September 21, 2010 17:16 Quantitative Finance methodsQFstyle Quantitative Finance, Vol. 00, No. 00, Sep 2010, 1–15 Numerical Solutions to an integro-differential parabolic problem arising in the pricing of financial options in a Levy market Ionut ¸ Florescu†, Maria Cristina Mariani * ‡ and Granville Sewell‡ †4 Dept. of Mathematical Sciences, Stevens Institute of Technology ‡ Department of Mathematical Sciences, The University of Texas at El Paso (September 21, 2010) We study the numerical solutions for an integro-differential parabolic problem modeling a process with jumps and stochastic volatility in Financial Mathematics. We present two general algorithms to calculate numerical solutions. The algorithms are implemented in PDE2D,a general purpose, partial differential equation solver. Keywords: integro-differential equation, numerical methods, option valuation 1. Introduction 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 ob- tained 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 par- ticular 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 hy- potheses 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 solu- tion. 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 * Corresponding author. Email: mcmariani@utep.edu Quantitative Finance ISSN 1469-7688 print/ISSN 1469-7696 online c ⃝ 2010 Taylor & Francis http://www.tandf.co.uk/journals DOI: 10.1080/1469768YYxxxxxxxx