Algorithms of Finite Difference for Pricing American Options under Fractional Diffusion Models

Abstract

It is well known that linear complementarity problem (LCP) involving partial integro differential equation (PIDE) arises from pricing American options under Lévy Models. In the case of infinite activity process, the integral part of the PIDE has a singularity, which is generally approximated by a small Brownian component plus a compound Poisson process, in the neighborhood of origin. The PIDE can be reformulated as a fractional partial differential equation (FPDE) under fractional diffusion models, including FMLS (finite moment log stable), CGMY (Carr-Madan-Geman-Yor), and KoBol (Koponen-Boyarchenko-Levendorskii). In this paper, we first present a stable iterative algorithm, which is based on the fractional difference approach and penalty method, to avoid the singularity problem and obtain numerical approximations of first-order accuracy. Then, on the basis of the first-order accurate algorithm, spatial extrapolation is employed to obtain second-order accurate numerical estimates. Numerical tests are performed to demonstrate the effectiveness of the algorithm and the extrapolation method. We believe that this can be used as necessary tools by the engineers in research.