Umberto Cherubini & Giovanni Della Lunga 
Fourier Transform Methods in Finance [EPUB ebook] 

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 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

Tabla de materias

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.

Sobre el autor

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 Ph D 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 Ph D in physics from NYU and his
scientific activity has been mainly in theoretical physics and
computer science.