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Smooth Solutions to Portfolio Liquidation Problems under Price-Sensitive Market Impact


  • Paulwin Graewe

    (Department of Mathematics - Humboldt Universität zu Berlin)

  • Ulrich Horst

    (Department of Mathematics - Humboldt Universität zu Berlin)

  • Eric Séré

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


We consider the stochastic control problem of a financial trader that needs to unwind a large asset portfolio within a short period of time. The trader can simultaneously submit active orders to a primary market and passive orders to a dark pool. Our framework is flexible enough to allow for price-dependent impact functions describing the trading costs in the primary market and price-dependent adverse selection costs associated with dark pool trading. We prove that the value function can be characterized in terms of the unique smooth solution to a PDE with singular terminal value, establish its explicit asymptotic behavior at the terminal time, and give the optimal trading strategy in feedback form.

Suggested Citation

  • Paulwin Graewe & Ulrich Horst & Eric Séré, 2018. "Smooth Solutions to Portfolio Liquidation Problems under Price-Sensitive Market Impact," Post-Print hal-01540537, HAL.
  • Handle: RePEc:hal:journl:hal-01540537
    DOI: 10.1016/
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    References listed on IDEAS

    1. Popier, A., 2006. "Backward stochastic differential equations with singular terminal condition," Stochastic Processes and their Applications, Elsevier, vol. 116(12), pages 2014-2056, December.
    2. Florian Klock & Alexander Schied & Yuemeng Sun, 2012. "Price manipulation in a market impact model with dark pool," Papers 1205.4008,, revised May 2014.
    3. Forsyth, P.A. & Kennedy, J.S. & Tse, S.T. & Windcliff, H., 2012. "Optimal trade execution: A mean quadratic variation approach," Journal of Economic Dynamics and Control, Elsevier, vol. 36(12), pages 1971-1991.
    4. Alexander Schied & Torsten Schoneborn & Michael Tehranchi, 2010. "Optimal Basket Liquidation for CARA Investors is Deterministic," Applied Mathematical Finance, Taylor & Francis Journals, vol. 17(6), pages 471-489.
    5. repec:wsi:ijtafx:v:14:y:2011:i:03:n:s0219024911006577 is not listed on IDEAS
    6. Peter Kratz, 2014. "An Explicit Solution of a Nonlinear-Quadratic Constrained Stochastic Control Problem with Jumps: Optimal Liquidation in Dark Pools with Adverse Selection," Mathematics of Operations Research, INFORMS, vol. 39(4), pages 1198-1220, November.
    7. Giorgio Fabbri & Fausto Gozzi & Andrzej Swiech, 2017. "Stochastic Optimal Control in Infinite Dimensions - Dynamic Programming and HJB Equations," Post-Print hal-01505767, HAL.
    8. Peter Kratz & Torsten Schöneborn, 2015. "Portfolio Liquidation In Dark Pools In Continuous Time," Mathematical Finance, Wiley Blackwell, vol. 25(3), pages 496-544, July.
    9. Alexander Schied, 2012. "A control problem with fuel constraint and Dawson-Watanabe superprocesses," Papers 1207.5809,, revised Dec 2013.
    10. Paulwin Graewe & Ulrich Horst & Jinniao Qiu, 2013. "A Non-Markovian Liquidation Problem and Backward SPDEs with Singular Terminal Conditions," Papers 1309.0461,, revised Jan 2015.
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    Cited by:

    1. repec:spr:finsto:v:22:y:2018:i:1:d:10.1007_s00780-017-0346-2 is not listed on IDEAS
    2. T Kruse & A Popier, 2015. "Minimal supersolutions for BSDEs with singular terminal condition and application to optimal position targeting," Papers 1504.01150,, revised Dec 2015.
    3. Ulrich Horst & Jinniao Qiu & Qi Zhang, 2014. "A Constrained Control Problem with Degenerate Coefficients and Degenerate Backward SPDEs with Singular Terminal Condition," Papers 1407.0108,, revised Jul 2015.
    4. Ulrich Horst & Xiaonyu Xia, 2018. "Continuous viscosity solutions to linear-quadratic stochastic control problems with singular terminal state constraint," Papers 1809.01972,
    5. Paulwin Graewe & Ulrich Horst & Jinniao Qiu, 2013. "A Non-Markovian Liquidation Problem and Backward SPDEs with Singular Terminal Conditions," Papers 1309.0461,, revised Jan 2015.

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    singular terminal value; stochastic optimal control; portfolio liquidation;

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