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An Optimal Execution Problem with S-shaped Market Impact Functions

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  • Takashi Kato

Abstract

In this study, we extend the optimal execution problem with convex market impact function studied in Kato (2014) to the case where the market impact function is S-shaped, that is, concave on $[0, \bar {x}_0]$ and convex on $[\bar {x}_0, \infty )$ for some $\bar {x}_0 \geq 0$. We study the corresponding Hamilton-Jacobi-Bellman equation and show that the optimal execution speed under the S-shaped market impact is equal to zero or larger than $\bar {x}_0$. Moreover, we provide some examples of the Black-Scholes model. We show that the optimal strategy for a risk-neutral trader with small shares is the time-weighted average price strategy whenever the market impact function is S-shaped.

Suggested Citation

  • Takashi Kato, 2017. "An Optimal Execution Problem with S-shaped Market Impact Functions," Papers 1706.09224, arXiv.org, revised Oct 2017.
    • Handle: RePEc:arx:papers:1706.09224
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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. Obizhaeva, Anna A. & Wang, Jiang, 2013. "Optimal trading strategy and supply/demand dynamics," Journal of Financial Markets, Elsevier, vol. 16(1), pages 1-32.
    3. Takashi Kato, 2017. "An Optimal Execution Problem in the Volume-Dependent Almgren-Chriss Model," Papers 1701.08972, arXiv.org, revised Aug 2017.
    4. Takashi Kato, 2009. "An Optimal Execution Problem with Market Impact," Papers 0907.3282, arXiv.org, revised Dec 2014.
    5. 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.
    6. 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.
    7. Mazur, Slava, 2013. "Modeling market impact and timing risk in volume time," Algorithmic Finance, IOS Press, vol. 2(2), pages 113-126.
    8. Bertsimas, Dimitris & Lo, Andrew W., 1998. "Optimal control of execution costs," Journal of Financial Markets, Elsevier, vol. 1(1), pages 1-50, April.
    9. Kensuke Ishitani & Takashi Kato, 2013. "Mathematical Formulation of an Optimal Execution Problem with Uncertain Market Impact," Papers 1301.6485, arXiv.org, revised Jun 2015.
    10. 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.
    11. Thierry Ané & Hélyette Geman, 2000. "Order Flow, Transaction Clock, and Normality of Asset Returns," Journal of Finance, American Finance Association, vol. 55(5), pages 2259-2284, October.
    12. Takashi Kato, 2011. "An Optimal Execution Problem with a Geometric Ornstein-Uhlenbeck Price Process," Papers 1107.1787, arXiv.org, revised Jul 2014.
    13. Takashi Kato, 2014. "An optimal execution problem with market impact," Finance and Stochastics, Springer, vol. 18(3), pages 695-732, July.
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