In a market with a rough or Markovian mean-reverting stochastic volatility thereis no perfect hedge. Here it is shown how various delta-type hedging strategies perform and canbe evaluated in such markets in the case of European options.A precise characterization of thehedging cost, the replication cost caused by the volatilityfluctuations, is presented in an asymptoticregime of rapid mean reversion for the volatility fluctuations. The optimal dynamic asset basedhedging strategy in the considered regime is identified as the so-called “practitioners” delta hedgingscheme. It is moreover shown that the performances of the delta-type hedging schemes are essentiallyindependent of the regularity of the volatility paths in theconsidered regime and that the hedgingcosts are related to a Vega risk martingale whose magnitude is proportional to a new market riskparameter. It is also shown via numerical simulations that the proposed hedging schemes whichderive from option price approximations in the regime of rapid mean reversion, are robust: the“practitioners” delta hedging scheme that is identified as being optimal by our asymptotic analysiswhen the mean reversion time is small seems to be optimal witharbitrary mean reversion times.