propertyNotice that eo, , , ,

. • ( - J.9 r(S)ds) _ P(O 0) We can then writeof discounted prices. Similarly, E e 0 -,.

the price of the option as

Co = P(O,T)pI (r(O) < r·) - K P(O,O)P2 (r(O) < r·) ,

We prove (cf. Exercise 36) that, if we set

a 2 (e"Y· o - 1)£1 = 2'" 'Y. (e"Y· o + 1) + (a 2 'ljJ (T - 0) + b·) (e"Y· o - 1)

anda2 (e"Y· o -1)

£2 - - -..,.---=--'-..,.----',-----,:----:-- 2 'Y. (e"Y· o+ 1) + b: (e"Y· o - 1)'

the law ofr(O) / £1 under PI (resp. r(O) / £2 under Ps) is a non-central chi-squaredlaw with 4a/a 2 degrees of freedom and parameter equal to (I (resp. (2), with

, 8r(Oh· 2e"Y· O . .

(1 = a2 (e"Y·O - 1) b·(e"Y· o + 1) + (a 2'ljJ (T - 0) + b·)(e"Y·o - 1))

and8r(Oh· 2e"Y· O

(2 = a2 (e"Y·O - 1) b·(e"Y· o + 1) + b·(e"Y·o - 1)) .

With these notations, introducing the distribution function Fo,( of the non-centralchi-squared law with fJ degrees offreedom and parameter (, we have consequently

Co = P(O,T)F4a/u 2' ( 1 (~:) - K P(O,O)F4a/u 2 ' ( 2 (~:) .

6.2.3 Other modeLs

The main drawback of the Vasicek model and the Cox-Ingersoll-Ross model liesin the fact that prices are explicit functions of the instantaneous 'spot' interestrate so that these models are unable to take the whole yield curve observed on themarket into account in the price structure.,

Some authors have resorted to a two-dimensional analysis to improve the modelsin terms of discrepancies between short and long rates, cf. Brennan and Schwartz(1979), Schaefer and Schwartz (1984) and Courtadon (1982). These more complexmodels do not lead' to explicit formulae and require the solution of partial differential equations. More recently, Ho and Lee (1986) have proposed a discrete-time

. \ ,

.model describing the behaviour of the whole yield curve. The continuous-timemodel we present now is based on the same' idea and has been introduced byHeath, Jarrow and Morton (1987) and Morton (1989).

First of all we define the forward interest rates f (t, s), for t ::; s, characterised

for any maturity u. So f(t, s) represents the instantaneous interest rate at time sas 'anticipated' by the market at time t. For each u, the process (J(t,u))o~t~umust then be an adapted process and it is natural to set f(t, t) = ret). Moreover,we constrain the map (t, s) H f(t, s), defined for t ~ s, to be continuous. Thenthe next step of the modelling consists in assuming that, for each maturity u, theprocess (J(t, u))o~t~u satisfies an equation of the following form:

135

dP(t, u)pet, u)

Some classicalmodels

and by the Ito formula

1ex, + "2d(X, x),

~ (f(" t) - ([Q(",)d,) + ~ ([U(f(",))d,)}t

-(lU

a(J(t, S))dS) dWt .

If the hypothesis (H) holds, we must have, from Proposition 6.1.3 and equalityf(t, t) = ret),

arq(t) =(lUa(t,S)dS) - ~(lU a(J(t,S))ds}2,

with ar =. :- (ftU

a(J(t, s))ds). Whence

l u

a(t, s)ds = ~ (lUa(J(t, S))dS) 2 - q(t) l u

a(J(t, s'))ds

and, differentiating with respect to u,

a(t,u) = a(J(t,u')) (l U

'a(J (t , s))ds - q(t)) .

Equation (6.20) becomes, if written in differential form,

df(t, u) = a(J(t, u)) (lU

a(J(t, S))dS) dt + a'(J(t, u)) dWt. (6.22)

The following theorem, by Heath, Jarrow and Morton (1987), gives some sufficientconditions such that equation (6.22) has a unique solution. I

Theorem 6.2.6 If the.function a is Lipschitz and bounded, for any continuousjunction ¢ from [0, T] to IR+ thereexists a uniquecontinuous processwithtwo indices (J(t, u))O~t~u~T suchthat,forall u, theprocess (J(t, u))o<t<u is adaptedand satisfies (6.22), with f(O, u) = ¢(u). - - ,

We see that, fo~ any continuous process (q(t)), it is then possible to build a modelof the form (6.20): take a solution of (6.22) and set .

~a(t, u) = a(J(t, u)) (lU

a(J(t, s))ds - q(t)) .

The striking feature of this model is that the law of forward rates under P" onlydepends on t~e function zr. This" is a consequence of equation (6.22), in whichonly a a~d (Wd appear. It follows that the price of the options only depends onthe function d. This situation is similar to Black-Scholes'. The case where a is aconstant is covered in Exercise 38. Note that the boundedness condition on a isessential since, for a(x) = x, there is nQ solution (cf. Heath, Jarrow and Morton(1987) and Morton (1989».

(6.20)

(6.19)


X; = l u(- f (s, S) + f (s, s) - f (t , s)) dS

-lu

f(s, s)ds +lu {is a(v, S)dV) ds

+l u (is a(J(v, S))dWv) ds

= -lu

f(s,s)ds +lu (lU

a(v:S)dS}dV

+l u

(lua(J(v, S))dS) dWv

= Xo + it f(S,s)dS-it (lU

a(V,S)dS}diJ'

-i~'(tU

a(J(v, s))dS) dWv ~ (6.21)

The fact that the integrals commute in equation (6.21) is justified in Exercise 37.

We then have 0

ex, = (f(t,t) -1~ a(t,S)dS) dt - (lU

a(J(t,S))dS) dWt

f(t, u) = f(O, u) + ,ita(v, u)dv + it a(J(v, u))dWv,

the process (a(t, u))o9~u being adapted, the map (t, u) H a(t, u) being continuous and a being a continuous map from IRinto IR (a could depend on time as

well, cf. Morton (1989».Then we have to make sure that this model is compatible with the hypothesis

(H). This gives some conditions on the coefficients a and a of the model. To findthem, we derive the differential dP(t, u)/ pet, u) and we compare it to equation(6.6).LetussetXt = - ft

U f(t,s)ds. WehaveP(t,u) = eX. and,fromequation

(6.20),

134

by the following equality:

P(t,u) = exp (-l U

f(t,S)dS)

Prove that s satisfies

2. For e < x < M, we set T:'M =T: A TM. Show that, for any t > 0, we have

137Exercises

/72 ~s ds-x- + (a - bx)- = 0.2 dx 2 dx

Deduce, taking the variance on both sides and using the fact that s' is boundedfrom below on the interval [e, M], that E (T:'M) < 00, which implies that T:,Mis finite a.s.

4. Using the previous question, show that under the probabilities whose densities

are respectively exp ( - J; r(s)ds) / P(O, ()) andexp ( - JoT r(s)ds) / P(O,T)with respect to P *, the random variable r( ()) is normal. Deduce an expressionfor the price of the option in the form Co = P(O,T)PI - K P(O, ())P2' for someparameters PI and P2 to be calculated.

Exercise 34 The aim of this exercise is to prove Proposition 6.2.4. For x, M > 0,we note TM the stopping time defined by TM= inf{t 2: °IXi = M}.1. Let s be the function defined on ]0,oo[ by

s(x) = jX e2by/u2 y-2a/u2dy.1. Show that T is a stopping time.

2. Using the optional sampling theorem, show that E (MT) = E (MT1{T=T}).Deduce that P ({\it E [O"T] M, > O}) = 1.

Exercise 31 Let (n,F, (Ft)o9~T,P) be a filtered space and let Q be a probability measure absolutely continuous with respect to P. We denote by L, thedensity of the restriction of Qto Ft. Let (~t)09~T ~e an adapted process. Showthat (Mt)09~T is a martingale under Q If and only If the process (LtMt)o~t~T

is a martingale under P.

Exercise 32 The notations are those of Section 6.1.3. L~t (Mt)O~t~T be a proce~s

adapted to the filtration (Ft ) . We suppose that (Mt ) IS a martingale under P .Using Exercise 31, show that there exists an adapted process (Ht)o9~T such that

J;{ Hldt < 00 a.s. and

6.3 Exercises

Exercise'30 Let (Mt)O~t~T be a continuous martingale such that, for any t E

[0, T], P(Mt > 0) = 1. We set

T = (inf{t E [O,T] IM, = O}) AT.

136

Notes: To price options on bonds with coupons, the reader is referred to Jamshidian(1989) and EI Karoui and Rochet (1989).

3. Show that if e < x <M, s(x) = s(e)P (T: < TM) + s(M)P (T: > TM)'4. We assume a 2: /72/2. Then prove that limx-..+~ s(x) = -00. Deduce that

P (TO < TM) =°for all M > 0, then that P (TO < 00) = 0.

5. We now assume that °~ a < /72/2 and we set s(O) == limx-..+o s(x). Showthat, for all M > x, we have s(x) = s(O)P (TO < TM) + s(M)P (TO> TM)and complete the proof of Proposition 6.2.4.

Exercise 35 Let d be an integer and let Xl, X 2 , .•. , X d , d be independentGaussian random variables with unit variance and respective means mI, m2,

... , md· Show that the random variable X = E~=I Xl follows a non-central

chi-squared law with d degrees of freedom and parameter ( = E~=I m~.

Exercise 36 Using Proposition 6.2.5, derive, for the Cox-Ingersoll-Ross model,the law of r(()) u.i'der the probabilities PI and P 2 introduced at the end of Section6.2.2. '

Exercise 37 Let (n, F, (Ft)O<t<T, P) be a filteredspace and let (Wt)O<t<Tbe a standard Brownian motion-with respect to (Ft ) . We consider a process'with two indices (H(t,s))O~t,s~T satisfying the following properties: for any w,the map (t, s) t-t H(t, s)(w) is continuous and for any s E [0, T], the process

dP ' e->'x

dP = E (e->'x)"I

Show that, under P, Y is normal and give its mean and variance.

3. Let (X, Y) be a Gaussian ~~ctor with values in .IR? tinde~ha probabtiltity:':~let P be a probability measure absolutely contmuous WIt respec 0 ,

density

for all t E [0, T].

E~ercise 33 We would like to price, at time 0, a call with mat~rity () and strikeprice K on a zero-coupon bond with maturity T > (), in the Vasicek model.

1. Show that the hypothesis of Proposition 6.1.6 does hold.

2. Show the option is exercised if and only if r( ()) < r" , where

(

a(T - ()) ) _ /72 (1 - e-a(T-II))

r* Roo 1 - 1 _ e-a(T II) . , 4a2 .

-log(K) (1 _e~a(T-II) ) .

138 Interest rate models Exercises 139

(H (t, s) )O$t~T is adapted. We would like to justify the equality

For simplicity, we assume that f: E (f: H2(t, s)dt) ds < 00 (which is sufficient

to justify equality (6.21». .

