# Volatility has to be rough

@article{Fukasawa2020VolatilityHT, title={Volatility has to be rough}, author={Masaaki Fukasawa}, journal={Quantitative Finance}, year={2020}, volume={21}, pages={1 - 8} }

Under power-law blow-up of the short ATM skew, volatility must be rough in a viable market for the underlying asset

## 26 Citations

### Short Communication: On the Weak Convergence Rate in the Discretization of Rough Volatility Models

- Computer Science, MathematicsSIAM J. Financial Math.
- 2022

The weak convergence rate in the discretization of rough volatility models is studied to give a sharper bound H + 1/2 under a linear model.

### Statistical Inference for Rough Volatility: Central Limit Theorems

- MathematicsSSRN Electronic Journal
- 2022

In recent years, there has been substantive empirical evidence that stochastic volatility is rough. In other words, the local behavior of stochastic volatility is much more irregular than…

### THE EAK R ATE IN THE D ISCRETIZATION OF R OUGH V OLATILITY

- Mathematics
- 2022

A bstract . We study the weak convergence rate in the discretization of rough volatility models. After showing a lower bound 2 H under a general model, where H is the Hurst index of the volatility…

### Short-time implied volatility of additive normal tempered stable processes

- EconomicsAnnals of Operations Research
- 2022

Empirical studies have emphasized that the equity implied volatility is characterized by a negative skew inversely proportional to the square root of the time-to-maturity. We examine the…

### Log-Modulated Rough Stochastic Volatility Models

- MathematicsSIAM J. Financial Math.
- 2021

We propose a new class of rough stochastic volatility models obtained by modulating the power-law kernel defining the fractional Brownian motion (fBm) by a logarithmic term, such that the kernel…

### Short-Time Expansion of Characteristic Functions in a Rough Volatility Setting With Applications

- MathematicsSSRN Electronic Journal
- 2022

We derive a higher-order asymptotic expansion of the conditional characteristic function of the increment of an Itô semimartingale over a shrinking time interval. The spot characteristics of the Itô…

### Statistical inference for rough volatility: Minimax Theory

- Computer Science
- 2022

This work establishes that the parameters of rough volatility models can be inferred with optimal accuracy in all regimes, and obtains an optimal speed of convergence of n−1/(4H+2) for estimating H based on n sampled data.

### Short-dated smile under rough volatility: asymptotics and numerics

- MathematicsQuantitative Finance
- 2021

In Friz et al. [Precise asymptotics for robust stochastic volatility models. Ann. Appl. Probab, 2021, 31(2), 896–940], we introduce a new methodology to analyze large classes of (classical and rough)…

### Local volatility under rough volatility

- Economics
- 2022

Several asymptotic results for the implied volatility generated by a rough volatility model have been obtained in recent years (notably in the small-maturity regime), providing a better understanding…

### Short-time asymptotics for non self-similar stochastic volatility models

- Mathematics
- 2022

We provide a short-time large deviation principle (LDP) for stochastic volatility models, where the volatility is expressed as a function of a Volterra process. This LDP holds under suitable…

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We study specific nonlinear transformations of the Black-Scholes implied volatility to show remarkable properties of the volatility surface. Model-free bounds on the implied volatility skew are…

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A rigorous analysis is presented for the case when the stationary stochastic volatility model is constructed in terms of a fractional Ornstein Uhlenbeck process to have such correlations.

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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…

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Abstract
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We document a surprising pattern in market prices of S&P 500 index options. When implied volatilities are graphed against a standard measure of moneyness, the implied volatility smirk does not…

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In a previous article we highlighted how traditional stochastic volatility and Jump/Lévy models impose structural constraints on how the short forward skew, the spot/vol correlation, and the term…

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Market impact is the link between the volume of a (large) order and the price move during and after the execution of this order. We show that in a quite general framework, under no‐arbitrage…

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This decomposition of option prices is used to develop first- and second-order approximation formulas for option prices and implied volatilities in the Heston volatility framework, as well as to study their accuracy for short maturities.

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- EconomicsThe 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,…

### Pricing under rough volatility

- Mathematics
- 2015

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…