# 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

## 23 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.

### On asymptotically arbitrage-free approximations of the implied volatility

- Mathematics
- 2022

Following-up Fukasawa and Gatheral (Frontiers of Mathematical Finance, 2022), we prove that the BBF formula, the SABR formula, and the rough SABR formula provide asymptotically arbitrage-free…

### 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…

### 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…

### A partial rough path space for rough volatility

- Mathematics, Economics
- 2022

We develop a variant of rough path theory tailor-made for the analysis of a class of ﬁnancial asset price models, the so-called rough volatility models. As an application, we prove a pathwise large…

### Functional quantization of rough volatility and applications to the VIX

- Computer Science
- 2021

A product functional quantization of rough volatility is developed, built on the insightful works by Luschgy and Pages, and becomes a strong competitor in the new arena of numerical tools for rough volatility.

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A small-time Edgeworth expansion of the density of an asset price is given under a general stochastic volatility model, from which asymptotic expansions of put option prices and at-the-money implied…

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Abstract
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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 under no-arbitrage assumption, the market impact function…

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We consider a local volatility model, with volatility taking two possible values, depending on the value of the underlying with respect to a fixed threshold. When the threshold is taken at the money,…

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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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- 2020

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.