Introduction to Stochastic Calculus Applied to Finance

~. r

Introduction to

I.

Stochastic Calculus Applied to Finance

I

Damien Loolberton L'Universite ffSfarne la Vallee France

and Bernard Lapeyre L'Ecole Nationale des Ponts et Chaussees France

Translated by Nicolas Rabeau Centre for Quantitative Finance Imperial College, London . and Merrill Lynch Int. Ltd., London

and

'

Francois Mantion Centrefor Quantitative Finance Imperial College London

CHAPMAN & HALUCRC Boca Raton London New York Washington, D.C.

L-G ~jj(5~3

L3(;/3

'9-96

Library of Congress Cataloging-in-Publication Data Catalog record is available from the Library of Congress

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Introduction Options Arbitrage and put/call parity Black-Scholes model and its extensions Contents of the book Acknowledgements 1

2

Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to .infringe,

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?

First edition 1996 First CRC reprint 2000 © 1996 by Chapman & Hall No claim to original U.S. Government works International Standard Book Number 0-412-71800-6 Printed in the United States of America 2 3 4 5 6 7 8 9 0 Printed on acid-free paper

3

Discrete-time models 1.1 Discrete-time formalism 1.2 Martingales and arbitrage opportunities 1.3 Complete markets and option pricing 1.4 Problem: Cox, Ross and Rubinstein model Optimal stopping problem and American options 2.1 Stopping time 2.2 The Snell envelope 2.3 Decomposition of supermartingales 2.4 Snell envelope and Markov chains 2.5 Application io American options 2.6 Exercises Brownian motion and stochastic differential equations 3.1 General comments on continuous-time processes 3.2 Brownian motion 3.3 Continuous-time martingales 3.4 Stochastic integral and Ito calculus 3.5 Stochastic differential equations 3.6 Exercises

vii Vll Vlll

IX X X

1 1 4 8 12

17 17 18 21 22 23

25 29 29 31 32 35

49 56

Contents

vi

4

5

The Black-Scholes model

63

4.1 4.2 4.3 4.4 4.5

63 65 67 72 77

Option pricing and partial differential equations 5.1 5.2 5.3 5.4

6

ModeIling principles Some classical models Exercises

Asset models with jumps 7.1 7.2 7.3 7.4

8

European option pricing and diffusions Solving parabolic equations numerically American options Exercises

Interest rate models 6.1 6.2 6.3

7

Description of the model Change of probability. Representation of martingales Pricing and hedging options in the Black-Scholes model American options in the Black-Scholes model Exercises

Poisson process Dynamics of the risky asset Pricing and hedging options Exercises

Simulation and algorithms for financial models 8.1 8.2 8.3

Simulation and financial models Some useful algorithms Exercises

Appendix Al Normal random variables A2 Conditional expectation A3 Separation of convex sets

Introduction

95 95 103 110 118 121 121 127 136

161 161 , 168 170

The objective of this book is to give an introduction to the probabilistic techniques required to understand the most widely used financial models. In the last few years, financial quantitative analysts have used more sophisticated mathematical concepts, such as martingales or stochastic integration, in order to describe the behaviour 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 "Theory of Speculation' , discovered what is now called Brownian motion. From 1973, the publications by Black and Scholes (1973) and Merton (1973) on option pricing and hedging gave a new dimension to the use of probability theory in finance. Since then, as the option markets have evolved, Black-Scholes and Merton results have developed to become clearer, more general and mathematicaIly more rigorous. The theory seems to be advanced enough to attempt to make it accessible to students.

173'

Options

141 141 143 150 159

173, 174 178

References

179

Index

183

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 of the relevance of applying stochastic calculus to finance. An option gives its holder the 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 a

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

viii

Introduction

• 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 is called a European option; • the exercise price which is the price at which the transaction is done if the option is exercised. The price of the option is the premium. When the option is traded on an organised market, the premium is quoted by the market. Otherwise, the problem is to price the option. Also, even if the option is traded on an-organised market, it can be interesting to detect some possible abnormalities in the market. Let us examine the case of a European call option on a stock, whose price at time t is denoted by St. Let us call T the expiration date and K the exercise price. Obviously, if K is greater than ST, the holder of the option has no interest whatsoever in exercising the option. But, if ST > K, the holder makes a profit of ST - K by exercising the option, i.e. buying the stock for K and selling it back on 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 the time of writing the option, which will be considered as the origin of time, Sr is unknown and therefore two questions have to be asked: . 1. How much should the buyer pay for the option? In other words, how should we price 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 necessary assumptions. The basic one, which is commonly accepted in every model, is the absence of arbitrage opportunity in liquid financial markets, i.e. there is no riskless profit available in the market. We will translate thatinto mathematical.terms in the first chapter. At this point, we will only show how we can derive formulae relating European put and call prices. Both the put and the call which have maturity T and exercise price K are contingent on the same underlying asset which is worth St at time t. We shall assume that it is possible to borrow or invest money at a constant rate r. Let us denote by Ct and P; respectively the prices of the call and the put at time t. Because of the absence of arbitrage opportunity, the following equation called

Introduction

ix

put/call parity is true for all t < T C t - Pt

= St -

K e-r(T-t).

To understand the notion of arbitrage, let us show how we could make a riskless profit 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 of the operation is

Ct -

Pt -

St.

If this amount is positive, we invest it at rate r until time T, whereas if it is negative we 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 and clear 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 finish with 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 mathematically the 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 are not sufficient in themselves for deriving pricing formulae. To achieve this, we need to model stock prices more precisely. Black and Scholes were the first to suggest 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 can hedge himself perfectly, and actually the call premium is the amount of money needed at time 0 to replicate exactly the payoff (ST - K)+ by following their dynamic hedging strategy until maturity. Moreover, the formula depends on only one non-directly observable parameter, the so-called volatility. It is by expressing the profit and loss resulting from a certain trading strategy as a stochastic integral that we can use stochastic calculus and, particularly, Ito formula, to obtain closed form results. In the last few years, many extensions of the Black-Scholes methods have been considered. From a thorough study of the Black-Scholesmodel, we will attempt to give to the reader the means to understand those extensions. r

x

Introduction

Introduction

Contents of the book

The first two chapters are devoted to the study of discrete time models. The link between the mathematical concept of martingale and the economic notion of arbitrage is brought to light. Also, the definition of complete markets and the pricing of options in these markets are given. We have decided to adopt the formalism of Harrison and Pliska (1981) and most of their results are stated in the first 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 time in subsequent chapters. Chapter 3 is an introduction to the main results in stochastic calculus that we will use in Chapter 4 to study the Black-Scholes model. As far as European options are concerned, this model leads to explicit formulae. But, in order to analyse American options or to perform computations within more sophisticated models, we need numerical methods based on the connection between option pricing and partial differential equations. These questions are addressed in Chapter 5. Chapter 6 is a relatively quick introduction to the main interest rate models and Chapter 7 looks at the problems of option pricing and hedging when the price of the underlying asset follows a simple jump process. In these latter cases,' perfect hedging is no longer possible and we must define a criterion to achieve optimal hedging. These models are rather less optimistic than the Black-Scholes model and seem to be closer to reality. However, their mathematical treatment is still a matter of research, in the framework of so-called incomplete markets. Finally, in order to help the student to gain a practical understanding, we have included a chapter dealing with the simulation of financial models and the use of computers in the pricing and hedging of options. Also, a few exercises and longer questions are listed at the end of each chapter. This book is only an introduction -to a field that has already benefited from considerable research. Bibliographical notes are given in some chapters to help the reader to find complementary information. We would also like to warn the reader that some important questions in financial mathematics are not tackled. Amongst them are the problems of optimisation and the questions of equilibrium for 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 is referred to Dudley (1989) and Williams (1991) for prerequisites. Ho~ever, some basic results are also proved in the Appendix. Acknowledgements

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

Xl

been possible without the encouragement ofN. Bouleau. Thanks to his dynamism, CERMA (Applied Mathematics.Institute of ENPC) started working on financial modelling as early as 1987, sponsored by Banque Indosuez and subsequently by Banque Intemationale de Placement. Since then, we have benefited from many stimulating discussions with G. Pages and other academics at CERMA, particularly O. Chateau and G. Caplain. A few people 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 advice and their motivating comments: N. El Karoui, T. Jeulin, J.E Le Gall and D. Talay.

,

"

i

t

1

Discrete-time models

The objective of this chapter is to present the main ideas related to option theory within 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 form of 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 in F: F o, F 1 , •.. , F N. F n can be seen as the information available at time nand is sometimes called the a-algebra of events up to time n. The horizon N will often correspond to the maturity of the options. From now on, we will assume that F o = {0,O}, FN = F = P(n) and Vw E 0, P ({w}) > O. The market consists in (d + 1) fiflanci;l assets, whose prices at time n are given by the non- negative random variables S~, S~, ... ,S~, measurable with respect to_:fn (investors know past arid present prices but obviously not the future ones). nie vector Sn = (S~, S~, .... , S~) is the vector of prices at time n. The asset indexed by 0 is the riskless asset and we have sg = 1. If the return of the riskless asset over one period is constant and equal to r, we will obtain S~ ~ (1 + rt.'The coefficient /3n = 1/ S~ is interpreted as the discount factor (from time n to time 0): if an amountAn is invested.in.the riskless a~et at time 0, then one dollarwill called risky assets. be available at time n. The assets indexed by i = 1 ... d are ----. . 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 where rPni denotes the number of c!!.~e)rP = rPn' ~n"'" ~n O~n~N In lR

Discrete-time models

2

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

3

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

~

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

Discrete-time formalism

n

Vn(if»

{ and, for n ~ 1:

= Vo(if» + L

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 until time n when new quotations are available.

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. The equivalence between (i) and (iii) follows from the fact that if>n,Sn = if>n+l,Sn if and only if = if>n+l.Sn. 0

«s;

The value ofthe portfolio at time n is the scalar product d

Vn(~) =

»;s: =

Lif>~S~. ;=0

Its discounted value is

Vn(if»

= {In (if>n,Sn) = «:s:

s:

with'{Jn = 1/ S~ and = (1, (JnS;, ... , (JnS~) is the v~tor of disco~nted prices. A strategy is called self-financing if the following equation is satisfied for all nE {O,I, ... ,N-I}

if>n,Sn

= if>n+l' S;".

This proposition shows that, if an investor follows a self-financing strategy, the discounted 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>~))O1, ... , if>d) is self-financing and its initial valueis Yo. " , Proof. The self-financing condition implies

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

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,

Vn(if»

if>~+if>~S~+"'+if>~S~~

(1 -1 + .... + if>j!::J.S d -d)j .

Vo + ~ L.J if>j!::J.Sj j=1

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

or to

Vn+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 is the net gain caused by the price changes between times nand n + I-:--Hence;-tI1e

«:

o

profit or loss realised by following a self-financing strategy is only due to the price moves. 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},

1.1.3 Admissible strategies and arbitrage

+L j=1

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

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

°

n

Vn(if» = Vo(if»

II' '

if>j . !::J.Sj,

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

The investor must be able to pay back his debts (in riskless or risky asset) at any time.. The notion of arbitrage (possibility of riskless profit) can be formalised as follows: Definition 1.1.5 An arbitrage strategy is an admissible strategy with zero initial value and non-zero final value.

Martingales and arbitrage opportunities

5

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 0

E (Xn+l - XnlFn) , E (Hn+lUv/n+l - Mn)IFn) = Hn+lE (Mn+1 - MnlFn) since Hn+l is Fn-measurable .;.' = O.

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 (without necessarily assuming that F N = F, nor F o = {0, D}). A sequence '(Xn)O::;n::;N ofrandom 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:

:S N - 1;

o

a martingale ifE (Mn+1IFn) = Mnfor all n

o

asupermartingale ifE (Mn+lIFn)

o

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

:S Mnforallri:S

N -1;

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

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.

'A E (Xn+1IFn) = E (XnIFn) That shows that (Xn ) is martingale ..

a

= X n. o

The following proposition is a very useful characterisation of martingales.

These definitions can be extended to the multidimensional case: for instance, a sequence (Mn)O 0, for all wEn. (b) The proof of the converse implication is more tricky. Let us call r the convex cone of strictly positive random variables. The market is viable if and only if for any admissible strategy ¢: Vo (¢) = 0 => VN (¢) ¢ r.

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

we define n = sup

admissi~le and

7

o}. Because ¢ is predictable and A is F n-

measurable, 'ljJ is also predictable. Moreover

Gj('ljJ)= { lA(G j(¢)-G1;l(¢))

~

n if j > n

if j

0,

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

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

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

There~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 (see

I,

the Appendix), there exists (oX (w)tEo such that: 1.

vx

E K,

L oX(w)X(w) > O. w

2. For any predictable ¢

martingales.

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

(¢n), (1.1.2) implies n

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*

t Recall

(VN (¢)) = E* (Vo (¢)).

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

p·({w}»o,

°

w

From Property 1: we deduce that oX(w) !, P* defined by '. P* ({w})

=

> 0 for all wEn, so that the probability .

'

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,

8

Discrete-time models

Complete markets and option pricing

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

(t ¢;6.8;)

9

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

= O.

The probability P* will appear to be the computing tool whereby we can derive closed-form pricing formulae and hedging strategies. Proof. (a) Let us assume that the market is viable and complete. Then, any non-neg~tlve, F N-~~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

J=I

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

=

)

h SO

1.3 Complete markets and option pricing 1.3.1 Complete markets

N

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 K will be defined by setting: h (S}y - K) +. A put on the same underlying asset

=

=

=

+.

(V

n

.

