Pricing Call Options under Stochastic Volatilities

Abstract

This paper derives a closed-form solution for the European call option price when the volatility of the underlying stock returns is governed by a diffusion process. The model uses the continuity property of a diffusion process and the martingale approach to valuation of assets under no arbitrage. The pricing formula differs from the Black-Scholes formula in that it needs a volatility adjustment. The volatility movement is allowed to be contemporaneously correlated with the stock price movement.