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Optimal Static Quadratic Hedging

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  • Tim Leung
  • Matthew Lorig

Abstract

We propose a flexible framework for hedging a contingent claim by holding static positions in vanilla European calls, puts, bonds, and forwards. A model-free expression is derived for the optimal static hedging strategy that minimizes the expected squared hedging error subject to a cost constraint. The optimal hedge involves computing a number of expectations that reflect the dependence among the contingent claim and the hedging assets. We provide a general method for approximating these expectations analytically in a general Markov diffusion market. To illustrate the versatility of our approach, we present several numerical examples, including hedging path-dependent options and options written on a correlated asset.

Suggested Citation

  • Tim Leung & Matthew Lorig, 2015. "Optimal Static Quadratic Hedging," Papers 1506.02074, arXiv.org, revised Nov 2015.
    • Handle: RePEc:arx:papers:1506.02074
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    References listed on IDEAS

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    1. Claude Bardos & Raphaël Douady & Andrei Fursikov, 2002. "Static Hedging Of Barrier Options With A Smile: An Inverse Problem," Post-Print hal-01477102, HAL.
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    8. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    9. Tim Leung & Ronnie Sircar, 2015. "Implied Volatility of Leveraged ETF Options," Applied Mathematical Finance, Taylor & Francis Journals, vol. 22(2), pages 162-188, April.
    10. Matthew Lorig & Stefano Pagliarani & Andrea Pascucci, 2017. "Explicit Implied Volatilities For Multifactor Local-Stochastic Volatility Models," Mathematical Finance, Wiley Blackwell, vol. 27(3), pages 926-960, July.
    11. Matthew Lorig & Stefano Pagliarani & Andrea Pascucci, 2013. "Analytical expansions for parabolic equations," Papers 1312.3314, arXiv.org, revised Nov 2014.
    12. Pagliarani, Stefano & Pascucci, Andrea, 2011. "Analytical approximation of the transition density in a local volatility model," MPRA Paper 31107, University Library of Munich, Germany.
    13. Tim Leung & Matthew Lorig & Andrea Pascucci, 2014. "Leveraged {ETF} implied volatilities from {ETF} dynamics," Papers 1404.6792, arXiv.org, revised Apr 2015.
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    Cited by:

    1. Peter Carr & Roger Lee & Matthew Lorig, 2021. "Robust replication of volatility and hybrid derivatives on jump diffusions," Mathematical Finance, Wiley Blackwell, vol. 31(4), pages 1394-1422, October.
    2. Georgios I. Papayiannis, 2022. "Static Hedging of Freight Risk under Model Uncertainty," Papers 2207.00862, arXiv.org.
    3. Peter Carr & Roger Lee & Matthew Lorig, 2015. "Robust replication of barrier-style claims on price and volatility," Papers 1508.00632, arXiv.org, revised Jan 2022.
    4. Fabien Le Floc’h, 2018. "Variance Swap Replication: Discrete or Continuous?," JRFM, MDPI, vol. 11(1), pages 1-15, February.
    5. Tim Leung & Brian Ward, 2020. "Tracking VIX with VIX Futures: Portfolio Construction and Performance," World Scientific Book Chapters, in: John B Guerard & William T Ziemba (ed.), HANDBOOK OF APPLIED INVESTMENT RESEARCH, chapter 21, pages 557-596, World Scientific Publishing Co. Pte. Ltd..
    6. Jun Deng & Bin Zou, 2020. "Quadratic Hedging for Sequential Claims with Random Weights in Discrete Time," Papers 2005.06015, arXiv.org, revised Dec 2020.
    7. Alvaro Cartea & Ryan Donnelly & Sebastian Jaimungal, 2019. "Hedging Non-Tradable Risks with Transaction Costs and Price Impact," Papers 1908.00054, arXiv.org, revised Mar 2020.
    8. Navratil, Robert & Taylor, Stephen & Vecer, Jan, 2022. "On the utility maximization of the discrepancy between a perceived and market implied risk neutral distribution," European Journal of Operational Research, Elsevier, vol. 302(3), pages 1215-1229.
    9. Álvaro Cartea & Ryan Donnelly & Sebastian Jaimungal, 2020. "Hedging nontradable risks with transaction costs and price impact," Mathematical Finance, Wiley Blackwell, vol. 30(3), pages 833-868, July.

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