  Recent empirical studies suggest that the volatilities associated with
financial time series exhibit short-range correlations. This entails that the
volatility process is very rough and its autocorrelation exhibits sharp decay
at the origin. Another classic stylistic feature often assumed for the
volatility is that it is mean reverting. In this paper it is shown that the
price impact of a rapidly mean reverting rough volatility model coincides with
that associated with fast mean reverting Markov stochastic volatility models.
This reconciles the empirical observation of rough volatility paths with the
good fit of the implied volatility surface to models of fast mean reverting
Markov volatilities. Moreover, the result conforms with recent numerical
results regarding rough stochastic volatility models. It extends the scope of
models for which the asymptotic results of fast mean reverting Markov
volatilities are valid. The paper concludes with a general discussion of
fractional volatility asymptotics and their interrelation. The regimes
discussed there include fast and slow volatility factors with strong or small
volatility fluctuations and with the limits not commuting in general. The
notion of a characteristic term structure exponent is introduced, this exponent
governs the implied volatility term structure in the various asymptotic
regimes.
