Delta Hedging European Options Under Fast Mean-Reverting Volatility

Summary

The document compares delta-based hedging strategies for European options when volatility is stochastic and mean-reverting. In this setting, volatility fluctuations prevent perfect replication, so the analysis focuses on the additional cost of hedging those fluctuations.

Using an asymptotic regime in which volatility mean reverts rapidly, the work characterizes hedging cost and identifies practitioners’ delta as the optimal dynamic asset-based strategy in that regime. It relates the cost to a vega-risk martingale and a market risk parameter. Numerical simulations suggest the strategy remains robust and may perform best even when mean reversion is not rapid. These conclusions are specific to the models and regimes studied; the excerpt provides no market calibration or out-of-sample trading evidence.

Key ideas

• Stochastic volatility makes perfect hedging of European options impossible in the setting considered.
• The analysis evaluates delta-type strategies when volatility mean reverts rapidly.
• Practitioners’ delta is identified as the optimal dynamic asset hedge in the rapid mean-reversion regime.
• Hedging costs are linked to vega risk and a market risk parameter.
• Simulations suggest the preferred strategy may remain effective beyond the asymptotic regime.

Tags

# Optimal hedging under fast-varying stochastic volatility


# Optimal hedging under fast-varying stochastic volatility









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.

Shown in full with attribution under the source's licence. Licence: abstract CC0

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