IDEAS home Printed from https://ideas.repec.org/p/hal/journl/hal-01247523.html
   My bibliography  Save this paper

Hedging of covered options with linear market impact and gamma constraint

Author

Listed:
  • 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

    (Monash University [Clayton], FiQuant - Chaire de finance quantitative - MICS - Mathématiques et Informatique pour la Complexité et les Systèmes - CentraleSupélec)

  • 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 ε-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-01247523, HAL.
    • Handle: RePEc:hal:journl:hal-01247523
    DOI: 10.1137/15m1054109
    Note: View the original document on HAL open archive server: https://hal.science/hal-01247523
    as

    Download full text from publisher

    File URL: https://hal.science/hal-01247523/document
    Download Restriction: no
    File URL: https://libkey.io/10.1137/15m1054109?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    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. 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.
    3. 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.
    4. B Bouchard & G Loeper & Y Zou, 2015. "Almost-sure hedging with permanent price impact," Working Papers hal-01133223, HAL.
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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. Benjamin Joseph & Gregoire Loeper & Jan Obloj, 2023. "The Measure Preserving Martingale Sinkhorn Algorithm," Papers 2310.13797, arXiv.org, revised Dec 2023.
    4. Gregoire Loeper & Jan Obloj & Benjamin Joseph, 2023. "Calibration of Local Volatility Models with Stochastic Interest Rates using Optimal Transport," Papers 2305.00200, arXiv.org.
    5. Bruno Bouchard & Xiaolu Tan, 2022. "Understanding the dual formulation for the hedging of path-dependent options with price impact," Post-Print hal-02398881, HAL.
    6. Mehdi Tomas & Iacopo Mastromatteo & Michael Benzaquen, 2021. "Cross impact in derivative markets," Working Papers hal-03378903, HAL.
    7. Damiano Brigo & Federico Graceffa & Eyal Neuman, 2022. "Price impact on term structure," Quantitative Finance, Taylor & Francis Journals, vol. 22(1), pages 171-195, January.
    8. Thai Nguyen & Mitja Stadje, 2020. "Utility maximization under endogenous pricing," Papers 2005.04312, arXiv.org, revised Mar 2024.
    9. 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.
    10. Mehdi Tomas & Iacopo Mastromatteo & Michael Benzaquen, 2021. "Cross impact in derivative markets," Papers 2102.02834, arXiv.org, revised Mar 2022.
    11. Bruno Bouchard & Xiaolu Tan, 2019. "Understanding the dual formulation for the hedging of path-dependent options with price impact," Working Papers hal-02398881, HAL.
    12. Mehdi Tomas & Iacopo Mastromatteo & Michael Benzaquen, 2022. "Cross impact in derivative markets," Post-Print hal-03378903, HAL.
    13. 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).
    14. 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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Bruno Bouchard & G. Loeper & Y. Zou, 2017. "Hedging of covered options with linear market impact and gamma constraint," Post-Print hal-01611790, HAL.
    2. B Bouchard & G Loeper & Y Zou, 2015. "Hedging of covered options with linear market impact and gamma constraint," Papers 1512.07087, arXiv.org.
    3. B Bouchard & G Loeper & Y Zou, 2015. "Hedging of covered options with linear market impact and gamma constraint," Working Papers hal-01247523, HAL.
    4. Olivier Guéant, 2016. "The Financial Mathematics of Market Liquidity: From Optimal Execution to Market Making," Post-Print hal-01393136, HAL.
    5. Bruno Bouchard & G Loeper & Y Zou, 2016. "Almost-sure hedging with permanent price impact," Post-Print hal-01133223, HAL.
    6. B. Bouchard & G. Loeper & Y. Zou, 2015. "Almost-sure hedging with permanent price impact," Papers 1503.05475, arXiv.org.
    7. B Bouchard & G Loeper & Y Zou, 2015. "Almost-sure hedging with permanent price impact," Working Papers hal-01133223, HAL.
    8. Dirk Becherer & Todor Bilarev & Peter Frentrup, 2017. "Stability for gains from large investors' strategies in M1/J1 topologies," Papers 1701.02167, arXiv.org, revised Mar 2018.
    9. Dirk Becherer & Todor Bilarev, 2018. "Hedging with physical or cash settlement under transient multiplicative price impact," Papers 1807.05917, arXiv.org, revised Jun 2023.
    10. Li, Zhe & Zhang, Weiguo & Zhang, Yue & Yi, Zhigao, 2019. "An analytical approximation approach for pricing European options in a two-price economy," The North American Journal of Economics and Finance, Elsevier, vol. 50(C).
    11. Frédéric Abergel & Grégoire Loeper, 2013. "Pricing and hedging contingent claims with liquidity costs and market impact," Working Papers hal-00802402, HAL.
    12. Sergey Lototsky & Henry Schellhorn & Ran Zhao, 2016. "A String Model of Liquidity in Financial Markets," Papers 1608.05900, arXiv.org, revised Apr 2018.
    13. Li, Zhe & Zhang, Wei-Guo & Liu, Yong-Jun, 2018. "Analytical valuation for geometric Asian options in illiquid markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 507(C), pages 175-191.
    14. Rossella Agliardi & Ramazan Gençay, 2012. "Hedging through a Limit Order Book with Varying Liquidity," Working Paper series 12_12, Rimini Centre for Economic Analysis.
    15. Alexandre Roch, 2011. "Liquidity risk, price impacts and the replication problem," Finance and Stochastics, Springer, vol. 15(3), pages 399-419, September.
    16. Li, Zhe & Zhang, Wei-Guo & Liu, Yong-Jun, 2018. "European quanto option pricing in presence of liquidity risk," The North American Journal of Economics and Finance, Elsevier, vol. 45(C), pages 230-244.
    17. Salvatore Federico & Paul Gassiat, 2014. "Viscosity Characterization of the Value Function of an Investment-Consumption Problem in Presence of an Illiquid Asset," Journal of Optimization Theory and Applications, Springer, vol. 160(3), pages 966-991, March.
    18. David German & Henry Schellhorn, 2012. "A No-Arbitrage Model of Liquidity in Financial Markets involving Brownian Sheets," Papers 1206.4804, arXiv.org.
    19. Gregoire Loeper, 2013. "Option pricing with linear market impact and non-linear Black and Scholes equations," Papers 1301.6252, arXiv.org, revised Aug 2016.
    20. Li, Zhe & Zhang, Wei-Guo & Liu, Yong-Jun & Zhang, Yue, 2019. "Pricing discrete barrier options under jump-diffusion model with liquidity risk," International Review of Economics & Finance, Elsevier, vol. 59(C), pages 347-368.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:hal:journl:hal-01247523. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: CCSD (email available below). General contact details of provider: https://hal.archives-ouvertes.fr/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.