Bergische Universität Wuppertal
Fachbereich Mathematik und Naturwissenschaften
Angewandte Mathematik - Numerische Analysis


Discrete Artificial Boundary Conditions for high-order methods to solve nonlinear Black-Scholes equation

Master's Thesis in Financial Mathematics, Halmstad University, Sweden



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 thesis 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 Ševčovič [Sev07, Sev08] and transforming the free boundary problem into a fully nonlinear nonlocal parabolic equation defined on a fixed, but unbounded domain. We will present the results of a split--step finite difference scheme for various volatility models including the Leland model, the Barles and Soner model and the Risk adjusted pricing methodology model.
As an possible outlook we will construct (discrete) artificial boundary conditions for the considered nonlinear parabolic partial differential equation, following the approach of Ehrhardt et al. [Ehr97, Ehr01, EhMi08].

Let us note that very recently using Lie group analysis Bordag [LB10] found families of exact solutions to a nonlinear risk-adjusted pricing methodology model (RAPM) of the BlackScholes equation. This risk-adjusted pricing methodology model (RAPM) incorporates both transaction costs and the risk from a volatile portfolio. This solution will be used as a reference for the numerical tests.



University of Wuppertal
Faculty of Mathematics and Natural Sciences
Department of Mathematics
Applied Mathematics & Numerical Analysis Group

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