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Paul Wilmott Introduces Quantitative Finance Paul Wilmott Introduces Quantitative Finance Second Edition Paul Wilmott Introduces Quantitative Finance Second Edition www.wilmott.com Copyright ≤ 2007 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone (+44) 1243 779777 Email (for orders and customer service enquiries): [email protected] Visit our Home Page on www.wiley.com Copyright ≤ 2007 Paul Wilmott All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP, UK, without the permission in writing of the Publisher. Requests to the Publisher should be addressed to the Permissions Department, John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England, or emailed to [email protected], or faxed to (+44) 1243 770620. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The Publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the Publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. Other Wiley Editorial Offices John Wiley & Sons Inc., 111 River Street, Hoboken, NJ 07030, USA Jossey-Bass, 989 Market Street, San Francisco, CA 94103-1741, USA Wiley-VCH Verlag GmbH, Boschstr. 12, D-69469 Weinheim, Germany John Wiley & Sons Australia Ltd, 42 McDougall Street, Milton, Queensland 4064, Australia John Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop #02-01, Jin Xing Distripark, Singapore 129809 John Wiley & Sons Canada Ltd, 6045 Freemont Blvd, Mississauga, ONT, L5R 4J3, Canada Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Anniversary Logo Design: Richard J. Pacifico Library of Congress Cataloging-in-Publication Data Wilmott, Paul. Paul Wilmott introduces quantitative finance.—2nd ed. p. cm. ISBN 978-0-470-31958-1 1. Finance—Mathematical models. 2. Options (Finance)—Mathematical models. 3. Options (Finance)—Prices— Mathematical models. I. Title. II Title: Quantitative finance. HG173.W493 2007 332—dc22 2007015893 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 978-0-470-31958-1 (PB) Typeset in 10/12pt Helvetica by Laserwords Private Limited, Chennai, India Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at least two trees are planted for each one used for paper production. To a rising Star contents Preface xxiii 1 Products and Markets: Equities, Commodities, Exchange Rates, Forwards and Futures 1 1.1 Introduction 2 1.2 Equities 2 1.2.1 Dividends 7 1.2.2 Stock splits 8 1.3 Commodities 9 1.4 Currencies 9 1.5 Indices 11 1.6 The time value of money 11 1.7 Fixed-income securities 17 1.8 Inflation-proof bonds 17 1.9 Forwards and futures 19 1.9.1 A first example of no arbitrage 20 1.10 More about futures 22 1.10.1 Commodity futures 23 1.10.2 FX futures 23 1.10.3 Index futures 24 1.11 Summary 24 2 Derivatives 27 2.1 Introduction 28 2.2 Options 28 2.3 Definition of common terms 33 2.4 Payoff diagrams 34 2.4.1 Other representations of value 34 2.5 Writing options 39 2.6 Margin 39 2.7 Market conventions 39 2.8 The value of the option before expiry 40 2.9 Factors affecting derivative prices 41 viii contents 2.10 Speculation and gearing 42 2.11 Early exercise 44 2.12 Put-call parity 44 2.13 Binaries or digitals 47 2.14 Bull and bear spreads 48 2.15 Straddles and strangles 50 2.16 Risk reversal 52 2.17 Butterflies and condors 53 2.18 Calendar spreads 53 2.19 LEAPS and FLEX 55 2.20 Warrants 55 2.21 Convertible bonds 55 2.22 Over the counter options 56 2.23 Summary 57 3 The Binomial Model 59 3.1 Introduction 60 3.2 Equities can go down as well as up 61 3.3 The option value 63 3.4 Which part of our ‘model’ didn’t we need? 65 3.5 Why should this ‘theoretical price’ be the ‘market price’? 65 3.5.1 The role of expectations 66 1 3.6 How did I know to sell 2 of the stock for hedging? 66 3.6.1 The general formula for 67 3.7 How does this change if interest rates are non-zero? 67 3.8 Is the stock itself correctly priced? 68 3.9 Complete markets 69 3.10 The real and risk-neutral worlds 69 3.10.1 Non-zero interest rates 72 3.11 And now using symbols 73 3.11.1 Average asset change 74 3.11.2 Standard deviation of asset price change 74 3.12 An equation for the value of an option 75 3.12.1 Hedging 75 3.12.2 No arbitrage 76 3.13 Where did the probability p go? 77 3.14 Counter-intuitive? 