Statistical inference for rough volatility: Minimax theory

Abstract

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 $\mathit{H}<1/2$. 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 ${\mathit{n}}^{-1/(4\mathit{H}\mathbf{+}2)}$ as n grows to infinity, extending results that were known only for $\mathit{H}>1/2$. 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 $\sqrt{\mathit{n}}$-consistent regular statistical models.