European option pricing under model involving slow growth volatility with jump

Abstract

In this paper, we suggest a new model for establishing a numerical study related to a European options pricing problem where assets' prices can be described by a stochastic equation with a discontinuous sample path (Slow Growth Volatility with Jump SGVJ model) which uses a non-standard volatility. A special attention is given to characteristics of the proposed model represented by its non-standard volatility defined by the parameters α and β. The mathematical modeling in the presence of jump shows that one has to resort to a degenerate partial integro-differential equation (PIDE) which the resolution of this one gives a price of the European option as a function of time, price of the underlying asset and the instantaneous volatility. However, in general, an exact or closed solution to this problem is not available. For this reason we approximate it using a finite difference method. At the end of the paper, we present some numerical and comparison results with some classical models known in the literature.