I. Prove that

f E(IfH(t"jdW,!) d,'; f [E(fH'(t,,)dt)r ds,

Deduce that the integral J: U: H(t, s)dWt) ds exists.

2. Let 0 = to < t l < : .. < t N =T be a partition of interval [0,T]. Remark that

[T (I: nu; s) (Wti+1 -.WtJ) dsJo .=0

= I: (iT H(ti,S)dS) (Wt;+l - Wti).=0 0 .

and justify why we can take the limit to obtain the desired equality.

Exercise 38 In the Heath-Jarrow-Morton model, we assume that the function a isa positive constant. We would like to price a call with maturity' 0 and strike priceK, on a zero-coupon bond with maturity T > O.

1. Show that the hypothesis of Proposition 6.1.6 holds.

2. Show that the solution of equation (6.22) is given by f(t, u) = f(O, u) +a2 t (u - t/2) + aWt . Deduce, that

P(ll T) = P(O,T) (_ (T' _ ll)TXT ., a

20T(T-0))

u'. P(O,O) exp a U HII . 2 .

3. Derive, for A E JR, E* (e -rJ' f: W.ds e,xWB) ~ Deduce the law of WII'under the

probability measures PI and P 2 with densities with respect to P* respectively

given by

where N is the standard normal distribution function and

d = a.JO(T - 0) _ log (KP(O,0)/ P(O,T))2 a.JO(T - 0)

dPl

e- JOB r(~)dSp(o, T)

dP* = P(O,T)

dP2

e- foB ~(s)dsand -=----

dP* P(O,O)

4.. Show that the price of a call at time 0 is given by

Co = P(O,T)N(d) - K P(O,O)N (d - a.JO(T - 0)) ,

7

Asset models with jumps

In the Black-Scholes model, the share price is a continuous function of time andthis property is one of the characteristics of the model. But some rare events(release of an unexpected economic figure, major political changes or even anatural disaster in a major economy) can lead to brusque variations in prices.To model this kind of phenomena, we have to introduce discontinuous stochasticprocesses.

Most of these models 'with jumps' have a striking feature that distinguishesthem from the Black-Scholes model: they are incomplete market models, andthere is no perfect hedging of options in this case. It is no longer possible to priceoptions using a replicating portfolio. A possible approach to pricing and hedgingconsists in defining a notion of risk and choosing a price and a hedge in order tominimise this risk.

In this chapter, we will study the simplest models with jumps. The descriptionof these models requires a review of the main properties of the Poisson process;this is the objective of the first section.

7.1 Poisson process

Definition 7.1.1 Let (Tik~l be a sequence of independent, identically exponentially distributed random variables with parameter A, i.e. their density is equal tol{x>o}Ae->.x. We s~t Tn = L~l t: We call Poisson process with intensity A theprocess N, defined by .

Nt = L l{Tn::;t} = L nl{Tn::;t<Tn+l}'

n~l n~l

Remark 7.1.2 N; represents the number of points of the sequence (Tn)n~1 whichare smaller than or equal to t. We have

Tn = inf{t ;to, N, = n}.

143Dynamics ofthe risky asset

• At time Tj, the jump of X, is given by

- \ 6,Xr , = X r , - X - = X -Uj ,. J J T j T j ,

be 'memoryless'. The independence of the increments is a consequence of thisproperty of exponential laws.

Remark 7.1.6 The law of a Poisson process with intensity Ais characterised byeither of the folIowing two properties:

• (Ntk~o is a right-continuous homogeneous Markov process with left-handlimit, such that

thus X r; =Xr~ (I + Uj ) .,

P(Nt =n) = e->'.t (At~n.n.

• (Nt)t>o is a process with independent and stationary increments, right-continuous, non-decreasing, the amplitude of the jumps being one.

For the first characterisation, cf. Bouleau (1988), Chapter III; for the second one,cf. Dacunha-Castelle and Duflo (1986), Section 6.3.

7.2 Dynamics of the risky asset

The objective of this section is to model a financial market in which there is oneriskless asset (with price Sr = e'", at time t) and one risky asset whose pricejumps in the proportions U1 , ... , Uj, ..., at some times Tl , ... , Tj, ... and which,between two jumps, folIows the Black-Scholes model. Moreover, we will assumethat. the Tj'S correspond to the jump times of a Poisson process.To be more rigorous, let us consider a probability space (11, A, P) on which we define a standardBrownian motion (Wdt~o, a Poisson process (Nt)t~o with intensity A and asequence (Uj)j~1 of independent, identicalIy distributed random variables takingvalues in ]-1, +00[. We will assume that the a-algebras generated respectivelyby.(Wdt~o, (Nt)t~o, (Uj)j~1 are independent.

For all t 2: 0, let us denote by Ft the a-algebra generated by the randomvariables Ws , N, for s ~ t and tt, 1{j~Nd for j 2: 1. It can be shown that(Wdt>o is a standard Brownian motion with respect to the filtration (Ft)t>o, that(Nt)t~o is a process adapted to this filtration and that, for all't > s, N, ~ N, isindependent of the zr-algebra Fs . Because the random variables UjJ{j~Nd areFrmeasurable, we deduce that, at timet, the relative amplitudes of the jumpstaking place before t are known. Note as well that the Tj'S are stopping times of(Fdt~o,.sinc~ {Tj ~ t} = {Nt 2: j} EFt. _

The dynamics of Xt, price of the risky asset at time t, can nowbe describedin the following manner, The process (Xt)t~O is an adapted, right-continuousprocess satisfying:

• On the time intervals h, Tj+l [

dXt = Xt(jldt + adWt).

o


(Att ->.t--e.n!=

=

thus the law of Tn = T 1 + ... + Tn is

E(e- ctr n ) =E(e-ct T l( = (A ~ 0:) n

We recognise the Laplace transform of the gamma law with parameters Aand n(cf. Bouleau (1986), Chapter VI, Section 7.12). Then we have, for n 2: 1

P(Nt = n) = P(Tn ~ t) - P(Tn+l ~ t)

t (\ )n-l it (AX)n{ Ae->'x AX dx _ Ae->'x --I-dxJo (n - I)! 0 n.

E (sN.) = exp {At (s - I)}.

Proof. First we notice rhatthe law of Tn is(AX)n-l .

l{x>O}.Ae->'x (n _ I)! dx,

: a gamma law with parameters Aand n. Indeed, the Laplace transform of T1 isI.e. ,A

E (e-ct T 1

) = A+ 0:'

Moreover, for s > °

P 't" 7 1 4 Let (N ) >0 be a Poisson process with intensity A and F t =roposl Ion . . t t d .a(N

s', s ~ t). The process (Ntk~o is a process with independent an stationary

increments, i.e. .• independence: if s > 0, N t+s - N, is independent of the a-algebra Ft.

• stationarity: the law ofNt+s - N; is identical to the law ofN, - No = N s·

Remark 7.1.5 It is easy to see that the jump times Tn are stopping tim~s. In~eed~{ Tn < t} = {Nt 2: n} E Ft. A random variable T with .expon~nuallaw sat~sfieP(T-2: t + siT 2: t) = P(T 2: s). The exponential variables are said to

In particular we have

E(Nt ) = At,

142 .The following proposition gives an explicit expression for the law of N; for a

given t. ., .Proposition 7.1.3 If (Ntk~o is a Poisson process Wlt~ intensity A then, for anyt > 0, the random variable N, follows a Poisson law witn parameter A

->.t (At)nP(Nt = n) = e -,-.n.

X T\ ,= X o(1 + U1)e(/L-<T2/2)T\ +<TWT \ •

Then, for t E [Tl,T2[,

00

Dynamics ofthe risky asset145

is independent of the a-algebra generated by the random variables N < dUjIUSN.}. Let A be a Borel subset of JRk, B aBorel subset f JRd u'dV,C- s anof the a-algebra a(N <)"r. ° an an event

u, V, _ s . we have, USIng the independence ofU d N dthat the Uj 's are independent and identically distributed, an an the fact

P ({(UN.+l, ... ,UN.+k) E A} n C n {(U1, .. . ,Ud) E B} n {d $ Ns

} )


and

consequently, the left-hand limit at Tl is given by

X _ = X oe(/L-<T2/2)T\ +<TWT \

T\

144

So we have, for t E [0, Tl [

X, = X e(/L-<T2/2)(t- T!l+<T(W, -wT \ )T\

= X -(1 + Ude(/L-<T2/2)(t-T!l+<T(W,-wT \ )

T\

= X o(1' + U1)e(/L-<T2/2)t+<TW,.

Repeating this scheme, we obtain

x, =Xo (fl(1 +Uj ) ) e(/L-<T2/2)t+<T~',J=1

with the convention I1~=1 = 1.The process (Xt)t>o is obviously right-continuous, adapted and has only finitely

many discontinuities-on each interval [0, t). We can also prove that it satisfies, for

all t ~ 0,t '. N, .

P a.s. x, = Xo +1x, (J.Lds + adWs) +L XTjUj • (7.1). 0 j=1 .

We will see that, for this kind of model, it is generally impossible to hedge theoptions perfectly. This difficulty is due to the fact that for T < +00, there areinfinitely many probabilities equivalent to P on :FT under which the discountedprice (e- rt Xt)O<t<T is a martingale. In the remainder, we will make the following assumption: under P, the process (e- rtXt)O<t<T is a martingale. Thisis a stringent-hypothesis, but it will allow us to determine simply some hedgingstrategies with minimal risk. When this hypothesis does not hold, the hedging ofoptions is rather tricky (see Schweizer (1989)). .

To derive E(Xtl:Fs) we will need the following lemma; which means intuitively that the relative amplitudes of the jumps which take place after time s areindependent of the a-algebra :F~.

Lemma 7.2.1 For all s ~ 0, the a-algebras

a (UN.H, UN.+2,·'·' UN.+k,"')

and :Fs are independent.

Proof. As thea-algebras W = a(W., s ~ O),N = a(N., s ~ 0) andU = a(Ui, i ~1) areindependent, it suffices toprovethatthea-algebra a (UN.+l, UN.• +2, ... , UN.+k,' .. )

P «U1, ... ,Uk) E A) LP «U1, ... , Ud) E B)P (Cn {Ns =p}).p=d .

From the last equality, we deduce (taking C = n and B - JRd) h(u U) l' - t at the vector

Ns+l, ... , N.+k ,olIowsthesamelawas(Ul U) and th th"", k, en at

P ({(UNs+l, ... , UN.+k) E A} n C n {(U1, ... , Ud) E B} n {d $ Ns

} )

= P«UN.+1, ... ,UN.+k E A)P(Cn{(U1,,,,,Ud E B)}n{d$ Ns

} ) .

