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Mathematical Problems in Engineering
Volume 2016, Article ID 5614950, 8 pages
http://dx.doi.org/10.1155/2016/5614950
Research Article

Numerical Methods for Pricing American Options with Time-Fractional PDE Models

1School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, Wenjiang 611130, China
2School of Economic Mathematics and School of Finance, Southwestern University of Finance and Economics, Chengdu, Wenjiang 611130, China

Received 26 September 2015; Accepted 22 December 2015

Academic Editor: George Tsiatas

Copyright © 2016 Zhiqiang Zhou and Xuemei Gao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper we develop a Laplace transform method and a finite difference method for solving American option pricing problem when the change of the option price with time is considered as a fractal transmission system. In this scenario, the option price is governed by a time-fractional partial differential equation (PDE) with free boundary. The Laplace transform method is applied to the time-fractional PDE. It then leads to a nonlinear equation for the free boundary (i.e., optimal early exercise boundary) function in Laplace space. After numerically finding the solution of the nonlinear equation, the Laplace inversion is used to transform the approximate early exercise boundary into the time space. Finally the approximate price of the American option is obtained. A boundary-searching finite difference method is also proposed to solve the free-boundary time-fractional PDEs for pricing the American options. Numerical examples are carried out to compare the Laplace approach with the finite difference method and it is confirmed that the former approach is much faster than the latter one.