Numerical method for pricing governing American options under fractional Black-Scholes model

Abstract

In this paper we develop a numerical approach to a fractional-order differential linear complementarity problem (LCP) arising in pricing European and American options under a geometric Lévy process. The (LCP) is first approximated by a penalized nonlinear fractional Black-Scholes (fBS) equation. To numerically solve this nonlinear (fBS), we use the horizontal method of lines to discretize the temporal variable and the spatial variable by means of Crank-Nicolson method and a cubic spline collocation method, respectively. This method exhibits a second order of convergence in space, in time and also has an acceptable speed in comparison with some existing methods. We will compare our results with some recently proposed approaches.

Keywords: Geometric Lévy process, fractional Black-Scholes, Crank-Nicolson scheme, Spline collocation, Free Boundary Value Problem.