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

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.