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 we 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, we will discuss a new integral technique 
to price basket options (multi asset options)
by using Mellin transform techniques. 
This approach results in 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 strategy is then generalized to price basket options.

Finally in the third part we will discuss the numerical solution
of nonlinear Black-Scholes models for European and American options.
Nonlinear Black--Scholes equations provide more accurate values by taking into
account more realistic assumptions, such as transaction costs, 
illiquid markets, risks from an unprotected portfolio or large investor's preferences, 
which may have an impact on the stock price, the volatility, 
the drift and the option price itself.

We will consider different models from the most relevant class of 
nonlinear Black--Scholes equations for American options with a volatility 
depending on different factors,  such as the stock price, 
the time, the option price and its derivatives.
We will analytically approach the option price by following the idea of Sevcovic 
and transforming the free boundary problem into a fully nonlinear nonlocal 
parabolic equation defined on a fixed domain. 

Finally, we will present the results of different numerical discretization schemes 
for various volatility models including the Leland model, 
the Barles and Soner model and the Risk adjusted pricing methodology model.

This is joint work with R.E. Mickens (Atlanta), D. Sevcovic (Bratislava,
in the scope of the german-slovakian 'FinDiffFin' project)
and two supervised diploma thesis of A. Wuerfel (TU Berlin)
and J. Ankudinova (TU Berlin).