41 Pages Posted: 15 Oct 2014
Date Written: October 13, 2014
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 model the log-volatility as a fractional Brownian motion with H<1/2; specifically we 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.
Keywords: High frequency data, volatility smoothness, fractional Brownian motion, fractional Ornstein-Uhlenbeck, long memory, volatility persistence, volatility forecasting, option pricing, volatility surface, Hawkes processes, high frequency trading, order splitting.
JEL Classification: C4, C5, C6
Suggested Citation: Suggested Citation