Fourier Transform Methods in Finance

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

Overview

Fourier transform methods in finance

“Fourier Transform Methods in Finance is rigorous, instructive, and loaded with useful examples. The authors have synthesized everything from the necessary underlying elements of complex analysis up through methods for derivative pricing. Almost anyone doing research or business applications in this area will want their own copy.”

—Professor Darrell Duffie, Dean Witter Distinguished Professor of Finance at The Graduate School of Business, Stanford University

“This excellent book is a selection of ready-to-use Fourier transform methods applied to option pricing and calibration, a technique that is indispensable today across different asset classes in a non Gaussian world.”

—Hélyette Geman, Professor of Finance, Birkbeck College, University of London & ESCP Europe

Product Details

ISBN-13:	9780470684924
Publisher:	Wiley
Publication date:	01/05/2010
Series:	The Wiley Finance Series , #524
Sold by:	Barnes & Noble
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.

List of Symbols.

1 Fourier Pricing Methods.

1.1 Introduction.

1.2 A general representation of option prices.

1.3 The dynamics of asset prices.

1.4 A generalized function approach to Fourier pricing.

1.5 Hilbert transform.

1.6 Pricing via FFT.

1.7 Related literature.

2 The Dynamics of Asset Prices.

2.1 Introduction.

2.2 Efficient markets and Lévy processes.

2.3 Construction of Lévy markets.

2.4 Properties of Lévy processes.

3 Non-stationary Market Dynamics.

3.1 Non-stationary processes.

3.2 Time changes.

3.3 Simulation of Lévy processes.

4 Arbitrage-Free Pricing.

4.1 Introduction.

4.2 Equilibrium and arbitrage.

4.3 Arbitrage-free pricing.

4.4 Derivatives.

4.5 Lévy martingale processes.

4.6 Lévy markets.

5 Generalized Functions.

5.1 Introduction.

5.2 The vector space of test functions.

5.3 Distributions.

5.4 The calculus of distributions.

5.5 Slow growth distributions.

5.6 Function convolution.

5.7 Distributional convolution.

5.8 The convolution of distributions in S.

6 The Fourier Transform.

6.1 Introduction.

6.2 The Fourier transformation of functions.

6.3 Fourier transform and option pricing.

6.4 Fourier transform for generalized functions.

6.5 Exercises.

6.6 Fourier option pricing with generalized functions.

7 Fourier Transforms at Work.

7.1 Introduction.

7.2 The Black–Scholes model.

7.3 Finite activity models.

7.4 Infinite activity models.

7.5 Stochastic volatility.

7.6 FFT at work.

Appendices.

A Elements of probability.

A.1 Elements of measure theory.

A.2 Elements of the theory of stochastic processes.

B Elements of Complex Analysis.

B.1 Complex numbers.

B.2 Functions of complex variables.

C Complex Integration.

C.1 Definitions.

C.2 The Cauchy–Goursat theorem.

C.3 Consequences of Cauchy's theorem.

C.4 Principal value.

C.5 Laurent series.

C.6 Complex residue.

C.7 Residue theorem.

C.8 Jordan's Lemma.

D Vector Spaces and Function Spaces.

D.1 Definitions.

D.2 Inner product space.

D.3 Topological vector spaces.

D.4 Functionals and dual space.

E The Fast Fourier Transform.

E.1 Discrete Fourier transform.

E.2 Fast Fourier transform.

F The Fractional Fourier Transform.

F.1 Circular matrix.

F.2 Toepliz matrix.

F.3 Some numerical results.

G Affine Models: The Path Integral Approach.

G.1 The problem.

G.2 Solution of the Riccati equations.

Bibliogrsphy.

Index.