Volume 10, Issue 4
Finite Element and Discontinuous Galerkin Methods with Perfect Matched Layers for American Options

Numer. Math. Theor. Meth. Appl., 10 (2017), pp. 829-851.

Published online: 2017-11

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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 option, Greeks, optimal exercise boundary, perfectly matched layer, finite element method, discontinuous Galerkin method.

91G60, 91G80, 65M6

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@Article{NMTMA-10-829, author = {}, title = {Finite Element and Discontinuous Galerkin Methods with Perfect Matched Layers for American Options}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2017}, volume = {10}, number = {4}, pages = {829--851}, 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.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2017.0020}, url = {http://global-sci.org/intro/article_detail/nmtma/10458.html} }
TY - JOUR T1 - Finite Element and Discontinuous Galerkin Methods with Perfect Matched Layers for American Options JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 829 EP - 851 PY - 2017 DA - 2017/11 SN - 10 DO - http://doi.org/10.4208/nmtma.2017.0020 UR - https://global-sci.org/intro/article_detail/nmtma/10458.html KW - American option, Greeks, optimal exercise boundary, perfectly matched layer, finite element method, discontinuous Galerkin method. AB -

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.

Haiming Song, Kai Zhang & Yutian Li. (2020). Finite Element and Discontinuous Galerkin Methods with Perfect Matched Layers for American Options. Numerical Mathematics: Theory, Methods and Applications. 10 (4). 829-851. doi:10.4208/nmtma.2017.0020
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