IDEAS home Printed from https://ideas.repec.org/a/spr/finsto/v28y2024i3d10.1007_s00780-024-00537-1.html
   My bibliography  Save this article

Reducing Obizhaeva–Wang-type trade execution problems to LQ stochastic control problems

Author

Listed:
  • Julia Ackermann

    (University of Wuppertal)

  • Thomas Kruse

    (University of Wuppertal)

  • Mikhail Urusov

    (University of Duisburg-Essen)

Abstract

We start with a stochastic control problem where the control process is of finite variation (possibly with jumps) and acts as integrator both in the state dynamics and in the target functional. Problems of such type arise in the stream of literature on optimal trade execution pioneered by Obizhaeva and Wang (J. Financ. Mark. 16:1–32, 2013) (models with finite resilience). We consider a general framework where the price impact and the resilience are stochastic processes. Both are allowed to have diffusive components. First we continuously extend the problem from processes of finite variation to progressively measurable processes. Then we reduce the extended problem to a linear–quadratic (LQ) stochastic control problem. Using the well-developed theory on LQ problems, we describe the solution to the obtained LQ one and translate it back to the solution for the (extended) initial trade execution problem. Finally, we illustrate our results by several examples. Among other things, the examples discuss the Obizhaeva–Wang model with random (terminal and moving) targets, the necessity to extend the initial trade execution problem to a reasonably large class of progressively measurable processes (even going beyond semimartingales), and the effects of diffusive components in the price impact process and/or the resilience process.

Suggested Citation

  • Julia Ackermann & Thomas Kruse & Mikhail Urusov, 2024. "Reducing Obizhaeva–Wang-type trade execution problems to LQ stochastic control problems," Finance and Stochastics, Springer, vol. 28(3), pages 813-863, July.
    • Handle: RePEc:spr:finsto:v:28:y:2024:i:3:d:10.1007_s00780-024-00537-1
    DOI: 10.1007/s00780-024-00537-1
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s00780-024-00537-1
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s00780-024-00537-1?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Paulwin Graewe & Ulrich Horst, 2016. "Optimal Trade Execution with Instantaneous Price Impact and Stochastic Resilience," Papers 1611.03435, arXiv.org, revised Jul 2017.
    2. Ulrich Horst & Xiaonyu Xia, 2019. "Multi-dimensional optimal trade execution under stochastic resilience," Finance and Stochastics, Springer, vol. 23(4), pages 889-923, October.
    3. Aurélien Alfonsi & José Infante Acevedo, 2014. "Optimal Execution and Price Manipulations in Time-varying Limit Order Books," Applied Mathematical Finance, Taylor & Francis Journals, vol. 21(3), pages 201-237, July.
    4. Aurélien Alfonsi & Alexander Schied, 2010. "Optimal trade execution and absence of price manipulations in limit order book models," Post-Print hal-00397652, HAL.
    5. Aur'elien Alfonsi & Antje Fruth & Alexander Schied, 2007. "Optimal execution strategies in limit order books with general shape functions," Papers 0708.1756, arXiv.org, revised Feb 2010.
    6. Aurelien Alfonsi & Antje Fruth & Alexander Schied, 2010. "Optimal execution strategies in limit order books with general shape functions," Quantitative Finance, Taylor & Francis Journals, vol. 10(2), pages 143-157.
    7. Kohlmann, Michael & Tang, Shanjian, 2002. "Global adapted solution of one-dimensional backward stochastic Riccati equations, with application to the mean-variance hedging," Stochastic Processes and their Applications, Elsevier, vol. 97(2), pages 255-288, February.
    8. Antje Fruth & Torsten Schöneborn & Mikhail Urusov, 2019. "Optimal trade execution in order books with stochastic liquidity," Mathematical Finance, Wiley Blackwell, vol. 29(2), pages 507-541, April.
    9. Antje Fruth & Torsten Schöneborn & Mikhail Urusov, 2014. "Optimal Trade Execution And Price Manipulation In Order Books With Time-Varying Liquidity," Mathematical Finance, Wiley Blackwell, vol. 24(4), pages 651-695, October.
    10. Obizhaeva, Anna A. & Wang, Jiang, 2013. "Optimal trading strategy and supply/demand dynamics," Journal of Financial Markets, Elsevier, vol. 16(1), pages 1-32.
    11. Peter Bank & Yan Dolinsky, 2018. "Continuous-time Duality for Super-replication with Transient Price Impact," Papers 1808.09807, arXiv.org, revised May 2019.
    12. Imkeller, Peter & Dos Reis, Gonçalo, 2010. "Path regularity and explicit convergence rate for BSDE with truncated quadratic growth," Stochastic Processes and their Applications, Elsevier, vol. 120(3), pages 348-379, March.
    13. René Carmona & Kevin Webster, 2019. "The self-financing equation in limit order book markets," Finance and Stochastics, Springer, vol. 23(3), pages 729-759, July.
    14. Dirk Becherer & Todor Bilarev & Peter Frentrup, 2018. "Optimal liquidation under stochastic liquidity," Finance and Stochastics, Springer, vol. 22(1), pages 39-68, January.
    15. Christopher Lorenz & Alexander Schied, 2013. "Drift dependence of optimal trade execution strategies under transient price impact," Finance and Stochastics, Springer, vol. 17(4), pages 743-770, October.
    16. Ulrich Horst & Evgueni Kivman, 2024. "Optimal trade execution under small market impact and portfolio liquidation with semimartingale strategies," Finance and Stochastics, Springer, vol. 28(3), pages 759-812, July.
    17. Julia Ackermann & Thomas Kruse & Mikhail Urusov, 2020. "Optimal trade execution in an order book model with stochastic liquidity parameters," Papers 2006.05843, arXiv.org, revised Apr 2021.
    18. Christopher Lorenz & Alexander Schied, 2012. "Drift dependence of optimal trade execution strategies under transient price impact," Papers 1204.2716, arXiv.org, revised Mar 2013.
    Full references (including those not matched with items on IDEAS)

