Pricing Under Rough Volatility

Quantitative Finance, Vol. 16, No. 6, 887-904, 2016.

42 Pages Posted: 25 Jan 2015 Last revised: 13 Jun 2016

See all articles by Christian Bayer

Christian Bayer

Weierstras Institute for Applied Analysis and Stochastics (WIAS)

Peter Friz

Technische Universität Berlin (TU Berlin)

Jim Gatheral

CUNY Baruch College

Date Written: September 18, 2015

Abstract

From an analysis of the time series of volatility using recent high frequency data, Gatheral, Jaisson and Rosenbaum previously showed that log-volatility behaves essentially as a fractional Brownian motion with Hurst exponent H of order 0.1, at any reasonable time scale. The resulting Rough Fractional Stochastic Volatility (RFSV) model is remarkably consistent with financial time series data. We now show how the RFSV model can be used to price claims on both the underlying and integrated volatility. We analyze in detail a simple case of this model, the rBergomi model. In particular, we find that the rBergomi model fits the SPX volatility markedly better than conventional Markovian stochastic volatility models, and with fewer parameters. Finally, we show that actual SPX variance swap curves seem to be consistent with model forecasts, with particular dramatic examples from the weekend of the collapse of Lehman Brothers and the Flash Crash.

Keywords: Fractional Brownian motion, volatility forecasting, option pricing, volatility surface, VIX, VVIX, variance swap curve, Flash Crash, Lehman Brothers collapse, Bergomi-Guyon expansion

JEL Classification: C2, C3, C4, C6

Suggested Citation

Bayer, Christian and Friz, Peter and Gatheral, Jim, Pricing Under Rough Volatility (September 18, 2015). Quantitative Finance, Vol. 16, No. 6, 887-904, 2016.. Available at SSRN: https://ssrn.com/abstract=2554754 or http://dx.doi.org/10.2139/ssrn.2554754

Christian Bayer

Weierstras Institute for Applied Analysis and Stochastics (WIAS) ( email )

Mohrenstr. 39
Berlin, 10117
Germany

Peter Friz

Technische Universität Berlin (TU Berlin) ( email )

Straße des 17
Juni 135
Berlin, 10623
Germany

Jim Gatheral (Contact Author)

CUNY Baruch College ( email )

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

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