Stochastic local volatility

Local volatility models are commonly used for pricing and hedging exotic options consistently with a ‘snap-shot’ of Black-Scholes implied volatilities from traded vanilla options. However, there is substantial evidence that local volatility models fail to capture the proper dynamics of implied volatilities and that their hedging performance is poor, even for vanilla options. This is a consequence of the assumption of a deterministic spot volatility for the asset price process, which implicitly requires a static local volatility surface. In this paper, we re-visit the definition of local volatility and show that this assumption can be relaxed, while still preserving the attractive properties of local volatility models. Our approach differs from other papers on stochastic volatility since here we explicitly model the stochastic evolution of the local volatility surface rather than of the spot volatility. The resulting dynamics for implied volatility are shown to be equivalent to a ‘market model ’ for implied volatility. KEY WORDS Option pricing and hedging, stochastic local volatility, implied volatility, smile-consistent models