DG framework for pricing European options under one-factor stochastic volatility models

Abstract

The modern theory of option pricing is based on models introduced almost 50 years ago. These models, however, are not able to capture real market behaviour sufficiently well. One line of extensions consists of introducing an additional variable into the model, the so-called stochastic volatility. Since such models lead to the (semi) closed-form solution only rarely, some form of a numerical approximation can be essential. In this paper we study a general one-factor stochastic volatility model for the pricing of European options. A standard mathematical approach to this problem leads to a degenerate partial differential equation completed by boundary and terminal conditions. We formulate this problem in a variational sense and prove the existence and the uniqueness of a weak solution. Further, a robust numerical procedure based on the discontinuous Galerkin approach is proposed to improve the numerical valuation process. The performance of the procedure is accompanied with theoretical results and documented using reference experiments with the emphasis on investigation of the behaviour of option values with respect to the different mesh sizes as well as polynomial orders of approximation.