The world of quantitative finance (QF) is one of the fastest
growing areas of research and its practical applications to
derivatives pricing problem. Since the discovery of the famous
Black-Scholes equation in the 1970's we have seen a surge in the
number of models for a wide range of products such as plain and
exotic options, interest rate derivatives, real options and many
others. Gone are the days when it was possible to price these
derivatives analytically. For most problems we must resort to some
kind of approximate method.
In this book we employ partial differential equations (PDE) to
describe a range of one-factor and multi-factor derivatives
products such as plain European and American options, multi-asset
options, Asian options, interest rate options and real options. PDE
techniques allow us to create a framework for modeling complex and
interesting derivatives products. Having defined the PDE problem we
then approximate it using the Finite Difference Method (FDM). This
method has been used for many application areas such as fluid
dynamics, heat transfer, semiconductor simulation and astrophysics,
to name just a few. In this book we apply the same techniques to
pricing real-life derivative products. We use both traditional (or
well-known) methods as well as a number of advanced schemes that
are making their way into the QF literature:
* Crank-Nicolson, exponentially fitted and higher-order schemes
for one-factor and multi-factor options
* Early exercise features and approximation using front-fixing,
penalty and variational methods
* Modelling stochastic volatility models using Splitting
methods
* Critique of ADI and Crank-Nicolson schemes; when they work and
when 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 to
set up FDM algorithms, how to map these algorithms to C++ as well
as several working programs for one-factor and two-factor models.
We also provide source code so that you can customize the
applications to suit your own needs.
