The Kolmogorov forward fractional partial differential equation for the CGMY-process with applications in option pricing

In this paper we derive the Kolmogorov forward fractional partial differential equation (FPDE) for the CGMY-process. The resulting FPDE is solved numerically with a second order method in space and Backward Differentiation Formula of order two in time. We price options by integrating the resulting probability density function multiplied by the pay-off function of the option. Hence, we only have to solve one FPDE to price several options. This is useful in practical applications where it is common to price many options simultaneously for the same underlying diffusion model. The traditional way to price options requires the solution of one FPDE per option. Since the FPDEs are of similar type and complexity, the advantage of our suggested method is obvious in the case of pricing multiple European type options on the same underlying asset when only the pay-off function differs. The numerical experiments verify that the extra cost of pricing several options compared to only one option is very small with our suggested method.