Whence the independence stated above. 0

Suppose now that E(lU1 1) < +00 and set X - «<x Tht - t- en

E(X,J.r.) ~ X.E (,(.-,-u',,*,-,)+U(w.-w.) . fi (1 +U;)j.r.)J=Ns+l

X\E (e(/L_r-<T2/2)(t_

S)+<T(W,_W.) Nrr'-Ns(1

+ U ')1). Ns+J :Fs

. j=1

= XsE (e(/L_r-<T2/2)(t_S)+<T(W,_WS) Nrr'-Ns( )

I+UNs+j ) ,

j=1

using Lemma 7.2.1 and the fact that Wt - Wand N N . d dthe a-algebra :F

s. Hence's t - s are In epen ent of

= Xse(/L-r)(t.,-s)e),(t-s)E(U!l ,

m-l N~i+l

ZP = L L <p(Ys"Vj).i=O j=N••+I

(7.2)

Using Lemma 7.2.1 and the fact that Y. . is F - bl ..A 2 5 f th A' 5, 8. measura e, we apply Proposition. . 0 e ppendix to see that

setting

Dynamics ofthe risky asset 147

Moreover IZPI < C(N. N ).. c 11d

. 2 - t - 5' It 10 ows that the convergence takes place in L1

an even In L . .We have

where ~i(Y) is defined by

~i(Y) =E (N';~N'; <p(y, VN.;+i)) .J=l

~i (y) is thus the expectation of a random sum and, from Exercise 40,

~i(Y) = A(Si+l - s.) Jav(z)<p(y, z).

Going back to equation (7.2), we deduce

E( Z'IF.) ~ E (~ ~.(Y.J IF.) ~ E (~ >'(s,+, - s.) I dv(z) 'I> (Y.;> z) IF.) .When the mesh of p tends to 0, we obtain

which proves that M; is a martingale. Now set ZP = "m-1 E(Z· IT)can wnte' L.",=o ,+ 1 .r5; • We

To a partition p = (80 =.8 < S'l < ... < 8m = t) of the interval [8, t], let us

associate

N.

Z ~ L <p(YTj ,Vj).j=N.+1

Proof. We assume first that <P is bounded and we set

N. 'It J 'Mt =L <p(YTj, Vj) - A ds v(dz)<p(Ys, z), :j=l 0

is a square-integrable martingale and't .

M; - A1ds JV(~Z)<p2(YSl z))

is a martingale.

Notice that by convention 2:~=1 = 1.

C = sup I<p(y, z)l·(y,z)EHtd xlR

UJ.en we have \2:7::1 <P(YTj,Vj)\ s CNt and II: I v(dz)~(Yslz)\ s Ct. So

M, is square-integrable. Let us fix 8 and t, with 8 < t, and set

The left-continuity of (Ydt~o and the continuity of <P with respect to y implythat ZP converges almost surely to Z when the mesh of the partition p tends to O.

To deal with the terms due to the jumps in the hedging schemes, we will need twomore lemmas, whose proofs can be omitted at first reading. We will denote by vthe common law of the random variables Vj's.

Lemma 7.2.2 Let <p(y,z) be a measurablefunctionfr~m ffid x ffi to ffi, suchthatfor any real number z the function y ~ <p(y, z) is continuous on ffid, and let(Yih>o be a left-continuous process, taking values in ffid. adapted to the filtration

(Ft)Qo, We assume thai.for ~ll t >0,' "

E (It

~8 Jv(dZ)<P2(:s,

z)) < +00.

Then the process M; defined by

146 Asset models with jumps

using Exercise 39.It is now clear that (X t ) is a martingale if and only if

f.L .=r: - AE(V1 ) ·

E ((Zp - 2p)2IFS) 2

~ E[(~ [Z;+, -E(Z;+,IF.JI) IF.]

= E (~[Z,+' - E(Z;+,IF.JI' IF.)

+2 LE ((Zi+1 - E(Zi+1IFs,)) (Zi+1 - E(Zi+dFsJ) Irs) .i<i

Taking the conditional expectation with respect to FSi and us~ng the fact that Zi+1

. F h ce F .-measurable, we see that the second sum IS 2. WhenceIS Si+1 en s, .

E ((Zp - 2 p)2IFs) = E (~1 (Zi+1 - E(Zi+lIFsi ))2IFs)

= E(~E ([Z,+' - E(Z,+dF.JI'1 F.;) IF}

148

Moreover,


,..~

Dynamics ofthe risky asset 149

and equality (7.3) implies that M? - Af; du f dv(z)~2(yu, z) is a martingale.If we do not assume that ~ is bounded, but instead

E (it ds / dv(z)~2(ys, z)) < +00,

for any t, we can introduce the (bounded) functions ~n's defined by ~n(y, z) =inf(n, sup(-n, ~(y, z))), and the martingales (M;')c?o defined by

It is easily seen that E (1; dsJ v(dz) (~n(Ys, z) - ~(Y., Z))2) tends to 0 as n

tends to infinity. It follows that the sequence (M;')n'21 is Cauchy in L2 and asM;' tends to .M; a.s., M; is square-integrable and taking the limit, the lemma issatisfied for ~. 0

Lemma 7.2.3 We keep the hypothesis and notations ofLemma 7.2.2. Let (Atk:~o

be an adapted process such that E (f; A;ds) < +00 for any t. We set L, =t .',fa AsdWs and, as in Lemma 7.2.2, ~

Using Lemma 7.2.1 once again

E [(Zi+1 - E(Zi+1IFs,))2IFsi] = V(ys,),

where the function V is defined by

V(y) = Var (N'i~N'i ~(y, UN.i+i))J ]=1

and, from Exercise 40,

V(y) = A(Si+l - Si)Jdv(z)~2(y,Z).

Therefore

E ((ZP:- 2p)2IFs) = E (~A(Si+1 - s.) JdV(Z)~2(YsilZ)IFs),

and so when the mesh of the partition p tends to 0,

'E [(M' - M,)'iF:j = E [A l duJdv(z)<l"(Y., Z)IF.] . (7.3)

Since (Mtk~.o is a square-integrable martingale, we obtain

E [(Mt - Ms)2IFs] = E (M; + M; - 2MtMs1Fs) = E (Mt2 - M;IFs)

! I

I

Then the product Li M, is a martingale.

Proof. It is sufficient to prove the lemma for ~ bounded (the general case isproved by approximating ~ by some ~n = inf(n, sup(-n, ~ )), as in the proof ofLemma 7.2.2). Let us fix s < t and denote by p = (so = s < S1 < ... < Sm = t)a partition of the interval [s, t]. We have

On the other hand, since (Ltk;~o and (Mtk:~o are martingales

E ((LSi+! MS i +1 - LSiMsJIFs,) = E ((L Si+1 - LSi)(MSi+1- Ms.)IFsi) .

Whence

withm-1

AI' = L (LSi+1 - Ls.)(Msi+! --; Ms,)., i=O

--j'p=

150 Asset models with jumps Pricing and hedging options 151

o

m-1

IAPI < (SUPO<i<m-1 IL';+1 - L., I) L IM',+I - M., Ii=O

(7.5)

Vo + it H?rer'ds + it H.X.(J.Lds + O'dW.)

N,

+'" HrUjX -.L.J 1 r,j=l

In the following, we fix a finite horizon T. A trading strategy will be defined,as in the Black-Scholes model, by an adapted process ¢ = ((HP, Hd )O<t<T,taking values in rn?, representing the amounts of assets held over time; b-ut~ totake the jumps into account, we will constrain the processes (HP) and (Hd to beleft-continuous. Since the process (Xd is itself right-continuous, this means, intuitively, that one can react to the jumps only after their occurrence. This conditionis the counterpart of the condition of predictability which is found in the discretemodels (cf. Chapter 1) and which is slightly more prickly to define in continuoustime.

The value at time t of the strategy ¢ is given by Vt = Hpert + Hi X, and thestrategy is said to be self-financing if

dVt = H?rertdt + HtdXt,

i.e., taking into account equation (7.1), dlit = Hprertdt + HtXt(J.Ldt + O'dWdbetween the jump times and at a jump time Tj, lit jumps by an amount ~Vrj =H rj~Xrj = H rjUjXr-:-. Precisely, the condition of self-financing can be written

1as

The random variablesuP.<u<t ILul is in L 2 (from the Doobinequality, c.f. C~apter3, Proposition 3.3.7), as welCas N; - N•. We deduce that AP tends to 0 in L ,and

consequently

-, (i}I'(Y", U;)I + ), f: du f dv(z)I~(Y., z)1)

.~ (SUPO~i~m-1) IL';+1 - L.,I (C(Nt - N.) + )"C(t - s)),

with C = sUPy

z 1(y,z)l. From the continuity of t H L t , we see that AP tendsalmost surely t~ 0 as the mesh of the partition p tends to O. Moreover

IAPI ~ 2 sup ILul (C(Nt - N.) + )"C(t - s))..~u9

We deduce

7.3 Pricing and hedging options

7.3.1 Admissible strategies1

Let us go back to the model introduced at the beginning of the previous section,assuming that the U/,s are square-integrable and that.

J.L = r - )"E(Ud = r - ).. Jzv(dz), (7.4)

which implie~ that the process (.it) t>O = (e-rt:Xtk:~o is a martingale. Notice) -

that

and consequently, from Exercise 39,

E (X;) = XJ exp ((0'2 + 2r)t) exp ()..tE(Ut)) .

Therefore the process' (.it) is a square-integrable martingale.t~O

For this equation to make sense, it suffices to impose the condition JOT IH~lds +J:: H:ds < 00, a.s. (it is easily seen that s H X. is almost surely bounded).Actually, for a specific reason to be discussed later, we will impose a strongercondition of integrability on the process (Hdo~t~T, by restricting the class ofadmissible strategies as follows:

Definition 7.3.1 An admissible strategy is defined by a process

o '¢ = ((Ht , Ht))O~t~T

adapted, left-continuous, with values in rn?, satisfying equality (7.5) a.s. for allt E [0, T] and such that

JOT IH~lds < +00 ~ ais. andE (JoT H;X;ds) < +00.Note that we do' not impose any condition of non-negativity on the value ofadmissible strategies. The following proposition is the counterpart of Proposition4.1.2 of Chapter 4.

Proposition 7.3.2 Let (Hdo9~T be an adapted, left-continuous process suchthat

E (iT H;X;dS) < 00,

and let Vo E nt., There exists a unique process (HP)O~t~T such that the pair

. \

o

153

which, taking into account equality (7.4), yields

Pricing and hedging options

= Vo + it HsXs((J.L - r)ds + adWs)

N,

+ LHrjUjXr-:-,j=1 '

7.3.2 Pricing

Remark 7.3.3 The condition E (J:{ H; X;ds) < 00 implies that the discounted

value (Vt ) of an admissible strategy is a square-integrable martingale. This resultsfrom the expression in Proposition 7.3.2 and Lemma 7.2.2, applied with thecontinuous process with left-hand limit defined by yt = (Ht , Xt - ) (note that inthe integral with respect to ds, one can substitute x, for x.. because there isonly finitely many discontinuities).

defines an admissible strategy with initial value Vo.

