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Optimal hedging under fast-varying stochastic volatility

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  • Josselin Garnier
  • Knut Solna

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.

Suggested Citation

  • Josselin Garnier & Knut Solna, 2018. "Optimal hedging under fast-varying stochastic volatility," Papers 1810.08337, arXiv.org, revised Mar 2020.
    • Handle: RePEc:arx:papers:1810.08337
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    References listed on IDEAS

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    1. Mingxin Xu, 2006. "Risk measure pricing and hedging in incomplete markets," Annals of Finance, Springer, vol. 2(1), pages 51-71, January.
    2. repec:cup:cbooks:9780521843584 is not listed on IDEAS
    3. Jean-Pierre Fouque & Ruimeng Hu, 2017. "Optimal Portfolio under Fast Mean-reverting Fractional Stochastic Environment," Papers 1706.03139, arXiv.org, revised Feb 2018.
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    Cited by:

    1. Giulia Di Nunno & Kk{e}stutis Kubilius & Yuliya Mishura & Anton Yurchenko-Tytarenko, 2023. "From constant to rough: A survey of continuous volatility modeling," Papers 2309.01033, arXiv.org, revised Sep 2023.
    2. Ranieri Dugo & Giacomo Giorgio & Paolo Pigato, 2024. "The Multivariate Fractional Ornstein-Uhlenbeck Process," CEIS Research Paper 581, Tor Vergata University, CEIS, revised 28 Aug 2024.

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