Duffy Finite Difference Methods in Financial Engineering

A Partial Differential Equation Approach
1. Auflage 2006
ISBN: 978-0-470-85883-7
Verlag: John Wiley & Sons
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 440 Seiten, E-Book

Reihe: Wiley Finance Series

The world of quantitative finance (QF) is one of the fastestgrowing areas of research and its practical applications toderivatives pricing problem. Since the discovery of the famousBlack-Scholes equation in the 1970's we have seen a surge in thenumber of models for a wide range of products such as plain andexotic options, interest rate derivatives, real options and manyothers. Gone are the days when it was possible to price thesederivatives analytically. For most problems we must resort to somekind of approximate method.
In this book we employ partial differential equations (PDE) todescribe a range of one-factor and multi-factor derivativesproducts such as plain European and American options, multi-assetoptions, Asian options, interest rate options and real options. PDEtechniques allow us to create a framework for modeling complex andinteresting derivatives products. Having defined the PDE problem wethen approximate it using the Finite Difference Method (FDM). Thismethod has been used for many application areas such as fluiddynamics, heat transfer, semiconductor simulation and astrophysics,to name just a few. In this book we apply the same techniques topricing real-life derivative products. We use both traditional (orwell-known) methods as well as a number of advanced schemes thatare making their way into the QF literature:
* Crank-Nicolson, exponentially fitted and higher-order schemesfor one-factor and multi-factor options
* Early exercise features and approximation using front-fixing,penalty and variational methods
* Modelling stochastic volatility models using Splittingmethods
* Critique of ADI and Crank-Nicolson schemes; when they work andwhen they don't work
* Modelling jumps using Partial Integro Differential Equations(PIDE)
* Free and moving boundary value problems in QF
Included with the book is a CD containing information on how toset up FDM algorithms, how to map these algorithms to C++ as wellas several working programs for one-factor and two-factor models.We also provide source code so that you can customize theapplications to suit your own needs.

Autoren/Hrsg.

Duffy, Daniel J.

Inhaltsverzeichnis

0 Goals of this Book and Global Overview 1
PART I THE CONTINUOUS THEORY OF PARTIAL DIFFERENTIAL EQUATIONS 5
1 An Introduction to Ordinary Differential Equations 7
2 An Introduction to Partial Differential Equations 13
3 Second-Order Parabolic Differential Equations 25
4 An Introduction to the Heat Equation in One Dimension 37
5 An Introduction to the Method of Characteristics 47
PART II FINITE DIFFERENCE METHODS: THE FUNDAMENTALS 61
6 An Introduction to the Finite Difference Method 63
7 An Introduction to the Method of Lines 79
8 General Theory of the Finite Difference Method 91
9 Finite Difference Schemes for First-Order Partial Differential Equations 103
10 FDM for the One-Dimensional Convection-Diffusion Equation 117
11 Exponentially Fitted Finite Difference Schemes 123
PART III APPLYING FDM TO ONE-FACTOR INSTRUMENT PRICING 135
12 Exact Solutions and Explicit Finite Difference Method for One-Factor Models 137
13 An Introduction to the Trinomial Method 147
14 Exponentially Fitted Difference Schemes for Barrier Options 153
15 Advanced Issues in Barrier and Lookback Option Modelling 165
16 The Meshless (Meshfree) Method in Financial Engineering 175
17 Extending the Black-Scholes Model: Jump Processes 183
PART IV FDM FOR MULTIDIMENSIONAL PROBLEMS 193
18 Finite Difference Schemes for Multidimensional Problems 195
19 An Introduction to Alternating Direction Implicit and Splitting Methods 209
20 Advanced Operator Splitting Methods: Fractional Steps 223
21 Modern Splitting Methods 229
PART V APPLYING FDM TO MULTI-FACTOR INSTRUMENT PRICING 237
22 Options with Stochastic Volatility: The Heston Model 239
23 Finite Difference Methods for Asian Options and Other 'Mixed' Problems 249
24 Multi-Asset Options 257
25 Finite Difference Methods for Fixed-Income Problems 273
PART VI FREE AND MOVING BOUNDARY VALUE PROBLEMS 285
26 Background to Free and Moving Boundary Value Problems 287
27 Numerical Methods for Free Boundary Value Problems: Front-Fixing Methods 295
28 Viscosity Solutions and Penalty Methods for American Option Problems 307
29 Variational Formulation of American Option Problems 315
PART VII DESIGN AND IMPLEMENTATION IN C++ 325
30 Finding the Appropriate Finite Difference Schemes for your Financial Engineering Problem 327
31 Design and Implementation of First-Order Problems 337
32 Moving to Black-Scholes 353
33 C++ Class Hierarchies for One-Factor and Two-Factor Payoffs 363
33.1 Introduction and objectives 363
Appendices 375
A1 An introduction to integral and partial integro-differential equations 375
A2 An introduction to the finite element method 393
Bibliography 409
Index 417

Über Autor(innen)

Daniel Duffy is a numerical analyst who has been working inthe IT business since 1979. He has been involved in the analysis,design and implementation of systems using object-oriented,component and (more recently) intelligent agent technologies tolarge industrial and financial applications. As early as 1993 hewas involved in C++ projects for risk management and optionsapplications with a large Dutch bank. His main interest is infinding robust and scalable numerical schemes that approximate thepartial differential equations that model financial derivativesproducts. He has an M.Sc. in the Finite Element Method first-orderhyperbolic systems and a Ph.D. in robust finite difference methodsfor convection-diffusion partial differential equations. Bothdegrees are from Trinity College, Dublin, Ireland.
Daniel Duffy is founder of Datasim Education and Datasim ComponentTechnology, two companies involved in training, consultancy andsoftware development.