(¢)) O' .. , SN. That is the case of the so-called Asian options where the strike price is equal to the average of the stock prices observed during a certain period of time before maturity..

discounted prices are martingales, then

_

= VN (¢) = Vo (¢) + L

':

((¢~'''''¢~))on)O vrnax) > 0, then

E(Uv)

= E(Mv) -

E(A v)

= E(Uo) -

E(A v) < E(Uo)

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

23

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

2.5 Application to American options

o

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

(fl,

F, (Fn)O:::;n:::;N ,P) and, as in Sections 1.3.1 and based on the filtered space 1.3.3 of Chapter 1, we will denote by P* the unique probability under which the discounted asset prices are martingales. 2.4 Snell envelope and Markov chains The aim of this section is to compute Snell envelopes in a Markovian setting. A sequence (Xn)n~O of random variables taking their values in a finite set E is called a Markov chain if, for any integer n 2: 1 and any elements xo, Xl,' .. , Xn-l, X, Y of E, we have

= 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)

2.5.1 Hedging American options In Section 1.3.3 of Chapter 1, we defined the value process (Un) of an American option described by the sequence (Zn), by the system

, UN { Un

P(Xn+l

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:

column indexed by E, then P f is indeed the product of the two matrices P and f. It can also be easily seen that a Markov chain, as defined at the beginning of the section, is a Markov chain with respect to.its natural filtration, definedby F n = a(Xo, ... ,Xn ) . ' . C The following proposition is an immediate consequence of the latter definition and the definition of a Snell envelope. ' Proposition 2.4.2 Let (Zn) be an adapted sequence defined by Zn = 'ljJ(n, X n), where (X n) is a homogeneous Markov chain with transition matrix P, taking values in E, and ib is afunctionfrom N x E to JR. Then, the Snell envelope (Un)

ZN

max (Zn, S~E* (Un+l/ S~+lIFn))

' ZT or AT > O. In both cases, the writer makes a profit VT(cP) - ZT = UT + AT - Zro which is positive.

c, ~ Cn.

25

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

and consequently

Hence

Exercises

+ r)-NE· ((SN - K)+IFn) > E· (SN - K(I + r)-NIFn) , (I

=

-N Sn-K(I+r) ,

s; -

using the martingale property of (Sn). Hence: ~n ~ K(I + r)-(N-n) ~ Sn - K, for r ~ O. As Cn .~ 0, we also have c., 2: (Sn - K) + and by Proposition 2.5.1,Cn = Cn. There is equality between the price of the European call and the price of the corresponding American call. , This property does not hold for the put, nor in the case of calls on currencies or dividend paying stocks. Notes: For further discussions .on the Snell envelope and optimal stopping, one may 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

(K - x)+. 5. An agent holds the American put at time 0. For which values of the spot So would he rather exercise his option immediately?

Vn (¢ ) = Vo(¢) + L

j=l

¢j.~Sj - L 1'j. j=l

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

n

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 we denote by P' the unique probability under which the assets discounted prices are martingales. Show that if the pair (¢, 1') defines a trading strategy with consumption, then (V~(¢)) is a supermartingale under P". 3. Let (Un) be an adapted sequence such that (Un) is a supermartingale under P". Using the Doob decomposition, show that there is a trading strategy with consumption (¢,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 for any n E {O,1, .. '. , N}. Show that there is at least one trading strategy with consumption that hedges (Zn), whose value is precisely the value (Un) of the American option. Also, prove that any trading strategy with consumption (¢, 1') hedging (Zn) satisfies Vn(¢) 2: Un, for any n E {O,1, ... , N}. '

Optimal stopping problem and American options

28

5. Let x be a non-negative number representing the investor's endowment and let 'Y = bnho is an increasing family of a-algebras included in A The o-algebra F t represents the information available at time t. We say that a process (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 the following 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 technical assumption is that if X = Y P a.s. and Y is Ft-measurable then we can show that X is also Ft-measurable. We can build a filtration generated by a process (Xt)t>o and we write F t = r7(X s , S ~ t). In general, this filtration does not satisfy -the previous condition. However, if we replace F t by :Ft which is the o-algebra generated by both F t and N (the o-algebra generated by all the P-null sets of A), we obtain a proper filtration satisfying the desired condition. We call it the natural filtration of the process (Xth~o. When we talk about a filtration without mentioning anything, it is assumed that we are dealing with the natural filtration of the process that we are considering. 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 than t. Formally, the definition is the following: .

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

3.2 Brownian motion A particularly important example of stochastic process is the Brownian motion. It will be the core of most financial models, whether we consider stocks, currencies or 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:

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

• continuity: P a.s. the map

S I--t

s). • stationary increments: if'S ~ t, X, - X; and X t - s - X o have the same probability law. This definition induces the distribution of the process X t , but the result is difficult to 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; - X o is a normal random variable with mean rt and variance r7 2 t, where rand o are constant real numbers. Remark 3;2.3 A Brownian motion is standard if X o = 0 P a.s.

E(Xd

= 0,

E

(Xi) = t.

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

(X

2)

1 --exp --

V2ii

where dx is the Lebesgue measure on IR. '

dx

2t'

32

Brownian motion and stochastic differential equations

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 result is the following: Theorem 3.2.4 If (Xdt?o is a Brownian motion and if 0 ~ (Xt1, ... , X t n ) is a Gaussian vector.

tt < ... < t«

then

The reader ought to consult the Appendix, page 173, to recall some properties of Gaussian vectors. Proof. Consider 0 ~ tl < ... < t«. then the random vector (X t1, 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, this vector is Gaussian and so is (Xt1, ... ,Xt n ) . · 0 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)-Brownian motion if it satisfies: ' • For any t

2:

33

Continuous-time martingales

3. exp (aX t - (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 expectation equal to zero, we have E(Xt - X s) = O. Hence the first assertion is proved. To show the second one, we remark that

E ((X t - X s)2

=

+ 2X s(Xt -

E ((X t - Xs)2IFs)

Xs)IFs)

+ 2X sE (Xt -

XsIFs) ,

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

Because the Brownian motion has independent and stationary increments, it follows that E ((X t - X s)2JFs ) = E (X't-s) t - s. The last equality is due to the fact that X, has a normal distribution with mean zero 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

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.

that

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

E (e-X9) On the other hand, if s

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 to o in probability.

J(H)t = J(Hnk

The process t ~ J(H)t is almost surely continuous, by definition. The extension property is satisfied by construction. We just need to prove the continuity property of J. To do so, we first notice that

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

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

U; H~du o an FtcBrownian motion. (Xt )05, t5,T is an IR-valuedIto process ifit can be written as

where

• X o is Fo-measurable. • (Kt)05,t5,T and (Ht)095,T are Fe-adapted processes. • f: IK.lds T

< +00 P a.s. < +00 P a.s.

• 2ds

We are about to summarise the conditions needed to define the stochastic integral with respect to a Brownian motion and we want to specify the assumptions that make it a martingale.

• fo IH.1 We can prove the following proposition (see Exercise 16) which underlines the of the previous decomposition. uniqueness ;J ,

Summary:

Proposition 3.4.9

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 T is a martingale ifE (fo H;ds) < +00. This condition is not necessary. Indeed, the inequality E

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

(SUp

05,t5:.T

(t Jo

If (M t )05, t5,T is a continuous martingale such that

T M t = i t K..ds, with P a.s.i IK.lds

< +00,

then

P a.s. 'lit

~

T,

u,

= O.

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

H.dW.)

2) < +00.

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-t f (Wd if f is twice continuously differentiable. The following example will simply show 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

x, then

= X o + i t K.ds 0: '

X o = x;

dP a.s.

H.

+

r

H.dW. =' Xb

+

Jo

= ti;

rK~ds + rH~dW.

Jo ds x dP a.e.

K.

Jo

= K;

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

ds x dPa.e.

+ f~ H.dW.,

then

We shall state Ito formula for continuous martingales. The interested reader should refer 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 a complete proof.

44

Stochastic integral and Ito calculus

Brownian motion and stochastic differential equations

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

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

x,

= X o + i t Ksds

+ i t HsdWs,

P a.s. ,

and f be a twice continuously differentiable function, then

I

It

+ i t f'(Xs)dX s + ~ i t j"(Xs)d(X,X)s

f(X t) = f(X o)

[

,~ I

whe re, by definition . (X,Xk and

it

Ii

it

log(St) = log(So)

It turns out that 2

Wt Since E

(J; W

2 s ds )

..:..

r + '2 '1

a WsdW s

+ it

(

1)

-2

Ss

a 2S;ds .

+ i t adWt,

+ (J.L 2/2)

a 2/2) t

t

+ aWt.

+ aWt)

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

St

=

f(t, Wd

=

f(O, W o)

+

it

f:(s, Ws)ds

+ i t f~(s, Ws)dWs +.~ it'f~/x(S, Ws)d(W, W)S'

io 2ds.

t t = 2 i WsdWs'

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

< +00, it confirms the fact that W? -

t is a martingale.

r

s, ~ x'o + i t s. (J.Lds + adW:).

(3.8)

+ adWt),

So

= xo·

. it. St = Xo + a SsJ.Lds

it

+ a

SsadWs"

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

This is often written in the symbolic form ~ St (J.Ldt

2 a

(J.L - a 2/2) dt

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

as,

+ -l i t

Ito formula is now applicable and yields

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

i

dS _s

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

3.4.3 Examples: Ito formula in practice

=2

t

a Ss

Yt = log(St) = log(So) Taking that into account, it seems that

roJ

2

i

St = Xo exp ((J.L - a

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

Wt

+

and finally

+ i t f:(s, Xs)ds

t

+ i t J.LSsds + i t aSsdWs.

Using (3.9), we get }Ii = Yo

Likewise, if(t, x) -+ f(t, x) is a function which is twice differentiable with respect to x and once with respect to t, and ifthese partial derivatives are continuous with respectto (t,x) (i.e. f is a junction of class C 1 ,2), ltd formulabecomes

°

Xo

II

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

f(O, X o)

s, =

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 least formally, because f(x) is not a C 2function!), and we obtain

tf

= iat H;ds,.

'it

45

(3.9)

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

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

46

Brownian motion and stochastic differential equations

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

In this case, we have

(X, Z)t =

t

Y.dX.

+ (X, Y)t

(1" X.adW., - 1" Z.adW.)t = -l,t a 2x.z.u.

Therefore

with the following convention

=

d(XtZt)

I H.H~ds.

+

t

=

(X, Y)t Proof. By

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

(X t + Yi)2 =

(Xo + YO)2 +2J;(X. + J;(H.

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

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

+ Y.)d(X. + Y.)

We have just proved the following theorem:

+ H~)2ds'

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,

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

=

2

Yo

r rt ,2 + 2 Jo Y.dY. + Jo H. ds. t

s, =

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

XtYi = XoYo +

I

t X.dY.

+

I

t Y"dX.

+

I H.H~ds.

+ aW t)

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

~~

= exp (( -J.L +a 2/2) t -q-Wt) ,

+ a 2 and a'

= -a, Then Zt = exp((J.L' - a,2/ 2)t verification that we have just Gone shows that

Zt

=1+

+ a'Wt)

+

I

t S. (J.Lds

+ adW.).

'0

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

Sc = Xo exp ( (J.L - a 2/2) t

Xo

This process is given by

t

o

J.L' =' -J.L

X tZt{(-J.L+a 2)dt-adWt) XtZt (J.Ldt + adWd - X tZta 2dt = o.

Hence, XtZ t is equal to XoZo, which implies that

Ito formula

, Zt =

47

Stochastic integral and Ito calculus

and the

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

Remark 3.4.14 • The process St that we just studied will model the evolution of a stock price in the Black-Scholes model. • When J.L = 0, St is actually a martingale (see Proposition 3.3.3) called the exponential martingale of Brownian motion.

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

f(Xt).~ f(;o) + I This result allows us to apply positive process.

t !'(X.)dX.

+~

I

t !"(X.)H;ds.

Ito formula to log(Xd for instance, if X t is a strictly "

.

1

,

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

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

Brownian motion and stochastic differential equations

48

Stochastic differential equations

49

• (J~ HsdWt , J~ H~dW/)t = 0 if i ;i j. • (J~ tt.aw], J~ H~dWDt = J~ HsH~ds. i

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

Definition 3.4.16 We call standard p-dimensional FrBrownian motion an lRP valued process (Wt = (Wl, . . . , Wi) k:~o adapted to F t, where all the (Wnt~O are independent standard FrBrownian motions.

This definition leads to the cross-variation stated in the previous proposition.

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

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

Definition 3.4.17

(Xt)09~T

is an Ito process

3.5 Stochastic differential equations

if

x, = x +

it

Xs(/Lds

+ adWs)'

We can now consider some more general equations of the type

x, = Z +

where: • K t and all the processes (Hi) are adapted to (Ft). • JoT IKslds

b(s,Xs)ds

+

it

(3.10)

a(s,Xs)dWs.

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 study some properties of the solutions to these equations.

< +00 ~ a.s.

• JoT (H;) 2ds

it

< +00 P a.s.

Ito formula becomes:

Proposition 3.4.18 Let (Xl, ... , X;') be n Ito processes X ti = Xi0

+

i

t

Kids s

P

+ '" L..J

o

3.5.I Ito theorem

it ,

Whatdowe mean by a solution of(3.1O)?

Hi,idWi s s

j:=l

0

then, if 1 is twice differentiable with respect to x and once differentiable with respect to t, with continuous partial derivatives in (t, x) 1(0,XJ, ... ,Xfn

+tit i=l

+~2 ..t

0

it ~~ (s,X~,;

.. ,X:)ds'

::.(s,x~, ... ,X:)dX;

it

',J=l

+

0

t

S

5

• d(X i , Xi) s =

,",P

L...,,-J==l

,",P L...,,-m=l

llb(s,Xs),ds

21 88 xixi

(s,X~, ... ,X:)d(Xi.,Xi)s .

Hi,idWj 5 5' Hi,m Hi,mds 5 s .

Remark3.4.19 If (Xs)Oo satisfies the Markov property if, for any bounded Borel function f and for any ~ and t such that s t, we have E (f (X t) IFs) = E (f (X t) IX s) .

s

We are going to state this property for a solution of equation (3.10). We shall denote by (X;'X, s ;:::- t) the solution of equation (3.10) starting from x at time t and by X" = Xo,x the solution starting from x at time O. For s ;::: t, X;'x satisfies s s b (u a (u 'Xt,X) Xt,x = x Xt,X) du + dWu· s " u u

~1

_1

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 is almost surely continuous with respect to these variables and is a solution of the previous equation. This result is difficult to prove and the interested reader should refer to Rogers and Williams (1987) for the proof. The Markov property is a consequence of the flow property of a solution of a stochastic differential equation which is itself an extension of the flow property of solutions of ordinary differential equations.

Lemma 3.5.6 Under the assumptions of Theorem 3.5.3,

if s ;::: t

55

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, for t ::; 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 with respect to the Brownian filtration (Ftk~o. Furthermore, for any bounded Borel function f we have P a.s. E (f (X t) IFs) = ¢(Xs), with ¢(x)

= E (f(X;·X)).