It is clear then that if Vo and (Ht) are given, the unique process (HP) such that((HP, Ht) )O$t<5,T is an admissible strategy with initial value Vo is given by

H? =. Vi - lftXt

it Nt

-u.x, +Vo + HsXsadWs +LHrjUjXr-:-o j=I'

-A1tdsHsXsJv(dz)z.

From this formula, we see that the process (HP) is adapted, has left-hand limit atany point and is such that HP = H~_. This last property is straightforward if t isnot a jump time Tj and if t is some Tj, we have

o 0 - - .Hr - H - = -HrD.Xr· + Hr·UjX - = o.

J T j J] } 'j

It is also obvious that foT IHPldt < 00 almost surely. Moreover, writing Hpert +HtXt = ert (Hp + HtXt) and integrating by parts as above we see that

((H~, Ht))O<t<T

'Let us consider'a European option with maturity T, defined by a random variableh, FT-measurable and square-integrable. To clarify, let us stand from the writer's

I

N, ( t ).L e-rrj + r(_re-rs)ds n.,u.x..j=1 ir, ,Nt

= Le-rrjHrjUjXrjj=1

N, rt .

.+ L in dsl{rj<5,s}( _re-rS)HrjU1Xrj-j=1 0 .

N," . t N .•

= ~ e-rrjHr,U1'X - +1ds(-re-rs)~ HrjUjXr-:-L.-t ] r , L.-t. }. '0 j=I'1=1. '.

N, r .= L e-rrjHrjUjXrj + io (-re-rs)Zsds.

j=1

Writing this in (7.6) and expressing dYs, we obtain

Vt Vo +1t(-re-rs)Vs~s +1t

H~rds +1tHsXs(J.Lds + adWs)

N,

+ LHrjUjX~j-j=1

= Vo -it r (H~ + HsXs) ds + 1tH~rds + It

HsX~(J.Lds + adWs)

Nt

+ LHrjUjXr~,j=1

152

((HP, Ht))O$t<5,T defines an admissible strategy with initial value Vo. The discounted value at time t ofthis strategy is given by

II, ~ Vo + J.' H,X,adW, + t,H.,uiX:;- - ~ J.' <isH,X, J;(dZ)Z.

Proof. If the pair (HP, Ht)O$t<5,T defines an admissible strattegy, its value at time tis given by Vi = yt + Zt, with yt = Vo+ f; H~rersds + fo HsXstude + adWs)

and Zt = EN~1 H; UjX -. Differentiating the product e-rtyt,J- ] ~ .

e-rtVi = Vo + 1t(-re-rs)Ysds +1t

e-rsdYs + e-rt

Zt." (7.6)

Moreover, the product crtZt can be written as follows:

,------

7.3.3 Prices ofcalls and puts

Before tackling the problem of hedging, we try to give an explicit expression forthe price of the call or the put with strike price K. We will assume therefore thath can be written as f(XT), with f(x) = (x - K)+ or f(x) = (K - x)+. As wesaw earlier, the price of the option at time t is given by

E (e-~(T-t) f(XT)IFt) '.

= E (e-r(T-t) f (Xte(l'-cr2 /2)(T-:-t)+cr~WT- W,) . IT, (1 + Uj ) ) Ft), .' ]=N,+l

= E (e-d T-.) I (X.e('-u'/2)(T-')+U(WT - W,t~r' (1 + UN,+j)) .r}From Lemma 7.2.1 and this equality, we deduce that

E (e-r(T-t)!(XT)!Ft) = F(t,Xt),

point of view. He sells the option at a price Vo at time 0 and' then follows anadmissible strategy between times 0 and T. From Proposition 7.3.2, this strategyis completely determined by the process (Ht)os;t5;T representing the amount ofthe risky asset. If Vi represents the value of this strategy at time t, the hedgingmismatch at maturity is given by h - VT. If this quantity is non-negative, the writerof the option loses money, otherwise he earns some. A way. of measuring the risk

is to introduce the quantity "

HI = E ((e-rT(h - VT))2) .

Since, from Remark 7.3.3, the discounted value (ft) is a martingale, we haveE (e-rTVT) = Vo. Applying the identity E(Z2) = (E(Z))2 +E ([Z :::- E(Z)]2)to the random variable Z = e-rT(h - VT), we obtain

HI = (E(e-rTh) - Vo)2 + E (e-rTh - E(e-rTh.) - (VT - Vo)f· (7.7)

Proposition 7.3.2 shows that the quantity VT' - Vo depends only on (Ht ) (andnot on Vo). If the writer of the option tries to minimise the risk RJ, he willask for a premium Vo = E(e-rTh). So it appears that E(e-rTh) is the initialvalue of any strategy designed to minimise the risk at maturity and this is whatwe will take as a definition of the price of the option associated with h. By asimilar argument, we see that an agent selling the option at time t > 0, who wants

to minimise the quantity R; = E ( (e-r(T-t)(h - VT) )2 1Ft ), will ask for a

premium Vi = E (e-r(T-t) hIFt). We will take this quantity to define the price

of the option at time t. '

F(t, x) = E (FO (t,xe-'(T-.)E(U,)If (1+ Uj ) ) ). . (7.8)

Since NT-t is a random variable independent of the Uj's, following a Poissonlaw with parameter >'(T - t), we can also write

F(t, x) ~ ~E (FO ~' xe-'(T-')E(U,)Q(1+ Uj ) ) ) e-'(T-')~~(T - t)"

Each term of this series can be computed numerically if we know how to simulatethe law of the Uj's. For some laws, the mathematical expectation in the formula'can be calculated explicitly (cf. Exercise 42).

155Pricing and hedging options

with

F( t, x) ~ E (e-"(T-,) f (xe('-u' I')(T-')+UWT_' If(1 + Uj)) )

~ E (e-"(T-.) f (xe("->E(U,)_U' I')(T-')+UWT_, If(1 + Uj ) ) ) .

Note that if we introduce the function

Fo(t,x) = E (e-r(T-t) f (xe(r-cr2/2)(T-t)+crwT_')) ,

which gives the price of the option for the Black-Scholes model, we have

7.3.4 Hedging ofcalls and puts

Let us examine the hedging problem for an option h= f(XT), with f(x) =(x - K)+ or f(x) = (K - x)+. We have seen that the initial value of anyadmissible strategy aiming at minimising the risk R6 at maturity is given byVo = E(e-rTh) = F(O; Xo). For such a strategy, equality (7.7) yields

RJ = E (e-rTh - VTf·Now we determine a process (Ht )Os;t5;T for the quantities of the risky asset to beheld in portfolio t? minimise RJ. To do so, we need the following proposition.

~~o.positio~7.3.4 Let Vi ~;;'he value at time t of an admissible strategy withinitial value Vo = E (e f(XT)) = F(O,Xo), determined by a process(Ht)Os;t5;T for the quantities of the risky asset. The quadratic risk at maturity

Asset models with jumps154

Nt

u, = L F(Tj, X r;) - F(Tj, X rj-)j=1

Alt dsJ(F(s,Xs(l+z))-F(s,Xs))dv(z)

is a square-integrable martingale. We also know that F(t, Xt ) is a martingale.Therefore the process F( t, X t ) - M, is also a martingale and, from equality(7.10), it is an Ito process. From Exercise 16 of Chapter 3, it can be written as astochastic integral. Whence

156 Asset models with jumps

RJ = E (e-.rT(f(XT) - VT))2 is given by the following formula:

T E ( rT (aF ( X) )2 - 2 2R& = Jo ax s, s - H; Xsu ds

+ J: AJv(dz)e-2rs (F(s, x.n+ z)) - F(s, X s) - HszXs)2 dS).

Proof. From Proposition 7.3.2, we have, for t ~ T,

F(t,x) = e-rtF(t,xert),

so that F(t, X t) = E (iiIFt ) . It emerges that F(t, X t) is the discounted price of

the option at timet. We deduce easily (exercise) from fonnula (7.8) that F(t, x)is C 2 on [0, T[xlR+ and, writing down the Ito formula between the jump times,we obtain

F(t, Xt ) =

r aF - r aF - -F(O, Xo)+ io 8;(s, Xs)ds + io ax (s,Xs)Xs(-AE(Vdds + udWs)

lit a2F - 2 - 2 N, - - - -+- -a2 (s,Xs)u Xsds +L F(Tj,XrJ - F(Tj,Xr~)· (7.10)2 0 x . .. . j=1 ]

Remark that the function F( t, x) is Lipschitz of order 1 with respect to z, since

ir==

Pricing and hedging options

E (it ds !v(dz) (F(s, Xs(l + z)) - F(s, X s)) 2)o t

s E (1 dsX; JV(dz)z2)

< +00,

which, from Lemma 7.2.2, implies that the process

- - it aFF(t, X t) - M, = F(O, Xo)+ 0 ax (s, Xs)XsudWs.

Gathering equalities (7.9) and (7.11), we get

ii - VT = M~1) + M¥),

with

and

157

(7.11)

IF(t,x) -(F(t'Y)1 ( NT_, )< E e-r(T-t) jxe(r--XE(Utl- u 2 /2)(T-t)+UWT_t U(1 + Vj)

= Ix -yl·It follows that

N,

= L (Fh,Xr;) - Fh,Xr~) - HrjVjXr~)j=1 ]]

-A I tds! dv(z) (F(s, Xs(l + z)) - F(s, X s) - HszXs) .

From Lemma 7.2:3, MP) Mt(2) is a martingale and consequently

E (MP) M12)) = J.!a 1) Ma2) = O.

Whence

E (ii - VT) =, E((M~1))2)+ E((M~2))2)

• 'E (J:{~~ (s,X.) - HrX;U'dS) +E((M~'»)'),

It follows that the minimal risk is obtained when H, satisfies P a.s.

and applying Lemma 7.2.2 again

E((Mf2))2) .

= E (A !:;ds Jv(dz) v- X s(l + z)) - F(s, X s) - HszXs) 2) .The risk at maturity is then given by

(aF - .) - 2 2ax (s, x.) - H, Xsa .

+ AJ v(dz) (F(s,Xs(l + z)) - F(s,Xs) - HszXs) zXs = O.

It suffices indeed to minimise the integrand with respect to ds. It yields, since

(Hdt~iJ must be left-continuous,

H, = 6.(s,.Xs - ) ,

In this way, we obtain a process which satisfies E (JoT H;X;ds) < +00 and

which determines therefore an admissible strategy minimising the risk at maturity.Note that if there is no jump (A = ,0), we recover the hedging formula for theBlack-Scholes model and, in this case, we know that the hedging is perfect, i.e,m= O. But, when there are jumps, the minimal risk is generally positive (cf.

Exercise 43 and Chateau (1990».Remark 7.3.5 The formulae we obtain indicate that calculations are still possiblefor models with jump. It remains to identify parameters and the law of the U, 'so Asfor the volatility in the Black-Scholes model, we can distinguish two approaches:(1) a statistical approach, from the historical data and (2) an implied approach,from the market data, in other words from the prices of options quoted on anorganised market. In the second approach, the models with jump, which-involve

several parameters, give a better 'fit' to the market prices.

