Volatility Is Rough

Quantitative Finance, Vol. 18, No. 6, 933-949, 2018.

50 Pages Posted: 15 Oct 2014 Last revised: 25 May 2018

See all articles by Jim Gatheral

Jim Gatheral

CUNY Baruch College

Thibault Jaisson

Pictet Asset Management

Mathieu Rosenbaum

Ecole Polytechnique, Palaiseau

Date Written: September 30, 2017


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.

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

Gatheral, Jim and Jaisson, Thibault and Rosenbaum, Mathieu, Volatility Is Rough (September 30, 2017). Quantitative Finance, Vol. 18, No. 6, 933-949, 2018., Available at SSRN: https://ssrn.com/abstract=2509457 or http://dx.doi.org/10.2139/ssrn.2509457

Jim Gatheral (Contact Author)

CUNY Baruch College ( email )

Department of Mathematics
One Bernard Baruch Way
New York, NY 10010
United States

Thibault Jaisson

Pictet Asset Management ( email )


Mathieu Rosenbaum

Ecole Polytechnique, Palaiseau ( email )

Route de Saclay
Palaiseau, 91128

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