Remark 3.5.8 The previous equality is often written as

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, we can 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 (see Friedman (1975)). If we use this result for fixed sand t we obtain X;·x (x, W s+u ~ w., u ;::: 0) and thus

x: = (X:, W s+u Proof. We are only going to sketch the proof of this lemma. For any x, we have s s P a.s. X;'x = x + b (u, du + dWu. a (u,

1

1

X~,X)

X~·X)

+ ~s b.(u,X~'Y) du + l

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

It follows that, P a.s. for any y E JR, X;·y = y

w.,

-

w.,

-

u;::: 0))1 F s )

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

(u,X~;Y) dWu,

=

E (f (X:·X))lx=x~ . -0

and also

1b(u,X~'X;)dU~ 1a(u,X~·X;)dWu. s

X;·x; =X:+

s

These results are intuitive, but they can be proved rigorously by using the continuity of H We can also notice that is also a solution of the previous equation.

y x»,

X:

The previous result can be extended to the case when we consider a function of the whole path of a diffusionafter time s. In particular, the following theorem is useful 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

Brownian motion and stochastic differential equations

56

measurable function. For t > s P a.s. E (e-

with

¢(x)

=E

Exercises

57

Exercise 9 Let S be a stopping time, prove that Sis Fs-measurable.

f.' r(u,X,,)du 1 (Xt) IFs)

Exercise 10 Let Sand T be two stopping times such that S thatFs eFT.

= ¢(Xs)

~

T P a.s. Prove

Exercise 11 Let S be a stopping time almost surely finite, and (Xt)t>o be an adapted process almost surely continuous. -

(e-f.' r(u,X:"~)du I(Xt'X)) .

l. Prove that, P a.s., for any s

It is also written as

E

(e- f.'

r(u,X,,)du

1 (Xt) IFs) = E

Remark 3.5.10 Actually, one can prove a more general result. Without getting into the technicalities, let us just mention that if ¢ is a function of the whole path of 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.'

E (J(X:;t)) = E

(J(X~'X)).

(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-f.'

r(X,,)du

1 (Xt) IFs) = E

lim

n-t+oo

2. Prove that the mapping

([0, t] x n, B([O, t]) x F t ) (s,w)

--+

(lR,B(lR))

f-----+

Xs(w)

3. Conclude that if S ~ t, X s is Fcmeasurable, and thus that X s is F smeasurable. Exercise 12 This exerciseis an introduction to the copcept of stochastic integration. We want to build an integral of the form l(s)dX s, where (Xt)t>o is an Ft-Brownian'motion and I(s) is a measurable function from (lR+, B(lR+)) into (lR, B(lR)) such that j2 (s )ds < +00. This type of integral is called Wiener integral 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' the inf{s 2: 0, W s function we have '

> A},

prove that if

f

is a bounded Borel

(i(Wt)I{T>'~t}) = E (I{T>'~t}¢(t - r>.)) , = E(f(Wu + A)). NoticethatE(f(Wu +A)) = E(f(-Wu+A)) E

where¢(u) and prove that

E (f(Wt)l{ T>'9

1} )

= E (i(2A -

Wt) l{ T>'9}) .

5. Show that if we write Wt = sUPs9 W s 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

61

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.). 7. Finally, check that the law of (W t , Wt) is given by l{o~y} l{x~y}

2(2y-x)

((2 Y

...,fi;i3 exp -

- X)2) dxdy.

2t

4

The Black-Scholes model

I

I·

I

Black and Scholes (1973) tackled the problem of pricing and hedging a European option (call or put) on a non-dividend paying stock. Their method, which is based on similar ideas to those developed in discrete-time in Chapter 1 of this book, leads to some formulae frequently used by practitioners, despite the simplifying character of the model. In this chapter, we give an up-to-date presentation of their work. The case of the American option is investigated and some extensions of the model 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 is a continuous-time model with one risky asset (a share with price St at time t) and a riskless asset (with price Sp at time t). We suppose the behaviour of Sp to be encapsulated by the following (ordinary) differential equation:

dSP = rSpdt,

(4.1)

where r is a non-negative constant. Note that r is an instantaneous interest rate and should not be confused with the one-period rate in discrete-time models. We set sg = 1, so that Sp = eft for t ~ O. . We assume that the behaviour of the stock price is determined by the following stochastic differential equation:

,

dS t

= 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 + aB t) ,

The Black-Scholes model

64

where So is the spot price observed at time O. One particular result from this model is 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 necessarily standard). 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), the relative increment (St - Su) / Su is independent of the O"-algebra O"(Sv, v :s u);

:s

• stationarity of the relative increments: if u identical to the law of (St-u - So)/ So·

t, the law of, (St - Su)/ Su is

These three properties express in concrete terms the hypotheses of Black and Scholes on the behaviour of the share price.

A strategy will be defined as it process ¢ = (¢t)09::;T=( (H?, H t)) with values in IR?, adapted to the natural filtration (Ft ) of the Brownian motion; the components H? and H, are the quantities of riskless asset and risky asset respectively, held in the portfolio at time t. The value of the portfolio at time t is then given by Vi (¢)

= Hf S?+tt.s;

This equality is extended to give the self-financing condition in the continuoustime case dVi (¢)

= HfdSf + HtdSt·

To give a meaning to this equality, we set the condition

< +00 a.s.

and

iT H;dt

< +00 a.s.

Then the integral

1T . iT 1 1 (H~St/-L) + 1 HfdSf =

0

Hfrertdt

is well-defined, as is the stochastic integral T "T HtdS t = dt f-t

I H~dS~ I n.as; t

2. Hfsf +

u.s, =

HgSg

+ tt-s; +

65

t

+

a.s.

for all t E [0, T]. We denote by St =' e- rt S, the discounted price of the risky asset. The following proposition is the counterpart of Proposition 1.1.2 of Chapter 1. Proposition 4.1.2 Let ¢ = ((H?, H t)')09::;T be an adapted process with vdlues inIR 2 , satisfying JoT IH?ldH JoT Hldt < +ooa.s. Weset: Vi(¢) = HfS?+HtSt and frt(¢) = e"':rtVi(¢). Then, ¢ defines a self-financing strategy if and 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 :

+ e-rtdVi(¢)

which results from the differentiation of the product of the processes (e~rt) arid (Vi (¢)) (the cross-variation term d{e- r . , (¢) kis null), we deduce dfrt(¢) . _re- rt (Hfe rt + HtSt) dt + e-rtHfd(e rt) + « sup E.[(Ke- -

E* ((ST - e-rTK)+IFr)

Uoo(x) = K - x

s;

~ (Sr - e- rr K) + .

We obtain the desired inequality by computing the expectation of both sides. , 0

Uoo(x) = (K - z")

In the case of the put, the American option price is not equal to the European one and there is no closed-form solution for the function u. One has to use numerical methods; we present some of-them in Chapter 5. In this section we will only use the formula .

- xexp (:--a 2(7 - t)/2 + a(Wr - Wd)) + .

. (4.10) to deduce some properties of the function u. To make our point clearer, we assume t = O. In fact, it is always possible to come down to this case by replacing Twith

(4.13)

(

X) x*

-"'I

for

x:::; z"

for

x > x"

=

K,/(l +,) and, = 2r/a 2 . with z" '. Proof. From formula (4.13) we deduce that the function u oo is convex, decreasing on [O,oo[ and satisfies: Uoo(x) ~ (K - x)+ and, for any T > 0, Uoo(x) ~ rT E(Ke- - xexp (aBT - (a 2T/2)))+, which implies: Uoo(x) > 0, for all x ~ O. Now we note x" = sup{x ~ Oluoo(x) = K - x}. From the properties of u oo we have just stated, it follows that "Ix:::; z"

4.4.2 Perpetual puts, critical price

(K~-r(r-t)

xexp (aB r - (a 27/2)))+ l{rO

E(e-a(T~t\t)exP(J-LBT~I\t-~2T~t\t)l {t 0 E (e-a(Tt: I\t)) = E

I n.e», + I t

Exercise 21 We consider an option described by ~ = f.(ST) and w~ note F the function of time and spot corresponding to the option pnce (cf. equation (4.7)).

.

Exercise 26 Let (Bt)Ot} a.s. The following property can be used: if 8 1 and 8 2 are two 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, then E ('~18l V 82) = E (XI82 ) , where 8 1 V 82 represents the o-algebragenerated by 8 1 and 8 2. . 3. Showthat there exists-no path-continuous process (X t ) such that for all t E [0,1]' P (M t = X t ) =,1 (remark that we would necessarily have

P ('Vt E [0, IJ M,

= X t ) = 1).

Deduce that the martingale (M t ) cannot be represented as a stochastic integral with 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.

The Black-Scholes model

Exercises

81

3. Prove that there exists a probability P" equivalent to P, under which the discounted 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 with maturity T and strike price K. Deduce that if

E (e

aWT1

x ~ JL

. (a

. } ) -exp -.

{WT~~,inf.::;TW.~A

2T

--

.

2

+ 2a),) N (2), - + aT) JL

1m

vT

(a) Let (H?, Hl) be a self-financing strategy, with value \It at time t. Show that if (\It/ SP) is a martingale under P" and if VT (ST - K)+, then

=

.

'TIt E [0, T) Vi = F(t, St),

2. Let H ~ K; we are looking for an analytic formula for

C

= E (e-rT(XT -

where F is the function defined by

K)+1{inf.::;T X.~H}) ,

F(!,

where X, = x exp ((r /2) t+ uWt) . Give a finan~iaUnterpretati.?n to this value and give an expression for the probability p that makes W t (r / o - o /2) t + W t ~ standard Brownian motion. _

x) = E" (x exp (J.T u(, )dW. - ~ J.T U'U)d,) -: K e- J.T .(.)d.) +

'u2

and (Wd is a standard Brownian motion under p ". (b) Give an expression for the function F and compare it to the Black-Scholes formula. (c) Construct a hedging strategy for the call (find H?and H t ; check the selffinancing c o n d i t i o n ) . l

3. Write C as the expectation under P of a random variable function only ofWT and sUP~~s~T Ws .

. ,.

4. Deduce an analytic formula for C.

Problem 2 Garman-Kohlhagen model

Problem 1 Black-Scholes model with time~dependent parameters ~e consider once again the Black-Scholes model, assuming that ~e ass~t p~ces are described by the following equations (we keep the same notations as In this J

ter)

dSP

ch~~.

= 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: . 1. Prove that

s,

= So exp

(~t JL(~)ds + ~t u(s)dB ~ ~t u

2(S)dS)

s _

I

.

. : JL(s)ds

dS t

.

S

.

= JLdt

t.

+ udWt,

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 generated by (WdtE[O,Tj and assume that F t represents the accumulated information up to time t '( . . ..'

J

You may consider the process

z, = s, exp --:-

!

The Garman-Kohlhagen model (1983) is the most commonly used model to price and hedge foreign-exchange options. It derives directly from the Black-Scholes model. To clarify, we shall concentrate on 'dollar-franc' options. For example, a European 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-per dollar. The behaviour of S, through time is modelled by the following stochastic differential equation:

+ ~t u(s)dB s

:-

~ ~t u

2(s)dS)

I

.

1. Express S, as a function of So, t and W t. Calculate the expectation of St.

2.

2. Show that if JL (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 random variable and calculate ItS varIance. 0

J~ u(s )dB s is anormal •

.

.

> 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 that Uc satisfies the following stochastic differential equation dU t (2 . = u - JL)dt - udWt.

u,

82

Exercises

The Black-Scholes model

83

Deduce that if 0 < J.L < a 2, both processes (St)tE[O,T] and (Ut)tE[O,TJ are submartingales. In what sense does it seem to be paradoxical?

(The symbol E stands for the expectation under the probability P.) 5. Show (through detailed calcUlation) that

II We would like to price and hedge a European call on one dollar, with maturity T and strike price K, using a Black-Scholes-type method. From his premium, which represents his initial wealth, the writer of the option elaborates a strategy, defining at any time t a portfolio made of HP francs and H, dollars, in order to create, at time 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, dollars is obviously Vt = H~ + (4.16) We suppose that French francs are invested or borrowed at the domestic rate TO and US dollars are invested or borrowed at the foreign rate TI' A self-financing strategy will thus be defined by an adaptedprocess ((HP, Ht»tE[O,Tj, such 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, an d

n.s;

dVt = ToHpdt + TIHtStdt

+ HtdSt

+ (a 2j2»(T a..;T=t

dl

10g(xjK) + (TO -

d2

10g(xjK)

-

(a) We set St = e h -ro)t St. Derive the equality

es, = aStdWt.

(4.17)

(b) Let

~ be the function defined by F(t, x)

F(t, St}. Derive the equality

_

-

TO t

a

is non-negative for all t and if SUPtE[O,Tj

Vi

(Vi) is square-integrable under

P. Show that the discounted value of an admissible strategy is a martingale under

P.

4. Show that if an admissible strategy replicates the call, in other words it is worth VT = (ST - K)+ at time T, then for any t ~ T the value of the strategy at time t is given by where

F(t,x)

=

E(xexP(-(TI.+(a 2j2»(T-t)+a(WT - Wt») - K e-~o(T-t)) . +

-rotc

t =

7. fors« down a put-call parity relationship, similar to the relations~iP we gave or stocksh' alndd grve an example of arbitrage opportunity when this relationship does not 0 .

Proble~ 3 Option to exchange one asset for another

+ UT t't't

is a standard Brownian motion. (b) A self-financing strategy is.said to be admissible if its discounted value

t - e

(c) renlicati that the c.all is replicable and give an explicit expression for the rep icatmg portfolio (( Hp, Ht».

To)dt + Hte-rotStadWt.

J.L + TI

, t

sc, = -aF ( t St)ae-rotS dW'; ax ' t t-

(a) Show that there exists a probability P, equivalent to P, under which the process t't't -

= e- rot F(t, xe(ro-r,jt) (F is the

f~nctl_on defined in Question 4). We set C, = F( t S) and G -

3.

TiT

(a 2j2»(T - t)

a..;r-=-t

Vi

+ TI

TI -

t)

6. The next step is to show that the option is effectively replicable.

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 = e-rotVt be the discounted value of the (self-financing) portfolio (HP, Ht}. Prove the equality

,dVt = Hte-rotSt(J.L

+ (TO -

TI

1 I'

I

.

W~ tO~~der a fin2anci~1 market in which there are two risky assets with respective pnces

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!

dS~

S! (J.Lldt

+ aldB!)

