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

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

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

Supervisor:

• Prof. Dr. Matthias Ehrhardt
• PD Dr. Daniel Ševčovič, Comenius University Bratislava, Slovakia

Description

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.

Keywords

References:

• [Ank08] J. Ankudinova, The numerical solution of nonlinear Black-Scholes equations, Diploma Thesis, TU Berlin, April 2008.
• [AnEh] J. Ankudinova and M. Ehrhardt, The numerical solution of nonlinear Black-Scholes equations, Comput. Math. Appl. 56 (2008), 799-812.
• [AE] J. Ankudinova and M. Ehrhardt, Fixed domain transformations and split-step finite difference schemes for Nonlinear Black-Scholes equations for American Options, Chapter 8 in M. Ehrhardt (ed.), Nonlinear Models in Mathematical Finance: New Research Trends in Option Pricing, Nova Science Publishers, Inc., Hauppauge, NY 11788, 2008, pp. 243-273.
• [LB10] L.A. Bordag, Study of the risk-adjusted pricing methodology model with methods of Geometrical Analysis Stochastics: An International Journal of Probability and Stochastic Processes, 2010.
• [DFJ03] B. Düring, M. Fournier and A. Jüngel, High order compact finite difference schemes for a nonlinear Black-Scholes equation, Int. J. Appl. Theor. Finance 7 (2003), 767-789.
• [DFJ04] B. Düring, M. Fournier and A. Jüngel, Convergence of a high-order compact finite difference scheme for a nonlinear Black-Scholes equation, Mod. Numer. Anal. 38 (2004), 359-369.
• [Due05] B. Düring, Black-Scholes Type Equations: Mathematical Analysis, Parameter Identification & Numerical Solution, Dissertation, University Mainz, 2005.
• [Ehr97] M. Ehrhardt, Discrete Transparent Boundary Conditions for Parabolic Equations, Z. Angew. Math. Mech. 77 (1997) S2, 543-544.
• [Ehr01] M. Ehrhardt, Discrete Artificial Boundary Conditions, Ph.D. Thesis, Technische Universität Berlin, 2001.
• [EhMi08] M. Ehrhardt and R.E. Mickens, A fast, stable and accurate numerical method for the Black-Scholes equation of American options, Int. J. Theoret. Appl. Finance 11 (2008), 471-501.
• [Rig94] A. Rigal, High order difference schemes for unsteady one-dimensional diffusion-convection problems, J. Comput. Phys. 114 (1994), 59-76.
• [Sev01] D. Ševčovič, Analysis of the free boundary for the pricing of an American call option, Europ. J. Appl. Math. 12 (2001), 25-37.
• [Sev07] D. Ševčovič, An iterative algorithm for evaluating approximations to the optimal exercise boundary for a nonlinear Black-Scholes equation, Canad. Appl. Math. Quarterly 15 (2007), 77-79.
• [Sev08] D. Ševčovič, Transformation Methods for Evaluating Approximations to the Optimal Exercise Boundary for Linear and Nonlinear Black-Scholes Equations, Chapter 6 in M. Ehrhardt (ed.), Nonlinear Models in Mathematical Finance: New Research Trends in Option Pricing, Nova Science Publishers, Inc., Hauppauge, NY 11788, 2008, pp. 173-218.