The Concepts and Practice of Mathematical Finance, Second Edition (Mathematics, Finance and Risk)

557 Pages·2008·6.58 MB·English

by  Mark S. Joshi

THE CONCEPTS AND PRACTICE OF
MATHEMATICAL FINANCE
Second Edition
Mathematics, Finance and Risk
Editorial Board
Mark Broadie, Graduate School of Business, Columbia University
Sam Howison, Mathematical Institute, University of Oxford
Neil Johnson, Centre for Computational Finance, University of Oxford
George Papanicolaou, Department of Mathematics, Stanford University
Modern finance in theory and practice relies absolutely on mathematical models
and analysis. It draws on and extends classical applied mathematics, stochastic
and probabilistic methods, and numerical techniques to enable models of financial
systems to be constructed, analysed and interpreted. This methodology underpins
applications to derivatives pricing for equities and fixed income products, asset-
liability modelling, volatility, risk management, credit risk, insurance analysis and
many more. This new series will consist of books that explain the processes and
techniques of the new applied mathematics, and how to use them to model finan-
cial systems and to understand the underlying phenomena and forces that drive
financial markets.
The audience for mathematical finance ranges from mathematics and probability
through econophysics to financial economics, and the series will reflect this breadth
of appeal, while maintaining a firm footing in the tradition of applied mathematics.
Books will be pedagogical in style, enabling them to be used for teaching in uni-
versities, business schools and financial institutions, and sufficiently self-contained
for stand-alone use. Mathematical techniques will be motivated by examples and
their use illustrated through applications, and complemented by computation.
THE CONCEPTS AND PRACTICE OF
MATHEMATICAL FINANCE
Second Edition
M. S. JOSHI
University of Melbourne
CAMBRIDGE
UNIVERSITY PRESS
CAMBRIDGE UNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo, Delhi
Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
www.cainbridge.org
Information on this title: www.cambridge.org/9780521514088
M. S. Joshi 2003
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without
the written permission of Cambridge University Press.
First published 2003
Reprinted 2004
Second Edition 2008
Printed in the United Kingdom at the University Press, Cambridge
A catalogue record for this publication is available from the British Library
Library of Congress Cataloguing in Publication data
Joshi, M. S. (Mark Suresh), 1969-
The concepts and practice of mathematical finance / M. S. Joshi.
p. cm. - (Mathematics, finance and risk)
Includes bibliographical references and index.
ISBN 0 521 82355 2
1. Derivative securities - Prices - Mathematical models. 2. Options (Finance) - Prices - Mathematical
models. 3. Interest rates - Mathematical models. 4. Finance - Mathematical models. 5. Investments -
Mathematics. 6. Risk management - Mathematical models. I. Title. II. Series.
HG6024.A3J67 2003
332'.01'51- dc22 2003055594
ISBN 978-0-521-51408-8 hardback
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.
To My Parents
Contents
Preface
Acknowledgements
Risk
1
What is risk?
1.1
1.2 Market efficiency
The most important assets
1.3
1.4 Risk diversification and hedging
The use of options
1.5
Classifying market participants
1.6
1.7 Key points
Further reading
1.8
1.9 Exercises
2 Pricing methodologies and arbitrage
2.1 Some possible methodologies
2.2 Delta hedging
2.3 What is arbitrage?
2.4 The assumptions of mathematical finance
2.5 An example of arbitrage-free pricing
2.6 The time value of money
2.7 Mathematically defining arbitrage
2.8 Using arbitrage to bound option prices