{ / Sl (J.L2dt + a2a B 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 anda2arerealnumbers abiesa11 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) e (Fd-Brownian motions and, for t ~ s, the vecto/ (~IO~];~ B2 ~ ~JO)?J :rrIndependent of F.. t·. , t . • IS .J'

The Black-Scholes model

84

85

I

where the function F is defined by

We study the pricing and hedging of an option giving the right to exchange one of the risky assets for the other at time T. 1. We set by

(h = (ILl - r) /0"1

u,

and

= exp (

Exercises

(h = (IL2 - r) /0"2, Show that the process defined

-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 the processes WI and W 2 defined by Wl = Bf +B1t and Wl = B; +B2t. Derive, under the probability P, the joint characteristic function of (Wl , Wl). Deduce that, for any t E [0, T], the random variables Wl and wl ar~ independent normal 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~T are (Ft)-indepe~dent st;n?~d Brownian motions and that, for t 2: s, the vector (Wl - W s1, W t - W s ) IS independent of F s •

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 the holder the right to exchange one unit of the asset 2 for cine unit of the as~et.1: do so we use the same method as in the Black-Scholes model. From hIS initial wealth, the premium, the writer ?f the opt~on builds a strategi' defini~g a~ any time t a portfolio made of Units of the riskless asset and H; and H; Units of the 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 processes HO, HI and H 2.

:0

HP

~ ,,2 ) +' F(t, Xl, X2) = E- ( x1eO'I (WIr - W•I) "'-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 strategy having this value will be proved later on. We will consider in the remainder that 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 :the put-call parity relationship previously seen and give an example of arbitrage opportunity when this relationship does not hold.

ill The objective of this section is to find an explicit expression for the function F defined 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 A be a real number. (a) Show that under the probability p(A), with density with respect to P given ._ by dP(A) 2

__ = eAgI-A

rtSl dW l -r d1Yt t 0"1 t - HIt e -

= e-rtVt is

2 + H t2e-rtS20" t 2 dW t .

2. Show that if the processes (HI )o~t~~ and (Hl)o:9~T of a self-financing strategy are uniformly bounded (which means that: 30 > 0, Vet,w) E [O,~] x fl, IHt(w)1 ::; 0, for i 1,2), then the. discounted value of the strategy IS a martirigale under P.

=

3. Prove that if a self-financing strategy satisfies the hypothesis of the previous question and has a terminal value equal to VT (St .:... Sf)+ then its value at any time t < T is given by

=

(4.18)

'

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

=eYI+A~/2N-(Yl -

+ A292))+

Y2 + A?) _ JA~+A~

·11 1. Define precisely th~ self- financing strategies' and prove that, if Vt the discounted value of a self-financing strategy, we have

/2

dP

.

eY2+A~/2N (Y1 -

Y2

-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 + f o J1dW; .: f0 7 ; dw ; + f~ Ksds and if it is a martingale under P, then K, = 0, dtdP-almost everywhere.

The Black-Scholes model

86

,

Exercises

show that condition (ii) is satisfied if and only if we have, for all t E [0, Tj.

4. Build a hedging scheme for the option.

t:t = Vo +

Problem 4 A study of strategies with consumption

We 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 model is studied on the time interval [0, Tj (0 ~ T < 00). In the following, (St)O~t~T is a stochastic process defined on a probability space (0, F, P), equipped with a filtration (Ft)o9~T' We assume that (Ft)O~t~T is the natural filtr~tion of a standard Brownian motion (Bt)o9~T and that the process (St)09~T IS adapted to this filtration. .

I

,

We want to study strategies in :ovhich consu~ption is allowed. The dynamic of (St)09~T is given by the Black-Scholes-model

dS t with JL E IR and a

= St(JLdt + adBt) ,

with

it

HudBu - i t c(u)du,

a.s.

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:" (H2 S2 + HtSt). Prove that the process (t:t)O~d}]

_ K1e-rIlN(d),

where

d = 10g(X/!1) + (r- a2 /2) 0

(u, Su)a(u)SudWu

is a martingale under probability P* . .

1

,

(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 standard ' , cumulative normal distribution. Prove that

G(O,x)

.

< T 1 , the compound option is 'worth G(T

with 9 bei~g a standard normal random variable.

6. Prove that the process defined by

ru

= K 1 has a unique solution Xl.

3.

I

. l't . eca}

> O.

.We wan~ to study an example of compound option. We consider a call option WI.th matunty T I E10, T[ and strike price K 1 on a call of maturity T and strike pnce K. The.value of this option at time T1 is equal to

1. Prove (using the price formulae written as expectations) that the, functions x I-t C1(t,x) and x I-t C 2(t,x) are convex.

s, =

89

the natural filtration generated by a standard Brownian motion (Bdo

+

-d}.

(r _~2) (0 + (1))

The Black-Scholes model

90

(d) From this, derive a formula for G(B, x) in terms of Nand N 2 the twodimensional cumulative normal distribution defined by

= P(g < y,g+ pg1

N 2(Y,Y1,P)

< yd

y,Y1,P E JR.

for

4. Show that we can replicate the compound option payoff by trading the underlying call and the riskless bond.

Problem 6 Behaviour of the critical price close to maturity

Exercises

91

Problem 7 Asian option We conside~ a ?nan~ial market offering two investment opportunities. The first traded secunty 1S a nskless asset whose price is equal to Sp = ert at time t (with r ~ 0) and the second security is risky and its price is denoted by St at time t 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 E (liminf t-+T . T - t t-+T

~ E' [,-d

K~) T - t

n.

Vi

=

aKg) +

We shall need Fatou l~mma: for any sequence (Xn)nEN of non-negative random variables, E(lim inf n -+ oo X n ) ~ lim inf n -+ oo E(X n ) .

T

- ')

(~ S.da -K

J;, Sudu ~ KT}, we have

e-r(T-t) it 1 ~ e-r(T-t) 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 (

4.

E·(St:Ke-rT)+ (a) Show that for any real number 1/,

(b) Deduce that

Vo

(b) Deduce that

.

t-+T

K-se(t)

VT -

t

~E·[e-rT(ST-K)+].

(Use conditional expectations given F t ) .

E(1/ - Kag)+ > 1/.

lim

{

. = hm

t-+T

K-s(t)

VT -

t

= +00.

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

The Black-Scholes model

92

< e-r(T-t)t (-l i t S du - K -

T

t

0

)+ + - iT e-r(T-u)Ct 1

T

U

t

duo

Sudu -

3. Prove the inequality

K) .

1. Show that (~t)O~t~T is the solution of the following stochastic differential

equation:

2. (a) Show that

v. ~ ;-'(T-'IS,E'

[((.+ ~ t S~dU) \1"] ,

with S~ = exp ((r - (12 /2)(u - t) (b) Conclude that

Vi

+ (1(Wu

-

Wt}) .

= 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 the function F introduced earlier is of class C 2 on [0, T[ x IR and we shall use Ito formula.

ill The purpose of this section is to suggest an approximation of Vo obtained by considering the geometric average as opposed to the arithmetic one. We define

Vo

- K) + ,

where 9 is a standard normal variable. Give a closed-form formulator Vt in terms of the normal distribution function. 0

We denote by (~t)09~T the process defined by t

Vo = e-rTE (So exp ((r - (12 /2)(T/2) + (1VT /39)

,u

n

~t = ~t (~ I

93

(b) Deduce that

at time u with strike price K. Prove the following inequality Vt

Exercises

~ 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 with zero mean and a variance equal to T 3/3.

Vo

-va, ~ Soe- rT (e

rT

rT- 1 - exp ((rT /2) - (12T /12) ) .

5

Option pricing and partial differential equations

In the previous chapter, we saw how we could derive a closed-form formula for the price of a European 'option in the Black-Scholes environment. But, if we are working 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 require numerical methods. The purpose of this chapter is to give an introduction to some concepts useful for computations. Firstly, we shall show how the problem of European option pricing is related to a parabolic partial differential equation (PDE). This link is basedon the concept of the infinitesimal generator of a diffusion. We shall also address the problem of solving the PDE numerically. ' The pricing of American options is rather difficult and we will not attempt to address it in its whole generality. We shall concentrate on the Black-Scholes model and, in particular, we shall underline the natural duality between the Snell envelope and a parabolic system of partial differential inequalities. We shall also explain how we can solve this kind of system numerically. We shall only use classical numerical methods and therefore we will just recall the few results that we need. However, an introduction to numerical methods to solve parabolic PDEs can be found in Ciarlet and Lions(1990) or Raiviart and Thomas (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 2/2)T+uW T S T -- x 0 e(r-u

.

96

Optionpricing and partial differential equations

In fact, we should point out that the pricing of a European option is only a particular case of the following problem. Let (Xtk~o be a diffusion in ffi., solution of

Europeanoption pricing and diffusions

97

Proof. Ito formula yields

(5.1) where band (1 are real-valued functions satisfying the assumptions of Theorem 3.5.3 in Chapter 3 and ret, x) is a bounded continuous function modelling the riskless interest rate. We generally want to compute

Vt

=E

Hence

(e-J.T r(s,X.)dsf(XT )IFt) .

f(Xo) + i t f'(X s)(1(X s )dWs

=

f(Xt)

+it

In the same way as in the Black-Scholes model, Vt can be written as

~nd ther~sult follows from the fact that the stochastic integral J~ f'(X s)(1(Xs )dW s IS a martingale, Indeed, ~ccording to Theorem 3.5.3 and since 1(1(x)1 is dominated by K(1 + Ix!), we obtain .

Vi = F(t, X t ) where

F(t,:x)

(e-J.T r(s,X;'%)ds f(X~X)) ,

=E

[~(12(Xs)JII(Xs)+ b(Xs)f'(Xs)] ds

,

:

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. The computation of Vt is therefore equivalent to the computation of F(t,x):'Under some regularity assumptions that we shall specify, this function F( t, x) is the unique solution of the following partial differential equatiori

{

"Ix E ffi.

u(T, x)

o Remark 5.1.2 If we denote by X{ the solution of (5.3) such that Xx = . 0 x, It follows from Proposition 5.1.1 that

= f(x) (5.2)

(au/at

+ Atu -

E (J (Xt)) = f(x)

+E (It Af (X:) dS) .

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 naturally when we solve stochastic differential equations.

Moreover, since the derivatives of f are bounded by a constant K] and since Ib(x)1 + 1(1(x)1 :::; K(1 + Ixl) we can say that

E(~~~ IAf(X:)I) s Kj (1 + E(~~~ IX:1 2) ) < +00. !heref?re, since x .H Af(x) and s H IS applicable and yields

5.1.1 Infinitesimal generator ofa diffusion

We assume that band (1 are time independent and we denote by (Xtk:~o the solution of ' dXt = b (Xt) dt + (1 (.Xt ) dWt. (5.3)

Proposition 5.1.1 Let f bea 0 2 function with bounded derivativesand A be the

differentialoperator that maps a 0 2 function f to Af such that

l

d"

dt E (f (Xt))lt=o

+ b(x)J'(x).

Then, the process M, = f(Xd - J~ Af(Xs)ds is an Ft-l'TIfrtingale.

(1 r Af(X:)dS) = Af(x).

= l~ E t 1

0

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 d some properties of the infinitesimal generator of a diffusion 0 s u Y J

(AI) (x) = (12 (x) f",(x) _ 2 '

X: ,are continuous, the Lebesgue theorem

.

•

The ProPositio,n 5.1.1 can also be extended to the time-dependent case. We assume that b ~nd (1 satisfy th.e assumptions ofTheorem 3.5.3 in Chapter 3 which guarantee the existence and unIqueness of a solution of equation (5.1).

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 t u, = u(t, X t) + Asu) (s, Xs)ds

I (~~

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 that we 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 of Proposition 5.1.3, and ifr(t, x) is a bounded continuous function defined on IR+ x IR, the process t = e- r(s,X.)dsu(t, Xt)-l e- r(v,Xu)dv + Asu - ru) (s, Xs)ds

u,

I;

(~~

I:

is a martingale. Proof. This proposition can be proved by using the integration by parts formula to differentiate the product (see Proposition 3.4.12 in Chapter 3)

e - Jof'r(s,X.)ds u (t , X t )i,

o

and then applying Ito formula to the process u(t, X t ) .

This result is still true in a multidimensional modeJ. Let us consider the stochastic

{

bl (t, X t) dt

=

bn (t, X t) dt

dXi'

+ :E~=I alj (t, X t) dW/

(5.4)

+ :E~=I anj (t, X t) dW/.

We assume that the assumptions of Theorem 3.5.5 are still satisfied. For any time 2 t we define the following differential operator At which maps a C function from . IRn to IR to a function characterised by

. 1

(At!) (x)

n

= -2 ..L

82 f ' a"j(t,x)8' -.8 (x)

.

.

x,

',J~I

XJ

8f

n

is a martingale.

The proof is based on the multidimensional Ito formula stated page 48.

Remark 5.1.6 The differential operator 8/ 8t + At is sometimes called the Dynkin operator 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 option and ~olvmg a parabolic partial differential equation. Let us consider (Xt)t>o a solution of system (5.4), f(x) a function from IRn to IR, and r(t, x) a bounded continuous function. We want to compute

Vt =E

r(s,X.)ds f(Xr) 1Ft) .'

where F(t,

z) = E (e-I,T

r(s,X;,Z)ds f(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 n

Vx E IR

u(T, x) = f(x),

and

(~~ + ~tU -

ru) (t,x) = 0

n

V(t,x)E [O,T) x IR

,

then

where (a,j (t, x)) is the matrix of components p

k=1

(e- r

In a similar way, as in the scalar case, we can prove that

0

a,j(t,x) = La'k(t,x)ajk(t,x)

where 0'* is the transpose of a(t, x) =

. Proposition 5.1.5 If(X t) is a solution ofsystem (5.4) andu(t, x) is a real-valued function of class C 1 ,2 defined on IR+ x IRn with bounded derivatives in x and also, r( t, x) is a continuous boundedfunction defined on IR+ x IR, then the proces; t M, = e - Io' r(s,X.)dsu(t, Xd-l e- Io' r(v,Xu)dv (~~ + Asu - ru) (s, Xs)ds

+ Lbj(t,x) 8x (x), . I J J=

= a(t, x)a* (t, x)

In other words a(t, x) (a,j(t, x)).

differential equation

d~: t =

99

European option pricing and diffusions

0'

V(t, x) E [0, T)

xIR

n

~(t, x)

= F(t, x) =I E

(e-r

r(s,X;,Z)ds f(X~X)) .

