Weak error estimates for rough volatility models

We consider a class of stochastic processes with rough stochastic volatility, examples of which include the rough Bergomi and rough Stein-Stein model, that have gained considerable importance in quantitative finance. A basic question for such (non-Markovian) models concerns efficient numerical schemes. While strong rates are well understood (order $H$), we tackle here the intricate question of weak rates. Our main result asserts that the weak rate, for a reasonably large class of test function, is essentially of order $\min \{3H+\frac{1}{2},1\}$ where $H\in (0,1\mathrm{/}2]$ is the Hurst parameter of the fractional Brownian motion that underlies the rough volatility process. Interestingly, the phase transition at $H=1\mathrm{/}6$ is related to the correlation between the two driving factors, and thus gives additional meaning to a quantity already of central importance in stochastic volatility modelling.Our results are complemented by a lower bound which show that the obtained weak rate is indeed optimal.