Exercises 159

Notes: The financial models with jumps were introduced by Merton (1976).The approach used in this chapter is based on Follmer and Sondermann (1986),CERMA (1988) and Bouleau and Lamberton (1989). The approach we have chosen.relies heavily on the assumption that the discounted stock price is a martingale.This assumption is rather arbitrary: Moreover, the use of variance as a measureof risk is questionable. Therefore, the reader is urged to consult the recent literature de.aling with incomplete markets, especially Follmer and Schweizer (1991),Schweizer (1992,1993,1994), El Karoui and Quenez (1995).

7.4 Exercises

~xercis~ 3~. Let (Vn)n~I be a sequence of non-negative, independent and identically distributed random variables and let N be a random variable with valuesin N, following a Poisson law with parameter A, independent of the sequence(Vn)n~I. Show that .

E (11 vn) ~ e'(E(V,j-')

~xercise40 Let (Vn)~~1 be a sequence of independent, identically distributed,Integrable random variables and let N be a random variable taking values in Nintegrable and independent of the sequence (V, ). We set S = ""N V, (with thea n L ..m=I n

convention En=I = 0).

. 1. Prove that S is integrable and that E(S) = E(N)E(Vi}

2. "!'Ie assume N and VI to be square-integrable. Then show that S is squareIntegrable and that its variance is Var(S) = E(N)Var(Vt) +Var(N) (E(Vt ))2.

3. Deduce that if N follows a Poisson law with parameter A, E(S) = AE(Vt)and Var(S) = AE (Vn.

Exercise 41 The hypothesis and notations are those of Exercise 40. We supposethat the Vi's take values in {a, ,8}, with a, ,8 E IR and we set p = P(VI = a) =1 - P(Vt = ,8). Prove that S has the same law as aNI + ,8N2 , where N I andN 2 are two independent random variables following a Poisson law with respectiveparameters AP and (1 - p)A. .

Exercise 42

1. W,e suppose, with the notations of Section 7.3, that UI takes values in {a, b},WIth P = P(UI = a) = 1 - P(UI = b). Write the price formula (7.8) as adouble series where each term is calculated from the Black-Scholes formulae(hint: use Exercise 41).

2. Now we suypose that UI has the same law as e9 - 1, where 9 is a normalvariablewithmean m and variance a 2 • Write the price formula (7.8) as a seriesof terms calculated from the Black-Scholes formulae (for some interest ratesand v?latilities to be given).


1 (2aF ( )a - s x(12 +A Jv(dz)z2 ax'

j. (d) (F(S,X(l+Z))-F(S,X)))+A v z z .x

( (_ )2T aF - - 2 2

E Jo ax (s, X s) - Hs X sa ds

+ JoT AJv(dz) (F(s, X s(l + z)) - F(s, X s) - HszXs) 2 dS) .

o

6.(s,x) =

m =

with

158


(8.1)

160

Exercise 43 The objective of this exercise is to show that there is no perfecthedging ofcalls and puts for the models with jumps we studied in this chapter.We consider a model in which a > 0, >. > aand P (U1 1= 0) > O.

1. From Proposition 7.3.4, show that if there is a perfect hedging scheme then,for ds almost every.s and for v almost every z, we have

P a.s. ZXs~~ (s,~s) = F(s,Xs(1 + z)) - F(s,Xs)'

2. Show that the law of X; has (for s > 0) a positive density on ]0, ~[. Itmay be worth noticing that if Y has a density 9 and i~ Z is a random vana~leindependent ofY with values in ]0,00[; the random vanable Y~ has the density

JdJ-L(z)(l/z)g(y/z),whereJ-L is the lawofZ. - .

3. Under the same assumptions as in the first question, ~ho"w that there eXl~ts

z 1= asuch that f~r s E [0,T[ and x E]0,00[,aF F(s,x(l+z))-F(s,x)ax (s,x) = zx .

Deduce (using the convexity of F with respect to x) that, for s E [0, T], the

function x t-t F (s; x) is linear. . . "

4. Conclude. It may be noticed that, in the case of the put, the function x t-t F(s, x)is non-negative and decreasing on ]0,00[.

o

8

Simulation and algorithmsfor financial models

8.1 Simulation and financial models

In this chapter, we describe some methods which can be used to simulate financialmodels and compute prices. When we can write the option price as the expectationof a random variable that can be simulated, Monte Carlo methods can be used.Unfortunately these methods are inefficient and are only used if there is no closedform solution for the price of the option. Simulations are also useful to evaluatecomplex hedging strategies (example: find the impact of hedging a portfolio everyten days instead of every day, see Exercise 46).

8.1.1 The Monte Carlo method

The problem of simulation can' be presented as follows. We consider a randomvariable with law J-L(dx) and we would like to generate a sequence of independenttrials, Xl, .. : ,Xn, . . . with common distribution J-L. Applying the law of largenumbers, we can assert that if f is a J-L-integrable function

lim N1

" f(Xn) =Jf(x)J-L(dx).N-t+oo L..Jl~n~N

To implement this method on a computer, we proceed as follows. We suppose thatwe know how to build a sequence of numbers (Un)n>1 which is the realisation

t.. -

of a sequence of independent, uniform random variables on the interval [0,1]and we look for a function (Ul' ... ,up) t-t F( Ul, ... ,up) such that the randomvariable F(Ul, ... ,Up) has the desired law J-L(dx). The sequence of randomvariables (Xn)n~1 where X n = F(U(n-l)p+l"" ,Unp) is then a sequence ofindependent random variables following the required law J-L. For example, we canapply (8.1) to the functions f(x) = x and f(x) = x2 to estimate the first andsecond-order moments of X (provided E(IXI 2 ) is finite).

The sequence (Un)n~l is obtained in practice from successive calls to a pseudorandom number generator. Most languages available on modern computers providea random' function, already coded, which returns either a pseudo-random number

{Xo = initial value E {O, 1, ... ,m - I}~n+I = aXn + b (modulo m),

a, b, m being integers to be chosen cautiously in order to obtain satisfactorycharacteristics for the sequence. Sedgewick (1987) advocates the following choice:

8.1.2 Simulation ofa uniform law on [0, 1]

We explain how to build random number generators because very often, thoseavailable with a certain compiler are not entirely satisfactory.

The simplest and most common method is to use the linear congruential generator. A sequence (Xn)n;?:O of integers between 0 andm - 1 is generated asfollows:

between 0 and 1, or a random integer in a fixed interval (this function is calledrand () in C ANSI, random in Turbo Pascal).

Remark 8.1.1 The function F can depend in some cases (in particular when itcomes to simulate stopping times), on the whole sequence (Un)n;?:I, and not onlyon a fixed number of Ui 'so The previous method can still be used if we can simulateX from an almost surely finite number of Ui 's, this number being possibly random.This is the case, for example, for the simulation of a Poisson random variable (seepage 163).

Simulation ofa Gaussian law

A classical method to simulate Gaussian random variables is based on the observation (see Exercise 44) that if (UI , U2 ) are two independent uniform randomvariables on [0, 1]

V-210g(Ud cos(27rU2 )

follows a standard Gaussian law (i.e. zero-mean and with variance 1).To simulate a Gaussian random variable with mean m and variance a, it suffices

to set X = m + ag, where 9 is a standard Gaussian random variable.function Gaussian(m, sigma : real) : real;begin

gaussian := m + sigma" sqrt(-2.0 " log(Random» " cos(2.0 " pi" Random);

Simulation and financial models 163

The previous generator provides reasonable results in common cases. Howeverit might happen that its period (here m = 108 ) is not big enough. Then it is possibleto create random number generators with an arbitrary long period by increasing m.The interested reader will find much information on random number generatorsand computer procedures in Knuth (1981) and L'Ecuyer (1990).

8.1.3 Simulation ofrandom variables

The probability laws we have used for financial models are mainly Gaussian laws(in the case of continuous models) and exponential and Poisson laws (in the caseof models with jumps). We give some methods to simulate each of these laws.

end;

314158211108 .

Simulation and algorithms for financial models162

This method enables us to simulate pseudo-random integers between 0 and m - 1;to obtain a random real-valued number between 0 and 1 we divide this randominteger by m.

constm 100000000;ml 10000;b 31415821;

var a : integer;

function Mult(p, q: integer) : integer;(" Multiplies p by q, avo i.d i nqjvover f Lows ' ")var pI, pO, ql, qO : integer; l.

begin. p l, := p div ml;pO := p mod ml;

ql := q div ml;ql :=.q mod ml;Mult := (((pO"ql + pI"qO) mod ml)"ml + pO"qO) mod m;

end;

Simulation ofan exponential law

We recall that a random variable X follows an exponential law with parameter f,Lif its law is

1{x;?:O}f,Le r- u» dx.

We can simulate X ~oticing ~hat, if U follows a uniform law on [0,1], 10g(U) / f,Lfollows an exponential law with parameter u,function exponential( mu : real) : real;begin

exponential := - log (Random) / mu;end;

Remark 8.1.2 This method of simulation of the exponential law is a particularcase of the so-called 'inverse distribution function' method (for this matter seeExercise 45). '

Simulation ofa Poisson random variable" -

oA Poisson random variable is a variable with values in N such that

AnP(X = n) = e->'" ifn 2: O.

n.

164 Simulation and algorithms for financial models

We have seen in Chapter 7 that if (Ti)i>l -iSa sequence of exponential random variables with parameter A, then the law oeNt = L:n>l nl {Tl +..+Tn9<T1+··+Tn+l}is a Poisson law with parameter At. Thus Ni h-as the same law as the variableX we want to simulate. On the other hand, it is always possible to write exponential variables T, 'as -log(Ui)/A, where the (Ui)i>l 's are independent randomvariables following the uniform law on [0, 1). N: can be written as

N: = Lnl{U1U2 ...Un+l~e-A<U1U2 ...Un}·n:2:1

This leads to the following algorithm to simulate a Poisson random variable.function poisson(larnbda : real) : integer;var

u : real;n : integer;

begina := exp(-larnbda);u := Random; .n := 0;while u > a do begin

u : = u * Random;n := n + 1

end;Poisson := n

end;

For the simulation of laws not mentioned above or for other methods of simulation of the previous laws, one may refer to Rubinstein (1981).

Simulation of Gaussian vectors

Multidimensional models will generally involve Gaussian processes with valuesin IR.". The problem in simulating Gaussian vectors (see Section A.l.2 of theAppendix for the definition of a Gaussian vector) is then essential. We give amethod of simulation for this kind of random variables.

