# Pricing under rough volatility

@article{Bayer2015PricingUR,
title={Pricing under rough volatility},
author={Christian Bayer and Peter K. Friz and Jim Gatheral},
journal={Quantitative Finance},
year={2015},
volume={16},
pages={887 - 904}
}
• Published 28 January 2015
• Mathematics
• Quantitative Finance
From an analysis of the time series of realized variance using recent high-frequency data, Gatheral et al. [Volatility is rough, 2014] previously showed that the logarithm of realized variance behaves essentially as a fractional Brownian motion with Hurst exponent H of order 0.1, at any reasonable timescale. 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…
302 Citations

### Volatility is rough

• 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

### Weak error rates for option pricing under linear rough volatility

• Mathematics
• 2020
In quantitative finance, modeling the volatility structure of underlying assets is vital to pricing options. Rough stochastic volatility models, such as the rough Bergomi model [Bayer, Friz,

### Rough volatility: Evidence from option prices

• Economics, Mathematics
IISE Transactions
• 2018
ABSTRACT It has been recently shown that spot volatilities can be closely modeled by rough stochastic volatility-type dynamics. In such models, the log-volatility follows a fractional Brownian motion

### Is Volatility Rough

• Economics, Mathematics
• 2019
Rough volatility models are continuous time stochastic volatility models where the volatility process is driven by a fractional Brownian motion with the Hurst parameter smaller than half, and have

### Perfect hedging in rough Heston models

• Economics
The Annals of Applied Probability
• 2018
Rough volatility models are known to reproduce the behavior of historical volatility data while at the same time fitting the volatility surface remarkably well, with very few parameters. However,

### Decoupling the Short- and Long-Term Behavior of Stochastic Volatility

• Economics
Journal of Financial Econometrics
• 2021
We introduce a new class of continuous-time models of the stochastic volatility of asset prices. The models can simultaneously incorporate roughness and slowly decaying autocorrelations, including

### The characteristic function of rough Heston models

• Mathematics
• 2016
It has been recently shown that rough volatility models, where the volatility is driven by a fractional Brownian motion with small Hurst parameter, provide very relevant dynamics in order to

### Short-time at-the-money skew and rough fractional volatility

The Black–Scholes implied volatility skew at the money of SPX options is known to obey a power law with respect to the time to maturity. We construct a model of the underlying asset price process

### Weak error rates for option pricing under the rough Bergomi model

• Mathematics
ArXiv
• 2020
In quantitative finance, modeling the volatility structure of underlying assets is a key component in the pricing of options. Rough stochastic volatility models, such as the rough Bergomi model

### Impact of rough stochastic volatility models on long-term life insurance pricing

• Economics
European actuarial journal
• 2022
The Rough Fractional Stochastic Volatility (RFSV) model of Gatheral et al. (Quant Financ 18(6):933–949, 2014) is remarkably consistent with financial time series of past volatility data as well as

## References

SHOWING 1-10 OF 21 REFERENCES

### Volatility is rough

• 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

### Realizing Smiles: Options Pricing with Realized Volatility

• Economics
• 2011
We develop a discrete-time stochastic volatility option pricing model exploiting the information contained in the Realized Volatility (RV), which is used as a proxy of the unobservable log-return

### 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

### 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

### Asymptotic analysis for stochastic volatility: martingale expansion

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

### The Volatility Surface: A Practitioner's Guide

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

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

• Mathematics, Economics
Finance 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,

### 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

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