A Second‐Order Variable Step‐Size IMEX Method for American Option Under Jump–Diffusion Model

ABSTRACT In this paper, we study stability and convergence of variable step‐size IMEX BDF2 method for solving numerically nonlinear partial integro‐differential equations (PIDEs) with a penalty term which describe the jump–diffusion American option pricing model in finance. To avoid the full matrix due to the discretization of nonlocal integral operators, we explicitly discretize the integral operator and implicitly discretize the rest of the operators. The monotonicity of the nonlinear operator plays key roles in showing the stability of the variable step‐size IMEX BDF2 method for abstract nonlinear PIDEs. Based on this stability result, the global error bounds for the variable step‐size IMEX BDF2 method are provided. By combining fixed‐point iteration and finite difference method for spatial discretization, the nonlinear PIDEs are effectively solved. Numerical results illustrate the effectiveness of the proposed method for American options under jump–diffusion models.