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Fourier Transform Methods in Finance
BY Umberto Cherubini, Giovanni Della Lunga, Sabrina Mulinacci, Pietro Rossi
This is the first book written on the application of Fourier transform to finance. Written by an academic and practitioner team, it is an accessible and practical guide to the subject providing an introduction to the mathematics and applications of Fourier transform. Starting with a thorough introduction to the topic, it then looks at recent developments within statistics and finance (jumps, time changes etc) and the problems these have raised in derivatives pricing and risk management. The book then moves onto the technical concepts and application of Fourier transform. It shows practitioners how to apply the tool to price financial instruments such as derivatives, (plain vanilla, barrier options, exotic options etc), to price credit risk applications and manage risk.
Chapter 1. From diffusive processes to L¨¦vy processes.
A review of the most recent developments of the statistics of financial markets, the inclusion of jumps, time changes and the like, and the problems raised for pricing and risk evaluation
Chapter 2. Calculus on complex domain : a primer
The reader is introduced to the main technical concepts from complex analysis that will be used throughout the book. Particularly, the reader will be introduced to the concept of path integral in the complex domain.
Chapter 3. Option pricing by Fourier transforms
The use of Fourier transform in asset pricing is introduced for the simplest products, namely digital products. This enables the reader to learn the basic principles of integration in the complex domain, using Heavyside step functions and Dirac delta function, and applying that directly to solve the basic pricing problem of a Arrow-Debreu securities
Chapter 4. Pricing plain vanilla options
Fourier transform methods are applied to the price of plain vanilla options. The reader is introduced to the problem first in the standard Black and Scholes model, and then in the Heston model, allowing for stochastic volatility. Some points that are left unexplained in the literature are also tackled.
Chapter 5. Pricing barrier options
The reader is introduced to frontier issues in the use of Fourier transforms for pricing barrier options. As in the previous chapter, the analysis is first performed in the standard model and then in a more general one. The chapter introduces the reader to Wiener-Hopf factorization techniques.
Chapter 6. Pricing exotic options
Several applications are proposed for standard classes of univariate exotic options, such as Asian options, lookback, and the like.
Chapter 7. Credit risk pricing applications
Factor copula models and the generating function technique is introduced, and the Fourier transform approach used in the market is investigated in detail.
Chapter 8. Multivariate pricing application
The chapter introduces frontiers issues in the topic. The analysis touches upon new results on multivariate analysis for Levy processes and introduces original research and results on the subject, that the group will be developing next year.
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.