Estimating the Hurst parameter from the zero vanna implied volatility and its dual

TL;DR

This paper establishes a theoretical link between implied volatility measures and the Hurst parameter in stochastic volatility models, providing a method to estimate the Hurst parameter from market data with numerical validation.

Contribution

It proves that the approximation error for covariance tends to zero in small time-to-maturity and relates the covariance to the slope of implied volatility, enabling Hurst parameter estimation.

Findings

Approximation error vanishes as time-to-maturity approaches zero.

A direct relation between covariance and implied volatility slope is established.

Numerical validation under the rough Bergomi model confirms the method's effectiveness.

Abstract

The covariance between the return of an asset and its realized volatility can be approximated as the difference between two specific implied volatilities. In this paper it is proved that in the small time-to-maturity limit the approximation error tends to zero. In addition a direct relation between the short time-to-maturity covariance and slope of the at-the-money implied volatility is established. The limit theorems are valid for stochastic volatility models with Hurst parameter $H \in(0, 1)$. An application of the results is to accurately approximate the Hurst parameter using only a discrete set of implied volatilities. Numerical examples under the rough Bergomi model are presented.