We present the discontinuous Galerkin method applied to valuation of European options assuming that the underlying follows a CGMY process. This special case of an infinite activity Lévy process has purely discontinuous paths with finite and/or infinite variation with respect to the density of Lévy measure. The corresponding CGMY model was proposed as an extension of geometric Brownian motion to overcome some of the limitations of the Black-Scholes approach. The evolution of the option prices under this model can be expressed in the form of a partial integro-differential equation, which involves both integrals and derivatives of an unknown option value function. With a localization to a bounded spatial domain, the pricing equation is discretized by the discontinuous Galerkin method over a finite element mesh and it is integrated in temporal variable by a semi-implicit Euler scheme. The special attention is paid to the proper discretization of jump components. The whole procedure is accompanied with preliminary practical results compared to reference values.

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