We will suppose that we want to simulate a Gaussian vector (Xl,' .. , X n)whose law is characterised by the vector of its means m = (mI, ... , m n ) =(E(X1 ) , ... , E(Xn)) and its variance matrix r = (O'ij h~i~n.1~j~n whererr., =E(XiXj) - E(Xi)E(Xj). The matrix r is positive definite and we will assumethat it is invertible. We can find the square root of I', in other words a matrix A,such that A x t A = r. As r is invertible so is A, and we can consider the vectorZ = A-I (X - m). It is easily verified that this vector is a Gaussian vector withzero-mean. Moreover, its variance matrix is given by

o

Simulation andfinancial models 165

Z is therefore a Gaussian vector with zero-mean and a variance matrix equal tothe identity. The law of the vector Z is the law of n independent standard normalvariables. The law of the vector X = m + AZ can then be simulated in thefollowing manner:

• Derive the square root of the matrix I', say A.

• Simulate n independent standard normal variables G = (gl' ... ,gn)'

• Compute m + AG.

Remark 8.1.3 To derive the square root of I', we may assume that A is uppertriangular; then there is a unique solution to the equation A x t A = r. This methodof calculation of the square root is called Cholesky's method (for a completealgorithm see Ciarlet (1988)).

8.1.4 Simulation ofstochastic processes

The methods delineated previously enable us to simulate a random variable, inparticular the value of a stochastic process at a given time. Sometimes we needto know how to simulate the whole path of a process (for example, when weare studying the dynamics through time of the value of a portfolio of options,see Exercise 47). This section suggests some simple tricks to simulate paths ofprocesses.

Simulation ofa Brownian motion

We distinguish two methods to simulate a Brownian motion (Wdt>o. The firstone consists in 'renormalising' a random walk. Let (Xi)i>O be a ~equence ofindependent, identically distributed random walks with law-P (Xi = 1) = 1/2,P (Xi = ,-:-~) = 1/2. Then we have E (Xi) = 0 and E (Xl) = 1. We setSn = Xl + .,. + X n; then we can 'approximate' the Brownian motion by theprocess (X;')t:2:o where

X;' = JnS[nt1

where [x) is the largest integer less than or equal to x. This method of simulationof the Brownian motion,is partially justified in Exercise 48.

In the second method, we notice that, if (gi)i:2:0 is a sequence of independentstandard normal random variables, if t1t > 0 and if we set

{So = 0

, Sn+l - Sn = g':'

then the law of (ViSJ,So, ViSJ,Sl,' .. , ViSJ,sn) is identical to the law of

~ (Wo, W~t, W2~t, ... ,Wn~t).

The Brownian motion can be approximated by X;' = ViSJ,S[t/ ~tl'

Simulation ofstochastic differential equations

CT being a constant depending only on T.

If Xl' = S[t/~t), (Xl')t~O approximates (Xt)t~O in the following sense:

We discretise time by a fixed mesh b.t. Then we can construct a discrete-timeprocess (Sn)n~O approximating the solution of the stochastic differential equationat times nb.t, setting

167

= x

Simulation and financial models

An application to the Black-Scholes model

In the case of the Black-Scholes model, we want to simulate the solution of theequation

{;;t : ~t(rdt + adWt).Two approaches are available. The first consists in using the Euler approximation.We set

{SOSn+l Sn(1 + rb.t + agn.,fl;i) ,

and simulate X; by Xl' = S[t/~t). The eithermethod consists in using the explicitexpression of the solution

Simulation ofmodels with jumps

We have investigated in Chapter 7 an extension of the Black-Scholes model withjumps; we describe now a method. to simulate this process. We take the notationsand the hypothesis of Chapter 7, Section 7.2. The process (Xt)t~O describing thedynamics, of the asset is . ., ,

x, = xexp (rt - ~2t + awt)

and simulating the Brownian motion by one of the methods presented previously.In the case where we simulate the Brownian motion by .,fl;i ~:::l gi, we obtain

s; = z exp ((, .; q' /2)nLlt +q~t, g,) . (8.2)

We always approximate X, by Xl' = S[t/~t).

Remark 8.1.6 We can also replace the Gaussian random variables gi by someBernouilli variables with values +1 or -1·with probability 1/2 in (8.2); we obtaina binomial-type model close to the Cox-Ross-Rubinstein model used in Section5.3.3 of Chapter 5.

Simulation and algorithms for financial models

= xb(Xt)dt + a(Xt)dWt.

x{b(Sn)b.t + a(Sn) (W(n+l)~t "- Wn~t)} .

{Xoex.

{SoSn+l - Sn

166

There are many methods, some of them very sophisticated, to simulate the solutionof a stochastic differential equation; the reader is referred to Pardoux and Talay(1985) or Kloeden and Platen (1992) for a review of these methods. Here wepresent only the basic method, the so-called 'Euler approximation' . The principleis the following: consider the stochastic differential equation

A proof of this result (as well as other schemes of discretisation of stochasticdifferential equations) can be found in Chapter 7 of Gard (1988) .

The law of the sequence (W(n+l)~t - Wn~t)n>O is the law of a sequenceof independent normal random variables with zero-mean ana variance b.t. In asimulation, we substitute gn.,fl;i to (W(n+l)~t - Wn~t) where (gn)n~O is asequence of independent standard normal variables. The approximating sequence(S~)n~O is in this case defined by

Theorem 8.1.4 For any T > 0

E (sup IXl' -.Xt12) ~ CTb.t,t<S.T ,

= xS~ + b.t b(S~) + a(S~)gn.,fl;i.

Remark 8.1.5 We can substitute to the sequence of independent Gaussian randomvariables (gi)i~O a sequence of independent random variables (Ui)i~O, such thatP(Ui = 1) = P(Ui = -1) = 1/2. Nevertheless, in this case, it must be noticedthat the convergence is different from that found in Theorem 8.1.4. There is stilla theorem of convergence, but it applies to the laws of the processes. Kushner(1977) and Pardoux and Talay (1985) can be consulted for some explanations onthis kind of convergence and many results on discretisation in law for stochasticdifferential equations.

(

N ' ) .x, ::;::: x . }1(i + ~j) e(Il-

CT2/

2)t+CTW, , (8.3)

where (Wt)t~O is a standard Brownian motion, (Nt)t>o is a Poisson processwith intensity .x, and (Uj) j~ 1 is a sequence of independent, identically distributedrandom variables, with values in ] -:- 1, +oo[ and law lJ(dx). The a-algebrasgenerated by (W~)t~O' (Nt)t~o, (Uj)j~l are supposed to be independent.

To simulate this process at times rust, we notice that

Xn~t ~ X X (X~t/x) x (X2~t/X~t) x ···x (Xn~t/X(n-l)~t).

Ifwe noteYk = (Xk~t/X(k-l)~t), we can prove, from the properties of (Nt)t~o,

FUNCTION PutObstacle(x : REAL;Opt : AmericanPut) :'REAL;VAR u : REAL;BEGIN

u := Opt,StrikePrice - exp(x);IF u > 0 THEN PutObstacle := u ELSE PutObstacle := 0,0;

END;

8.2.2 Implementation of the Brennan and Schwartz method

The following program prices an American put using the method described inChapter 5, Section 5.3.2: we make a logarithmic change of variable, we discretisethe parabolic inequality using a totally implicit method and finally solve theinequality in infinite dimensions using the algorithm described on page 116,CONST

PriceStepNb = 200;'TimeStepNb = 200;Accuracy = 0,01;DaysInYearNb = 360;

Some useful algorithms

opposed to an exponential. If x > 0

169

annual riskless interest rate *)annual volatility *)initial value of the SDE *)

Cl = 0.196854C2 = 0.115194C3 = 0.000344C4 = 0.019527

1 'N(x) ::::: 1 - 2(1 + CIX+ C2x2 + C3X,3 + C4x4)-4.

r

TYPEDate = INTEGER;Amount = REAL;AmericanPut = RECORD

ContractDate : Date; (* in days *)MaturityDate : Date; (* in days *)StrikePrice : Amount;END;

vector = ARRAY[l, ,PriceStepNbI OF REAL;Model = RECORD

REAL; (*

REAL; (*

i REAL; (*

sigmaxOEND;I '" ,,2 dx

N(x) = P(X :::; x) = e--: T rs:':-00 y 27r

where X is astandard Gaussian random variable: Due to the importance of thisfunction in the pricing of options, we give two approximation formulae fromAbramowitz and Stegun (1970) ,

The first approximation is accurate to 10-7 , but it uses the exponential function,

Ifx>O

is identical to the law 'of Yl .

8.2 Some useful algorlthms

In this section, we have gathered some widely used algorithms for the pricing ofoptions.

8.2.1 Approximation ofthe distributionfunction ofa Gaussian variable

We saw in 'Chapter 4 that the pricing of many classical options requires thecalculation of


(Wdt2:o and (Uj ) j2:l that (Yk h2:l is a sequence of independent random variableswith the same law. Since Xn~t = xYl . , . Yn, the simulation of X at timesn.6.t comes down to the simulation of the sequence (Yk h 2: l . This sequencebeing independent and identically distributed, it sufficesto know how to simulateYl = X~tlx. Then' we operate as follows:

• We simulate a 'standard Gaussian random variable g.

• We simulate a Poisson random variable with parameter )".6.t: N.

• If N = n, we simulate n random variables following the law J.L(dx): Ul , ... , Un'

All these variables are assumedto be independent. Then, from equation (8.3), itis clear that the law of

p -' 0.231641900b1 = 0.319381530b2 = -0.356563782b3 = 1.781477937

-b4 ~1.821255978b5 1.330274429t = 1/(I+px)

1 ,,2 2 3 4 5N(x) ::::: 1 - rrce-T (bit + b2t + b3t + b4t + b5tv ) .

, y27r

The second approximation is accurate to 10-3 but it involves only a ratio as

FUNCTION Price(t : Date; x : Amount;option : AmericanPut; 'model : Model) : REAL;

( *prices the 'option' for the 'model'at time 't' if the price o~th~ underlying at 'this timeis "x ".

*)VAR

Obst,A,B,C,G : vector;alpha, beta ;gamina, h , k , VV, temp, r , Y» del t a , Time', 1 : REAL;Index,PriceIndex,TimeIndex : INTEGER;

BEGIN '""Time := (option,MaturityDate - ~) / Days InYearNb;k := Time / TimeStepNb;r := model.r;

170 Simulation and algorithms for financial models Exercises 171

vv := ~odel.sigma * model. sigma;

1 := (model.sigma * sqrt(Temps) * sqrt(ln(l/Accuracy» + abs(r - vv / 2) *Time) ;

h := 2 * 1 / PriceStepNb;writeln(1:5:3.'·' ,In(2) :5:3);

f(x)dx. We setF(u) = I::oo f(x)dx. Prove thatifU is a uniform random variableon [0,1], then the law of F-l (U) is f(x)dx. Deduce a method of simulation ofX.

