Local volatility under rough volatility

Several asymptotic results for the implied volatility generated by a rough volatility model have been obtained in recent years (notably in the small-maturity regime), providing a better understanding of the shapes of the volatility surface induced by rough volatility models, supporting their calibration power to SP500 option data. Rough volatility models also generate a local volatility surface, via the so-called Markovian projection of the stochastic volatility. We complement the existing results on implied volatility by studying the asymptotic behavior of the local volatility surface generated by a class of rough stochastic volatility models, encompassing the rough Bergomi model. Notably, we observe that the celebrated "1/2 skew rule" linking the short-term at-the-money skew of the implied volatility to the short-term at-the-money skew of the local volatility, a consequence of the celebrated "harmonic mean formula" of [Berestycki et al. (2002). Quantitative Finance, 2, 61-69], is replaced by a new rule: the ratio of the at-the-money implied and local volatility skews tends to the constant 1/(H+3/2)$1/(H + 3/2)$ (as opposed to the constant 1/2), where H is the regularity index of the underlying instantaneous volatility process.