Finite Difference Methods in Financial Engineering: A Partial Differential Equation Approach

by Daniel J. Duffy

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.

440 pages, ebook

First published March 30, 2006