Many smile consistent volatility models are scale invariant, including jump diffusions, standard stochastic volatility models, mixture models and sticky delta local volatility models. Sticky tree local volatility models and the SABR model are not scale invariant. The short-comings of scale invariant models motivates the specification of a general parametric stochastic local volatility model which we show is equivalent to the market model of implied volatilities introduced by Schönbucher (1999).
When volatility is scale invariant the price sensitivities are model free, the only differences between the models being their quality of fit to the market. In stochastic volatility models where price-volatility correlation is non-zero we show how this model free price sensitivity is adjusted to obtain the correct delta. Similar adjustments to obtain the delta for sticky tree and stochastic local volatility models are derived. Our theoretical and empirical results illustrate the inferior hedging performance of mixture models and sticky delta local volatility in equity index markets, even compared with the Black-Scholes model. The best hedging results are obtained with stochastic (local) volatility models.
The last part of the talk introduces the GARCH Jump model as the continuous limit of normal mixture GARCH, a discrete time model that provides the most flexible and intuitive view of skew dynamics and the closest fit to historical data in both equity and FX markets. This is a stochastic local volatility model, but not one with parameter diffusions. The parameters simply jump (occasionally, and simultaneously) between two states.
This highlights the fact that the hedging failure of mixture models can be attributed to the fixed parameters that are commonly applied. By introducing parameter uncertainty the GARCH Jump model provides a tractable, flexible and intuitive tool for capturing regime specific mean-reversion and leverage mechanisms and a skew term structure that persists into long maturities. However, its hedging performance has yet to be studied.