Volatility is Rough

J. Gatheral, T. Jaisson and M. Rosenbaum

Chapter 8 in Options — 45 Years since the Publication of the Black–Scholes–Merton Model:The Gershon Fintech Center Conference, 2023, pp 127-172 from World Scientific Publishing Co. Pte. Ltd.

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 adopt the fractional stochastic volatility (FSV) model of Comte and Renault [21]. 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 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.

Keywords: Options; Call; Put; Stock; Equity; Bond; Debt; Dividend; Investment; Diversification; Volatility; Black–Scholes; Merton Model; Stochastic; Swap; Commodity; Index; Contingent Claims; Exotic Option (search for similar items in EconPapers)
JEL-codes: C C02 G1 G11 G12 G15 (search for similar items in EconPapers)
Date: 2023
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