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A. Jüngel:
"Some mathematical results for nonlinear Black-Scholes-type equations for financial derivatives";
Keynote Lecture: International Workshop on Kinetic Theory and Socio-economical Equilibria Modelling, Orleans; 2007-03-15.

Standard financial derivatives like European options are priced
by the famous Black-Scholes model which has the form of a
linear parabolic equation. The Black-Scholes equation is derived
under quite restrictive assumptions, e.g., no transaction costs
occur and the market is complete.
Without these conditions the resulting models may become
nonlinear due to feedback effects, for instance.

In this talk some nonlinear Black-Scholes-type equations are
discussed. The first model including the effect of transaction
costs is a Black-Scholes equation with a volatility depending
on the second derivatives of the solution. It has been
derived by Barles and Soner in 1998. The equation is discretized
using a higher order compact finite difference scheme and some
numerical convergence results are given. The solutions are compared
to those from the standard Black-Scholes model.

The second model describes the optimal value function in incomplete
markets and has been derived by Leitner in 2001. It gives
information on the transaction of shares the investor should make
in order to maximise her or his profit. The market is allowed
to possess non-tradable state variables like the employee income,
weather parameter etc. Mathematically, the model is a parabolic
equation with quadratic gradients. Existence and uniqueness results
and numerical simulations are presented in order to show the
influence of the nontradable state variables.

Siehe englisches Abstract.

Black-Scholes equation