The Smile in Stochastic Volatility Models

We consider general stochastic volatility models with no local volatility component and derive the general expression of the volatility smile at order two in volatility-of-volatility. We show how, at this order, the smile only depends on three dimensionless numbers whose precise expressions as functionals of the model's spot/variance and variance/variance covariance functions we provide.Finally we assess the accuracy of our order two expansion using realistic levels of volatility-of-volatility.