Numerical Valuation of European and American Options under Fractional Black-Scholes Model

Abstract

In this paper, we investigate the numerical valuation of European and American options under the time fractional Black-Scholes model. We first apply a coordinate stretching transformation to the asset price so that the spatial region can focus on the vicinity of singularities, which are usually found in the payoff function. The radial basis function finite difference method is used for the spatial discretization, and the improved L1 method is used to deal with the reduced order of convergence for the nonsmooth initial data. We use the operator splitting method for solving the linear complementary problem of American options. The proposed scheme leads to a sparse linear system which is trivial to solve. Moreover, the stability of the proposed numerical scheme is analyzed using Fourier analysis. Numerical experiments demonstrate the accuracy and efficiency of the proposed method.