Fourier Transform Methods in Finance
Published on 2. January 2010
256 pages
978-0-470-68822-9 (ISBN)
Description
In recent years, Fourier transform methods have emerged as one ofthe major methodologies for the evaluation of derivative contracts,largely due to the need to strike a balance between the extensionof existing pricing models beyond the traditionalBlack-Scholes setting and a need to evaluate pricesconsistently with the market quotes.
Fourier Transform Methods in Finance is a practical andaccessible guide to pricing financial instruments using Fouriertransform. Written by an experienced team of practitioners andacademics, it covers Fourier pricing methods; the dynamics of assetprices; 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 aFast Fourier Transform (FFT) technique
* characterise the price dynamics on a market in terms of thecharacteristic function, allowing for both diffusive processes andjumps
* apply the concept of characteristic function to non-stationaryprocesses, in particular in the presence of stochastic volatilityand more generally time change techniques
* perform a change of measure on the characteristic function inorder to make the price process a martingale
* recover a general representation of the pricing kernel of theeconomy in terms of Hilbert transform using the theory ofgeneralised functions
* apply the pricing formula to the most famous pricing models,with stochastic volatility and jumps.
Junior and senior practitioners alike will benefit from thisquick reference guide to state of the art models and marketcalibration techniques. Not only will it enable them to write analgorithm for option pricing using the most advanced models,calibrate a pricing model on options data, and extract the impliedprobability distribution in market data, they will also understandthe most advanced models and techniques and discover how thesetechniques have been adjusted for applications in finance.
ISBN 978-0-470-99400-9
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Umberto Cherubini | Giovanni Della Lunga | Sabrina Mulinacci
Fourier Transform Methods in Finance
Book
12/2009
Wiley
€119.00
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Persons
UMBERTO CHERUBINI is Associate Professor of FinancialMathematics at the University of Bologna. He is fellow of theFinancial Econometrics Research Center, FERC, University of Warwickand Ente Einaudi, Bank of Italy, and member of the ScientificCommittee of the Risk Management Education program of the ItalianBanking Association (ABI). He has published in internationaljournals in economics and finance, and he is co-author of the booksCopula Methods in Finance, John Wiley & Sons, 2004, andStructured Finance: The Object Oriented Approach, John Wiley& Sons, 2007.
GIOVANNI DELLA LUNGA is a quantitative analyst atPrometeia Consulting. Prior to this he was head of Market RiskMethodologies at Prometeia and acted as Principal at PolyhedronComputational Finance, a Florence-based consulting company inmathematical models for financial firms and software companies. Healso lectures at the University of Bologna in computational financefor undergraduates and runs courses in computational finance at theBank of Italy. Giovanni is a member of the scientific committee ofAbiformazione, the educational branch of the Italian BankingAssociation and manages the charge of screen-based educationalprogram. His research background covers physics, chemistry andfinance, and he co-authored Structured Finance: The ObjectOriented Approach, John Wiley & Sons, 2007.
SABRINA MULINACCI is a Professor of Mathematical Methodsfor Economics and Finance at the University of Bologna, Italy.Prior to this Sabrina was Associate Professor of MathematicalMethods for Economics and Actuarial Sciences at the CatholicUniversity of Milan. She has a PhD in Mathematics from theUniversity of Pisa and has published a number of research papers ininternational journals in probability and mathematical finance.
PIETRO ROSSI is a Senior Financial Analyst within theMarket Risk Group at Prometeira Consulting, specializing in thedevelopment of analytical tractable approximations for exoticoptions. Prior to this, he worked as senior scientist at ENEA inthe high performance computing division and was also Director ofthe Parallel Computing Group at the Center for Advanced Studies,Research and Development in Sardinia (CRS4), working on highperformance computing and large scale computational problems forcompanies such as FIAT. He has a PhD in physics from NYU and hisscientific activity has been mainly in theoretical physics andcomputer science.
Content
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.
