An efficient variable step-size method for options pricing under jump-diffusion models with nonsmooth payoff function

Abstract

We develop an implicit–explicit midpoint formula with variable spatial step-sizes and variable time step to solve parabolic partial integro-differential equations with nonsmooth payoff function, which describe the jump-diffusion option pricing model in finance. With spatial differential operators being treated by using finite difference methods and the jump integral being computed by using the composite trapezoidal rule on a non-uniform space grid, the proposed method leads to linear systems with tridiagonal coefficient matrices, which can be solved efficiently. Under realistic regularity assumptions on the data, the consistency error and the global error bounds for the proposed method are obtained. The stability of this numerical method is also proved by using the Von Neumann analysis. Numerical results illustrate the effectiveness of the proposed method for European options under jump-diffusion models.