Fixed domain transformations and split-step finite difference schemes for Nonlinear Black-Scholes equations for American Options

Author(s) : Julia Ankudinova , Matthias Ehrhardt

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 7-2008

MSC 2000

35A35 Theoretical approximation to solutions

65N99 None of the above, but in this section

91B26 Market models

Abstract :
Due to transaction costs, illiquid markets, large investors or risks from an unprotected portfolio the assumptions in the classical Black-Scholes model become unrealistic and the model results in strongly or fully nonlinear, possibly degenerate, parabolic diffusion-convection equations, where the stock price, volatility, trend and option price may depend on the time, the stock price or the option price itself.
In this chapter we will be concerned with several 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 ideas proposed by �evčovič and transforming the free boundary problem into a fully nonlinear nonlocal parabolic equation defined on a fixed, but unbounded domain. Finally, we will present the results of a split-step finite difference schemes for various volatility models including the Leland model, the Barles and Soner model and the Risk adjusted pricing methodology model.

Keywords : nonlinear Black-Scholes models, fixed domain transformation, split-step methods, American options