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Optimal Trade Execution Under Endogenous Pressure to Liquidate: Theory and Numerical Solutions

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  • Pavol Brunovsk'y
  • Alev{s} v{C}ern'y
  • J'an Komadel

Abstract

We study optimal liquidation of a trading position (so-called block order or meta-order) in a market with a linear temporary price impact (Kyle, 1985). We endogenize the pressure to liquidate by introducing a downward drift in the unaffected asset price while simultaneously ruling out short sales. In this setting the liquidation time horizon becomes a stopping time determined endogenously, as part of the optimal strategy. We find that the optimal liquidation strategy is consistent with the square-root law which states that the average price impact per share is proportional to the square root of the size of the meta-order (Bershova and Rakhlin, 2013; Farmer et al., 2013; Donier et al., 2015; T\'oth et al., 2016). Mathematically, the Hamilton-Jacobi-Bellman equation of our optimization leads to a severely singular and numerically unstable ordinary differential equation initial value problem. We provide careful analysis of related singular mixed boundary value problems and devise a numerically stable computation strategy by re-introducing time dimension into an otherwise time-homogeneous task.

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  • Pavol Brunovsk'y & Alev{s} v{C}ern'y & J'an Komadel, 2017. "Optimal Trade Execution Under Endogenous Pressure to Liquidate: Theory and Numerical Solutions," Papers 1707.07284, arXiv.org.
    • Handle: RePEc:arx:papers:1707.07284
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    Cited by:

    1. Li, Xin & Liu, Haibo & Tang, Qihe & Zhu, Jinxia, 2020. "Liquidation risk in insurance under contemporary regulatory frameworks," Insurance: Mathematics and Economics, Elsevier, vol. 93(C), pages 36-49.
    2. Sadoghi, Amirhossein & Vecer, Jan, 2022. "Optimal liquidation problem in illiquid markets," European Journal of Operational Research, Elsevier, vol. 296(3), pages 1050-1066.
    3. Ramirez, H & Sanchez, J. F, 2023. "Optimal liquidation with temporary and permanent price impact, an application to cryptocurrencies," Documentos de Trabajo 20669, Universidad del Rosario.
    4. Josef Jablonský & Michal Černý & Juraj Pekár, 2022. "The last dozen of years of OR research in Czechia and Slovakia," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 30(2), pages 435-447, June.
    5. Amirhossein Sadoghi & Jan Vecer, 2022. "Optimal liquidation problem in illiquid markets," Post-Print hal-03696768, HAL.
    6. Hugo E. Ramirez & Juli'an Fernando Sanchez, 2023. "Optimal liquidation with temporary and permanent price impact, an application to cryptocurrencies," Papers 2303.10043, arXiv.org.

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