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Robust Strategies for Optimal Order Execution in the Almgren--Chriss Framework

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  • Alexander Schied

Abstract

Assuming geometric Brownian motion as unaffected price process , Gatheral and Schied (2011; Optimal trade execution under geometric Brownian motion in the Almgren and Chriss framework, International Journal of Theoretical and Applied Finance, 14, pp. 353--368) derived a strategy for optimal order execution that reacts in a sensible manner on market changes but can still be computed in closed form. Here, we will investigate the robustness of this strategy with respect to misspecification of the law of . We prove the surprising result that the strategy remains optimal whenever is a square-integrable martingale. We then analyse the optimization criterion of Gatheral and Schied (2011) in the case in which is any square-integrable semimartingale and we give a closed-form solution to this problem. As a corollary, we find an explicit solution to the problem of minimizing the expected liquidation costs when the unaffected price process is a square-integrable semimartingale. The solutions to our problems are found by stochastically solving a finite-fuel control problem without assumptions of Markovianity.

Suggested Citation

  • Alexander Schied, 2013. "Robust Strategies for Optimal Order Execution in the Almgren--Chriss Framework," Applied Mathematical Finance, Taylor & Francis Journals, vol. 20(3), pages 264-286, July.
    • Handle: RePEc:taf:apmtfi:v:20:y:2013:i:3:p:264-286
    DOI: 10.1080/1350486X.2012.683963
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    Citations

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    Cited by:

    1. Phillip Monin, 2014. "Hedging Market Risk in Optimal Liquidation," Working Papers 14-08, Office of Financial Research, US Department of the Treasury.
    2. Alvaro Cartea & Luhui Gan & Sebastian Jaimungal, 2018. "Trading Cointegrated Assets with Price Impact," Papers 1807.01428, arXiv.org.
    3. Brunovský, Pavol & Černý, Aleš & Komadel, Ján, 2018. "Optimal trade execution under endogenous pressure to liquidate: Theory and numerical solutions," European Journal of Operational Research, Elsevier, vol. 264(3), pages 1159-1171.
    4. Chiara Benazzoli & Luca Di Persio, 2017. "Optimal execution strategy in liquidity framework," Cogent Economics & Finance, Taylor & Francis Journals, vol. 5(1), pages 1364902-136, January.
    5. 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).
    6. Lokka, A. & Xu, Junwei, 2020. "Optimal liquidation trajectories for the Almgren-Chriss model," LSE Research Online Documents on Economics 106977, London School of Economics and Political Science, LSE Library.
    7. 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.
    8. 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.
    9. Ulrich Horst & Xiaonyu Xia & Chao Zhou, 2019. "Portfolio liquidation under factor uncertainty," Papers 1909.00748, arXiv.org.
    10. Álvaro Cartea & Sebastian Jaimungal & Damir Kinzebulatov, 2016. "Algorithmic Trading With Learning," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(04), pages 1-30, June.
    11. Julien Vaes & Raphael Hauser, 2018. "Optimal Trade Execution with Uncertain Volume Target," Papers 1810.11454, arXiv.org, revised Sep 2021.
    12. Weston Barger & Matthew Lorig, 2019. "Optimal Liquidation Under Stochastic Price Impact," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(02), pages 1-28, March.
    13. Takashi Kato, 2017. "An Optimal Execution Problem in the Volume-Dependent Almgren-Chriss Model," Papers 1701.08972, arXiv.org, revised Aug 2017.
    14. 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.
    15. Arne Lokka & Junwei Xu, 2020. "Optimal liquidation trajectories for the Almgren-Chriss model with Levy processes," Papers 2002.03376, arXiv.org, revised Sep 2020.
    16. Erhan Bayraktar & Alexander Munk, 2017. "Mini-Flash Crashes, Model Risk, and Optimal Execution," Papers 1705.09827, arXiv.org, revised Aug 2018.

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