Optimal hedging under fast-varying stochastic volatility

TL;DR

This paper analyzes the performance of delta-type hedging strategies for European options in markets with fast mean-reverting stochastic volatility, identifying an optimal scheme and demonstrating its robustness through asymptotic analysis and simulations.

Contribution

It provides a rigorous asymptotic characterization of hedging costs and identifies the practitioners' delta hedging scheme as optimal in rapid mean reversion regimes.

Findings

Practitioners' delta hedging is asymptotically optimal in fast mean reversion.

Hedging costs are linked to a vega risk martingale proportional to a new market risk parameter.

Hedging schemes are robust across different volatility path regularities.

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 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.