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A Market Impact Game Under Transient Price Impact

Author

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  • Alexander Schied

    (Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario N2L 3G1 Canada; and Department of Mathematics, University of Mannheim, 68131 Mannheim, Germany)

  • Tao Zhang

    (Department of Mathematics, University of Mannheim, 68131 Mannheim, Germany)

Abstract

We consider a Nash equilibrium between two high-frequency traders (HFTs) in a simple market impact model with transient price impact and additional quadratic transaction costs. We prove existence and uniqueness of the Nash equilibrium and show that, for small transaction costs, the HFTs engage in a “hot potato game,” in which the same asset position is sold back and forth. We then identify a critical value for the size of the transaction costs above, for which all oscillations disappear and strategies become buy only or sell only. Numerical simulations show that, for both traders, the expected costs can be lower with transaction costs than without. Moreover, the costs can increase with the trading frequency if there are no transaction costs but decrease with the trading frequency if transaction costs are sufficiently high. We argue that these effects occur due to the need for protection against predatory trading in the regime of low transaction costs.

Suggested Citation

  • 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.
    • Handle: RePEc:inm:ormoor:v:44:y:2019:i:1:p:102-121
    DOI: 10.1287/moor.2017.0916
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    References listed on IDEAS

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    8. 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.
    9. Bruce Ian Carlin & Miguel Sousa Lobo & S. Viswanathan, 2007. "Episodic Liquidity Crises: Cooperative and Predatory Trading," Journal of Finance, American Finance Association, vol. 62(5), pages 2235-2274, October.
    10. Alexander Schied & Tao Zhang, 2017. "A State-Constrained Differential Game Arising In Optimal Portfolio Liquidation," Mathematical Finance, Wiley Blackwell, vol. 27(3), pages 779-802, July.
    11. Florian Klöck & Alexander Schied & Yuemeng Sun, 2017. "Price manipulation in a market impact model with dark pool," Applied Mathematical Finance, Taylor & Francis Journals, vol. 24(5), pages 417-450, September.
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    Citations

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    Cited by:

    1. Masamitsu Ohnishi & Makoto Shimoshimizu, 2024. "Trade execution games in a Markovian environment," Papers 2405.07184, arXiv.org.
    2. Samuel Drapeau & Peng Luo & Alexander Schied & Dewen Xiong, 2019. "An FBSDE approach to market impact games with stochastic parameters," Papers 2001.00622, arXiv.org.
    3. Francesco Cordoni & Fabrizio Lillo, 2022. "Transient impact from the Nash equilibrium of a permanent market impact game," Papers 2205.00494, arXiv.org, revised Mar 2023.
    4. Guanxing Fu & Ulrich Horst & Xiaonyu Xia, 2020. "Portfolio Liquidation Games with Self-Exciting Order Flow," Papers 2011.05589, arXiv.org.
    5. Guanxing Fu & Ulrich Horst & Xiaonyu Xia, 2022. "Portfolio liquidation games with self‐exciting order flow," Mathematical Finance, Wiley Blackwell, vol. 32(4), pages 1020-1065, October.
    6. Yan Dolinsky & Shir Moshe, 2021. "Utility Indifference Pricing with High Risk Aversion and Small Linear Price Impact," Papers 2111.00451, arXiv.org, revised Jan 2022.
    7. Fabrizio Lillo & Andrea Macr`i, 2024. "Deviations from the Nash equilibrium and emergence of tacit collusion in a two-player optimal execution game with reinforcement learning," Papers 2408.11773, arXiv.org.
    8. Masamitsu Ohnishi & Makoto Shimoshimizu, 2022. "Optimal Pair–Trade Execution with Generalized Cross–Impact," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 29(2), pages 253-289, June.
    9. Moritz Vo{ss}, 2019. "A two-player portfolio tracking game," Papers 1911.05122, arXiv.org, revised Jul 2022.
    10. Puru Gupta & Saul D. Jacka, 2023. "Portfolio Choice In Dynamic Thin Markets: Merton Meets Cournot," Papers 2309.16047, arXiv.org.
    11. Guanxing Fu & Paul P. Hager & Ulrich Horst, 2023. "Mean-Field Liquidation Games with Market Drop-out," Papers 2303.05783, arXiv.org, revised Sep 2023.
    12. Eyal Neuman & Moritz Voß, 2023. "Trading with the crowd," Mathematical Finance, Wiley Blackwell, vol. 33(3), pages 548-617, July.
    13. Eyal Neuman & Moritz Vo{ss}, 2021. "Trading with the Crowd," Papers 2106.09267, arXiv.org, revised Mar 2023.
    14. 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.
    15. Michail Anthropelos & Constantinos Stefanakis, 2024. "Continuous-time Equilibrium Returns in Markets with Price Impact and Transaction Costs," Papers 2405.14418, arXiv.org.
    16. Guanxing Fu & Paul P. Hager & Ulrich Horst, 2024. "A Mean-Field Game of Market Entry: Portfolio Liquidation with Trading Constraints," Papers 2403.10441, arXiv.org.
    17. Moritz Voß, 2022. "A two-player portfolio tracking game," Mathematics and Financial Economics, Springer, volume 16, number 6, December.

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