Option pricing and partial differential equations

100

Pr~of. Let us prove the equality u(t, x)

= F(t, x) at time t = 0. By Proposition

5.1.5, we know that the process

- eMt-

f

0

= sinceu(T,x)

E

101

The operator At is now time independent and is equal to (12 2 a a A t = A b. =-x -+ rx2 2 ax ax' 2

r(s,X?,Z)ds u.(t , Xo,X) t

is a martingale. Therefore the relation E(Mo)

u(O,x) '= E

European option pricing and diffusions

= E(MT) yields

It is straightforward to check that the call price given by F(t x) = xN(d ) _ - (1vT _ t) with ' I

K e-r(T-t) N(dl

(e- JoT r(s,X?'Z)dSU(T,X~'X)) (e- JoT r(s,X?,Z)ds j(X~'X))

= j(X). The proof runs similarly fort > 0.

N(d)

o

=

log(x/ K)

=

1. --

+ (r + (12 /2)(T - t) (1'1/'T - t

I V2i

d

e- X 2 /2dx

-00

'

is solution of the equation

Remark 5.1.8 Obviously, Theorem 5.1.7 suggests the following method to price the option. In order to fompute

F(t,x) for

a given

=E

(e- J,T r(s,X:,Z)ds j(X~~))

The same type of result holds for the put.

au at + A,u _ ru ~ ° in_ [0, T] n

u(T, x) = j(x), "Ix E IR

x JR" (5.5)

ex, =

Problem (5.5) is a parabolic equation with afinal condition (as soon as the function

since S - S e(r-0'2/ 2)t+O'w, I t -

For the problem to be well defined, we need to work in a very specific function space (see Raviart and Thomas (1983)). Then we can apply some theorems of existence and uniqueness, and if the solution u of (5.5) is smooth enough to satisfy the assumptions of Proposition 5.1.4 we can conclude that F = u. Generally speaking, 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 IR

"1(6,.·.,

Note th~t th~ operato~ A b. doe~ not satisfy the ellipticity condition (5.6). However, the tnck IS to consider the diffusion X, = log (St), which is solution of

.

u(T,.) is given).

n

°

.'

~n) E IRn ~ aij(t, X)~i~j ~ C (t ~?) >=1

.

(5.6)

(r - ~2)

.-

2

Ab.-log

= (12 a + 2 ax 2

' .

It is clearly elliptic because We write

(12

dt.+ (1dWt , ,',

. fini . ts m mtesimal generator can be written as

"

>

(r _2

(12)

.

~ ax'

°and, moreover, it has constant coefficients.

a2 + ( r ~ ~2) ax 'a = '"2 aX 2'

'(12 Ab.-Iog

..

Q

in [0, T]x ]0, +oo[

, u(T, z) = (z - K)+, "Ix E]O, +00[.

j, we just ~eed to find u such tha~

{

au '. : '{ at +Ab,u - ru ~ °

- 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 highlighted as follows:.lf we ~ant to compute the price F(t, x) at time t and for a spot price x of an option paying off j (ST) at time T, we need to find a regular solution v of

5.1.3 Application to the Black-Scholes model

:~ (~' x) +Ab.-1ogv(t, x) = °

We are working under probability p ". The process (Wt)t::::o is a standard Brownian motion and the asset price S; satisfies

{

(5.8)

v(T, x) ,= j(e then F(t, z)

in [0, TJ x IR

= v(t, log(x)).

X

) ,

"Ix E IR,

/

Optionpricing and partial differential equations

102

5.1.4

Dnrtial ditterentiai equationson a boundedopen set and computationof su:

Solving parabolic equationsnumerically is a bounded stopping time, because r" = T; 1\ T: 1\ T where

r u.

trI = inf {O ,

E

r

Z

•

0 on the event

[t, T], X;,x It O} ;

consequently

u(O,x)

[0, T] x 0

T

= u(T, x) and u(-rx , X~~X) =

=

au (t x) at '

-J.T° r(s,X~,X)ds u (T , XO,X)) -r r(s,X~,Z)ds ( XO,X)) X:,zllO}e °

X:,xEO}e

{3sE[t,Tj,

(5.9)

"Ix E nt.

s: ':(S'X~'X)dSu("X,X~~X))

E(1 {\lsE[t,Tj, +E(1

r(x)f(x).

on [O,T] x nt

Xt,x = I} s

and indeed T{ is a stopping time according to Proposition 3.3.6. By applying the optional sampling theorem between 0 and r", we get E(Mo) = E(Mrx), thus by noticing that if s E [0, "X], Af(X~'X)= 0, it follows that

Equation (5.5) becomes

~~(t,~) + Au(t,x) = 0

103

Xo,Z)ds ° ,) = E ( 1{\lsE[t,TJ, X:,xEO}e - J.T° r(s'. f(XT'X).

That completes the proof for t

= O..

o

u(t,a) = u(t,b) = 0 . "It S T u(T,x)

= f(x)

Remark 5.1.10 An option on the FT-measurable random variable "Ix E O.

As we are about to explain, a regular solution of (5.10) can a~so be expr~ssed in terms of the diffusion Xt,x which is the solution of (5.3) starting at x at orne. t.

a

Theorem 5.1.9 Let u be C 1 ,2function withboundedderivativein x that satisfies

equation (5.10). We then have V(t,x) E [O,T] x 0, u(t,x)

= E ( 1{\lsE[t,T],.x:,xEo}e

' - I,T r(X:,Z)ds f(Xt,X))

M«

=

- J.'r(XO,X)ds (t Xo,X) eo' u, t

i° t

-

-

e

f r(X~,X)dv °

(au at

+ Au - ru) (s, X~,X)ds'

is a martingale. Moreover "x

o x d o} = inf { 0 S s S T , X s'-'f'

or T if this set is empty

- J.T (X"Z)d ,r

•

.

t X

s f(Xi

)

is called extinguishable. Indeed, as soon as the asset price exits the open set 0, the option 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 Cox and Rubinstein (1985) and exercise 27 for the pricing of Down and Out options).

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 C 1,2 . We shall continue to denote by u such an extension. From Proposition 5.1.4, we know that

, 1{\lsE[t,T], x:,xEo}e

5.2 Solving parabolic equations numerically , (

We saw under which conditions the option price coincided with the solution of the partial differential equation (5.9); We now want to address the problem of solving a PDE such as (5.9) numerically and we shall see how we can approximate its solution using the so-called finite difference method. This method is obviously useless in the Black-Scholes model since we are able to derive a closed-form solution, but it proves to be useful when we are dealing with more general diffusion models. We shall only state the most important results, but the reader can referto Glowinsky, Lions and Tremolieres (1976) or Raviart and Thomas (1983) for a detailed analysis.

Option pricing and partial differential equations

104

Solving parabolic equations numerically

5.2.1 Localisation Problem (5.9) is set on ffi. In order to discretise, we will have to work on a bounded open set VI =]- I, l[, where I is a constant to be chosen carefully in 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)). If we specify Dirichlet boundary conditions, the PDE becomes

=

105

Thus

lu(t, x) - UI(t, x)1

-

+ Au(t, x)

< MP (suPO:S;s:S;T Ix + O'Wsl 2: I -lr'T/) . By Proposition 3.3.6we know that if we define T. = inf {s > 0 W - } th E(exp(-ATa )) = exp(-.J2>:la/). It infers that f~r any a> 0, a~d f~r:n~ ~ en

= 0 on [0,T] x VI

P (sup s:S;T

u(t, I) = u(t, -I) = 0 if t E [0, T] u(T, x)

w, 2: a)

= P (Ta

Minimising with respect to

= f(x) if x EVI.

~ T) s e~TE (e-~To) _< e~T e -a.,;'2I .

>. yields

w, 2: a) ~ exp (_ a -sr T 2

P (sup

We are going to show how we can estimate the error that we make if we restrict our state space to VI. We shall work in a Black-Scholes environment and, thus, the logarithm of the asset price solves the following stochastic differential equation

)

,

'

and therefore

P (SUP(x

" dX t = (r - O' 2/2)dt + O'dWt.

We want to compute the price of an option whose payoff can be written as f(ST) = f(Soe XT). We write f(x) = !(e). To simplify, we adopt Dirichlet boundary conditions. We can prove in that case that the solution u of (5.9) and the solutions UI of (5.10) are smooth enough to be able to say that

MP (suPO:5S~T_t Ix + O'Wsl 2: I -lr'T/)

=

=

8u(t x) 8;

< MP (sUPt~s;5T [z + O'(Ws - Wt)1 2: I -lr'TI)

s:S;T

+ O'Ws) 2: a) < exp -

(

ja - X I2) 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 ,

and

UI'(t,X) =

E(1 {'v'sE[t,TJ, IX;,zl'xOt if x ~ x"

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 to the equation f(x) = x.

[O,T]x]O,+oo[ .

(U_(K_X)+).(~~(t,x)+AbSU(t,X)) u(T, x) =

+oof

and u~(t, x) = (au.(t, x)jax).

5 Derive the existence of a solution to (5.17). . '. . to a roximate the Black-Scholes American ~ut pnce Exercise 29 We arIel :I;~ is a ~~lution of the partial differential inequality u(t,x). Let us reca a .1

a.e. in

]0,

K-u.(O,x)

f(x) =

'Show that for sufficiently small p, Sp is a contraction.

u (t, x) ~ (K - x)+

a.e. in

Write down the equations satisfied by >. and a so that v is continuous with continuous derivative at z" . Deduce that if v is continuously differentiable then z" is a solution of f(x) = x where

~:

aU(t,x)+AbBu(t,x).::;O a.e.in at .

=0

]0, +oo[

+ T Absv(x) = O.

vex) =

(M X - G, V ., X) ~ O..

(Y - X

a.e. in

4. We look for a continuous solution of (5.18) with a continuous derivative at x"

1. Show that this is equivalent to find X ~ F such that

VV > F

x)+ - u.(O,x)

(5.18) 3. Find the unique negative value for a such that vex) = x" is a solution of

(Mi -G,X - F) =0.

F

]0, +oo[

(v(x) - ¢(x)) ( -vex) + T AbSv(x))

:::G

VV ~

= (K -

a.e. in

8. From the previousresults, write an algorithm in Pascal to compute the American put price.

[O,T]x]O,+oo[

The algorithm that we have just studied is a marginally different version of the MacMillan algorithm (see MacMillan (1986) and Barone-Adesi and Whaley (1987».

x)+

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-Scholes model for stock options. In this chapter, we will present the main features of interest rate modelling (following essentially Artzner and Delbaen (1989», study three 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 assumed to be constant. In the real world, it is observed that the loan interest rate depends both 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 pay back an amount F(t, T) at time T, which is equivalent to an average interest rate R(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 u > t are not known. Nevertheless, intuitively, it makes sense to believe that there should be some relationships between the different rates; the aim of the modelling is to determine them. Essentially, the issue is to price bond options. We call 'zero-coupon bond' a security paying 1 dollar at a maturity date T and we note P(t, T) the value of this security at time t. Obviously we have P(T, T) = 1 and in a world-where the future is certain

P(t, T)

123

u E [0, T], the process (F(t, u))oStSu defined by

= e- J.T r(s)ds.

is a martingale. This hypothesis has some interesting conse erty under P* leads to, using the equality p(~~~)e~ I;,deed, the martingale prop-

. .

~(t,U) = E* (F(u,u)/Ft)

= E*

(e- fa"r(S)ds!Ft)

and, eliminating the discounting,

P(t,u)

(e- J."r(S)ds!;:,)

= E*

.

(6.1)

.

(6.2)

t·

This equality, which could be compared to fo

P(t, u) only depend on the behaviour of th rmula (6.1), shows that the prices d he process (r(s))OSsST under the probability P*. The hypothesis w e rna e on t e filtration (;:, ) . 11 express the density of the probabilit P* ith t 09ST a ows us to

6.1.2 Yield curvefor an uncertainfuture For an uncertain future, one must think of the instantaneous rate in terms of a random process: between times t and t + dt, it is possible to borrow at the rate r(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 0 d T - 1 - L o and, forP" is t, To obtain the 'formula (6~~), we :p.sp'tynth m~~efgenera11Y P(L t > O} = 1 for any . ( e 0 ormula to the log function To do . so, we need to chec.k that P "It E [0 T] L + r t H d W ) f hi ~ ~ , , 0 Jo s s > 0 = 1 The proof OtIS }act relies in a crucial way on the martin a . . . I' g le property and It IS the purpose of Exercise 30 Then the ItA ~ . 0 tormu a yields

10g('L t)

.

=

r L HsdWs"':' ~2 Jr £22. H 2ds 1

Jo

s

o which leads to equality (6.3) with q(t) = HtiL . t

s

s

a.s.

o

Interest rate models

124

. > Corollary 6.1.2 The price at time t of the zero-coupon bond of maturity u -

Modelling principles

t

riskier. Furthermore, the term r(t) - arq(t) corresponds intuitively to the average yield (i.e. in expectation) of the bond at time t (because increments of Brownian motion have zero expectation) and the term -a;:q(t) is the difference between the 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 = W t - f; q(s)ds is a standard Brownian motion (Girsanov theorem), and we have

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)

6.1.1 and fro.m the following P roo f . This follows immediately from Proposition . d able X' ' formula which is easy to derive for any non-negative ran om van.. . '

E· (XIFt ) =

E (XLTI F t ) L t

125

dP(t, u) P(t, u)

(65)

.

.

U

-

= r(t)dt + at dWt.

(6.7)

For this reason the probability P" is often called the 'risk neutral' probability.

o

The following proposition gives lm ec~nomic interpretation of the process (q(t))

6.1.3 Bond options

(cf. following Remark 6.1.4). ~ . U 't' 613 For'each maturity u there is an adapted process (at )o9~u ' P,roposl Ion . "

To make things clearer, let us first consider a European option with maturity 0 on the zero-coupon bond with maturity equal to the horizon T. If it is a call with strike price K, the value of the option at time 0 is obviously (P(O, T) - K)+ and it 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, H t) )O~t~T with values in rn?, Hp representing the quantity of riskless asset and H, the number of bonds with maturity T held in the portfolio at time t. The value of the portfolio at time t is given by

such that, on [0, uj, dP(t, u) = (r(t) _ afq(t))dt P(t, u) ,

+ afdWt.

(6.6)

Proof. Since the process (F(t, u))o9~u is a martingale under P~, (F()t u)L~09~U is a martingale under P (see Exercise 31). Moreo~er~ we have: P(t, U .t. > 6 ~.si' f ofPr?po~ltI2on ->: ' uj Then , using the same rationaleUas In the proof or aII t E [0, . h h t t ( OU) dt < 00 we see that there exists an adapted process (Ot )o~t~u SUC t a Jo t

L

and

.

