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Mean-Field Liquidation Games with Market Drop-out

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  • Guanxing Fu
  • Paul P. Hager
  • Ulrich Horst

Abstract

We consider a novel class of portfolio liquidation games with market drop-out ("absorption"). More precisely, we consider mean-field and finite player liquidation games where a player drops out of the market when her position hits zero. In particular round-trips are not admissible. This can be viewed as a no statistical arbitrage condition. In a model with only sellers we prove that the absorption condition is equivalent to a short selling constraint. We prove that equilibria (both in the mean-field and the finite player game) are given as solutions to a non-linear higher-order integral equation with endogenous terminal condition. We prove the existence of a unique solution to the integral equation from which we obtain the existence of a unique equilibrium in the MFG and the existence of a unique equilibrium in the $N$-player game. We establish the convergence of the equilibria in the finite player games to the obtained mean-field equilibrium and illustrate the impact of the drop-out constraint on equilibrium trading rates.

Suggested Citation

  • Guanxing Fu & Paul P. Hager & Ulrich Horst, 2023. "Mean-Field Liquidation Games with Market Drop-out," Papers 2303.05783, arXiv.org, revised Sep 2023.
    • Handle: RePEc:arx:papers:2303.05783
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    References listed on IDEAS

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    1. Paulwin Graewe & Ulrich Horst, 2016. "Optimal Trade Execution with Instantaneous Price Impact and Stochastic Resilience," Papers 1611.03435, arXiv.org, revised Jul 2017.
    2. Alfonsi Aurélien & Alexander Schied & Alla Slynko, 2012. "Order Book Resilience, Price Manipulation, and the Positive Portfolio Problem," Post-Print hal-00941333, HAL.
    3. Kruse, T. & Popier, A., 2016. "Minimal supersolutions for BSDEs with singular terminal condition and application to optimal position targeting," Stochastic Processes and their Applications, Elsevier, vol. 126(9), pages 2554-2592.
    4. 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.
    5. Antje Fruth & Torsten Schöneborn & Mikhail Urusov, 2014. "Optimal Trade Execution And Price Manipulation In Order Books With Time-Varying Liquidity," Mathematical Finance, Wiley Blackwell, vol. 24(4), pages 651-695, October.
    6. Obizhaeva, Anna A. & Wang, Jiang, 2013. "Optimal trading strategy and supply/demand dynamics," Journal of Financial Markets, Elsevier, vol. 16(1), pages 1-32.
    7. Alexander Schied & Tao Zhang, 2019. "A Market Impact Game Under Transient Price Impact," Mathematics of Operations Research, INFORMS, vol. 44(1), pages 102-121, February.
    8. Fu, Guanxing & Horst, Ulrich & Xia, Xiaonyu, 2022. "Portfolio Liquidation Games with Self-Exciting Order Flow," Rationality and Competition Discussion Paper Series 327, CRC TRR 190 Rationality and Competition.
    9. Alexander Schied & Elias Strehle & Tao Zhang, 2015. "High-frequency limit of Nash equilibria in a market impact game with transient price impact," Papers 1509.08281, arXiv.org, revised May 2017.
    10. Philippe Casgrain & Sebastian Jaimungal, 2020. "Mean‐field games with differing beliefs for algorithmic trading," Mathematical Finance, Wiley Blackwell, vol. 30(3), pages 995-1034, July.
    11. Peter Kratz, 2014. "An Explicit Solution of a Nonlinear-Quadratic Constrained Stochastic Control Problem with Jumps: Optimal Liquidation in Dark Pools with Adverse Selection," Mathematics of Operations Research, INFORMS, vol. 39(4), pages 1198-1220, November.
    12. Paulwin Graewe & Ulrich Horst & Jinniao Qiu, 2013. "A Non-Markovian Liquidation Problem and Backward SPDEs with Singular Terminal Conditions," Papers 1309.0461, arXiv.org, revised Jan 2015.
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    Cited by:

    1. Dianetti, Jodi & Ferrari, Giorgio & Tzouanas, Ioannis, 2023. "Ergodic Mean-Field Games of Singular Control with Regime-Switching (extended version)," Center for Mathematical Economics Working Papers 681, Center for Mathematical Economics, Bielefeld University.
    2. Dianetti, Jodi, 2023. "Linear-Quadratic-Singular Stochastic Differential Games and Applications," Center for Mathematical Economics Working Papers 678, Center for Mathematical Economics, Bielefeld University.

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