On the local convexity of the implied volatility curve in uncorrelated stochastic volatility

In this paper we give an alternative proof of the convexity of the implied volatility curve as a function of the strike, for stochastic volatility models in the uncorrelated case. Our method is based on the computation of the corresponding first and second derivatives, and on Malliavin calculus techniques. We prove that the implied volatility is a locally convex function of the strike, with a minimum at the forward price of the stock, recovering the previous results by Renault and Touzi (1996). Moreover, we obtain an expression for the short-time limit of the smile in terms of the Malliavin derivative of the volatility process. Our analysis only needs some general integrability and regularity conditions in the Malliavin calculus sense and does not need the volatility to be Markovian nor a diffusion process, as we can see in the examples.