E-book

# Fourier Transform Methods in Finance

ISBN: 978-0-470-68492-4
256 pages
January 2010

## Description

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.

• 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 of generalised 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 for option 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.

ISBN 978-0-470-99400-9

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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.

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## Author Information

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.

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