# Rough volatility: evidence from option prices

## TL;DR

This paper confirms that implied volatility from short-term S&P 500 options supports the rough volatility model, with a Hurst parameter around 0.3, indicating roughness in market volatility dynamics.

## Contribution

It provides empirical evidence from option prices supporting the rough volatility model, complementing previous findings based on historical data.

## Key findings

- Implied volatility approximations confirm rough volatility.
- Hurst parameter estimated around 0.3 from options data.
- Smoothing effect explains slight difference from historical estimates.

## Abstract

It has been recently shown that spot volatilities can be very well modeled by rough stochastic volatility type dynamics. In such models, the log-volatility follows a fractional Brownian motion with Hurst parameter smaller than 1/2. This result has been established using high frequency volatility estimations from historical price data. We revisit this finding by studying implied volatility based approximations of the spot volatility. Using at-the-money options on the S&P500 index with short maturity, we are able to confirm that volatility is rough. The Hurst parameter found here, of order 0.3, is slightly larger than that usually obtained from historical data. This is easily explained from a smoothing effect due to the remaining time to maturity of the considered options.