Efficient operator splitting and spectral methods for the time-space fractional Black–Scholes equation

Abstract

In this paper, we aim at developing improved L1 operator splitting method and spectral method for Black–Scholes differential systems with fractional derivatives in both time and space. We address the challenge of slow convergence in time and provide efficient approaches to handle complementarity conditions in the differential system. We design an operator splitting method to decouple the constraints together with an improved L1 approximation to accommodate nonsmooth initial data. In the space dimension, a spectral collocation method based on Legendre–Gauss–Lobatto points was employed. To show the efficiency and accuracy, we benchmark our improved L1 method with the original L1 method within the operator splitting framework. For European options, both the L1 method and improved L1 method provide the convergence order of 2−γ, where γ is fractional order in time. For American options, it is necessary to apply the improved L1 method to achieve the order of 2−γ.