Exerci.se 46 We model a risky asset S, by the stochastic differential equation

where (Wt)t>o is a standard Brownian motion, a the volatility and r is the risklessinterest rate. Propose a method of simulation to approximate

alpha := k * (- vv / (2.0 * h * h) + (r - vv / 2'.0) / (2.0 * h»;beta := 1 + k * (r + vv / (h * h»;gamma := k * (- vv / (2.0 * h * h) - (r - vv / 2.0) / (2.0 * h»;FOR PriceIndex:=l TO PriceStepNb DO BEGIN

A[PriceIndex] := alpha;B[PriceIndex] := beta;C[PriceIndex] := gamma;END;

B[l] := beta + alpha;B[PriceStepNb] := beta + gamma;G[PriceIndex] := 0.0;

{as,So = x,

Give an interpretation for the final value in terms of option.

Exercise 47 The aim of this exercise is to study the influence of the hedgingfrequency on the variance of a portfolio of options. The underlying asset of theoptions is described by the Black-Scholes model

( ,We ,90 not hedge: we sell the option, get the premium, we wait for threemonths, we take into account the exercise of the option sold and we evaluatethe portfolio.

We hedge immediately after selling the option, then we do nothing.

(Wtk~o represents a standard Brownian motion, a the annual volatility and r theriskless interest rate. Further on we will fix r = lO%jyear, a = 20%/Jyear = 0.2and x = 100.

Being 'delta neutral' means that we compensate the total delta of the portfolioby trading the adequate amount of underlying asset.

In the following, the options have 3 months to maturity and are contingent onone unit of asset. We will choose one of the following combinations of options:

• Bull spread: long a call with strike price 90 (written as 90 call) and short a 110call with same maturity.

• Strangle: short a 90 put and short a 110 call.

• Condor: short a 90 call, long a 95 call and a 105 call and finally short a 110call.

• Put ratio backspread: short a 110 put and long 3 90 puts.

First we suppose that f.L = r . Write a program which:

• Simulates the asset described previously.

• Calculates the mean and variance of the discounted final value of the portfolioin the following cases:

x,{as,So

Exercise 44 Let X and Y be two standard Gaussian random variables; derive! the joint law of (JX2 + Y2,arctg(Y/X)). Deduce that, if U1 and U2

are two independent uniform random variables on [0,1], the random variablesJ -210g(Ul ) cos(27l'U2 ) and J -210g(Ud sin(27l'U2 ) are independent and fol-Iowa standard Gaussian law. . 0

Exercise 45 Let f be a function from JRto JR, such that f(x) > 0 for all x, and

such that r: f(x)dx = 1. We want to simulate a random variable X with-law

B[PriceStepNbl := B[PriceStepNb];FOR PriceIndex:=PriceStepNb-l DOWNTO 1 DO

B[PriceIndex] := B[Pri'ceIndex] - C[PriceIndex] * A[PriceIndex+l] /B[PriceIndex+l] ;

FOR PriceIndex:= i TO PriceStepNb DO A[PriceIndex] := A[PriceIndex] /"B[PriceIndex] ;

FOR PriceIndex:= 1 TO PriceStepNb - 1 DO C[PriceIndex] := C[Pricelndex] /B[PriceIndex+l] ;

G[l] := G[l] / at i i ,FOR ,PriceIndex:=2 TO PriceStepNb DO BEGIN

G[PriceIndex] : = G[PriceIndexl / B[PriceIndex] - A[Pric'~IndexJ * G [PriceIndex-l] ;

temp := Obst[PriceIndex];IF G[PriceIndex] < temp THEN G[PriceIndex] := temp;

END;END;Index := PriceStepNb DIV 2;delta := (G[Indice+l] - G[Index]) / h;Prix':= G[Index]+ delta*(Index * h - 1);

END;

y := In(x);FOR PriceIndex:=l TO PriceStepNb DO Obst(PriceIndex] := PutObstacle(y - 1 +

PriceIndex * h , option ); .FOR PriceIndex:=l TO PriceStepNb DO G[PriceIndex] := ,0bst[priceIndex];

8.3 Exercises.:..

FOR TimeIndex:=l TO TimeStepNb DO BEGINFOR PriceIndex := PriceStepNb-l DOWNTO 1 DO

G[PriceIndex) := G[PriceIndex] - '. C[PriceIndex] * G[PriceIndex+1];

_ We hedge immediately after seIling the option, then every month.

_ We hedge immediately after selling the option, then every 10 days .

. _ We hedge immediately after selling the option, then every day.

Investigate the influence of the discretisation frequency.Now consider the previous simulation assuming that J.L =I r (take values of J.L

bigger and smaller than r). Are there arbitrage opportunities?

Exercise 48 We suppose that (Wt)t>o is a standard Brownian motion and that(Ui)i>l is a sequence of independent random variables taking values +1 or -1with probability 1/2. We set Sn = Xl + .,. + X n.

1. Prove that, if X? = S[ntj/"fii, X? converges in law to Wt.

2. Let t and s be non-negative.using the fact that the random variable X?+s - X?is independent of X?, prove that the pair (X?+s' X?) converges in law to

(Wt+s, Wt).3. IfO < tl < ... < t p,show that (X~, . . . ,X~) converges in law to (Wt l , · .. , Wt p ) .

o

~

I

~I

Appendix

A.I Normal random variables

In this section, we recall the main properties of Gaussian variables. The followingresults are proved in Bouleau (1986), Chapter VI, Section 9.

A.i.i Scalar normal variables

A real random variable X is a standard normal variable if its probability densityfunction is equal to -

n(x) = _1_ exp (_ X

2)

•.J2; 2

If X is a standard normal variable and m and a are two real numbers, then thevariable Y = m + aX is normal with mean m and variance 0'2. Its law is denotedby N(m, 0'2) (it does not depend on the sign of a since X and - X have the samelaw). If a i= .0, the density of Y is

_1_ exp ( (x-m)2)J27fO'2 20'2'

If a = 0, the law of Y is the Dirac measure in m and therefore it does not have adensity. It is sometimes called 'degenerate normal variable'.

If X is a standard normal variable, we can prove that for any complex numberz, we have . .

E (e z X) = e4 .

Thus,the characteristic function of X is given by ifJx(u) = e-u 2/

2 and for Y,ifJy(u) = eiUTne-u2(j2/2. It is sometimes useful to know that if X is a standardnormal variable, we have P(IXI > 1,96,..) = 0,05 and P(IXI > 2,6 ...) =0,01. For-large values of t > 0, the following approximation is handy:

. 1 100 1 100_t

2/2P(X > t) = _rrc e'--x

2/2dx$ _rrc xe-x2/2dx = _e__ .

, v 27f t tv 27f t . t.J2;

Finally, one.should know that there exist very good approximations of the cumulative normal distribution (cf. Chapter 8) as well as statistical tables.

A.I.2 Multivariate normal variables

Definition Ad.I A random variable X = (XI, ... ,Xd) in lRd is a Gaussianvector iffor any sequence ofreal numbers at. ... , ad, the scalar random variable2:~=1 aiX is normal.

The components Xl •...• X d of a Gaussian vector are obviously normal, but thefact that each component of a vector is a normal random variable does not implythat the vector is normal. However. if Xl, X 2 • . . . , X d are real-valued, normal.independent random variables. then the vector (Xl, ... ,Xd) is normal.

The covariance matrix of-a random vector X = (Xl, ... , X d ) is the matrixI'(X) = (aij h~i,j~d whose coefficients are equal to

aij = cov(Xi, Xj) = E [(Xi - E(Xi))(Xj - E(Xj))].

It is well known that if the random variables Xl'•... , X d are independent. thematrix I'(X) is diagonal. but the converse is generally wrong. except in theGaussian case:

Theorem A.I.2 Let X = (Xl,' .. , X d) be a Gaussian vector in lRd. The randomvariables Xl, ... , X d are independent if and only if the covariance matrix X isdiagonal. . ,

The reader should consult Bouleau (1986), Chapter VI. p. 155, for a proof ofthisresult.

RemarkA.I.3 The importance of normal random variables in modelling comespartly from the central limit theorem (cf. Bouleau (1986), Chapter VI,I, Section 4).The reader ought to refer to Dacunha-Castelle and Dufto (1986) (Chapter 5) forproblems of estimation and to Chapter 8 for problems of simulation.

Conditional expectation174 Appendix175

Let us now consider a random variable X defined o!!-(O, A) with values in am~able ~~~.JE, E).Th~ ~~l~bra generated~X isthesmallest a-algebraf~~~~.Ich X ~~~~~~~ It IS denoted by a(X). It isolWiouslyincluaea-in Aand It IS easy to show that .~--.

-'-'---'-~(X)~ {A E AI3B E E,A = X-I(B) = {X E B}}.

We can prove that a random variable Y from (0, A) to (F,:F) is a(X)-measurableif and only if it can be written as . .

Y = foX,

where fJs a me~~b~~~aP._~~~(E~E) toJF,:F). (cf. Bouleau (1986), p.101-102). In other words, a(X)-measurable random variables are the measurablefunctions of X.

A.2.2 Properties. ofthe co~dition~l expectation

Let (0, A, P) be a probability space and B a a-algebra in~luded in A. Thedefinition of the conditional expectation is based on the following theorem (referto Bouleau (1986), Chapter 8):

!heorem A.2.1 For any real integrable random variable X, there exists a realIntegrable Btmeasurable random variable Y such that

VB E B E(XIB) = E(YIB).

If Y is another randomvar~ablewith these properties then Y = Y P a.s.

Y is th~ cond~tional expectation of X given B and it is denoted by E(XIB).If B IS a finite sub-a-algebra, with atoms B I , ... , Bn , . .

. E(XIB) =L E(XIB;)/P(Bi)IB;,

A.2 Conditional expectation

A.2.I Examples ofa-algebras

.Let us consider a~~Race (P,A) and a P~.Q'I. B 2 • • • • , ~n; withn events in A. The set B containing the elements of ~ which are ei~pty orthat can be written as Bit U B i2 U··· U B ik, where i~, ... , ik E {I, ... , n}, is afinite sub-a-alg;bra of A. It is-the a-algebra generated by the sequence of B;

" Conversely, to any finite s.!Jb~a-=algebraI3 of .A,we can associ~ition(B I , ... , B n ) of 0 where Bis·g~.!'erated by theelements B, of A: B, are the non-

_.-"""'-~ ----. ---. ~~ ,........-empty elements ofB whichcontain onIy-tnemselves and the empty set. They are.--...--:""_.._-~._----- .,.r:---.-.-.- ' ----- .'called--atoms of B. There is a one-to-one mapping-fffiffi-tfie ser'OCfinite sub-a-algeoras ofAonto the set of partitions of 0 by elements of A. One should noticethat if B is a sub-a-algebra of A, a map from 0 to lR (and its Borel a-algebra) isB-measurable if and only. if it is constant on each atom of B.~----=-----~~--_.~

where we sum on the atoms with strictly positive probability..Consequently, oneach atom B i, E(XIB) is the mean value of X on Bi, As far as the trivial a-algebrais concerned (B = {0, OJ), we have E(XIB) = E(X). . .

The ~omputationsinvolving conditional expectations are based on the followingproperties:

~!.; If£~~~l![~ble,E(XIB) = X, a.s.

.~ E (E (XIB)) = E (X).

