Journal of Applied Mathematics and Stochastic Analysis
Volume 2009 (2009), Article ID 782572, 37 pages
doi:10.1155/2009/782572
Research Article

Spectral Approximation of Infinite-Dimensional Black-Scholes Equations with Memory

Mou-Hsiung Chang1 and Roger K. Youree2

1Mathematical Sciences Division, U. S. Army Research Office, P.O. Box 12211, RTP, NC 27709, USA
2Instrumental Sciences Inc., P.O. Box 4711, Huntsville, AL 35811, USA

Received 2 September 2009; Accepted 2 December 2009

Academic Editor: Kambiz Farahmand

Copyright © 2009 Mou-Hsiung Chang and Roger K. Youree. 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

This paper considers the pricing of a European option using a (B,S)-market in which the stock price and the asset in the riskless bank account both have hereditary price structures described by the authors of this paper (1999). Under the smoothness assumption of the payoff function, it is shown that the infinite dimensional Black-Scholes equation possesses a unique classical solution. A spectral approximation scheme is developed using the Fourier series expansion in the space C[−h,0] for the Black-Scholes equation. It is also shown that the nth approximant resembles the classical Black-Scholes equation in finite dimensions.