Fourier Transform Methods in Finance (The Wiley Finance Series)

258 Pages·2010·2.12 MB·English

by  Umberto Cherubini

Fourier Transform Methods in Finance
For other titles in the Wiley Finance series
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Fourier Transform Methods in Finance
Umberto Cherubini
Giovanni Della Lunga
Sabrina Mulinacci
Pietro Rossi
A John Wiley and Sons, Ltd., Publication
This edition first published 2010
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Library of Congress Cataloging-in-Publication Data
Fourier transform methods in finance / Umberto Cherubini ... [et al.].
p. cm.
Includes bibliographical references and index.
ISBN 978-0-470-99400-9 (cloth)
1. Options (Finance)–Mathematical models. 2. Securities–Prices–Mathematical models.
3. Finance–Mathematical models. 4. Fourier analysis. I. Cherubini, Umberto.
HG6024.A3F684 2010
332.63(cid:2) 2042–dc22
2009043688
A catalogue record for this book is available from the British Library.
ISBN 978-0-470-99400-9
Typeset in 10/12pt Times by Aptara Inc., New Delhi, India
Printed in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire
Contents
Preface xi
List of Symbols xiii
1 Fourier Pricing Methods 1
1.1 Introduction 1
1.2 A general representation of option prices 1
1.3 The dynamics of asset prices 3
1.4 A generalized function approach to Fourier pricing 6
1.4.1 Digital payoffs and the Dirac delta function 7
1.4.2 The Fourier transform of digital payoffs 8
1.4.3 The cash-or-nothing option 9
1.4.4 The asset-or-nothing option 10
1.4.5 European options: the general pricing formula 11
1.5 Hilbert transform 12
1.6 Pricing via FFT 14
1.6.1 The sampling theorem 15
1.6.2 The truncated sampling theorem 17
1.6.3 Why bother? 21
1.6.4 The pricing formula 21
1.6.5 Application of the FFT 23
1.7 Related literature 26
2 The Dynamics of Asset Prices 29
2.1 Introduction 29
2.2 Efficient markets and Le´ vy processes 30
2.2.1 Random walks and Brownian motions 30
2.2.2 Geometric Brownian motion 31
2.2.3 Stable processes 31
2.2.4 Characteristic functions 32
2.2.5 Le´ vy processes 34
2.2.6 Infinite divisibility 36
2.3 Construction of Le´ vy markets 39
vi Contents
2.3.1 The compound Poisson process 39
2.3.2 The Poisson point process 41
2.3.3 Sums over Poisson point processes 42
2.3.4 The decomposition theorem 45
2.4 Properties of Le´vy processes 49
2.4.1 Pathwise properties of Le´ vy processes 49
2.4.2 Completely monotone Le´ vy densities 53
2.4.3 Moments of a Le´vy process 54
3 Non-stationary Market Dynamics 57
3.1 Non-stationary processes 57
3.1.1 Self-similar processes 57
3.1.2 Self-decomposable distributions 58
3.1.3 Additive processes 60
3.1.4 Sato processes 63
3.2 Time changes 63
3.2.1 Stochastic clocks 64
3.2.2 Subordinators 64
3.2.3 Stochastic volatility 66
3.2.4 The time-change technique 67
3.3 Simulation of Le´ vy processes 73
3.3.1 Simulation via embedded random walks 74
3.3.2 Simulation via truncated Poisson point processes 74
4 Arbitrage-Free Pricing 79
4.1 Introduction 79
4.2 Equilibrium and arbitrage 79
4.3 Arbitrage-free pricing 80
4.3.1 Arbitrage pricing theory 80
4.3.2 Martingale pricing theory 81
4.3.3 Radon–Nikodym derivative 82
4.4 Derivatives 83
4.4.1 The replicating portfolio 83
4.4.2 Options and pricing kernels 84
4.4.3 Plain vanilla options and digital options 86
4.4.4 The Black–Scholes model 88
4.5 Le´ vy martingale processes 89
4.5.1 Construction of martingales through Le´vy processes 89
4.5.2 Change of equivalent measures for Le´ vy processes 90
4.5.3 The Esscher transform 91
4.6 Le´vy markets 92
5 Generalized Functions 95
5.1 Introduction 95
5.2 The vector space of test functions 95
5.3 Distributions 97
5.3.1 Dirac delta and other singular distributions 98
Contents vii
5.4 The calculus of distributions 99
5.4.1 Distribution derivative 100
5.4.2 Special examples of distributions 100
5.5 Slow growth distributions 103
5.6 Function convolution 104
5.6.1 Definitions 104
5.6.2 Some properties of convolution 104
5.7 Distributional convolution 105
5.7.1 The direct product distributions 105
5.7.2 The convolution of distributions 106
5.8 The convolution of distributions in S 108
6 The Fourier Transform 113
6.1 Introduction 113
6.2 The Fourier transformation of functions 113
6.2.1 Fourier series 113
6.2.2 Fourier transform 117
6.2.3 Parseval theorem 120
6.3 Fourier transform and option pricing 120
6.3.1 The Carr–Madan approach 120
6.3.2 The Lewis approach 122
6.4 Fourier transform for generalized functions 123
6.4.1 The Fourier transforms of testing functions of rapid descent 123
6.4.2 The Fourier transforms of distributions of slow growth 124
6.5 Exercises 125
6.6 Fourier option pricing with generalized functions 127
7 Fourier Transforms at Work 129
7.1 Introduction 129
7.2 The Black–Scholes model 130
7.3 Finite activity models 132
7.3.1 Discrete jumps 132
7.3.2 The Merton model 133
7.4 Infinite activity models 134
7.4.1 The Variance Gamma model 135
7.4.2 The CGMY model 137
7.5 Stochastic volatility 138
7.5.1 The Heston model 141
7.5.2 Vanilla options in the Heston model 142
7.6 FFT at Work 146
7.6.1 Market calibration 147
7.6.2 Pricing exotics 147
Appendices 153
A Elements of Probability 155
A.1 Elements of measure theory 155
viii Contents
A.1.1 Integration 157
A.1.2 Lebesgue integral 158
A.1.3 The characteristic function 160
A.1.4 Relevant probability distributions 161
A.1.5 Convergence of sequences of random variables 167
A.1.6 The Radon–Nikodym derivative 167
A.1.7 Conditional expectation 168
A.2 Elements of the theory of stochastic processes 169
A.2.1 Stochastic processes 169
A.2.2 Martingales 170
B Elements of Complex Analysis 173
B.1 Complex numbers 173
B.1.1 Why complex numbers? 173
B.1.2 Imaginary numbers 174
B.1.3 The complex plane 175
B.1.4 Elementary operations 176
B.1.5 Polar form 177
B.2 Functions of complex variables 179
B.2.1 Definitions 179
B.2.2 Analytic functions 179
B.2.3 Cauchy–Riemann conditions 180
B.2.4 Multi-valued functions 181
C Complex Integration 185
C.1 Definitions 185
C.2 The Cauchy–Goursat theorem 186
C.3 Consequences of Cauchy’s theorem 187
C.4 Principal value 190
C.5 Laurent series 193
C.6 Complex residue 196
C.7 Residue theorem 197
C.8 Jordan’s Lemma 199
D Vector Spaces and Function Spaces 201
D.1 Definitions 201
D.2 Inner product space 203
D.3 Topological vector spaces 205
D.4 Functionals and dual space 205
D.4.1 Algebraic dual space 206
D.4.2 Continuous dual space 206
E The Fast Fourier Transform 207
E.1 Discrete Fourier transform 207
E.2 Fast Fourier transform 208