Novel numerical techniques for the finite moment log stable computational model for European call option

Abstract

Option pricing models are often used to describe the dynamic characteristics of prices in financial markets. Unlike the classical Black-Scholes (BS) model, the finite moment log stable (FMLS) model can explain large movements of prices during small time steps. In the FMLS, the second-order spatial derivative of the BS model is replaced by a fractional operator of order which generates an-stable Lévy process. In this paper, we consider the finite difference method to approximate the FMLS model. We present two numerical schemes for this approximation: the implicit numerical scheme and the Crank-Nicolson scheme. We carry out convergence and stability analyses for the proposed schemes. Since the fractional operator routinely generates dense matrices which often require high computational cost and storage memory, we explore three methods for solving the approximation schemes: the Gaussian elimination method, the bi-conjugate gradient stabilized method (Bi-CGSTAB) and the fast Bi-CGSTAB (FBi-CGSTAB) in order to compare the cost of calculations. Finally, two numerical examples with exact solutions are presented where we also use extrapolation techniques to achieve higher-order convergence. The results suggest that the proposed schemes are unconditionally stable and convergent, and the FMLS model is useful for pricing options.

Review of the Fractional Black-Scholes Equations and Their Solution Techniques

“…Furthermore, they employed the bi-conjugate gradient stabilized method [96] in conjunction with the fast Fourier transform to effectively solve the resulting linear systems, leading to a significant reduction in storage requirements and computational cost. Other related studies on the FMLS equation were explored in [97,98].…”