Option Pricing with Stochastic Volatility: A Closed-Form Solution Using the Fourier Transform

Abstract

The Black and Scholes (1973) option pricing model was developed starting from the hypothesis of constant volatility. However, many empirical studies, have argued that the mentioned hypothesis is subject to debate. A few authors, among who - Stein and Stein (1991), Heston (1993), Bates (1996) and Bakshi et al.(1997, 2000) - suggested the use of the Fourier transform for the density of the underlying return or for the risk-neutral probabilities, in order to evaluate the fair price of an option. In this paper we propose a stochastic valuation model using the Fourier transform for option price. This model can be used for the valuation of European options, characterized by two state variables: the price of the underlying asset and its volatility. We model the stochastic processes described by the two variables and we obtain a partial derivatives equation of which the solution is the price of the derivative. We propose a solution to this partial derivatives equation using the Fourier transform. When we apply the Fourier transform, we demonstrate that a second order partial derivatives equation is solved as an ordinary differential equation. We consider a correlation between the underlying asset price and its volatility and two sources of risk: return and volatility. The first part of the paper describes the hypotheses of the model. After describing the Fourier transforms, we propose a formula for the valuation of European options with stochastic volatility. In the second part, we present a few empirical results on the pricing of CAC 40 index call options.