3. For any bounded! B-measurablerandom variable Z,E (ZE(XIB)) = E(ZX).4. Linearity:' . ''-' ~ ,'--

E,: (~~ + JLYIB) =AE (XIB) + JLE (YIB) a.s.I ~_

5. Positivity: if X 2: 0, then E(XIB) 2: aa.s. and more generally, X 2: Y =>E(XIB) .~ E(YIB) a.s. It follows from this property that

~ IE (XIB)I $ E (IXIIB) a.s.

,

o

Appendix

is a Borelfunction on (E, E) and we have

Proof. Let us denote by P y the law of Y; We have

177Conditional expectation

which completes the proof. 0

In other words, under the previous assumptions, we can compute E ( (X, Y) IB)as if X was a-constant. ' .., .

E ( (X, Y)IB) = cp(X) a.s.

Vx E E cp(x) =;: E ((x, Y))

and the measurability of cp is a consequence of the,Fubini theorem. Let Z be anon-negative B-measurable random variable (for example Z = IB, with B E B).Ifwe denote by P X,Z the law of (X, Z), it follows from the independence betweenY and (X, Z) that,

E ((X, Y)Z) = / / (x, y)zdPx,z(x, z)dPy(y)

/ (/ (x, y)dPy(y)) zdPx,z(x, z)

= / cp(x)zdPx,z(x,z)

= E (cp(X)Z) ,

Proposition A.2.S Let us consider a B-measurable random variable X takingvalues in (E, E) and Y, a random variable independent of B with values in(F, F). For any Borelfunction non-negative (or bounded) on (E x F, E I8i F),the function cp defined by ,

cp(x) = i (x, y)dPy(y)

Remark A.2.6 In the Gaussian case, the computation of a conditional expectation is particularly simple. Indeed, if (Y, Xl, X 2 , ..• ,Xn ) is a normal vector (inIR

n +I) , the conditional expectation Z' == E (YIXI , . . . ,Xn ) has the following

form " '

n

Z = Co + LCiXi,i=l

where c, are real constant numbers. This means that the function of Xi whichapproximates.Y in the least-square sense is linear. On top of that, we can computeZ by p~?jecting the random variable Y in L2 on the linear subspace generated byI and the X/s (cf. Bouleau (1986), Chapter 8, Section 5).

This equality means that the characteristic function of X is identical under measureP and measure Q where thedensity of Q with respectto P is equal to IB /P(B).The equality of characteristic functions implies the equality of probability lawsand consequently ,

E {l(X:~p~~)) =.E (f(X)),

for any bo~nded Borelfunction j, hence the independence.

Remark A.2.3 If X is square integrable, so is E(XIB), and E(XIB) coincideswith the orthogonal projection of X on L2(n,B, P), which is a closed subspaceof L2 (n, A,P), together with the scalar product (X, Y) H E(XY) (cf. Bouleau(1986), CnapterVIII, Section 2). The conditional expectation of X given B is theleast-square-best B-measurable predictor of X. In particular, if B is the a-algebragenerated by a random variable €, the conditional expectation E(XIB) is notedE(XI€), and it is the best approximation of X by a function of €, since a(€)measurable random variables are the measurable functions of €- Notice that byPythagoras' theorem, we know that IIE(XIB)IIL2Cfl) ~ IIXII£2Cfl).

Remark A.2.4 We can define E(XIB) for any non-negative random variableX (without integrability condition). Then E(XZ) = E (E(XIB)Z), for any Bmeasurable non-negative random variable Z. The rules are basically the same asin the integrable case (see Dacunha-Castelle and Duflo (1982), Chapter 6).

:J

176

and therefore II E(XIB)II£lCfl) ~ IIXII£lCfl).6. If C is a sub-a-algebra of B, then

E (E (XIB) IC) = E (XIC) a.s.

7. If Z is B-measurable and bounded, E (ZXIB) = ZE (XIB) a.s.

8. If X is independent of B then E (XIB) = E (X) a.s.

The converse property is not true but we have the following result.

Proposition A.2.2 Let X be a real random variable. X is independent of thea-algebra B if and only if

VU'E IR E (eiUXIB) ='E (e iuX) a:!. (A.l)

Proof. Given the Property 8..above,we just need to prove that (A.l) implies thatX is independent of B.

If E (e iuXIB) = E (e iUX) then, by definition of the conditional expectation,for all B E B,E (e iuXIB) = E (eiuX) P(B). If P(B) =j:. 0, we can write '

E (eiUX~) = E (eiuX)P(B) -, ~ .

A.2.3 Computations ofconditional expectations

, The following proposition is crucial and is used quite often in this book.

I /'\

178 Appendix

A.3 Separation of convex sets

. In this section, we state the theorem of separation of convex sets that we use inthe first chapter. For more details, the diligent reader can refer to Dudley (1989)

p. 152 or Minoux (1983).

Theorem A.3.1 Let C be a closed convex set which does not contain the origin.Then there exists a real linear functional ( defined on IRn and 0: > 0 such that

'<Ix E C ((x) 2: 0:.

In particular, the hyperplane ((x) = 0 does not intersect C.

Proof. Let>' be anon-negative real number such that the closed ball B(>') withcentre at the origin and radius>' intersects C. Let Xo be the point where the mapx ~ Ilxll achieves its minimum (where 11·11 is the Euclidean norm) on the compactset C n B(>'). It follows immediately that . ,

'<Ix E C IIxll 2: IIxoll·The vector Xo is nothing but the projection of the origin on the closed convex setC. If we consider x E C, then for all t E [0,1], Xo + t(x - xo) E C, since Cisconvex. By expanding the following inequality " .

IIxo + t(x - xo)I12 2: Ilxoll 2,

it yields xo.x 2: IIxoll2 > 0 for any x E C, where xo.x denotes the scalar productof Xo and x. This completes the proof.' 0

Theorem A.3.2 Let us consider a compact convex set K and a vector subspaceVof IRn

. If V and K are disjoint, there exists a linear functional ( defined onIRn

, satisfying the following conditions:

J. '<Ix E K ((x) > O.

2. '<Ix E V ((x) = O.Therefore, thesubspace V is included in a hyperplane that does not intersect K.

Proof. The set

C=K-V={xEIRn 13(y,z)EKxV,x=y-z}

is convex, closed (because V is closed and IS is compact) and does, not containthe origin. By Theorem A.3.1" we can find a linearfunctional ( defined on IR

n

and a certain 0: > 0 such that

'<Ix E C ((x) 2: 0:.

Hence'<Iy EX, '<Iz E V, ((y) - ((z) 2: 0:. (A.2)

For a fixed y, we can apply (A.2) to >.z, with>. E IR to obtainxvz E V, ((z) = 0,

thus '<Iy E K, ((y) 2: 0:, 0

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..... "

o

Index'

Adapted,4Algorithm

Brennan and Schwartz, 116, 169Cox, Ross and Rubinstein, 117

American callpricing, 25

American optionprice, 11

American puthedging, 27pricing, 26

Arbitrage, viiiAsset '

financial, vii, 1riskless, 1, 122risky, 1underlying, vii

Atom, 174Attainable, 8

Bachelier, viiBessel function, 131Black, viiBlack-Scholes formulae, 70Black-Scholes model, ix, 12Bond

pricing, 124, 129, 132Bond option

pricing, 125,' 136Brownian motion, vii, 31

Simulation of process, 165

Call, viipricing, 154

CalIon bondpricing, 133, 138

Complete, 8Conditional expectation

Gaussian case, 177of a non-negative random variable, 176orthogonal projection, 176w.r.t. a random variable, 176

Contingent claim, 8Continuous-time process, 29Cox-Ross-Rubinstein model, 12Crank-Nicholson scheme, 108Critical price, 77

Delta, 72Diffusion, 49Doob decomposition, 21Doob inequality, 35Dynkin operator, 99

Equivalent probabilities, 66Equivalent probability, 6European call

pricing, 70, 74European option

pricing, 68, 154European put

pricing, 70Exercise price, viiiExpectation, 5

184

conditional, 174Expiration, viiiExponential martingale, 33

Filtration, I, 30Forward interest rate, 133

Gamma, 72Girsanov theorem, 66, 77Greeks

delta, 72gamma, 72theta, 72vega, 72

Hedging, viiia can, 14cans and puts, 71no replication, 160of cans and puts, 155-159

Infinitesimal generator, 112Ito calculus, 42Ito formula, 42

multidimensional, 47Ito processes, 43

Lawchi-square, 131exponential, 141, 142gamma, 142lognormal, 64

Marketcomplete, 8incomplete, 141, 160

Markov Property; 54, 56Martingale, 4

continuous-time, 32exponential, 47optional sampling theorem, 33

Martingale transform, 5Martingales representation, 66Merton, viiMethod

Index

Monte Carlo, 161Model

Black-Scholes, 12,47,63,80simulation, 167

Cox-Ingersoll-Ross, 129-133Cox-Ross-Rubinstein, 12discrete-time, 1interest rate, 121-139Vasicek, 127-129with jumps, 141-160

simulation, 167Yield curve, 133

Natural filtration, 30Normal variable, 173

degenerate, 173standard, 173

Numerical methodsBrennan and Schwartz algorithm,116algorithm of Brennan and Schwartz, 169Cox, Ross and Rubinstein method, 117distribution function of a gaussian law,s 168finite differences, 106Gauss method, 109inequality in finite dimension, 116Mac Millan and Waley, 118partial differential inequality, 113

OptionAmerican, viiiAsian, 8European, viii, ,8replicable, 68 I

Optional sampling theorem; 33

Parityput/call, 13

Partial differential equationnumerical solution, 106on a bounded open set. 102parabolic, 95, 99

Partial differential inequalities.vl ll, 113Perpetual put, 75

pricing, 75Poisson process, 141

Index

Portfoliovalue, 2

Positionshort, 3

Predictable, 5Premium, viiiPricing, viiiProcess

continuous-time, 29Omstein-Ulhenbeck, 52Poisson, 141

Put, viipartial differential inequalities, 113pricing, 154

Put/Call parity, ix

Radon-Nikodym, 66Random number generators, 162Replicating strategy, 14

Scholes, viiSeparation of convex sets, 178Short- selling, 3Sigma-algebra, 174Simulation of processes, 165

Black-Scholes model, 167Brownian motion, 165model with jumps, 167stochastic differential equations, 166

Simulation of random variables, 163exponential variable, 163Gaussian, 163Gaussian vector, 164

. Poisson variable, 163Snell envelope, 18Stochastic differential equations, 49, 52, 96Stopped sequence, 18 'Stopping time, 17,30

hitting time, 34optimal,20

Strategy, 1admissible, 3, 68, 125, 151consumption, 27,73self-financing, 2, 64, 125, 151

Strike price, viiSubmartingale, 4Supermartingale, 4

Theta, 72

Vega, 72Viable, 6Volatility, ix, 70

implied,71

Wiener integral, 57

Yield curve, 121, 133

Zero coupon bond, 122

185

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