A Course in Financial Calculus

A Course in

Financial Calculus

Alison Etheridge University of Oxford

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521890779 © Cambridge University Press 2002 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2002 eBook (EBL) ISBN-13 978-0-511-33725-3 ISBN-10 0-511-33725-6 eBook (EBL) ISBN-13 ISBN-10 paperback 978-0-521-89077-9 paperback 0-521-89077-2

Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Contents

Preface 1 Single period models Summary 1.1 Some deﬁnitions from ﬁnance 1.2 Pricing a forward 1.3 The one-step binary model 1.4 A ternary model 1.5 A characterisation of no arbitrage 1.6 The risk-neutral probability measure Exercises Binomial trees and discrete parameter martingales Summary 2.1 The multiperiod binary model 2.2 American options 2.3 Discrete parameter martingales and Markov processes 2.4 Some important martingale theorems 2.5 The Binomial Representation Theorem 2.6 Overture to continuous models Exercises Brownian motion Summary 3.1 Deﬁnition of the process 3.2 L´ vy’s construction of Brownian motion e 3.3 The reﬂection principle and scaling 3.4 Martingales in continuous time Exercises Stochastic calculus Summary

page vii 1 1 1 4 6 8 9 13 18 21 21 21 26 28 38 43 45 47 51 51 51 56 59 63 67 71 71 v

2

3

4

vi

contents

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8

Stock prices are not differentiable Stochastic integration Itˆ ’s formula o Integration by parts and a stochastic Fubini Theorem The Girsanov Theorem The Brownian Martingale Representation Theorem Why geometric Brownian motion? The Feynman–Kac representation Exercises

72 74 85 93 96 100 102 102 107 112 112 112 118 122 126 131 132 134 139 139 139 141 144 149 150 154 159 159 160 163 175 181 185 189 191 193

5

The Black–Scholes model Summary 5.1 The basic Black–Scholes model 5.2 Black–Scholes price and hedge for European options 5.3 Foreign exchange 5.4 Dividends 5.5 Bonds 5.6 Market price of risk Exercises Different payoffs Summary 6.1 European options with discontinuous payoffs 6.2 Multistage options 6.3 Lookbacks and barriers 6.4 Asian options 6.5 American options Exercises Bigger models Summary 7.1 General stock model 7.2 Multiple stock models 7.3 Asset prices with jumps 7.4 Model error Exercises Bibliography Notation Index

6

7

Preface

Financial mathematics provides a striking example of successful collaboration between academia and industry. Advanced mathematical techniques, developed in both universities and banks, have transformed the derivatives business into a multi-trillion-dollar market. This has led to demand for highly trained students and with that demand comes a need for textbooks. This volume provides a ﬁrst course in ﬁnancial mathematics. The inﬂuence of Financial Calculus by Martin Baxter and Andrew Rennie will be obvious. I am extremely grateful to Martin and Andrew for their guidance and for allowing me to use some of the material from their book. The structure of the text largely follows Financial Calculus, but the mathematics, especially the discussion of stochastic calculus, has been expanded to a level appropriate to a university mathematics course and the text is supplemented by a large number of...