2.9 Conclusion
2.10 Key points
2.11 Further reading
2.12 Exercises
3 Trees and option pricing
3.1 A two-world universe
3.2 A three-state model
vii
viii Contents
3.3 Multiple time steps 50
3.4 Many time steps 53
3.5 A normal model 55
3.6 Putting interest rates in 58
3.7 A log-normal model 60
3.8 Consequences 68
3.9 Summary 70
3.10 Key points 70
3.11 Further reading 71
3.12 Exercises 71
4 Practicalities 73
4.1 Introduction 73
4.2 Trading volatility 73
4.3 Smiles 74
4.4 The Greeks 77
4.5 Alternative models 85
4.6 Transaction costs 90
4.7 Key points 90
4.8 Further reading 91
4.9 Exercises 91
5 The Ito calculus 97
5.1 Introduction 97
5.2 Brownian motion 97
5.3 Quadratic variation 100
5.4 Stochastic processes 102
5.5 Ito's lemma 106
5.6 Applying Ito's lemma 111
5.7 An informal derivation of the Black-Scholes equation 114
5.8 Justifying the derivation 116
5.9 Solving the Black-Scholes equation 119
5.10 Dividend-paying assets 121
5.11 Key points 123
5.12 Further reading 125
5.13 Exercises 125
6 Risk neutrality and martingale measures 127
6.1 Plan 127
6.2 Introduction 128
6.3 The existence of risk-neutral measures 129
6.4 The concept of information 140
6.5 Discrete martingale pricing 145
Contents ix
6.6 Continuous martingales and filtrations 154
6.7 Identifying continuous martingales 156
6.8 Continuous martingale pricing 157
6.9 Equivalence to the PDE method 161
6.10 Hedging 162
6.11 Time-dependent parameters 164
6.12 Completeness and uniqueness 166
6.13 Changing numeraire 167
6.14 Dividend-paying assets 171
6.15 Working with the forward 172
6.16 Key points 175
6.17 Further reading 176
6.18 Exercises 176
7 The practical pricing of a European option 181
7.1 Introduction 181
7.2 Analytic formulae 182
7.3 Trees 183
7.4 Numerical integration 187
7.5 Monte Carlo 191
7.6 PDE methods 195
7.7 Replication 196
7.8 Key points 198
7.9 Further reading 198
7.10 Exercises 199
8 Continuous barrier options 202
8.1 Introduction 202
8.2 The PDE pricing of continuous barrier options 205
8.3 Expectation pricing of continuous barrier options 207
8.4 The reflection principle 208
8.5 Girsanov's theorem revisited 210
8.6 Joint distribution 213
8.7 Pricing continuous barriers by expectation 216
8.8 American digital options 219
8.9 Key points 220
8.10 Further reading 220
8.11 Exercises 220
9 Multi-look exotic options 222
9.1 Introduction 222
9.2 Risk-neutral pricing for path-dependent options 223
9.3 Weak path dependence 225
x Contents
9.4 Path generation and dimensionality reduction 226
9.5 Moment matching 231
9.6 Trees, PDEs and Asian options 233
9.7 Practical issues in pricing multi-look options 234
9.8 Greeks of multi-look options 236
9.9 Key points 239
9.10 Further reading 239
9.11 Exercises 240
10 Static replication 243
10.1 Introduction 243
10.2 Continuous barrier options 244
10.3 Discrete barriers 247
10.4 Path-dependent exotic options 249
10.5 The up-and-in put with barrier at strike 251
10.6 Put-call symmetry 252
10.7 Conclusion and further reading 256
10.8 Key points 258
10.9 Exercises 259
11 Multiple sources of risk 260
11.1 Introduction 260
11.2 Higher-dimensional Brownian motions 261
11.3 The higher-dimensional Ito calculus 263
11.4 The higher-dimensional Girsanov theorem 267
11.5 Practical pricing 272
11.6 The Margrabe option 273
11.7 Quanto options 275
11.8 Higher-dimensional trees 277
11.9 Key points 280
11.10 Further reading 281
11.11 Exercises 281
12 Options with early exercise features 284
12.1 Introduction 284
12.2 The tree approach 287
12.3 The PDE approach to American options 289
12.4 American options by replication 291
12.5 American options by Monte Carlo 293
12.6 Upper bounds by Monte Carlo 295
12.7 Key points 297
12.8 Further reading 297
12.9 Exercises 298