  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.
