  In a market with a rough or Markovian mean-reverting stochastic volatility
there is no perfect hedge. Here it is shown how various delta-type hedging
strategies perform and can be evaluated in such markets in the case of European
options. A precise characterization of the hedging cost, the replication cost
caused by the volatility fluctuations, is presented in an asymptotic regime of
rapid mean reversion for the volatility fluctuations. The optimal dynamic asset
based hedging strategy in the considered regime is identified as the so-called
`practitioners' delta hedging scheme. It is moreover shown that the
performances of the delta-type hedging schemes are essentially independent of
the regularity of the volatility paths in the considered regime and that the
hedging costs are related to a vega risk martingale whose magnitude is
proportional to a new market risk parameter. It is also shown via numerical
simulations that the proposed hedging schemes which derive 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 analysis when the mean reversion time is small seems to be optimal
with arbitrary mean reversion times.
