A Finite Elements Strategy for Spread Contract Valuation Via Associated PIDE

We study an efficient strategy based on finite elements to value spread options on commodities whose underlying assets follow a dynamic described by a certain class of two-dimensional Levy models by solving their associated partial integro-differential equation (PIDE). To this end we consider a Galerkin approximation in space along with an implicit scheme for time evolution. Diffusion and drift in the associated operator are discretized using an exact Gaussian quadrature, while the integral part corresponding to jumps is approximated using the symbol method recently introduced in the literature. A system with blocked Toeplitz with Toeplitz blocks (BTTB) matrix is efficiently solved via biconjugate stabilized gradient method (BICSTAB) with a circulant pre-conditioner at each time step. The technique is applied to the pricing of \textit{crack} spread options between the prices of futures RBOB gasoline (reformulated blendstock for oxygenate blending) and West Texas Intermediate(WTI) oil in NYMEX.