Fourier Transform Methods in Finance (PDF)

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

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

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