High order Semi-IMEX BDF schemes for nonlinear partial integro-differential equations arising in finance

In this paper, semi-implicit–explicit (Semi-IMEX) and semi-implicit multistep methods are proposed to solve nonlinear partial integro-differential equations (PIDEs), which describe the option pricing models with transaction costs or illiquid markets under Merton’s jump-diffusion process. After spatial differential operators are treated by using finite difference methods and the jump integral is computed by using the composite trapezoidal rule, the consistency error and global error bounds for the semi-IMEX and semi-implicit multistep methods for abstract PIDEs are provided when the nonlinear operator satisfies the boundedness and coercivity conditions. A numerical study is carried out for different option pricing models based on the convergence properties of the schemes and the comparison of the different Greek values. Numerical experiments verify our theoretical results and illustrate the intrinsic nature of the proposed option pricing models under jump-diffusion models with transaction costs or illiquid markets.