Statistical inference for rough volatility: Minimax Theory

TL;DR

This paper provides a rigorous statistical analysis of rough volatility models, establishing minimax lower bounds and optimal estimation procedures for the Hurst parameter, extending previous results to all regimes.

Contribution

It introduces minimax bounds and wavelet-based procedures for optimal inference of the Hurst parameter in rough volatility models across all regimes.

Findings

Optimal convergence rate of n^{-1/(4H+2)} for estimating H

Establishment of minimax lower bounds for parameter estimation

Extension of results to all Hurst parameter regimes

Abstract

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.