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Finite Element and Discontinuous Galerkin Methods with Perfect Matched Layers for American Options

Part of: Mathematical finance

Published online by Cambridge University Press:  12 September 2017

Haiming Song ,
Kai Zhang  and
Yutian Li
Haiming Song
Affiliation:
Department of Mathematics, Jilin University, Changchun 130012, China
Kai Zhang*
Affiliation:
Department of Mathematics, Jilin University, Changchun 130012, China
Yutian Li*
Affiliation:
Department of Mathematics, and Institute of Computational and Theoretical Studies, Hong Kong Baptist University, Kowloon, Hong Kong
*
*Corresponding author. Email addresses:[email protected] (K. Zhang), [email protected] (Y. T. Li)
*Corresponding author. Email addresses:[email protected] (K. Zhang), [email protected] (Y. T. Li)
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Abstract

This paper is devoted to the American option pricing problem governed by the Black-Scholes equation. The existence of an optimal exercise policy makes the problem a free boundary value problem of a parabolic equation on an unbounded domain. The optimal exercise boundary satisfies a nonlinear Volterra integral equation and is solved by a high-order collocation method based on graded meshes. This free boundary is then deformed to a fixed boundary by the front-fixing transformation. The boundary condition at infinity (due to the fact that the underlying asset's price could be arbitrarily large in theory), is treated by the perfectly matched layer technique. Finally, the resulting initial-boundary value problems for the option price and some of the Greeks on a bounded rectangular space-time domain are solved by a finite element method. In particular, for Delta, one of the Greeks, we propose a discontinuous Galerkin method to treat the discontinuity in its initial condition. Convergence results for these two methods are analyzed and several numerical simulations are provided to verify these theoretical results.

Keywords

American optionGreeksoptimal exercise boundaryperfectly matched layerfinite element methoddiscontinuous Galerkin method

MSC classification

Secondary: 91G60: Numerical methods (including Monte Carlo methods) 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems)
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 10 , Issue 4 , November 2017 , pp. 829 - 851
Copyright
Copyright © Global-Science Press 2017 

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