Numerical Method for American Option Pricing under the Time‐Fractional Black–Scholes Model

Abstract

The fractional Black–Scholes model has had limited applications in financial markets. Instead, the time‐fractional Black–Scholes equation has attracted much research interest. However, it is difficult to obtain the analytic expression for American option pricing under the time‐fractional Black–Scholes model. This paper will present an operator‐splitting method to price the American options under the time‐fractional Black–Scholes model. The fractional partial differential complementarity problem (FPDCP) that the American option price satisfied is split into two subproblems: a linear boundary value problem and an algebraic system. A high‐order compact (HOC) scheme and a grid stretching (GS) method are considered for the linear boundary problem. Furthermore, numerical results show that the HOC scheme with a GS method gives an accurate numerical solution for American options under the time‐fractional Black–Scholes model.