77 3.15 The binomial tree 78 3.16 The asset price distribution 78 3.17 Valuing back down the tree 80 3.18 Programming the binomial method 85 3.19 The greeks 86 3.20 Early exercise 88 3.21 The continuous-time limit 90 3.22 Summary 90 contents ix 4 The Random Behavior of Assets 95 4.1 Introduction 96 4.2 The popular forms of ‘analysis’ 96 4.3 Why we need a model for randomness: Jensen’s inequality 97 4.4 Similarities between equities, currencies, commodities and indices 99 4.5 Examining returns 100 4.6 Timescales 105 4.6.1 The drift 107 4.6.2 The volatility 108 4.7 Estimating volatility 109 4.8 The random walk on a spreadsheet 109 4.9 The Wiener process 111 4.10 The widely accepted model for equities, currencies, commodities and indices 112 4.11 Summary 115 5 Elementary Stochastic Calculus 117 5.1 Introduction 118 5.2 A motivating example 118 5.3 The Markov property 120 5.4 The martingale property 120 5.5 Quadratic variation 120 5.6 Brownian motion 121 5.7 Stochastic integration 122 5.8 Stochastic differential equations 123 5.9 The mean square limit 124 5.10 Functions of stochastic variables and Ito’sˆ lemma 124 5.11 Interpretation of Ito’sˆ lemma 127 5.12 Itoˆ and Taylor 127 5.13 Itoˆ in higher dimensions 130 5.14 Some pertinent examples 130 5.14.1 Brownian motion with drift 131 5.14.2 The lognormal random walk 132 5.14.3 A mean-reverting random walk 134 5.14.4 And another mean-reverting random walk 135 5.15 Summary 136 6 The Black–Scholes Model 139 6.1 Introduction 140 6.2 A very special portfolio 140 6.3 Elimination of risk: delta hedging 142 6.4 No arbitrage 142 6.5 The Black–Scholes equation 143 6.6 The Black–Scholes assumptions 145 6.7 Final conditions 146 x contents 6.8 Options on dividend-paying equities 147 6.9 Currency options 147 6.10 Commodity options 148 6.11 Expectations and Black–Scholes 148 6.12 Some other ways of deriving the Black–Scholes equation 149 6.12.1 The martingale approach 149 6.12.2 The binomial model 149 6.12.3 CAPM/utility 149 6.13 No arbitrage in the binomial, Black–Scholes and ‘other’ worlds 150 6.14 Forwards and futures 151 6.14.1 Forward contracts 151 6.15 Futures contracts 152 6.15.1 When interest rates are known, forward prices and futures prices are the same 153 6.16 Options on futures 153 6.17 Summary 153 7 Partial Differential Equations 157 7.1 Introduction 158 7.2 Putting the Black–Scholes equation into historical perspective 158 7.3 The meaning of the terms in the Black–Scholes equation 159 7.4 Boundary and initial/final conditions 159 7.5 Some solution methods 160 7.5.1 Transformation to constant coefficient diffusion equation 160 7.5.2 Green’s functions 161 7.5.3 Series solution 161 7.6 Similarity reductions 163 7.7 Other analytical techniques 163 7.8 Numerical solution 164 7.9 Summary 164 8 The Black–Scholes Formulæ and the ‘Greeks’ 169 8.1 Introduction 170 8.2 Derivation of the formulæ for calls, puts and simple digitals 170 8.2.1 Formula for a call 175 8.2.2 Formula for a put 179 8.2.3 Formula for a binary call 181 8.2.4 Formula for a binary put 182 8.3 Delta 182 8.4 Gamma 184 8.5 Theta 187 8.6 Speed 187 8.7 Vega 188 8.8 Rho 190 8.9 Implied volatility 191 8.10 A classification of hedging types 194 8.10.1 Why hedge? 194 contents xi 8.10.2 The two main classifications 194 8.10.3 Delta hedging 195 8.10.4 Gamma hedging 195 8.10.5 Vega hedging 195 8.10.6 Static hedging 195 8.10.7 Margin hedging 196 8.10.8 Crash (Platinum) hedging 196 8.11 Summary 196 9 Overview of Volatility Modeling 203 9.1 Introduction 204 9.2 The different types of volatility 204 9.2.1 Actual volatility 204 9.2.2 Historical or realized volatility 204 9.2.3 Implied volatility 205 9.2.4 Forward volatility 205 9.3 Volatility estimation by statistical means 205 9.3.1 The simplest volatility estimate: constant volatility/moving window 205 9.3.2 Incorporating mean reversion 206 9.3.3 Exponentially weighted moving average 206 9.3.4 A simple GARCH model 207 9.3.5 Expected future volatility 207 9.3.6 Beyond close-close estimators: range-based estimation of volatility 209 9.4 Maximum likelihood estimation 211 9.4.1 A simple motivating example: taxi numbers 211 9.4.2 Three hats 211 9.4.3 The math behind this: find the standard deviation 213 9.4.4 Quants’ salaries 214 9.5 Skews and smiles 215 9.5.1 Sensitivity of the straddle to skews
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