F(t, u)Lt

Ie'(O~·)2ds = P(O,u)e Ie'oO"dW._l ' . 2

0

•

P(O,u) exp (it r(s)ds

+ fat (O~ -

dVi = HPdS~

.

q(s))d~s

r(t)dt

+ (Of -

q(t))dWt -

~((Of)2 - ~(t)2)dt

=

+ q(t)2 -

which gives the equality (6.6) with ar

if it is self-

Proposition 6.1.6 We assume sUPO',I'(t) =

a2A (e1't _ 1) +

2

= Jb + 2a

2JL.

)

b)) + 2JL (e1't - 1)

bt

.

-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 t re a d y, (1990)) If additivity property of the process condition x (cf. Ikeda and Watanabe (l?81) , p. 225, ~evuz an or ., for A and JL fixed, we consider the function F(t, x) defined by F(t,x):= E (e->.X;e-I'_J: X;dS) , it is

(6.17)

n~tural t~ look for F as a soiution of the problem . {

2 a 2F aF _ ='~x-2 at. Q . ax

., v

.

+ (a -

F(O,x)=e

aF . bx)- - JLxF ax ->.x

.

Indeed, if F satisfies these equations and has bounded derivatives, the Ito formula shows that, for any T, the process (Mt)o9~T, defined by .r

, XT.ds Mt.='e -I' 1. ~ F(T 0

x

t, X t )

2AA~ 1) .

This function is the Laplace transform of the non-central chi-square law with 8 degrees of freedom and parameter ( (see Exercise 35 for this matter). The density of this law is given by the function [s.c. defined by .

i: b + e1'tCT + b)

.

Abe- bt ) -x a2 /2A(1 _ e-bt) + b

AL() 2AL + 1

96,((A) = (2A +11)6/2 exp ( -

and

(

with L = a (1 - e- ) and ( = 4xb/(a 2 (e bt - 1)). With these notations, the Laplace transform of Xi / L is given by the function 94a/u2,(, where 96,( is defined by

where thefunctions·¢~,1' and 'lj;>',1' are given by ¢>',I'(t)

=

¢'(t)

Solving these two differential equations gives the desired expressions for ¢ and 'lj;. 0

I If a > a 2/2, we have P(T~ = 00) = 1, for all x> O. - a < a 2/2 and b ~ 0, we have P( TOx < 00) -- 1, for all x > O. 2. If 0 < 3: If 0 ~ a < a 2/2 and b < 0, we have Ph) < 00) E ]0, 1[,jor all x> O.

(x

131

is a martingale and the equality E(MT) = M o leads to (6.17). If F can be written as F(t, x) = e-aq,(t)-xt/J(t) , the equations above become ¢(O) = 0, 'lj;(0) = Aand

= inf{t ~ 0IX: = O}

T~

with, as usual, inf

Some classical models

.

)

(r::;()

-(/2

I () e J6,( x - 2(6/4-1/2 e -x/2 x 6/4-1/2I6/2-1 V zt,

lor x > 0 ,

l'

where Iv is the first-order modified Bessel function with index u, defined by

Iv(x) = ~

(~)v 2

f: n=O

(x/2)2n 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 and Stegun (1970)rChapters 9 and 26. Let us go back to the Cox-Ingersoll-Ross model. From the hypothesis on the processes (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
J

We prove (cf. Exercise 36) that, if we set

)

2'Y· e ........,.-

e-

dP· -

(6.18)

where the functions ¢ and 'ljJ are given by the following formulae 2

_

and

a2 (e"Y· o -1) -..,.---=--'-..,.----',-----,:----:- 2 'Y. (e"Y· o + 1) + b: (e"Y· o - 1)'

£2 - -

and

. 2(e"Y· t - l ) .'ljJ(t) ,= 'Y. _ b· + e"Y·tb· + b·)

.

with b"

= b + atx and 'Y • -P(t, T)

J(b·)2 + 2a 2. The price at time t is given by

= exp (-a¢(T -

t) - r(t)'ljJ(T - t)) .

the law of r(O) / £1 under PI (resp. r(O) / £2 under Ps) is a non-central chi-squared law 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)) and

8r(Oh· 2e"Y· O (2 = a2 (e"Y·O - 1) b·(e"Y· o + 1) + b·(e"Y· o - 1)) .

Let us now price a European call with maturity 0 and exercise pr~ce K, on a ~e~o­ coupon bond with maturity ~. We .can sho~ that the hypothesIs of Proposition 6.1.6 holds; the call price at time

Co

=

E·

°ISthus given by

With these notations, introducing the distribution function Fo,( of the non-central chi-squared law with fJ degrees of freedom and parameter (, we have consequently

0

[e-J09r(s)ds (P(O,~) - K)+]

,

• [ _J.9 r(s)ds ( e _a(T_O)-r(O).p(T-O) _ K) ] =Eeo . + =

Co

E. (e- J: r(S)dS p(O,T)1{r(O) 0) = 1. We set T

= (inf{t E [O,T] I M, = O}) AT.

Exercise 34 The aim of this exercise is to prove Proposition 6.2.4. For x, M we note TM the stopping time defined by TM = inf{t 2: I Xi = M}.

°

1. Let s be the function defined on

1. Show that T is a stopping time.

2. Using the optional sampling theorem, show that E (MT) Deduce that P ({\it E [O"T] M, > O}) = 1.

s(x)

= E (MT1{T=T}).

= jX e2by/u2 y-2a/u 2dy.

/72

~s

- x -2 2 dx 2. For e

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 00

]0,oo[ by

Prove that s satisfies

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, the density of the restriction of Qto Ft. Let (~t)09~T ~e an adapted process. Show that (Mt)09~T is a martingale under Q If and only If the process (LtMt)o~t~T is a martingale under P.

J;{ Hldt
0,

< x < M, we set T:'M

ds

+ (a - bx)= 0. dx

= T: A TM. Show that, for any t > 0, we have

Deduce, taking the variance on both sides and using the fact that s' is bounded from below on the interval [e, M], that E (T:'M) < 00, which implies that T:,M is finite a.s.

a.s. and

< x <M, s(x) = s(e)P (T: < TM) + s(M)P (T: > TM)' 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. We now assume that ~ a < /72/2 and we set s(O) == limx-..+o s(x). Show that, for all M > x, we have s(x) = s(O)P (TO < TM) + s(M)P (TO> T M) and complete the proof of Proposition 6.2.4.

3. Show that if e for all t

4.

E [0, T].

E~ercise 33

We would like to price, at time 0, a call with mat~rity () and strike price 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.

< r" , where _ /72 (1 - e-a(T-II))

2. Show the option is exercised if and only if r( ()) a(T - ())

r*

Roo ( 1 - 1 _ -log(K)

)

e-a(T II)

(1 _e~a(T-II)

. , 4a2

.

I

°

Exercise 35 Let d be an integer and let Xl, X 2 , .•. , X d , d be independent Gaussian random variables with unit variance and respective means mI, m2, ... , md· Show that the random variable X = E~=I Xl follows a non-central E~=I m~. chi-squared law with d degrees of freedom and parameter (

=

) .

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

dP

5.

°

' e->'x

dP = E (e->'x)"

Show that, under P, Y is normal and give its mean and variance.

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 Section ' 6.2.2. Exercise 37 Let (n, F, (Ft)O 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) a 2 t (u - t/2) + aWt . Deduce, that ll

T)

P( u'.

= P(O,T)

(_ (T' _ P(O,O) exp a

3. Derive, for A E JR, E*

(e

-rJ'

ll)TXT U

.,

a

HII

f: W.ds e,xWB) ~ Deduce

20T(T.

2

=

f(O, u)

0))

+

.

the law of WII'under the

probability measures PI and P 2 with densities with respect to P* respectively given by

dP

l

dP* =

e-

JOB r(~)dSp(o, T) P(O,T)

and

dP e- foB ~(s)ds 2 -= ---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 and this property is one of the characteristics of the model. But some rare events (release of an unexpected economic figure, major political changes or even a natural disaster in a major economy) can lead to brusque variations in prices. To model this kind of phenomena, we have to introduce discontinuous stochastic processes. Most of these models 'with jumps' have a striking feature that distinguishes them from the Black-Scholes model: they are incomplete market models, and there is no perfect hedging of options in this case. It is no longer possible to price options using a replicating portfolio. A possible approach to pricing and hedging consists in defining a notion of risk and choosing a price and a hedge in order to minimise this risk. In this chapter, we will study the simplest models with jumps. The description of 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 (Tik~l be a sequence of independent, identically exponentially distributed random variables with parameter A, i. e. their density is equal to l{x>o}Ae->.x. We s~t Tn L~l t: We call Poisson process with intensity A the process N, defined by .

Definition 7.1.1 Let

=

Nt

=L n~l

l{Tn::;t}

= L nl{T

n::;tO}.Ae->'x (n _ I)! dx, :I.e. a gamma law with parameters , A and n. Indeed, the Laplace transform of

E thus the law of Tn

(e-

T1

is

A

ct T 1 )

= T 1 + ... + Tn is E(e- ctr n ) = E(e-

= A + 0:'

ct T l (

= (A ~ 0:) n

We recognise the Laplace transform of the gamma law with parameters A and 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)

=

)n-l { Ae->'x AX dx _ Jo (n - I)!

=

(Att ->.t -e . n!

t (\

it 0

(AX)n Ae->'x - - I-dx n.

o

t 't" 7 1 4 Let (Nt ) t >0 be a Poisson process with intensity A d and .F = Proposl Ion . . a(N ', s ~ t). The process (Ntk~o is a process with independent an stationary

The objective of this section is to model a financial market in which there is one = e'", at time t) and one risky asset whose price riskless asset (with price jumps in the proportions U1 , ... , Uj, ..., at some times Tl , ... , Tj, ... and which, between two jumps, folIows the Black-Scholes model. Moreover, we will assume that. 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 standard Brownian motion (Wdt~o, a Poisson process (Nt)t~o with intensity A and a sequence (Uj)j~1 of independent, identicalIy distributed random variables taking values in ]-1, +00[. We will assume that the a-algebras generated respectively by. (Wdt~o, (Nt)t~o, (Uj)j~1 are independent. For all t 2: 0, let us denote by F t the a-algebra generated by the random variables 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, is independent of the zr-algebra F s . Because the random variables UjJ{j~Nd are Frmeasurable, we deduce that, at timet, the relative amplitudes of the jumps taking 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 described in the following manner, The process (Xt)t~O is an adapted, right-continuous process satisfying:

Sr

• On the time intervals

s

increments, i.e. • independence:

dX t

.

if s > 0, N t+ s

• stationarity: the law of N t+ s

-

-

h, Tj+l [

N, is independent of the a-algebra Ft.

N; is identical to the law of N, - 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~sfie P(T- 2: t + siT 2: t) = P(T 2: s). The exponential variables are said to

• At time

Tj,

the jump of X, is given by

- \ 6,Xr .

thus X r;

= Xt(jldt + adWt).

= Xr~, (I + Uj ) .

J

,

=X

r J, -

XT j

=X

Tj

-U , j

,

Asset models with jumps

144

Dynamics ofthe risky asset

145

So we have, for t E [0, Tl [

is independent of the a-algebra generated by the random variables N < d UjIUSN.}. Let A be a Borel subset of JRk, B aBorel subset f JRd u'dV,C- s an r. an of the a-algebra a(Nu, V, '(s,+, -

s.)

I

dv(z) 'I> (Y.;> z)

When the mesh of p tends to 0, we obtain

N.

L

O = -

)

- )..

J

zv(dz),

that

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 stronger condition of integrability on the process (Hdo~t~T, by restricting the class of admissible strategies as follows:

Definition 7.3.1 An admissible strategy is defined by a process

o

'

¢ = ((Ht , Ht))O~t~T

(7.4)

(e-rt:Xtk:~o is a martingale. Notice

+

adapted, left-continuous, with values in t E [0, T] and such that JOT

IH~lds < +00

~ ais.

rn?,

satisfying equality (7.5) a.s. for all

andE (JoT H;X;ds)

< +00.

Note that we do' not impose any condition of non-negativity on the value of admissible strategies. The following proposition is the counterpart of Proposition 4.1.2 of Chapter 4.

and consequently, from Exercise 39,

E (X;) Therefore the process'

= XJ exp ((0'2 + 2r)t) exp ()..tE(Ut)) .

(.it)

is a square-integrable martingale. t~O

Proposition 7.3.2 Let (Hdo9~T be an adapted, left-continuous process such that

E (iT H;X;dS)

< 00,

and let Vo E nt., There exists a unique process (HP)O~t~T such that the pair

Asset models with jumps

152

((HP, H t))O$tE(U,)_U' I')(T-')+UWT_,

VT))2) .

(1 +

(1 + Uj

Uj)) ) )) )

.

Note that if we introduce the function

Since, from Remark 7.3.3, the discounted value (ft) is a martingale, we have E (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

Fo(t,x) = E (e-r(T-t) f (xe(r-cr 2/2)(T-t)+crwT_')) ,

=

HI = (E(e-rTh) -

Vo)2 + E (e-rTh - E(e-rTh.) - (VT - Vo)f·

which gives the price of the option for the Black-Scholes model, we have (7.7)

Proposition 7.3.2 shows that the quantity VT' - Vo depends only on (Ht ) (and he will not on Vo). If the writer of the option tries to minimise the risk ask for a premium Vo E(e-rTh). So it appears that E(e-rTh) is the initial value of any strategy designed to minimise the risk at maturity and this is what we will take as a definition of the price of the option associated with h. By a similar argument, we see that an agent selling the option at time t > 0, who wants

=

to minimise the quantity premium

R; = E ( (e-r(T-t)(h -

Vi = E (e-r(T-t) hIFt).

VT) )2 1Ft ) , will ask for a '

Before tackling the problem of hedging, we try to give an explicit expression for the price of the call or the put with strike price K. We will assume therefore that h can be written as f(XT), with f(x) = (x - K)+ or f(x) = (K - x)+. As we saw earlier, the price of the option at time t is given by

(e-~(T-t) f(XT)IFt)

=

E (e-r(T-t) f (Xte(l'-cr2

E (e- d T -.) I

).

.

(7.8)

Since NT-t is a random variable independent of the Uj's, following a Poisson law with parameter >'(T - t), we can also write

F(t, x)

~ ~E (FO ~' xe-'(T-')E(U,)

Q

(1+ Uj ) )

)

e-'(T-')

~~(T -

t)"

/2)(T-:-t)+cr~WT - W,) . IT, .'