    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. Julia Ackermann & Thomas Kruse & Mikhail Urusov, 2021. "Self-exciting price impact via negative resilience in stochastic order books," Papers 2112.03789, arXiv.org, revised Jul 2022.
    2. Julia Ackermann & Thomas Kruse & Mikhail Urusov, 2022. "Reducing Obizhaeva-Wang type trade execution problems to LQ stochastic control problems," Papers 2206.03772, arXiv.org, revised Sep 2023.
    3. 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.
    4. Tao Chen & Mike Ludkovski & Moritz Vo{ss}, 2022. "On Parametric Optimal Execution and Machine Learning Surrogates," Papers 2204.08581, arXiv.org, revised Oct 2023.
    5. Julia Ackermann & Thomas Kruse & Mikhail Urusov, 2020. "Optimal trade execution in an order book model with stochastic liquidity parameters," Papers 2006.05843, arXiv.org, revised Apr 2021.
    6. Ulrich Horst & Evgueni Kivman, 2024. "Optimal trade execution under small market impact and portfolio liquidation with semimartingale strategies," Finance and Stochastics, Springer, vol. 28(3), pages 759-812, July.
    7. Julia Ackermann & Thomas Kruse & Mikhail Urusov, 2020. "C\`adl\`ag semimartingale strategies for optimal trade execution in stochastic order book models," Papers 2006.05863, arXiv.org, revised Jul 2021.
    8. Ulrich Horst & Evgueni Kivman, 2021. "Optimal trade execution under small market impact and portfolio liquidation with semimartingale strategies," Papers 2103.05957, arXiv.org, revised Jul 2023.
    9. 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.
    10. Dirk Becherer & Todor Bilarev & Peter Frentrup, 2015. "Optimal Asset Liquidation with Multiplicative Transient Price Impact," Papers 1501.01892, arXiv.org, revised Apr 2017.
    11. Xinman Cheng & Guanxing Fu & Xiaonyu Xia, 2024. "Long Time Behavior of Optimal Liquidation Problems," Papers 2405.14177, arXiv.org.
    12. Christopher Lorenz & Alexander Schied, 2013. "Drift dependence of optimal trade execution strategies under transient price impact," Finance and Stochastics, Springer, vol. 17(4), pages 743-770, October.
    13. Marcel Nutz & Kevin Webster & Long Zhao, 2023. "Unwinding Stochastic Order Flow: When to Warehouse Trades," Papers 2310.14144, arXiv.org.
    14. Aurélien Alfonsi & José Infante Acevedo, 2014. "Optimal execution and price manipulations in time-varying limit order books," Post-Print hal-00687193, HAL.
    15. Aurélien Alfonsi & Florian Klöck & Alexander Schied, 2016. "Multivariate Transient Price Impact and Matrix-Valued Positive Definite Functions," Mathematics of Operations Research, INFORMS, vol. 41(3), pages 914-934, August.
    16. Alexander Schied & Elias Strehle & Tao Zhang, 2015. "High-frequency limit of Nash equilibria in a market impact game with transient price impact," Papers 1509.08281, arXiv.org, revised May 2017.
    17. Olivier Guéant & Charles-Albert Lehalle, 2015. "General Intensity Shapes In Optimal Liquidation," Mathematical Finance, Wiley Blackwell, vol. 25(3), pages 457-495, July.
    18. Aur'elien Alfonsi & Alexander Schied & Florian Klock, 2013. "Multivariate transient price impact and matrix-valued positive definite functions," Papers 1310.4471, arXiv.org, revised Sep 2015.
    19. Ningyuan Chen & Steven Kou & Chun Wang, 2018. "A Partitioning Algorithm for Markov Decision Processes with Applications to Market Microstructure," Management Science, INFORMS, vol. 64(2), pages 784-803, February.
    20. Olivier Gu'eant, 2013. "Permanent market impact can be nonlinear," Papers 1305.0413, arXiv.org, revised Mar 2014.

    More about this item

    Keywords

    Optimal trade execution; Stochastic price impact; Stochastic resilience; Finite-variation stochastic control; Continuous extension of cost functional; Progressively measurable execution strategy; Linear–quadratic stochastic control; Backward stochastic differential equation;
    All these keywords.

    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • G10 - Financial Economics - - General Financial Markets - - - General (includes Measurement and Data)
    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions

    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:spr:finsto:v:28:y:2024:i:3:d:10.1007_s00780-024-00537-1. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.