Relative Pricing of Options with Stochastic Volatility

This paper offers a new approach for pricing options on assets with stochastic volatility. We start by taking as given the prices of a few simple, liquid European options. More specifically, we take as given the “surface†of Black-Scholes implied volatilities for European options with varying strike prices and maturities. We show that the Black-Scholes implied volatilities of at-the-money options converge to the underlying asset’s instantaneous (stochastic) volatility as the time to maturity goes to zero. We model the stochastic processes followed by the implied volatilities of options of all maturities and strike prices as a joint diffusion with the stock price. In order for no arbitrage opportunities to exist in trading the stock and these options, the drift of the processes followed by the implied volatilities is constrained in such a way that it is fully characterized by the volatilities of the implied volatilities. Finally, we suggest how to use the arbitrage-free joint process for the stock price and its volatility to price other derivatives, such as standard but illiquid options as well as exotic options, using numerical methods. Our approach simply requires as inputs the stock price and the implied volatilities at the time the exotic option is to be priced, as well as estimates of the volatilities of the implied volatilities.