An Approximation Scheme for Black-Scholes Equations with Delays * Mou-Hsiung Chang † Tao Pang ‡ Moustapha Pemy § August 9, 2009 Abstract This paper addresses a finite difference approximation for an infinite dimensional Black-Scholes equation obtained in Chang and Youree [5]. The equation arises from a consideration of an European option pricing problem in a market in which stock prices and the riskless asset prices have hereditary structures. Under a general condition on the payoff function of the option, it is shown that the pricing function is the unique viscosity solution of the infinite dimensional Black-Scholes equation. In addition, a finite difference approximation of the viscosity solution is provided and the convergence results are proved. Keywords: Finite difference, Black-Scholes equation, stochastic functional differential equations, viscosity solutions. AMS 2000 subject classifications: primary 90A09, 60H30; secondary 60G44, 90A16. * The research of this paper is partially supported by a grant W911NF-04-D-0003 from the U. S. Army Research Office † Mathematics Division, U. S. Army Research Office, P. O. Box 12211, RTP, NC 27709, USA, mouhsi- ung.chang@us.army.mil ‡ Department of Mathematics, North Carolina State University, Raleigh, NC 27695 USA, tpang@ncsu.edu § Department of Mathematics, Towson University, Towson, MD 21252 USA, mpemy@towson.edu 1