# Pricing Under Rough Volatility

@article{Bayer2015PricingUR, title={Pricing Under Rough Volatility}, author={Christian Bayer and Peter K. Friz and Jim Gatheral}, journal={ERN: Other Econometric Modeling: Derivatives (Topic)}, year={2015} }

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…

## 171 Citations

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## References

SHOWING 1-10 OF 22 REFERENCES

Volatility Is Rough

- Mathematics, Economics
- 2014

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…

Affine fractional stochastic volatility models

- Mathematics
- 2012

By fractional integration of a square root volatility process, we propose in this paper a long memory extension of the Heston (Rev Financ Stud 6:327–343, 1993) option pricing model. Long memory in…

Asymptotic analysis for stochastic volatility: martingale expansion

- MathematicsFinance Stochastics
- 2011

A general class of stochastic volatility models with jumps is considered and an asymptotic expansion for European option prices around the Black–Scholes prices is validated in the light of Yoshida’s…

On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility

- Mathematics, EconomicsFinance Stochastics
- 2007

Abstract
In this paper we use Malliavin calculus techniques to obtain an expression for the short-time behavior of the at-the-money implied volatility skew for a generalization of the Bates model,…

The Volatility Surface: A Practitioner's Guide

- Economics
- 2006

List of Figures. List of Tables. Foreword. Preface. Acknowledgments. Chapter 1: Stochastic Volatility and Local Volatility. Stochastic Volatility. Derivation of the Valuation Equation, Local…

Realized Volatility: A Review

- Mathematics
- 2006

This article reviews the exciting and rapidly expanding literature on realized volatility. After presenting a general univariate framework for estimating realized volatilities, a simple discrete time…

Rough fractional diffusions as scaling limits of nearly unstable heavy tailed Hawkes processes

- Mathematics
- 2015

We investigate the asymptotic behavior as time goes to infinity of Hawkes processes whose regression kernel has $L^1$ norm close to one and power law tail of the form $x^{-(1+\alpha)}$, with…

Smile Dynamics IV

- Computer Science
- 2009

This paper introduces a new quantity, which is called the Skew Stickiness Ratio, and shows how, at order one in the volatility of volatility, it is linked to the rate at which the at-the-money-forward skew decays with maturity.

Multivariate High-Frequency-Based Volatility (HEAVY) Models

- Economics
- 2011

This paper introduces a new class of multivariate volatility models that utilizes high-frequency data. We discuss the models dynamics and highlight their di¤erences from multivariate GARCH models. We…