Evaluation finite moment log-stable option pricing by a spectral method

The classical Black-Scholes pricing model is based on standard geometric Brownian motion, and the log-returns of this model are independent and Gaussian. However, most of the recent researches on the statistical properties of the log-returns make this hypothesis not always consistent. One of the ongoing issues of mathematical finance today is to design an efficient numerical algorithm for the pricing model, which might be modified from the standard Black-Scholes diffusion equation and would have favorable empirical results. Of those financial models that have been already proposed, the most interesting include the Finite Moment Log-Stable (FMLS) process model and its fractional partial integral-differential equation. In this paper, we consider to use Gauss-Jacobi spectral method on a two-dimensional computation domain in order to discretize the FMLS fractional partial integral-differential equation, and further illustrate the flexibility and accuracy of the method by comparing the first order finite difference scheme for the pricing examples of European and American-styled options. Our results suggest that the global character of the Gauss-Jacobi method makes them well-suited to fractional partial integral-differential equations and can naturally take the global behavior of the solution into account and thus do not lead to an extra computational cost when moving from a second-order to a fractional-order diffusion model.