Error analysis of finite difference scheme for American option pricing under regime-switching with jumps

Abstract

This paper mainly focuses on evaluating American options under regime-switching jump-diffusion models (Merton’s and Kou’s models). An efficient numerical method is designed for the concerned problems. The problem of American option pricing under regime-switching jump-diffusion models can be described as a free-boundary problem or a complementarity problem with integral and differential terms on an unbounded domain. By analyzing the relation of optimal exercise boundaries among several options, we truncate the solving domain of regime-switching jump-diffusion options, and present reasonable boundary conditions. For the integral terms of the truncated model, a composite trapezoidal formula is applied, which guarantees that the integral discretized matrix is a Toeplitz matrix. Meanwhile, a finite difference scheme is proposed for the resulting system, which leads to a linear complementary problem (LCP) with a unique solution. Moreover, we also prove the stability, monotonicity, and consistency of the discretization scheme and estimate the convergence order. In consideration of the characteristics of the discrete matrix, a projection and contraction method is suggested to solve the discretized LCP. Numerical experiments are carried out to verify the efficiency of the proposed scheme.