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Pricing and hedging contingent claims with liquidity costs and market impact

Author

Listed:
  • Frédéric Abergel

    (MAS - Mathématiques Appliquées aux Systèmes - EA 4037 - Ecole Centrale Paris, BNP Paribas)

  • Grégoire Loeper

    (MAS - Mathématiques Appliquées aux Systèmes - EA 4037 - Ecole Centrale Paris)

Abstract

We study the influence of taking liquidity costs and market impact into account when hedging a contingent claim, first in the discrete time setting, then in continuous time. In the latter case and in a complete market, we derive a fully non-linear pricing partial differential equation, and characterizes its parabolic nature according to the value of a numerical parameter naturally interpreted as a relaxation coefficient for market impact. We then investigate the more challenging case of stochastic volatility models, and prove the parabolicity of the pricing equation in a particular case.

Suggested Citation

  • Frédéric Abergel & Grégoire Loeper, 2013. "Pricing and hedging contingent claims with liquidity costs and market impact," Working Papers hal-00802402, HAL.
    • Handle: RePEc:hal:wpaper:hal-00802402
    DOI: 10.2139/ssrn.2239498
    Note: View the original document on HAL open archive server: https://hal.science/hal-00802402v4
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    References listed on IDEAS

    as
    1. P. Weber & B. Rosenow, 2005. "Order book approach to price impact," Quantitative Finance, Taylor & Francis Journals, vol. 5(4), pages 357-364.
    2. Eckhard Platen & Martin Schweizer, 1998. "On Feedback Effects from Hedging Derivatives," Mathematical Finance, Wiley Blackwell, vol. 8(1), pages 67-84, January.
    3. Leland, Hayne E, 1985. "Option Pricing and Replication with Transactions Costs," Journal of Finance, American Finance Association, vol. 40(5), pages 1283-1301, December.
    4. 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..
    5. Frédéric Abergel & Nicolas Millot, 2011. "Nonquadratic Local Risk-Minimization for Hedging Contingent Claims in Incomplete Markets," Post-Print hal-00620843, HAL.
    6. 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.
    7. Frédéric Abergel & Nicolas Millot, 2011. "Non Quadratic Local Risk-Minimization for Hedging Contingent Claims in the Presence of Transaction Costs," Working Papers hal-00621256, HAL.
    8. 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.
    9. 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.
    10. Gregoire Loeper, 2013. "Option pricing with linear market impact and non-linear Black and Scholes equations," Papers 1301.6252, arXiv.org, revised Aug 2016.
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    Citations

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    Cited by:

    1. Rongju Zhang & Nicolas Langren'e & Yu Tian & Zili Zhu & Fima Klebaner & Kais Hamza, 2016. "Dynamic portfolio optimization with liquidity cost and market impact: a simulation-and-regression approach," Papers 1610.07694, arXiv.org, revised Jun 2019.
    2. Taiga Saito, 2017. "Hedging and pricing illiquid options with market impacts," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 4(02n03), pages 1-37, June.
    3. Damiano Brigo & Federico Graceffa & Eyal Neuman, 2022. "Price impact on term structure," Quantitative Finance, Taylor & Francis Journals, vol. 22(1), pages 171-195, January.
    4. Rongju Zhang & Nicolas Langren'e & Yu Tian & Zili Zhu & Fima Klebaner & Kais Hamza, 2018. "Local Control Regression: Improving the Least Squares Monte Carlo Method for Portfolio Optimization," Papers 1803.11467, arXiv.org, revised Sep 2018.

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    Keywords

    Market impact; partial differential equations; liquidity costs;
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