On the curvature of the smile in stochastic volatility models

In this paper we compute analytically the at-The-money second derivative of the implied volatility curve as a function of the strike price, for correlated stochastic volatility models. We also obtain an expression for the short-Time limit of this second derivative in terms of the first and second Malliavin derivatives of the volatility process and the correlation parameter. Our analysis does not need the volatility to be Markovian and can be applied to the case of fractional volatility models, both with H <1/2 and H >1/2.