Volatility is rough

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 timescale. This leads us to adopt the fractional stochastic volatility (FSV) model of Comte and Renault [Long memory in continuous-time stochastic volatility models. Math. Finance, 1998, 8(4), 291–323]. We call our model Rough FSV (RFSV) to underline that, in contrast to FSV, . 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 a long memory process 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.