Rough volatility models have gained considerable interest in the quantitative finance community in recent years. In this paradigm, the volatility of the asset price is driven by a fractional Brownian motion with a small value for the Hurst parameter $H$. In this work, we provide a rigorous statistical analysis of these models. To do so, we establish minimax lower bounds for parameter estimation and design procedures based on wavelets attaining them. We notably obtain an optimal speed of convergence of $n^{-1/(4H+2)}$ for estimating $H$ based on n sampled data, extending results known only for the easier case $H>1/2$ so far. We therefore establish that the parameters of rough volatility models can be inferred with optimal accuracy in all regimes.

翻译：近年来，粗糙波动率模型在量化金融领域引起了广泛关注。在该范式下，资产价格的波动率由Hurst参数$H$取值较小的分数布朗运动驱动。本文对此类模型进行了严谨的统计分析。为此，我们建立了参数估计的极小极大下界，并设计了基于小波的方法以达到该下界。我们特别证明了基于$n$个采样数据估计$H$的最优收敛速度为$n^{-1/(4H+2)}$，将此前仅适用于$H>1/2$这一较简单情形的结果进行了扩展。因此，我们确立了粗糙波动率模型的所有参数可在各情形下以最优精度进行推断。