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Pricing and hedging contingent claims with liquidity costs and market impact

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  • Frédéric Abergel

    ()
    (FiQuant - Chaire de finance quantitative - Ecole Centrale Paris, MAS - Mathématiques Appliquées aux Systèmes - EA 4037 - Ecole Centrale Paris)

  • Grégoire Loeper

    (FiQuant - Chaire de finance quantitative - Ecole Centrale Paris)

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    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.

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    Bibliographic Info

    Paper provided by HAL in its series Working Papers with number hal-00802402.

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    Date of creation: 19 Mar 2013
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    Handle: RePEc:hal:wpaper:hal-00802402

    Note: View the original document on HAL open archive server: http://hal.archives-ouvertes.fr/hal-00802402
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    Related research

    Keywords: Market impact; partial differential equations; liquidity costs;

    This paper has been announced in the following NEP Reports:

    References

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    1. Eckhard Platen & Martin Schweizer, 1998. "On Feedback Effects from Hedging Derivatives," Mathematical Finance, Wiley Blackwell, vol. 8(1), pages 67-84.
    2. Gregoire Loeper, 2013. "Option pricing with market impact and non-linear Black and Scholes pde's," Papers 1301.6252, arXiv.org.
    3. 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.
    4. Hayne E. Leland., 1984. "Option Pricing and Replication with Transactions Costs," Research Program in Finance Working Papers 144, University of California at Berkeley.
    5. 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.
    6. P. Weber & B. Rosenow, 2005. "Order book approach to price impact," Quantitative Finance, Taylor & Francis Journals, vol. 5(4), pages 357-364.
    7. Frédéric Abergel & Nicolas Millot, 2011. "Nonquadratic Local Risk-Minimization for Hedging Contingent Claims in Incomplete Markets," Post-Print hal-00620843, HAL.
    8. H. Mete Soner & Umut Cetin & Nizar Touzi, 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. Umut Çetin & Robert Jarrow & Philip Protter, 2004. "Liquidity risk and arbitrage pricing theory," Finance and Stochastics, Springer, vol. 8(3), pages 311-341, 08.
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