An Alternative Stochastic Volatility Model

Abstract

Stochastic volatility modelling is of fundamental importance in financial risk management. Among the most popular existing models in the literature are the Heston and the CEV stochastic models. Each of these models has some advantages that the other one lacks. For example, the CEV model and the Heston model have different relative properties concerning the leverage as well as the smile effects. In this work we deal with the hybrid stochastic volatility model that is based on the CEV and the Heston models combined. This alternative model is expected to perform better than any of the two previously mentioned models in terms of dealing with both the leverage and the smile effects. We deal with the pricing and hedging problems for European options. We first find the set of equivalent martingale measures (E.M.M.). The market is found to be incomplete within this framework since there are infinitely many of E.M.M. We then find the targeted E.M.M. by minimizing the entropy. Using Ito calculus and risk-neutral method enable us to find the partial differential equation (P.D.E.) corresponding to the option price. Moreover, we use Clark-Ocone formula to obtain a hedging strategy that minimizes the distance between the payoff and the value of the hedged portfolio at the maturity. This hedging strategy is among the most efficient available strategies.