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

This paper considers the pricing of a European option using a ( ð µ , 𠑆 ) -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 ð ¶ [ − â„Ž , 0 ] for the Black-Scholes equation. It is also shown that the ð ‘› th approximant resembles the classical Black-Scholes equation in finite dimensions.