(1 + Uj ) )

Ft)

]=N,+l

(X .e('-u'/2)(T-')+U(WT - W,t~r' (1 +

From Lemma 7.2.1 and this equality, we deduce that

E (e-r(T-t)!(XT)!Ft)

7.3.4 Hedging of calls and puts Let us examine the hedging problem for an option h= f(XT), with f(x) = (K - x)+. We have seen that the initial value of any (x - K)+ or f(x) at maturity is given by admissible strategy aiming at minimising the risk Vo = E(e-rTh) = F(O; X o). For such a strategy, equality (7.7) yields

=

'.

,

(1+ Uj ) )

Each term of this series can be computed numerically if we know how to simulate the law of the Uj's. For some laws, the mathematical expectation in the formula' can be calculated explicitly (cf. Exercise 42).

7.3.3 Prices of calls and puts

=

If

We will take this quantity to define the price

of the option at time t.

E

F(t, x) = E (FO (t,xe-'(T-.)E(U,)

RJ,

= F(t,Xt),

R6

RJ = E (e-rTh UN,+

j))

.r}

VT

f·

Now we determine a process (H t )Os;t5;T for the quantities of the risky asset to be To do so, we need the following proposition. held in portfolio t? minimise

RJ.

~~o.positio~ 7.3.4 Let Vi ~;;'he value at time t of an admissible strategy with f(XT)) = F(O, X o), determined by a process initial value Vo = E (e (Ht)Os;t5;T for the quantities of the risky asset. The quadratic risk at maturity

ir== 156

RJ = R&T

Asset models with jumps

Pricing and hedging options

157

E (e-.rT(f(X T) - VT)) 2 is given by the following formula:

E ( Jo r T (aF ax (s, X) s

=

-

- 2 2 ds H; ) 2 Xsu

E (it ds !v(dz) (F(s, Xs(l

o

+ J: A J v(dz)e- 2rs (F(s,

x.n + z)) -

Proof. From Proposition 7.3.2, we have, for t

~

s

F(s, X s) - H szXs)2 dS).

+ z)) - F(s, X s)) 2)

(1 dsX; JV(dz)z2) t

E

< +00,

T,

which, from Lemma 7.2.2, implies that the process Nt

u,

L

=

j=1

F(Tj, X r;) - F(Tj, X rj-)

Al t ds F(t,x) = e-rtF(t,xe rt), 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 ) = F(O, X o)+ lit

+-

2

0

aF r aF - ior 8;(s, Xs)ds + io ax (s,Xs)X s(-AE(Vdds + udWs)

aF 2

-

2 - 2

-a 2 (s,Xs)u Xsds x . .. .

+ L F(Tj,XrJ - F(Tj,Xr~)· N,

j=1

-

-

-

-

]

(7.10)

Remark that the function F( t, x) is Lipschitz of order 1 with respect to z, since

J

(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 a stochastic integral. Whence

--

F(t, X t) - M,

= F(O, X o) +

i 0

t aF ax (s, Xs)XsudWs.

(7.11)

Gathering equalities (7.9) and (7.11), we get

ii - VT = M~1) + M¥), with

and N,

U

IF(t,x) -(F(t'Y)1 ( N _, ) T < E e-r(T-t) jxe(r--XE(Utl- u 2 /2)(T-t)+UWT_t (1 + V j)

=

L (Fh,Xr;) j=1

I !

Fh,Xr~) - HrjVjXr~) ] ]

t

-A

ds

dv(z) (F(s, Xs(l

+ z)) - F(s, X s) - HszX s) .

From Lemma 7.2:3, MP) M t(2) is a martingale and consequently

E (MP) M1

2))

= J.!a 1) Ma2) = O.

Whence

E

=

Ix -yl·

It follows that

(ii - VT)

=,

E((M~1))2) + E((M~2))2)

• 'E

(J:{~~

(s, X.) -

HrX;U'dS) + (M~'»)'), E(

Asset models with jumps

158

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 measure of 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).

and applying Lemma 7.2.2 again

E(( M f2))2) =

E

.

(A !:;ds J v(dz) v- X s(l + z)) -

The risk at maturity is then given by

m

=

E

_

2) .

)2 - s a ds

T aF Jo ax (s, X s) ( (

+ JoT A J v(dz)

F(s, X s) - HszX s)

Hs

X

2

159

Exercises

2

7.4 Exercises

(F(s, X s(l

+ z)) - F(s, X s) - HszX s) 2 dS) .

o It follows that the minimal risk is obtained when H, satisfies P a.s.

~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 values in N, following a Poisson law with parameter A, independent of the sequence (Vn)n~I. Show that .

E

aF .) -2 2 ( ax (s, x.) - H, Xsa

.

(11 vn) ~

e'(E(V,j-')

It suffices indeed to minimise the integrand with respect to ds. It yields, since

~xercise 40 Let (Vn)~~1 be a sequence of independent, identically distributed, Integrable random variables and let N be a random variable taking values in N integrable and aindependent of the sequence (V,n ). We set S = ""N V, (with the L ..m =I n

(Hdt~iJ must be left-continuous,

convention En=I = 0).

+

AJ v(dz) (F(s,X s(l

+ z)) - F(s,Xs) - HszX s) zX s = O.

H,

= 6.(s,.X s - ) ,

. 1. Prove that S is integrable and that E(S) = E(N)E(Vi}

with

6.(s,x)

=

aF (2 a - (s x ) (12 +A J v(dz)z2 ax' 1

+A

. j. v (d)z z (F(S,X(l+Z))-F(S,X))) x

(JoT

In this way, we obtain a process which satisfies E 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 the Black-Scholes model and, in this case, we know that the hedging is perfect, i.e, = O. But, when there are jumps, the minimal risk is generally positive (cf.

m

Exercise 43 and Chateau (1990». Remark 7.3.5 The formulae we obtain indicate that calculations are still possible for models with jump. It remains to identify parameters and the law of the U, 'so As for 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 an organised market. In the second approach, the models with jump, which-involve several parameters, give a better 'fit' to the market prices.

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) and Var(S) = AE (Vn.

= AE(Vt)

Exercise 41 The hypothesis and notations are those of Exercise 40. We suppose that 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 and N 2 are two independent random variables following a Poisson law with respective parameters 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 a double 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 e 9 - 1, where 9 is a normal variablewithmean m and variance a 2 • Write the price formula (7.8) as a series of terms calculated from the Black-Scholes formulae (for some interest rates and v?latilities to be given).

Asset models with jumps 160 Exercise 43 The objective of this exercise is to show that there is no perfect hedging of calls and puts for the models with jumps we studied in this chapter. We consider a model in which a > 0, >. > a and 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,X s)'

2. Show that the law of X; has (for s > 0) a positive density on ]0, ~[. It may be worth noticing that if Y has a density 9 and i~ Z is a random vana~le independent ofY with values in ]0,00[; the random van able 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= a such that f~r s E [0,T[ and x E]0, 00[,

aF ax (s,x)

=

F(s,x(l+z))-F(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[.

8

Simulation and algorithms for financial models

8.1 Simulation and financial models In this chapter, we describe some methods which can be used to simulate financial models and compute prices. When we can write the option price as the expectation of 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 evaluate complex hedging strategies (example: find the impact of hedging a portfolio every ten 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 random variable with law J-L(dx) and we would like to generate a sequence of independent trials, Xl, .. : ,Xn, . . . with common distribution J-L. Applying the law of large numbers, we can assert that if f is a J-L-integrable function 1

lim N N-t+oo

o

" L..J

l~n~N

f(Xn)

=

J

f(x)J-L(dx).

(8.1)

To implement this method on a computer, we proceed as follows. We suppose that we know how to build at sequence of numbers (Un)n>1 which is the realisation . . 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 random variable F(U l, ... ,Up) has the desired law J-L(dx). The sequence of random variables (Xn)n~1 where X n = F(U(n-l)p+l"" ,Unp) is then a sequence of independent random variables following the required law J-L. For example, we can apply (8.1) to the functions f(x) = x and f(x) = x 2 to estimate the first and second-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 provide a random' function, already coded, which returns either a pseudo-random number

Simulation and algorithms for financial models

162

between 0 and 1, or a random integer in a fixed interval (this function is called rand () in C ANSI, random in Turbo Pascal).

Remark 8.1.1 The function F can depend in some cases (in particular when it comes to simulate stopping times), on the whole sequence (Un)n;?:I, and not only on a fixed number of Ui 'so The previous method can still be used if we can simulate X 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 (see page 163). 8.1.2 Simulation of a uniform law on [0, 1] We explain how to build random number generators because very often, those available 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 as follows: 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 satisfactory characteristics for the sequence. Sedgewick (1987) advocates the following choice:

163

The previous generator provides reasonable results in common cases. However it might happen that its period (here m = 108 ) is not big enough. Then it is possible to create random number generators with an arbitrary long period by increasing m. The interested reader will find much information on random number generators and computer procedures in Knuth (1981) and L'Ecuyer (1990).

8.1.3 Simulation of random 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 case of models with jumps). We give some methods to simulate each of these laws.

Simulation ofa Gaussian law A classical method to simulate Gaussian random variables is based on the observation (see Exercise 44) that if (U I , U2 ) are two independent uniform random variables on [0, 1]

V- 2 10g(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» " Random); end;

31415821 1

108 . 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 random integer by m. const m ml b

Simulation and financial models

100000000; 10000; 31415821;

" cos(2.0 " pi

Simulation ofan exponential law We recall that a random variable X follows an exponential law with parameter f,L if its law is 1{x;?:O}f,Le u» dx. r-

We can simulate X ~oticing ~hat, if U follows a uniform law on [0,1], 10g(U) / f,L follows an exponential law with parameter u,

var a : integer;

function exponential( mu : real) : real; begin exponential := - log (Random) / mu; end;

function Mult(p, q: integer) : integer; (" Multiplies p by q, avo i.d i nqjvove r 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;

Remark 8.1.2 This method of simulation of the exponential law is a particular case of the so-called 'inverse distribution function' method (for this matter see Exercise 45). ' Simulation ofa Poisson random variable

"

o

-

A Poisson random variable is a variable with values in N such that

P(X

An

= n) = e->'" n.

ifn 2: O.

164

Simulation and algorithms for financial models

We have seen in Chapter 7 that if (Ti )i>l -iS a sequence of exponential random variables with parameter A, then the law oeNt = L:n>l nl {Tl +..+Tn9o. The first one consists in 'renormalising' a random walk. Let (Xi)i>O be a ~equence of independent, 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 set Sn = Xl + .,. + X n; then we can 'approximate' the Brownian motion by the process (X;')t:2:o where

X;'

= JnS[nt 1

where [x) is the largest integer less than or equal to x. This method of simulation of 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 independent standard 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

o

~

(W o, W~t, W2~t, ... , Wn~t).

The Brownian motion can be approximated by X;' = ViSJ,S[t/ ~tl'

Simulation and algorithms for financial models

166

Simulation and financial models

167

Simulation ofstochastic differential equations

An application to the Black-Scholes model

There are many methods, some of them very sophisticated, to simulate the solution of 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 we present only the basic method, the so-called 'Euler approximation' . The principle is the following: consider the stochastic differential equation

In the case of the Black-Scholes model, we want to simulate the solution of the equation

{

=

Xo

ex.

x

b(Xt)dt + a(Xt)dWt.

We discretise time by a fixed mesh b.t. Then we can construct a discrete-time process (Sn)n~O approximating the solution of the stochastic differential equation at times nb.t, setting

{;;t

Two approaches are available. The first consists in using the Euler approximation. We set

= x SO { Sn+l Sn(1 + rb.t + agn.,fl;i) , and simulate X; by Xl' = S[t/ ~t). The either method consists in using the explicit expression of the solution

x, =

x

So { Sn+l - Sn

{b(Sn)b.t

+ a(Sn)

(W(n+l)~t "- Wn~t)} .

If Xl' = S[t/~t), (Xl')t~O approximates (Xt)t~O in the following sense:

Theorem 8.1.4 For any T

t<S.T

s; = z exp ((, .; ,

C T being a constant depending only on T. A proof of this result (as well as other schemes of discretisation of stochastic differential 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 sequence of independent normal random variables with zero-mean ana variance b.t. In a simulation, we substitute gn.,fl;i to (W(n+l)~t - Wn~t) where (gn)n~O is a sequence of independent standard normal variables. The approximating sequence (S~)n~O is in this case defined by

=

x S~

xexp (rt -

We always approximate X, by Xl'

q'

/2)nLlt +

q~t, g,) .

(8.2)

= S[t/~t).

Remark 8.1.6 We can also replace the Gaussian random variables gi by some Bernouilli variables with values + 1 or -1·with probability 1/2 in (8.2); we obtain a binomial-type model close to the Cox-Ross-Rubinstein model used in Section 5.3.3 of Chapter 5.

Simulation of models with jumps We have investigated in Chapter 7 an extension of the Black-Scholes model with jumps; we describe now a method. to simulate this process. We take the notations and the hypothesis of Chapter 7, Section 7.2. The process (Xt)t~O describing the dynamics, of the asset is . ., ,

x, : ;: : x

+ b.t b(S~) + a(S~)gn.,fl;i.

Remark 8.1.5 We can substitute to the sequence of independent Gaussian random variables (gi)i~O a sequence of independent random variables (Ui)i~O, such that P(Ui = 1) = P(Ui = -1) = 1/2. Nevertheless, in this case, it must be noticed that the convergence is different from that found in Theorem 8.1.4. There is still a 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 on this kind of convergence and many results on discretisation in law for stochastic differential equations.

~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

>0

E (sup IXl' -.Xt12) ~ CTb.t,

~t(rdt + adWt).

:

}1 (i + ~j) N'

. (

)

e(Il-

CT2 /

2

.

)t+CTW, ,

(8.3)

where (Wt)t~O is a standard Brownian motion, (Nt)t>o is a Poisson process with intensity and (Uj) j~ 1 is a sequence of independent, identically distributed random variables, with values in ] -:- 1, +oo[ and law lJ(dx). The a-algebras generated 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

.x,

Xn~t ~ X If we noteYk

X

(X~t/x)

x

(X2~t/X~t)

x ···x

(Xn~t/X(n-l)~t).

