Volatility is Rough

41 Pages Posted: 15 Oct 2014  

Jim Gatheral

CUNY Baruch College

Thibault Jaisson

Ecole Polytechnique, Paris

Mathieu Rosenbaum

Université Paris VI Pierre et Marie Curie - Laboratoire de Probabilités et Modèles Aléatoires (LPMA)

Date Written: October 13, 2014

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 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

Gatheral, Jim and Jaisson, Thibault and Rosenbaum, Mathieu, Volatility is Rough (October 13, 2014). 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

Ecole Polytechnique, Paris ( email )

1 rue Descartes
Paris, 75005
France

Mathieu Rosenbaum

Université Paris VI Pierre et Marie Curie - Laboratoire de Probabilités et Modèles Aléatoires (LPMA) ( email )

Couloir 16-26, 1er étage
4, Place Jussieu
Paris, 75005
France

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