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Optimal execution with dynamic risk adjustment

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  • Xue Cheng
  • Marina Di Giacinto
  • Tai-Ho Wang

Abstract

This paper considers the problem of optimal liquidation of a position in a risky security in a financial market, where price evolution are risky and trades have an impact on price as well as uncertainty in the filling orders. The problem is formulated as a continuous time stochastic optimal control problem aiming at maximizing a generalized risk-adjusted profit and loss function. The expression of the risk adjustment is derived from the general theory of dynamic risk measures and is selected in the class of $g$-conditional risk measures. The resulting theoretical framework is nonclassical since the target function depends on backward components. We show that, under a quadratic specification of the driver of a backward stochastic differential equation, it is possible to find a closed form solution and an explicit expression of the optimal liquidation policies. In this way it is immediate to quantify the impact of risk-adjustment on the profit and loss and on the expression of the optimal liquidation policies.

Suggested Citation

  • Xue Cheng & Marina Di Giacinto & Tai-Ho Wang, 2019. "Optimal execution with dynamic risk adjustment," Papers 1901.00617, arXiv.org, revised Jul 2019.
    • Handle: RePEc:arx:papers:1901.00617
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    References listed on IDEAS

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    1. Obizhaeva, Anna A. & Wang, Jiang, 2013. "Optimal trading strategy and supply/demand dynamics," Journal of Financial Markets, Elsevier, vol. 16(1), pages 1-32.
    2. Domenico Cuoco & Hua He & Sergei Isaenko, 2008. "Optimal Dynamic Trading Strategies with Risk Limits," Operations Research, INFORMS, vol. 56(2), pages 358-368, April.
    3. Xue Cheng & Marina Di Giacinto & Tai-Ho Wang, 2017. "Optimal execution with uncertain order fills in Almgren–Chriss framework," Quantitative Finance, Taylor & Francis Journals, vol. 17(1), pages 55-69, January.
    4. Jim Gatheral, 2010. "No-dynamic-arbitrage and market impact," Quantitative Finance, Taylor & Francis Journals, vol. 10(7), pages 749-759.
    5. Kiseop Lee, 2008. "Risk minimization under budget constraints," Journal of Risk Finance, Emerald Group Publishing, vol. 9(1), pages 71-80, January.
    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. Kiseop Lee, 2008. "Risk minimization under budget constraints," Journal of Risk Finance, Emerald Group Publishing Limited, vol. 9(1), pages 71-80, January.
    8. Brian Bulthuis & Julio Concha & Tim Leung & Brian Ward, 2017. "Optimal execution of limit and market orders with trade director, speed limiter, and fill uncertainty," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 4(02n03), pages 1-29, June.
    9. Bertsimas, Dimitris & Lo, Andrew W., 1998. "Optimal control of execution costs," Journal of Financial Markets, Elsevier, vol. 1(1), pages 1-50, April.
    10. Robert Almgren, 2003. "Optimal execution with nonlinear impact functions and trading-enhanced risk," Applied Mathematical Finance, Taylor & Francis Journals, vol. 10(1), pages 1-18.
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    Cited by:

    1. Marina Di Giacinto & Claudio Tebaldi & Tai-Ho Wang, 2021. "Optimal order execution under price impact: A hybrid model," Papers 2112.02228, arXiv.org, revised Aug 2022.
    2. Meng Wang & Tai-Ho Wang, 2023. "Relative entropy-regularized robust optimal order execution," Papers 2311.06476, arXiv.org.

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