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Hedging of covered options with linear market impact and gamma constraint

Author

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  • Bruno Bouchard

    (CEREMADE - CEntre de REcherches en MAthématiques de la DEcision - Université Paris Dauphine-PSL - PSL - Université Paris Sciences et Lettres - CNRS - Centre National de la Recherche Scientifique)

  • G. Loeper
  • Y. Zou

    (CEREMADE - CEntre de REcherches en MAthématiques de la DEcision - Université Paris Dauphine-PSL - PSL - Université Paris Sciences et Lettres - CNRS - Centre National de la Recherche Scientifique)

Abstract

Within a financial model with linear price impact, we study the problem of hedging a covered European option under gamma constraint. Using stochastic target and partial differential equation smoothing techniques, we prove that the super-replication price is the viscosity solution of a fully non-linear parabolic equation. As a by-product, we show how "-optimal strategies can be constructed. Finally, a numerical resolution scheme is proposed.

Suggested Citation

  • Bruno Bouchard & G. Loeper & Y. Zou, 2017. "Hedging of covered options with linear market impact and gamma constraint," Post-Print hal-01611790, HAL.
    • Handle: RePEc:hal:journl:hal-01611790
    Note: View the original document on HAL open archive server: https://hal.science/hal-01611790
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    References listed on IDEAS

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    1. Umut Çetin & Robert A. Jarrow & Philip Protter, 2008. "Liquidity risk and arbitrage pricing theory," World Scientific Book Chapters, in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 8, pages 153-183, World Scientific Publishing Co. Pte. Ltd..
    2. Umut Çetin & H. Soner & Nizar Touzi, 2010. "Option hedging for small investors under liquidity costs," Finance and Stochastics, Springer, vol. 14(3), pages 317-341, September.
    3. B Bouchard & G Loeper & Y Zou, 2015. "Almost-sure hedging with permanent price impact," Working Papers hal-01133223, HAL.
    4. Soner, H. Mete & Cetin, Umut & Touzi, Nizar, 2010. "Option hedging for small investors under liquidity costs," LSE Research Online Documents on Economics 28992, London School of Economics and Political Science, LSE Library.
    5. B. Bouchard & G. Loeper & Y. Zou, 2015. "Almost-sure hedging with permanent price impact," Papers 1503.05475, arXiv.org.
    6. RØdiger Frey, 1998. "Perfect option hedging for a large trader," Finance and Stochastics, Springer, vol. 2(2), pages 115-141.
    7. Liu, Hong & Yong, Jiongmin, 2005. "Option pricing with an illiquid underlying asset market," Journal of Economic Dynamics and Control, Elsevier, vol. 29(12), pages 2125-2156, December.
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    Citations

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    Cited by:

    1. Emilio Said, 2019. "How Option Hedging Shapes Market Impact," Papers 1910.05056, arXiv.org, revised Nov 2019.
    2. Emilio Said & Ahmed Bel Hadj Ayed & Damien Thillou & Jean-Jacques Rabeyrin & Frédéric Abergel, 2020. "Market Impact: A Systematic Study of the High Frequency Options Market," Post-Print hal-02014248, HAL.
    3. Gregoire Loeper & Jan Obloj & Benjamin Joseph, 2023. "Calibration of Local Volatility Models with Stochastic Interest Rates using Optimal Transport," Papers 2305.00200, arXiv.org.
    4. Bruno Bouchard & Xiaolu Tan, 2022. "Understanding the dual formulation for the hedging of path-dependent options with price impact," Post-Print hal-02398881, HAL.
    5. Damiano Brigo & Federico Graceffa & Eyal Neuman, 2022. "Price impact on term structure," Quantitative Finance, Taylor & Francis Journals, vol. 22(1), pages 171-195, January.
    6. Thai Nguyen & Mitja Stadje, 2020. "Utility maximization under endogenous pricing," Papers 2005.04312, arXiv.org, revised Mar 2024.
    7. Emilio Said, 2019. "How Option Hedging Shapes Market Impact," Working Papers hal-02310080, HAL.
    8. Bruno Bouchard & Xiaolu Tan, 2019. "Understanding the dual formulation for the hedging of path-dependent options with price impact," Working Papers hal-02398881, HAL.
    9. Mehdi Tomas & Iacopo Mastromatteo & Michael Benzaquen, 2023. "Cross impact in derivative markets," Post-Print hal-03378903, HAL.
    10. Anderegg, Benjamin & Ulmann, Florian & Sornette, Didier, 2022. "The impact of option hedging on the spot market volatility," Journal of International Money and Finance, Elsevier, vol. 124(C).
    11. Benjamin Joseph & Gregoire Loeper & Jan Obloj, 2023. "The Measure Preserving Martingale Sinkhorn Algorithm," Papers 2310.13797, arXiv.org, revised May 2024.
    12. Mehdi Tomas & Iacopo Mastromatteo & Michael Benzaquen, 2021. "Cross impact in derivative markets," Working Papers hal-03378903, HAL.
    13. Ivan Guo & Grégoire Loeper & Shiyi Wang, 2022. "Calibration of local‐stochastic volatility models by optimal transport," Mathematical Finance, Wiley Blackwell, vol. 32(1), pages 46-77, January.
    14. Mehdi Tomas & Iacopo Mastromatteo & Michael Benzaquen, 2021. "Cross impact in derivative markets," Papers 2102.02834, arXiv.org, revised Mar 2022.
    15. Bruno Bouchard & Xiaolu Tan, 2019. "Understanding the dual formulation for the hedging of path-dependent options with price impact," Papers 1912.03946, arXiv.org, revised Jan 2020.
    16. Dirk Becherer & Todor Bilarev, 2018. "Hedging with physical or cash settlement under transient multiplicative price impact," Papers 1807.05917, arXiv.org, revised Jun 2023.

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