Stochastic volatility’s orderly smiles

Lorenzo Bergomi and Julien Guyon derive an expansion of the volatility surface of general stochastic volatility models at second order in volatility of volatility that is accurate for a wide range of strikes. They characterise the shape of stochastic volatility smiles in terms of three effective quantities that compactly summarise the joint dynamics of spot and volatilities in the model

Stochastic volatility models generate an implied volatility surface as well as its associated dynamics. While Monte Carlo simulation is always an option, a fast and accurate approximation of the volatility surface is a useful implement for assessing any given model. In this article, we obtain such an approximation at second order in the volatility of volatility for a general class of stochastic