Complete Models with Stochastic Volatility

The paper proposes an original class of models for the continuous time price process of a nancial security with non-constant volatility. The idea is to deene instantaneous volatility in terms of exponentially-weighted moments of historic log-price. The instantaneous volatility is therefore driven by the same stochastic factors as the price process, so that unlike many other models of non-constant volatility, it is not necessary to introduce additional sources of randomness. Thus the market is complete and there are unique, preference-independent options prices. We nd a partial diierential equation for the price of a European Call Option. Smiles and skews are found in the resulting plots of implied volatility. Acknowledgement. It is a pleasure to thank the referees of an earlier draft of this paper whose perceptive comments have resulted in many improvements.