Efficient Variable-step Algorithm And Posterior Error Estimation For European Option Pricing Under The Jump-diffusion Model

With the rapid development of the financial basic market,the use of options and other derivatives to manage risk has become an inherent demand for market development.The jumpdiffusion option pricing has become one of the hotspots in the mathematical research of finance.Its mathematical model is a partial integro-differential equation with non-local integral term.The existence of non-local integral terms brings certain difficulties to the numerical computation of the model.This article mainly studies two variable step-sizes implicit-explicit time methods for European jump-diffusion option pricing model.Divided into two parts.First,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.Next,we study a posteriori error estimates of the IMEX BDF2 scheme for time discretizations of solving parabolic partial integro-differential equations.Due to the nonsmoothness of the payoff function,a posteriori error control and adaptivity will be crucial in solving numerically this type of equations.To derive optimal order a posteriori error estimates,quadratic reconstructions for the IMEX BDF2 method are introduced.By using these continuous,piecewise time reconstructions,the upper and lower error bounds depending only on the discretization parameters and the data of the problems are derived.Based on these a posteriori error estimates,we further develop a time adaptive algorithm.The numerical implementations are performed with both nonuniform partitions and adaptivity in time.The adaptive algorithm reduce the computational cost substantially and provides efficient error control for the solution.