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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 semi-group property (Bellman principle) and characterise the value function as a viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation. We introduce some examples where the forms of the optimal strategies change 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.

Suggested Citation

  • Takashi Kato, 2009. "An Optimal Execution Problem with Market Impact," Papers 0907.3282, arXiv.org, revised Dec 2014.
  • Handle: RePEc:arx:papers:0907.3282
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    File URL: http://arxiv.org/pdf/0907.3282
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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. 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. 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.
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    Cited by:

    1. Takashi Kato, 2014. "VWAP Execution as an Optimal Strategy," Papers 1408.6118, arXiv.org, revised Jan 2017.
    2. 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.
    3. Takashi Kato, 2017. "An Optimal Execution Problem with S-shaped Market Impact Functions," Papers 1706.09224, arXiv.org, revised Oct 2017.
    4. 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.
    5. Masashi Ieda, 2015. "A dynamic optimal execution strategy under stochastic price recovery," Papers 1502.04521, arXiv.org.

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