  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.
