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Smooth solutions to portfolio liquidation problems under price-sensitive market impact

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  • Paulwin Graewe
  • Ulrich Horst
  • Eric S'er'e

Abstract

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.

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  • Paulwin Graewe & Ulrich Horst & Eric S'er'e, 2013. "Smooth solutions to portfolio liquidation problems under price-sensitive market impact," Papers 1309.0474, arXiv.org, revised Jun 2017.
    • Handle: RePEc:arx:papers:1309.0474
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    1. Stefan Ankirchner & Monique Jeanblanc & Thomas Kruse, 2013. "BSDEs with singular terminal condition and control problems with constraints," Papers 1305.6541, arXiv.org, revised Jun 2013.
    2. Florian Klock & Alexander Schied & Yuemeng Sun, 2012. "Price manipulation in a market impact model with dark pool," Papers 1205.4008, arXiv.org, 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. Jim Gatheral & Alexander Schied, 2011. "Optimal Trade Execution Under Geometric Brownian Motion In The Almgren And Chriss Framework," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 14(03), pages 353-368.
    5. Peter Kratz & Torsten Schöneborn, 2015. "Portfolio Liquidation In Dark Pools In Continuous Time," Mathematical Finance, Wiley Blackwell, vol. 25(3), pages 496-544, July.
    6. Popier, A., 2006. "Backward stochastic differential equations with singular terminal condition," Stochastic Processes and their Applications, Elsevier, vol. 116(12), pages 2014-2056, December.
    7. 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, arXiv.org, revised Jul 2015.
    8. 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.
    9. 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.
    10. Giorgio Fabbri & Fausto Gozzi & Andrzej Swiech, 2017. "Stochastic Optimal Control in Infinite Dimensions - Dynamic Programming and HJB Equations," Post-Print hal-01505767, HAL.
    11. Alexander Schied, 2012. "A control problem with fuel constraint and Dawson-Watanabe superprocesses," Papers 1207.5809, arXiv.org, revised Dec 2013.
    12. Paulwin Graewe & Ulrich Horst & Jinniao Qiu, 2013. "A Non-Markovian Liquidation Problem and Backward SPDEs with Singular Terminal Conditions," Papers 1309.0461, arXiv.org, revised Jan 2015.
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    Cited by:

    1. Ulrich Horst & Xiaonyu Xia, 2018. "Continuous viscosity solutions to linear-quadratic stochastic control problems with singular terminal state constraint," Papers 1809.01972, arXiv.org, revised Apr 2020.
    2. Dirk Becherer & Todor Bilarev & Peter Frentrup, 2018. "Optimal liquidation under stochastic liquidity," Finance and Stochastics, Springer, vol. 22(1), pages 39-68, January.
    3. Paulwin Graewe & Ulrich Horst & Jinniao Qiu, 2013. "A Non-Markovian Liquidation Problem and Backward SPDEs with Singular Terminal Conditions," Papers 1309.0461, arXiv.org, revised Jan 2015.
    4. Horst, Ulrich & Xia, Xiaonyu & Zhou, Chao, 2021. "Portfolio Liquidation under Factor Uncertainty," Rationality and Competition Discussion Paper Series 274, CRC TRR 190 Rationality and Competition.
    5. Elliott, Robert & Qiu, Jinniao & Wei, Wenning, 2022. "Neumann problem for backward SPDEs with singular terminal conditions and application in constrained stochastic control under target zone," Stochastic Processes and their Applications, Elsevier, vol. 148(C), pages 68-97.
    6. Julia Ackermann & Thomas Kruse & Mikhail Urusov, 2021. "Càdlàg semimartingale strategies for optimal trade execution in stochastic order book models," Finance and Stochastics, Springer, vol. 25(4), pages 757-810, October.
    7. T Kruse & A Popier, 2015. "Minimal supersolutions for BSDEs with singular terminal condition and application to optimal position targeting," Papers 1504.01150, arXiv.org, revised Dec 2015.
    8. Yang, Qing-Qing & Ching, Wai-Ki & Gu, Jia-wen & Wong, Tak Kwong & Zhu, Dong-Mei, 2024. "Viscosity solution for optimal liquidation problems with randomly-terminated horizon," Finance Research Letters, Elsevier, vol. 61(C).
    9. Graewe, Paulwin & Popier, Alexandre, 2021. "Asymptotic approach for backward stochastic differential equation with singular terminal condition," Stochastic Processes and their Applications, Elsevier, vol. 133(C), pages 247-277.
    10. 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, arXiv.org, revised Jul 2015.
    11. Ulrich Horst & Xiaonyu Xia & Chao Zhou, 2019. "Portfolio liquidation under factor uncertainty," Papers 1909.00748, arXiv.org.

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