Fourier Transform Methods in Finance

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John Wiley & Sons, Jan 5, 2010 - Business & Economics - 256 pages
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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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Contents

The Dynamics of Asset Prices
29
vi
39
ArbitrageFree Pricing
79
Generalized Functions
95
The Fourier Transform
113
Fourier Transforms at Work
129
xiii
144
Appendices
153
12
177
Complex Integration
185
Vector Spaces and Function Spaces
201
E The Fast Fourier Transform
207
14
210
F The Fractional Fast Fourier Transform
215
The Path Integral Approach
225
26
230

11
155
Contents
157
B Elements of Complex Analysis
173

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About the author (2010)

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.

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