Volatility is rough

TL;DR

This paper shows that log-volatility behaves like a rough fractional Brownian motion with a low Hurst exponent, leading to a new Rough FSV model that better fits financial data and explains the perceived long memory in volatility.

Contribution

It introduces the Rough FSV model with H<1/2, demonstrating its consistency with data and providing a microstructure-based explanation for volatility roughness.

Findings

Rough FSV model fits financial data well

Improved volatility forecasting with RFSV

Classical tests misinterpret rough volatility as long memory

Abstract

Estimating volatility from recent high frequency data, we revisit the question of the smoothness of the volatility process. Our main result is that log-volatility behaves essentially as a fractional Brownian motion with Hurst exponent H of order 0.1, at any reasonable time scale. This leads us to adopt the fractional stochastic volatility (FSV) model of Comte and Renault. We call our model Rough FSV (RFSV) to underline that, in contrast to FSV, H<1/2. We demonstrate that our RFSV model is remarkably consistent with financial time series data; one application is that it enables us to obtain improved forecasts of realized volatility. Furthermore, we find that although volatility is not long memory in the RFSV model, classical statistical procedures aiming at detecting volatility persistence tend to conclude the presence of long memory in data generated from it. This sheds light on why long memory of volatility has been widely accepted as a stylized fact. Finally, we provide a quantitative market microstructure-based foundation for our findings, relating the roughness of volatility to high frequency trading and order splitting.