Estimating the roughness exponent of stochastic volatility from discrete observations of the integrated variance

TL;DR

This paper introduces a simple, efficient, pathwise estimator for the roughness exponent of stochastic volatility, applicable to discrete data, with proven convergence and robustness in rough volatility models.

Contribution

A new pathwise estimator for the roughness exponent of volatility processes, with strong consistency and robustness, applicable directly to observed price trajectories.

Findings

Estimator converges under broad pathwise conditions.

It performs well in numerical simulations.

Robust to proxy errors between integrated and realized variance.

Abstract

We consider the problem of estimating the roughness of the volatility process in a stochastic volatility model that arises as a nonlinear function of fractional Brownian motion with drift. To this end, we introduce a new estimator that measures the so-called roughness exponent of a continuous trajectory, based on discrete observations of its antiderivative. The estimator has a very simple form and can be computed with great efficiency on large data sets. It is not derived from distributional assumptions but from strictly pathwise considerations. We provide conditions on the underlying trajectory under which our estimator converges in a strictly pathwise sense. Then we verify that these conditions are satisfied by almost every sample path of fractional Brownian motion (with drift). As a consequence, we obtain strong consistency theorems in the context of a large class of rough volatility models, such as the rough fractional volatility model and the rough Bergomi model. We also demonstrate that our estimator is robust with respect to proxy errors between the integrated and realized variance, and that it can be applied to estimate the roughness exponent directly from the price trajectory. Numerical simulations show that our estimation procedure performs well after passing to a scale-invariant modification of our estimator.