Fourier Transform Methods in Finance

by Umberto Cherubini, Giovanni Della Lunga, Sabrina Mulinacci, Pietro Rossi

Overview

In recent years, Fourier transform methods have emerged as one of the major methodologies for the evaluation of derivative contracts, largely due to the need to strike a balance between the extension of existing pricing models beyond the traditional Black-Scholes setting and a need to evaluate prices consistently with the market quotes.

Fourier Transform Methods in Finance is a practical and accessible guide to pricing financial instruments using Fourier transform. Written by an experienced team of practitioners and academics, it covers Fourier pricing methods; the dynamics of asset prices; non stationary market dynamics; arbitrage free pricing; generalized functions and the Fourier transform method.

Readers will learn how to:

compute the Hilbert transform of the pricing kernel under a Fast Fourier Transform (FFT) technique

characterise the price dynamics on a market in terms of the characteristic function, allowing for both diffusive processes and jumps

apply the concept of characteristic function to non-stationary processes, in particular in the presence of stochastic volatility and more generally time change techniques

perform a change of measure on the characteristic function in order to make the price process a martingale

recover a general representation of the pricing kernel of the economy in terms of Hilbert transform using the theory generalized functions

apply the pricing formula to the most famous pricing models, with stochastic volatility and jumps.

Junior and senior practitioners alike will benefit from this quick reference guide to state of the art models and market calibration techniques. Not only will it enable them to write an algorithm foroption pricing using the most advanced models, calibrate a pricing model on options data, and extract the implied probability distribution in market data, they will also understand the most advanced models and techniques and discover how these techniques have been adjusted for applications in finance.

Product Details

ISBN-13:	9780470684924
Publisher:	Wiley
Publication date:	01/05/2010
Series:	The Wiley Finance Series , #524
Sold by:	JOHN WILEY & SONS
Format:	eBook
Pages:	256
File size:	6 MB

About the Author

UMBERTO CHERUBINI is Associate Professor of Financial Mathematics at the University of Bologna. He is fellow of the Financial Econometrics Research Center, FERC, University of Warwick and Ente Einaudi, Bank of Italy, and member of the Scientific Committee of the Risk Management Education program of the Italian Banking Association (ABI). He has published in international journals in economics and finance, and he is co-author of the books Copula Methods in Finance, John Wiley & Sons, 2004, and Structured Finance: The Object Oriented Approach, John Wiley & Sons, 2007.

GIOVANNI DELLA LUNGA is a quantitative analyst at Prometeia Consulting. Prior to this he was head of Market Risk Methodologies at Prometeia and acted as Principal at Polyhedron Computational Finance, a Florence-based consulting company in mathematical models for financial firms and software companies. He also lectures at the University of Bologna in computational finance for undergraduates and runs courses in computational finance at the Bank of Italy. Giovanni is a member of the scientific committee of Abiformazione, the educational branch of the Italian Banking Association and manages the charge of screen-based educational program. His research background covers physics, chemistry and finance, and he co-authored Structured Finance: The Object Oriented Approach, John Wiley & Sons, 2007.

SABRINA MULINACCI is a Professor of Mathematical Methods for Economics and Finance at the University of Bologna, Italy. Prior to this Sabrina was Associate Professor of Mathematical Methods for Economics and Actuarial Sciences at the Catholic University of Milan. She has a PhD in Mathematics from the University of Pisa and has published a number of research papers in international journals in probability and mathematical finance.

PIETRO ROSSI is a Senior Financial Analyst within the Market Risk Group at Prometeira Consulting, specializing in the development of analytical tractable approximations for exotic options. Prior to this, he worked as senior scientist at ENEA in the high performance computing division and was also Director of the Parallel Computing Group at the Center for Advanced Studies, Research and Development in Sardinia (CRS4), working on high performance computing and large scale computational problems for companies such as FIAT. He has a PhD in physics from NYU and his scientific activity has been mainly in theoretical physics and computer science.

Table of 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 Lévy 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 Lévy processes 34

2.2.6 Infinite divisibility 36

2.3 Construction of Lévy markets 39

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 Lévy processes 49

2.4.1 Pathwise properties of Lévy processes 49

2.4.2 Completely monotone Lévy densities 53

2.4.3 Moments of a Lévy 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 volatility66

3.2.4 The time-change technique 67

3.3 Simulation of Levy 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 Lévy martingale processes 89

4.5.1 Construction of martingales through Lévy processes 89

4.5.2 Change of equivalent measures for Lévy processes 90

4.5.3 The Esscher transform 91

4.6 Lévy 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

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

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.l.3 The complex plane 175

B.1.4 Elementary operations 176

B.l.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 Functional 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

F The Fractional Fast Fourier Transform 215

F.1 Circular matrix 216

F.1.1 Matrix vector multiplication 218

F.2 Toepliz matrix 219

F.2.1 Embedding in a circular matrix 219

F.2.2 Applications to pricing 220

F.3 Some numerical results 221

F.3.1 The Variance Gamma model 221

F.3.2 The Heston model 223

G Affine Models: The Path Integral Approach 225

G.1 The problem 225

G.2 Solution of the Riccati equations 227

Bibliography 229

Index 233