Short-time near-the-money skew in rough fractional volatility models

We consider rough stochastic volatility models where the driving noise of volatility has fractional scaling, in the rough regime of Hurst parameter $H < 1/2$. This regime recently attracted a lot of attention both from the statistical and option pricing point of view. With focus on the latter, we sharpen the large deviation results of Forde-Zhang (2017) in a way that allows us to zoom-in around the money while maintaining full analytical tractability. More precisely, this amounts to proving higher order moderate deviation estimates, only recently introduced in the option pricing context. This in turn allows us to push the applicability range of known at-the-money skew approximation formulae from CLT type log-moneyness deviations of order $t^{1/2}$ (recent works of Al`{o}s, Le{o}n & Vives and Fukasawa) to the wider moderate deviations regime.