A high-order compact method for nonlinear Black-Scholes option pricing equations with transaction costs


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


Supervisor

Description

Over the last several years there were numerous discussions about the Black-Scholes model and its appliance to the nonlinear case. In the linear model there exists a lot of restrictions such as frictionless, liquid and complete market. But all these assumptions are not fulfilled in reality. One of the most important issues that has a great influence on the option pricing strategy are transaction costs. Several authors proposed relaxing hedging conditions dealing with transaction costs, e.g. Leland, Boyle & Vorst, Barles & Soner or the Risk Adjusted Pricing Methodology (RAPM) proposed by Kratka in [17] and improved by Jandacka and Sevcovic in [14]. Although the Leland model has been criticized by Kabanov, it has played a significant role in financial mathematics.

In this work we will consider compact schemes for European and American options and work closer with the transaction cost model of Barles and Soner. Instead of solving the singular differential equation of Barles and Soner we propose to use some fixpoint properties of the Barles/Soner volatlity function psi described recently in [6]. As compact schemes cannot be directly applied to American type options and multi-dimensional problems, we will try to employ them using a fixed domain transformation explained in [2]. We will also consider the method of Liao and Khaliq [19] for solving nonlinear Black-Scholes equation with transaction costs.

References:

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  2. 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, pp. 243-273.
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mailbox ehrhardt@math.uni-wuppertal.de