ABSTRACT
In this chapter, we study the at-the-money curvature of the implied volatility, that is, its second derivative as a function of the strike price. Moreover, we explicitly compute its short-end limit in terms of the Malliavin derivatives of the volatility process and the correlation parameter. As a particular example, we study this limit for classical local and diffusion volatility models as well as for fractional volatilities. In particular, our results allow us to derive a condition for the at-the-money local convexity of the implied volatility, in terms of the correlation parameter and the Malliavin derivatives of the volatility process. As particular examples, we see that short-end curvature is positive for the CEV and the rough Bergomi models, as well for the Heston and the SABR under some conditions on the correlation. Moreover, we see that, in the case of rough volatilities, the ATM short-time curvature blows-up at a higher rate than the ATM skew, which implies that, for short maturities, the smile effect is more relevant than the skew effect, according to real market data