= (Xk~t/X(k-l)~t), we can prove, from the properties of (Nt)t~o,

Simulation and algorithms for financial models

168

(Wdt2:o and (Uj ) j2:l that (Yk h2:l is a sequence of independent random variables with the same law. Since Xn~t = xYl . , . Yn , the simulation of X at times

Some useful algorithms

169

Cl

n.6.t comes down to the simulation of the sequence (Yk h 2: l . This sequence

C2

being independent and identically distributed, it sufficesto know how to simulate Yl = X~tlx. Then' we operate as follows:

C3 C4

• We simulate a 'standard Gaussian random variable g.

1

N(x) ::::: 1 - 2(1

• 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

, ... ,

8.2 Some useful algorlthms In this section, we have gathered some widely used algorithms for the pricing of options. 8.2.1 Approximation ofthe distributionfunction of a Gaussian variable We saw in 'Chapter 4 that the pricing of many classical options requires the calculation of ,,2 dx N(x) = P(X :::; x) = e--: T rs:':

I

'"

-00

a

y 27r

where X is standard Gaussian random variable: Due to the importance of this function in the pricing of options, we give two approximation formulae from Abramowitz and Stegun (1970) , The first approximation is accurate to 10- 7 , but it uses the exponential function, Ifx>O p -' 0.231641900 b1 = 0.319381530 b2 = -0.356563782 b3 = 1.781477937 -b 4 ~ 1.821255978 b5 1.330274429

,

N(x) ::::: 1 -

1

= ,,2

rrce-T (bit

y27r

0.196854 0.115194 0.000344 0.019527 '

+ CIX + C2x2 + C3X,3 + C4x4)-4.

8.2.2 Implementation of the Brennan and Schwartz method The following program prices an American put using the method described in Chapter 5, Section 5.3.2: we make a logarithmic change of variable, we discretise the parabolic inequality using a totally implicit method and finally solve the inequality in infinite dimensions using the algorithm described on page 116, CONST PriceStepNb = 200;' TimeStepNb = 200; Accuracy = 0,01; DaysInYearNb = 360;

.

t

= = = =

Un'

All these variables are assumedto be independent. Then, from equation (8.3), it is clear that the law of

is identical to the law 'of Yl

>0

opposed to an exponential. If x

1/(I+px)

+ b2t 2 + b3t3 + b4t4 + b5t5v ) .

The second approximation is accurate to 10- 3 but it involves only a ratio as

TYPE Date = INTEGER; Amount = REAL; AmericanPut = RECORD ContractDate : Date; (* in days *) MaturityDate : Date; (* in days *) StrikePrice : Amount; END; vector = ARRAY[l, ,PriceStepNbI OF REAL; Model = RECORD r REAL; (* annual riskless interest rate sigma REAL; (* annual volatility *) xO i REAL; (* initial value of the SDE *) END;

*)

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;

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 time is "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;

Simulation and algorithms for financial models

170 vv :=

~odel.sigma

171

I::

* model. sigma; + abs(r - vv / 2)

1 := (model.sigma * sqrt(Temps) * sqrt(ln(l/Accuracy» Time) ; h := 2 * 1 / PriceStepNb; writeln(1:5:3.'·' ,In(2) :5:3); alpha := k * (- vv / (2.0 * h * h) + (r - vv / 2'.0) / beta := 1 + k * (r + vv / (h * h»; gamma := k * (- vv / (2.0 * h * h) - (r - vv / 2.0) / FOR PriceIndex:=l TO PriceStepNb DO BEGIN A[PriceIndex] := alpha; B[PriceIndex] := beta; C[PriceIndex] := gamma;

Exercises

*

f(x)dx. We setF(u) = f(x)dx. Prove thatifU is a uniform random variable oo on [0,1], then the law of F-l (U) is f(x)dx. Deduce a method of simulation of X. Exerci.se 46 We model a risky asset S, by the stochastic differential equation

as, { So

(2.0 * h»; (2.0 * h»;

=

x,

where (Wt)t>o is a standard Brownian motion, a the volatility and r is the riskless interest rate. Propose a method of simulation to approximate

END;

B[l] B[PriceStepNb] G[PriceIndex]

:= := :=

beta + alpha; beta + gamma; 0.0;

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] ; 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]; FOR TimeIndex:=l TO TimeStepNb DO BEGIN FOR PriceIndex := PriceStepNb-l DOWNTO 1 DO G[PriceIndex) := G[PriceIndex] - '. C[PriceIndex] * G[PriceIndex+1];

G[l] := G[l] / at i i , FOR ,PriceIndex:=2 TO PriceStepNb DO BEGIN G [PriceIndex] : = G [PriceIndexl / B [PriceIndex] - A[Pric'~IndexJ * G [Price Index-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;

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 hedging frequency on the variance of a portfolio of options. The underlying asset of the options is described by the Black-Scholes model

as, { So

x,

(Wtk~o represents a standard Brownian motion, a the annual volatility and r the riskless interest rate. Further on we will fix r = lO%jyear, a = 20%/ Jyear = 0.2 and x = 100. Being 'delta neutral' means that we compensate the total delta of the portfolio by trading the adequate amount of underlying asset. In the following, the options have 3 months to maturity and are contingent on one 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 110 call 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 110 call. • Put ratio backspread: short a 110 put and long 3 90 puts.

8.3 Exercises

..

.:

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 variables -210g(Ul ) cos(27l'U2 ) and -210g(Ud sin(27l'U2 ) are independent and folIowa standard Gaussian law. . 0

J

J

Exercise 45 Let f be a function from JR to JR, such that f(x) > 0 for all x, and such that f(x)dx = 1. We want to simulate a random variable X with-law

r:

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 portfolio in the following cases: (

,

We ,90 not hedge: we sell the option, get the premium, we wait for three months, we take into account the exercise of the option sold and we evaluate the portfolio. We hedge immediately after selling the option, then we do nothing.

172

Simulation and algorithms for financial models

_ 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.

Appendix

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 -1 with 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

A.I Normal random variables

(Wt+ s, Wt). 3. IfO < tl < ... < t p, show that (X~, . . . ,X~) converges in law to (W t l , · .. , Wt p ) .

In this section, we recall the main properties of Gaussian variables. The following results are proved in Bouleau (1986), Chapter VI, Section 9.

A.i.i Scalar normal variables ~

I

~

A real random variable X is a standard normal variable if its probability density function is equal to n(x)

= _1_ exp .J2;

2)

(_ X

•

2 If X is a standard normal variable and m and a are two real numbers, then the variable Y = m + a X is normal with mean m and variance 0'2. Its law is denoted by N( m, 0'2) (it does not depend on the sign of a since X and - X have the same law). If a i= .0, the density of Y is

(x-m)2) 20'2'

_1_ exp (

I

J27fO'2

If a = 0, the law of Y is the Dirac measure in m and therefore it does not have a density. It is sometimes called 'degenerate normal variable'. If X is a standard normal variable, we can prove that for any complex number z, we have . .

E

(e z X )

= e4 . u2

o

Thus,the characteristic function of X is given by ifJx(u) = e- / 2 and for Y, ifJy(u) = eiUTne-u2(j2/2. It is sometimes useful to know that if X is a standard normal 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 P(X > t) = _rrc , v 27f

1

00

t

e'--x

2/2dx

$

1 _rrc tv 27f

1

00

t

2

/2

xe- x2/2dx = _e__ . _t

.

t.J2;

174

Appendix

Finally, one.should know that there exist very good approximations of the cumulative normal distribution (cf. Chapter 8) as well as statistical tables.

=

Definition Ad.I A random variable X (XI, ... ,Xd) in lR d is a Gaussian vector if for any sequence ofreal numbers at. ... , ad, the scalar random variable 2:~=1 aiX is normal. The components Xl •...• X d of a Gaussian vector are obviously normal, but the fact that each component of a vector is a normal random variable does not imply that 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 matrix I'(X) = (aij h~i,j~d whose coefficients are equal to

=

= cov(Xi, Xj) = E [(Xi -

E(Xi))(Xj - E(Xj))].

It is well known that if the random variables Xl'•... , X d are independent. the matrix I'(X) is diagonal. but the converse is generally wrong. except in the Gaussian case: Theorem A.I.2 Let X (Xl,' .. , X d) be a Gaussian vector in lRd. The random variables Xl, ... , X d are independent if and only if the covariance matrix X is . , diagonal.

=

The reader should consult Bouleau (1986), Chapter VI. p. 155, for a proof ofthis result.

RemarkA.I.3 The importance of normal random variables in modelling comes partly 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) for problems of estimation and to Chapter 8 for problems of simulation.

A.2 Conditional expectation A.2.I Examples of a-algebras .Let us consider a ~~Race (P,A) and a P~.Q'I. B 2 • • • • , ~n; with n events in A. The set B containing the elements of ~ which are ei~pty or that can be written as Bit U B i2 U··· U B ik, where i~, ... , ik E {I, ... , n}, is a finite sub-a-alg;bra of 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 is·g~.!'erated by theelements B, of A: B, are the non_.-"""'-~ ----. ---. ~ ~ ,........ empty elements ofB whichcontain onIy-tnemselves and the empty set. They are 'called--atoms of B. There is a one-to-one mapping-fffiffi-tfie ser'OCfinite sub-aalgeoras ofAonto the set of partitions of 0 by elements of A. One should notice that if B is a sub-a-algebra of A, a map from 0 to lR (and its Borel a-algebra) is B-measurable if and only. if it is constant on each atom of B.

A.

.--.. --:""_..

_-~._-----

B .,.r:---.-.-.-

175

Let us now consider a random variable X defined o!!-(O, A) with values in a m~able ~~~.JE, E). Th~ ~~l~bra generated ~X isthesmallest a-algebra f~~~~.Ich X ~~~~~~~ It IS denoted by a(X). It isolWiouslyincluaea-in A .~--.

and It IS easy to show that

A.I.2 Multivariate normal variables

aij

Conditional expectation

'

-----

~----=----- ~~--_.~

-.

-'-'---'-~(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)-measurable if 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 measurable functions 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. The definition of the conditional expectation is based on the following theorem (refer to Bouleau (1986), Chapter 8):

!heorem A.2.1 For any real integrable random variable X, there exists a real Integrable Btmeasurable random variable Y such that VB E B

E(XI B) = E(YI B).

If Y is another randomvar~able with 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 , ... , B n , . . . E(XIB)

=L

E(XIB;)/P(Bi)IB;,

where we sum on the atoms with strictly positive probability..Consequently, on each atom B i, E(XIB) is the mean value of X on Bi, As far as the trivial a-algebra is concerned (B = {0, OJ), we have E(XIB) = E(X). . . The ~omputationsinvolving conditional expectations are based on the following properties:

~!.; If £~~~l![~ble,E(XIB) = X, a.s. .~ E (E (XIB)) = E (X). 3. For any bounded! B-measurablerandom variable Z,E (ZE(XIB)) 4. L i n e a r i t y : ' . ''-' ~

E,: (~~ I

+ JLYIB) =AE (XIB) + JLE (YIB)

= E(ZX). ,'--

a.s.

~_

5. Positivity: if X 2: 0, then E(XIB) 2: a a.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.

=>

, 176

Appendix

Conditional expectation

and therefore II E(XIB)II£lCfl) ~ IIXII£lCfl). 6. If C is a sub-a-algebra of B, then

Proposition A.2.S Let us consider a B-measurable random variable X taking values 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 ,

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 the a-algebra B if and only if VU'E IR

E

(eiUXIB) ='E (e iuX) a:!.

Vx E E

E

(eiUX~) = E P(B) -, ~

{l(X:~p~~))

((x, Y))

= cp(X) a.s.

In other words, under the previous assumptions, we can compute E ( (X, Y) IB) as if X was a-constant. ' .., .

(A.l)

Proof. Let us denote by P y the law of Y; We have

cp(x) =

.

i

(x, y)dPy(y)

and the measurability of cp is a consequence of the,Fubini theorem. Let Z be a non-negative B-measurable random variable (for example Z = IB, with B E B). If we denote by P X,Z the law of (X, Z), it follows from the independence between Y and (X, Z) that,

E ((X, Y)Z) =.E (f(X)),

for any bo~nded Borelfunction j, hence the independence.

=;: E

E ( (X, Y)IB)

This equality means that the characteristic function of X is identical under measure P 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 laws , and consequently E

cp(x)

is a Borelfunction on (E, E) and we have

Proof. Given the Property 8..above, we just need to prove that (A.l) implies that X is independent of B. If E (e iuX IB) = E (e iUX) then, by definition of the conditional expectation, for all B E B, E (e iuX IB) = E (e iuX) P(B). If P(B) =j:. 0, we can write '

(e iuX)

177

= / /

o

/

Remark A.2.3 If X is square integrable, so is E(XIB), and E(XIB) coincides with the orthogonal projection of X on L2(n, B, P), which is a closed subspace of L 2 (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 the least-square-best B-measurable predictor of X. In particular, if B is the a-algebra generated by a random variable €, the conditional expectation E(XIB) is noted E(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 by Pythagoras' theorem, we know that IIE(XIB)IIL2Cfl) ~ IIXII£2Cfl).

(x, y)zdPx,z(x,

( / (x, y)dPy(y))

z)dPy(y) zdPx,z(x, z)

= / cp(x)zdPx,z(x,z) = E (cp(X)Z) , which completes the proof.

0

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 (in n I IR + ) , the conditional expectation Z' == E (YIX I , . . . ,Xn ) has the following form " '

Remark A.2.4 We can define E(XIB) for any non-negative random variable X (without integrability condition). Then E(X Z) = E (E(XIB)Z), for any Bmeasurable non-negative random variable Z. The rules are basically the same as in the integrable case (see Dacunha-Castelle and Duflo (1982), Chapter 6).

n

Z

= Co + LCiXi, i=l

:J

where c, are real constant numbers. This means that the function of Xi which approximates.Y in the least-square sense is linear. On top of that, we can compute Z by p~?jecting the random variable Y in L 2 on the linear subspace generated by I and the X/s (cf. Bouleau (1986), Chapter 8, Section 5).

A.2.3 Computations of conditional expectations , The following proposition is crucial and is used quite often in this book. I

/'\

Appendix

178

A.3 Separation of convex sets . In this section, we state the theorem of separation of convex sets that we use in the first chapter. For more details, the diligent reader can refer to Dudley (1989) p. 152 or Minoux (1983).

References

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 '' be anon-negative real number such that the closed ball B(>') with centre at the origin and radius>' intersects C. Let Xo be the point where the map x ~ Ilxll achieves its minimum (where 11·11 is the Euclidean norm) on the compact set C n B(>'). It follows immediately that ., ' 0 such that

'. E IR to obtainxvz E V, ((z) thus '