Numerical Solutions to an integro-differential parabolic problem arising in the pricing of financial options in a Levy market

In Financial Mathematics the old problem of finding the price of derivatives (options, futures, etc.) leads to the study of Partial Differential Equations. The standard type of equations obtained are of parabolic type. In recent years, the complexity of the models used has increased and in turn this lead to more and more complicated equations for the derivative prices. Of particular interest in a type of differential equations containing an integral term. These equations aptly named Partial Integro-Differential Equations (PIDE) are difficult to solve and numerical methods specially constructed for them are not easy to find. In Florescu and Mariani (2010) we study these type of problems and we prove the existence of the solution under general hypotheses about the integral term. In the present work we are extending the work on PIDE by providing a completely novel algorithm which is suggested in the proof of existence of the solution. Additionally, we present another algorithm which is a more classical finite element scheme coupled with a discretization of the integral term. In our numerical applications the two schemes are convergent to the same solution. The work is structured as follows. In Sections 1 and 2 we