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An optimal execution problem with market impact

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  • Takashi Kato

Abstract

We study an optimal execution problem in a continuous-time market model that considers market impact. We formulate the problem as a stochastic control problem and investigate properties of the corresponding value function. We find that right-continuity at the time origin is associated with the strength of market impact for large sales; otherwise the value function is continuous. Moreover, we show the semigroup property (Bellman principle) and characterise the value function as a viscosity solution of the corresponding Hamilton–Jacobi–Bellman equation. We present some examples where the form of the optimal strategy changes completely, depending on the amount of the trader’s security holdings, and where optimal strategies in the Black–Scholes type market with nonlinear market impact are not block liquidation but gradual liquidation, even when the trader is risk-neutral. Copyright Springer-Verlag Berlin Heidelberg 2014

Suggested Citation

  • Takashi Kato, 2014. "An optimal execution problem with market impact," Finance and Stochastics, Springer, vol. 18(3), pages 695-732, July.
    • Handle: RePEc:spr:finsto:v:18:y:2014:i:3:p:695-732
    DOI: 10.1007/s00780-014-0232-0
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. 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.
    4. Holthausen, Robert W. & Leftwich, Richard W. & Mayers, David, 1987. "The effect of large block transactions on security prices: A cross-sectional analysis," Journal of Financial Economics, Elsevier, vol. 19(2), pages 237-267, December.
    5. Merton, Robert C., 1971. "Optimum consumption and portfolio rules in a continuous-time model," Journal of Economic Theory, Elsevier, vol. 3(4), pages 373-413, December.
    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. Ajay Subramanian & Robert A. Jarrow, 2001. "The Liquidity Discount," Mathematical Finance, Wiley Blackwell, vol. 11(4), pages 447-474, October.
    8. Gur Huberman & Werner Stanzl, 2005. "Optimal Liquidity Trading," Review of Finance, Springer, vol. 9(2), pages 165-200, June.
    9. Alfonsi Aurélien & Alexander Schied & Alla Slynko, 2012. "Order Book Resilience, Price Manipulation, and the Positive Portfolio Problem," Post-Print hal-00941333, HAL.
    10. 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.
    11. Merton, Robert C, 1969. "Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case," The Review of Economics and Statistics, MIT Press, vol. 51(3), pages 247-257, August.
    12. Bertsimas, Dimitris & Lo, Andrew W., 1998. "Optimal control of execution costs," Journal of Financial Markets, Elsevier, vol. 1(1), pages 1-50, April.
    13. Aurélien Alfonsi & Alexander Schied, 2010. "Optimal trade execution and absence of price manipulations in limit order book models," Post-Print hal-00397652, HAL.
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    Citations

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    Cited by:

    1. Xiaoyue Li & John M. Mulvey, 2023. "Optimal Portfolio Execution in a Regime-switching Market with Non-linear Impact Costs: Combining Dynamic Program and Neural Network," Papers 2306.08809, arXiv.org.
    2. 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).
    3. Kensuke Ishitani & Takashi Kato, 2015. "Theoretical and Numerical Analysis of an Optimal Execution Problem with Uncertain Market Impact," Papers 1506.02789, arXiv.org, revised Aug 2015.
    4. Goldys, Beniamin & Wu, Wei, 2019. "On a class of singular stochastic control problems driven by Lévy noise," Stochastic Processes and their Applications, Elsevier, vol. 129(9), pages 3174-3206.
    5. Takashi Kato, 2017. "An Optimal Execution Problem with S-shaped Market Impact Functions," Papers 1706.09224, arXiv.org, revised Oct 2017.
    6. Saerom Park & Jaewook Lee & Youngdoo Son, 2016. "Predicting Market Impact Costs Using Nonparametric Machine Learning Models," PLOS ONE, Public Library of Science, vol. 11(2), pages 1-13, February.
    7. Takashi Kato, 2014. "VWAP Execution as an Optimal Strategy," Papers 1408.6118, arXiv.org, revised Jan 2017.
    8. Masashi Ieda, 2015. "A dynamic optimal execution strategy under stochastic price recovery," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 2(04), pages 1-24, December.
    9. Masashi Ieda, 2015. "A dynamic optimal execution strategy under stochastic price recovery," Papers 1502.04521, arXiv.org.

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    More about this item

    Keywords

    Optimal execution; Market impact; Liquidity problems; Hamilton–Jacobi–Bellman (HJB) equation; Viscosity solutions; 91G80; 93E20; 49L20; G33; G11;
    All these keywords.

    JEL classification:

    • G33 - Financial Economics - - Corporate Finance and Governance - - - Bankruptcy; Liquidation
    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions

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