A market model for stochastic smile: a conditional density approach

Abstract

The purpose of this paper is to introduce a new approach that allows to construct no-arbitrage market models of for implied volatility surfaces (in other words, stochastic smile models). That is to say, the idea presented here allows us to model prices of liquidly traded vanilla options as separate stochastic quantities. The main reason why market models of implied volatilities need to be constructed is that they can capture the stochastic nature of an implied volatility surface. More to the point, market models have a potential of improved pricing of forward volatility depending products, such as compound options. Besides, this framework allows to match the initial vanilla market by construction and hedge with simple call and put options in a natural way. The modelling approach presented in this paper relies on taking a deterministic smile model as a backbone around which a stochastic smile model can be constructed without violating no-arbitrage constraints