Title: Finite differences for Financial derivative models

Abstract:

The famous Black--Scholes equation is an effective model for option pricing.

The standard approach for the scalar Black--Scholes equation for
American options results after a standard transformation in a
diffusion equation posed on an unbounded domain with a free boundary.

Usually finite differences are used to discretize the equation
and artificial boundary conditions are
introduced in order to confine the computational domain.
While the numerical treatment of the free boundary has attracted a lot
of attention and different strategies were developed
less attention was payed to the accurate treatment of the
artificial boundary.
In fact, many textbooks propose to use a homogeneous Dirichlet
boundary condition at some finite distance.

In the first part of the talk i will explain how
our new artificial boundary condition is designed on a purely discrete level 
directly for the chosen scheme.
With this strategy the stability is conserved and
numerical reflections at these boundaries do not occur.

Secondly, I will introduce Mellin transform techniques in order 
to derive an integral representation for the free boundary and the 
option price amenable for an efficient numerical evaluation 
in case of European, American and perpetual options.
This methodology is then generalized to price basket options
that amount to solving multidimensional (free) boundary problems.

Finally we discuss the adequate solution of
nonlinear Black-Scholes models for American options
using the Landau fixed domain transformation
and an iterative operator splitting algorithm.
We will consider different models 
(Leland, Barles/Soner, Frey/Stremme and Kratka/Jandacka/Sevcovic)
where the volatility may depend on the time to expiry T-t, the asset price S
and the second derivative of the option price V_SS.

This is joint work with R.E. Mickens (Atlanta), 
a supervised diploma thesis of A. Wuerfel (TU Berlin).
and the nonlinear part is a proposed bilateral DAAD-project
with D. Sevcovic (Bratislava, Slovakia), see http://www.black-scholes.de.vu