Interior and exterior cooperative penalty method: a new solution strategy for fractional partial differential complementarity problems

This paper is aimed to combine the interior penalty method with the exterior penalty method to propose a new solution strategy for solving the fractional partial differential complementarity problems governing American option under a geometric Lévy process. The fractional partial differential complementarity problem is first reformulated as a fractional partial differential variational inequality problem by choosing the appropriate representation of the fractional derivative. A penalty equation is constructed by introducing a new penalty function term to approximate the variational inequality problem. The existence and convergence of the solution for the penalized problem are established. A finite difference scheme to numerically solving the resulted penalty equation is also designed. Finally, the new solution strategy is applied to pricing a call American option and a put American option respectively. Comparisons with the single interior penalty method and the exterior penalty method, the usefulness and effectiveness for this method are numerically illustrated.