Statistical inference for rough volatility: Minimax theory
本文首次严格分析了从资产价格历史观测中估计粗糙波动率模型Hurst指数H的统计问题，建立了极小极大下界并设计了基于小波的自适应最优估计程序，证明了最优收敛速度。
In recent years, rough volatility models have gained considerable attention in quantitative finance. In this paradigm, the stochastic volatility of the price of an asset has quantitative properties similar to that of a fractional Brownian motion with small Hurst index H<1/2. In this work, we provide the first rigorous statistical analysis of the problem of estimating H from historical observations of the underlying asset. We establish minimax lower bounds and design optimal procedures based on adaptive estimation of quadratic functionals based on wavelets. We prove in particular that the optimal rate of convergence for estimating H based on price observations at n time points is of order n−1/(4H+2) as n grows to infinity, extending results that were known only for H>1/2. Our study positively answers the question whether H can be inferred, although it is the regularity of a latent process (the volatility); in rough models, when H is close to 0, we even obtain an accuracy comparable to usual n-consistent regular statistical models.