IDEAS home Printed from https://ideas.repec.org/p/arx/papers/1006.0768.html
   My bibliography  Save this paper

Numerical methods for an optimal order execution problem

Author

Listed:
  • Fabien Guilbaud
  • Mohamed Mnif
  • Huy^en Pham

Abstract

This paper deals with numerical solutions to an impulse control problem arising from optimal portfolio liquidation with bid-ask spread and market price impact penalizing speedy execution trades. The corresponding dynamic programming (DP) equation is a quasi-variational inequality (QVI) with solvency constraint satisfied by the value function in the sense of constrained viscosity solutions. By taking advantage of the lag variable tracking the time interval between trades, we can provide an explicit backward numerical scheme for the time discretization of the DPQVI. The convergence of this discrete-time scheme is shown by viscosity solutions arguments. An optimal quantization method is used for computing the (conditional) expectations arising in this scheme. Numerical results are presented by examining the behaviour of optimal liquidation strategies, and comparative performance analysis with respect to some benchmark execution strategies. We also illustrate our optimal liquidation algorithm on real data, and observe various interesting patterns of order execution strategies. Finally, we provide some numerical tests of sensitivity with respect to the bid/ask spread and market impact parameters.

Suggested Citation

  • Fabien Guilbaud & Mohamed Mnif & Huy^en Pham, 2010. "Numerical methods for an optimal order execution problem," Papers 1006.0768, arXiv.org.
    • Handle: RePEc:arx:papers:1006.0768
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/1006.0768
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Alexander Schied & Torsten Schöneborn, 2009. "Risk aversion and the dynamics of optimal liquidation strategies in illiquid markets," Finance and Stochastics, Springer, vol. 13(2), pages 181-204, April.
    2. Vathana Ly Vath & Mohamed Mnif & Huyên Pham, 2007. "A model of optimal portfolio selection under liquidity risk and price impact," Finance and Stochastics, Springer, vol. 11(1), pages 51-90, January.
    3. He, Hua & Mamaysky, Harry, 2005. "Dynamic trading policies with price impact," Journal of Economic Dynamics and Control, Elsevier, vol. 29(5), pages 891-930, May.
    4. Bertsimas, Dimitris & Lo, Andrew W., 1998. "Optimal control of execution costs," Journal of Financial Markets, Elsevier, vol. 1(1), pages 1-50, April.
    5. Fabrizio Lillo & J. Doyne Farmer & Rosario N. Mantegna, 2003. "Master curve for price-impact function," Nature, Nature, vol. 421(6919), pages 129-130, January.
    6. Potters, Marc & Bouchaud, Jean-Philippe, 2003. "More statistical properties of order books and price impact," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 324(1), pages 133-140.
    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. Olivier Guéant & Charles-Albert Lehalle, 2015. "General Intensity Shapes In Optimal Liquidation," Mathematical Finance, Wiley Blackwell, vol. 25(3), pages 457-495, July.
    2. Sophie Laruelle & Charles-Albert Lehalle & Gilles Pag`es, 2011. "Optimal posting price of limit orders: learning by trading," Papers 1112.2397, arXiv.org, revised Sep 2012.

    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. Olivier Guéant, 2016. "The Financial Mathematics of Market Liquidity: From Optimal Execution to Market Making," Post-Print hal-01393136, HAL.
    2. Zhao, Jingdong & Zhu, Hongliang & Li, Xindan, 2018. "Optimal execution with price impact under Cumulative Prospect Theory," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 490(C), pages 1228-1237.
    3. Mauricio Junca, 2011. "Stochastic impulse control on optimal execution with price impact and transaction cost," Papers 1103.3482, arXiv.org, revised Jan 2013.
    4. Takashi Kato, 2014. "An optimal execution problem with market impact," Finance and Stochastics, Springer, vol. 18(3), pages 695-732, July.
    5. 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.
    6. Olivier Guéant & Charles-Albert Lehalle, 2015. "General Intensity Shapes In Optimal Liquidation," Mathematical Finance, Wiley Blackwell, vol. 25(3), pages 457-495, July.
    7. Kashyap, Ravi, 2020. "David vs Goliath (You against the Markets), A dynamic programming approach to separate the impact and timing of trading costs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 545(C).
    8. repec:dau:papers:123456789/7391 is not listed on IDEAS
    9. J. Doyne Farmer & Austin Gerig & Fabrizio Lillo & Henri Waelbroeck, 2013. "How efficiency shapes market impact," Quantitative Finance, Taylor & Francis Journals, vol. 13(11), pages 1743-1758, November.
    10. Aurélien Alfonsi & Alexander Schied, 2010. "Optimal trade execution and absence of price manipulations in limit order book models," Post-Print hal-00397652, HAL.
    11. Ludovic Moreau & Johannes Muhle-Karbe & H. Mete Soner, 2014. "Trading with Small Price Impact," Papers 1402.5304, arXiv.org, revised Mar 2015.
    12. Martin D. Gould & Mason A. Porter & Stacy Williams & Mark McDonald & Daniel J. Fenn & Sam D. Howison, 2010. "Limit Order Books," Papers 1012.0349, arXiv.org, revised Apr 2013.
    13. Olivier Gu'eant & Charles-Albert Lehalle & Joaquin Fernandez Tapia, 2011. "Optimal Portfolio Liquidation with Limit Orders," Papers 1106.3279, arXiv.org, revised Jul 2012.
    14. Nikolay Andreev, 2019. "Robust Portfolio Optimization in an Illiquid Market in Discrete-Time," Mathematics, MDPI, vol. 7(12), pages 1-16, November.
    15. Siu, Chi Chung & Guo, Ivan & Zhu, Song-Ping & Elliott, Robert J., 2019. "Optimal execution with regime-switching market resilience," Journal of Economic Dynamics and Control, Elsevier, vol. 101(C), pages 17-40.
    16. Alexander Schied & Torsten Schöneborn, 2009. "Risk aversion and the dynamics of optimal liquidation strategies in illiquid markets," Finance and Stochastics, Springer, vol. 13(2), pages 181-204, April.
    17. Yan, Tingjin & Chiu, Mei Choi & Wong, Hoi Ying, 2023. "Portfolio liquidation with delayed information," Economic Modelling, Elsevier, vol. 126(C).
    18. Erhan Bayraktar & Thomas Cayé & Ibrahim Ekren, 2021. "Asymptotics for small nonlinear price impact: A PDE approach to the multidimensional case," Mathematical Finance, Wiley Blackwell, vol. 31(1), pages 36-108, January.
    19. Qixuan Luo & Shijia Song & Handong Li, 2023. "Research on the Effects of Liquidation Strategies in the Multi-asset Artificial Market," Computational Economics, Springer;Society for Computational Economics, vol. 62(4), pages 1721-1750, December.
    20. Enzo Busseti & Fabrizio Lillo, 2012. "Calibration of optimal execution of financial transactions in the presence of transient market impact," Papers 1206.0682, arXiv.org.
    21. Colaneri, Katia & Eksi, Zehra & Frey, Rüdiger & Szölgyenyi, Michaela, 2020. "Optimal liquidation under partial information with price impact," Stochastic Processes and their Applications, Elsevier, vol. 130(4), pages 1913-1946.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    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:arx:papers:1006.0768. 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: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    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.