Fourier Transform Methods in Finance

by Umberto Cherubini (University of Bologna); Giovanni Della Lunga (Prometeia); Sabrina Mulinacci (University of Bologna); Pietro Rossi (Prometeira Consulting)

• ISBN13: 9780470688229
• ISBN10: 047068822X
• Format: eBook
• Copyright: 2010-01-01
• Publisher: Wiley

Summary

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.Contents 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 evaluationChapter 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 securitiesChapter 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.

Table of Contents

Preface

List of Symbols

Fourier Pricing Methods

Introduction

A general representation of option prices

The dynamics of asset prices

A generalized function approach to Fourier pricing

Hilbert transform

Pricing via FFT

Related literature

The Dynamics of Asset Prices

Introduction

Efficient markets and Lévy processes

Construction of Lévy markets

Properties of Lévy processes

Non-stationary Market Dynamics

Non-stationary processes

Time changes

Simulation of Lévy processes

Arbitrage-Free Pricing

Introduction

Equilibrium and arbitrage

Arbitrage-free pricing

Derivatives

Lévy martingale processes

Lévy markets

Generalized Functions

Introduction

The vector space of test functions

Distributions

The calculus of distributions

Slow growth distributions

Function convolution

Distributional convolution

The convolution of distributions in S

The Fourier Transform

Introduction

The Fourier transformation of functions

Fourier transform and option pricing

Fourier transform for generalized functions

Exercises

Fourier option pricing with generalized functions

Fourier Transforms at Work

Introduction

The Black-Scholes model

Finite activity models

Infinite activity models

Stochastic volatility

FFT at work

Appendices

Elements of probability

Elements of measure theory

Elements of the theory of stochastic processes

Elements of Complex Analysis

Complex numbers

Functions of complex variables

Complex Integration

Definitions

The Cauchy-Goursat theorem

Consequences of Cauchy's theorem

Principal value

Laurent series

Complex residue

Residue theorem

Jordan's Lemma

Vector Spaces and Function Spaces

Definitions

Inner product space

Topological vector spaces

Functionals and dual space

The Fast Fourier Transform

Discrete Fourier transform

Fast Fourier transform

The Fractional Fourier Transform

Circular matrix

Toepliz matrix

Some numerical results

Affine Models: The Path Integral Approach

The problem

Solution of the Riccati equations

Bibliogrsphy

Index

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