A positivity-preserving numerical scheme for option pricing model with transaction costs under jump-diffusion process

Option pricing model and numerical method with transaction costs under jump-diffusion process of Merton is studied in this paper. Partial integro-differential equation satisfied by the option value is derived by delta-hedge method, which is a nonlinear Black-Scholes equation with an infinite integral, and it is difficult to obtain the analytic solution. Based on a nonstandard approximation of the second partial derivative, a double discretization strategy is introduced for the unbounded part of the spatial domain and a positivity-preserving numerical scheme is developed. The scheme is not only unconditionally stable and positive, but also allows us to solve the discrete equation explicitly, and after modifying it becomes consistent. The numerical results for European call option and digital call option are compared to the standard finite difference scheme. It turns out that the proposed scheme is efficient and reliable.