# Volatility has to be rough

**Authors:** Masaaki Fukasawa  
**arXiv:** [2002.09215](https://arxiv.org/abs/2002.09215)  
**Date:** 2020-02-21  
**Summary page:** https://www.summarizepaper.com/en/paper/2002.09215v1/

## Abstract

First, we give an asymptotic expansion of short-dated at-the-money implied volatility that refines the preceding works and proves in particular that non-rough volatility models are inconsistent to a power law of volatility skew. Second, we show that given a power law of volatility skew in an option market, a continuous price dynamics of the underlying asset with non-rough volatility admits an arbitrage opportunity. The volatility therefore has to be rough in a viable market of the underlying asset of which the volatility skew obeys a power law.