Modeling in Option Pricing with Memory in Assets and Stochastic Volatility of Hobson and Rogers

Abstract

This paper presents a new class of models for continuous time price process of financial assets. The dynamics of asset returns traded on spot markets are based on Jump-Telegraph-DiffusionDrift processes (JTDD-process) and also on the Stochastic Volatility of Hobson and Rogers. Thus, the main contribution of this paper is the inclusion of memory not only into the price but also in the volatility of underlying assets of European and American options. In this framework, the models are formulated as a differential system. The variational formulations, their respective finite element approximations, as well as the numerical results via simulations are presented.