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Volatility is Rough

Jim Gatheral, Thibault Jaisson, Mathieu Rosenbaum

2014
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Social Science Research Network
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Volatility is rough 3 Some elements about volatility modeling Building the Rough FSV model The structure of volatility in the RFSV model Application of the RFSV model : Volatility prediction Microstructural foundations for the RFSV model Main classes of volatility models Prices are often modeled as continuous semi-martingales of the form dP t = P t (µ t dt + σ t dW t ). The volatility process σ s is the most important ingredient of the model. Practitioners consider essentially three classes of
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... y three classes of volatility models : Deterministic volatility (Black and Scholes 1973), Local volatility (Dupire 1994), Stochastic volatility (Hull and White 1987, Heston 1993, Hagan et al. 2002,...). In term of regularity, in these models, the volatility is either very smooth or with a smoothness similar to that of a Brownian motion. J. Gatheral, T. Jaisson, M. Rosenbaum Volatility is rough 4 Some elements about volatility modeling Building the Rough FSV model The structure of volatility in the RFSV model Application of the RFSV model : Volatility prediction Microstructural foundations for the RFSV model Long memory in volatility Definition A stationary process is said to be long memory if its autocovariance function decays slowly, more precisely : +∞ t=1 Cov[σ t+x , σ x ] = +∞. Power law long memory for the volatility : Cov[σ t+x , σ x ] ∼ C /t γ , with γ < 1, is considered a stylized fact and has been notably reported in Ding and Granger 1993 (using extra day data) and Andersen et al., 2001 and 2003 (using intra day data). J. Gatheral, T. Jaisson, M. Rosenbaum Volatility is rough 5 J. Gatheral, T. Jaisson, M. Rosenbaum Volatility is rough 7 Some elements about volatility modeling Building the Rough FSV model The structure of volatility in the RFSV model Application of the RFSV model : Volatility prediction Microstructural foundations for the RFSV model Long memory volatility models Some models have been built using fractional Brownian motion with Hurst parameter H > 1/2 to reproduce the long memory property of the volatility : Comte and Renault 1998 (FSV model) : d log(σ t ) = νdW H t + α(m − log(σ t ))dt. Comte, Coutin and Renault 2012, where they define a kind of fractional CIR process. J. Gatheral, T. Jaisson, M. Rosenbaum Volatility is rough 8 J. Gatheral, T. Jaisson, M. Rosenbaum Volatility is rough 9

doi:10.2139/ssrn.2509457
fatcat:c42w5j3rmrfg7fxwlwc7u7lksu