Title: Optimal hedging under fast-varying stochastic volatility,
Abstract: 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. 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.