COMPLETE MODELS WITH STOCHASTIC VOLATILITY

The paper proposes an original class of models for the continuous-time price process of a financial security with nonconstant volatility. Th e idea is to define 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 proces s, so that, unlike many Other models of nonconstant volatility, it is not necessary to introduce additional sources of randomness. Thus the market is complete and there are unique, preference-independent option s prices. We find a partial differential equation for the price of a E uropean call option. Smiles and skews are found in the resulting plots of implied volatility.