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Mean‐field games with differing beliefs for algorithmic trading

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  • Philippe Casgrain
  • Sebastian Jaimungal

Abstract

Even when confronted with the same data, agents often disagree on a model of the real world. Here, we address the question of how interacting heterogeneous agents, who disagree on what model the real world follows, optimize their trading actions. The market has latent factors that drive prices, and agents account for the permanent impact they have on prices. This leads to a large stochastic game, where each agents performance criteria are computed under a different probability measure. We analyze the mean‐field game (MFG) limit of the stochastic game and show that the Nash equilibrium is given by the solution to a nonstandard vector‐valued forward–backward stochastic differential equation. Under some mild assumptions, we construct the solution in terms of expectations of the filtered states. Furthermore, we prove that the MFG strategy forms an ε‐Nash equilibrium for the finite player game. Finally, we present a least square Monte Carlo based algorithm for computing the equilibria and show through simulations that increasing disagreement may increase price volatility and trading activity.

Suggested Citation

  • 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.
    • Handle: RePEc:bla:mathfi:v:30:y:2020:i:3:p:995-1034
    DOI: 10.1111/mafi.12237
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    References listed on IDEAS

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    1. Olivier Guéant & Pierre Louis Lions & Jean-Michel Lasry, 2011. "Mean Field Games and Applications," Post-Print hal-01393103, HAL.
    2. Bruno Bouchard & Masaaki Fukasawa & Martin Herdegen & Johannes Muhle-Karbe, 2018. "Equilibrium returns with transaction costs," Finance and Stochastics, Springer, vol. 22(3), pages 569-601, July.
    3. Bruno Bouchard & Masaaki Fukasawa & Martin Herdegen & Johannes Muhle-Karbe, 2017. "Equilibrium Returns with Transaction Costs," Papers 1707.08464, arXiv.org, revised Apr 2018.
    4. Bruno Bouchard & Masaaki Fukasawa & Martin Herdegen & Johannes Muhle-Karbe, 2018. "Equilibrium Returns with Transaction Costs," Post-Print hal-01569408, HAL.
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    Citations

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    Cited by:

    1. Masaaki Fujii & Akihiko Takahashi, 2021. "Equilibrium Price Formation with a Major Player and its Mean Field Limit," CARF F-Series CARF-F-509, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    2. Arumugam, Devika & Prasanna, P. Krishna & Marathe, Rahul R., 2023. "Do algorithmic traders exploit volatility?," Journal of Behavioral and Experimental Finance, Elsevier, vol. 37(C).
    3. Bastien Baldacci & Philippe Bergault & Dylan Possamai, 2022. "A mean-field game of market-making against strategic traders," Papers 2203.13053, arXiv.org.
    4. Rama Cont & Alessandro Micheli & Eyal Neuman, 2022. "Fast and Slow Optimal Trading with Exogenous Information," Papers 2210.01901, arXiv.org, revised Jun 2023.
    5. Guanxing Fu & Ulrich Horst & Xiaonyu Xia, 2020. "Portfolio Liquidation Games with Self-Exciting Order Flow," Papers 2011.05589, arXiv.org.
    6. Olivier Féron & Peter Tankov & Laura Tinsi, 2020. "Price Formation and Optimal Trading in Intraday Electricity Markets with a Major Player," Risks, MDPI, vol. 8(4), pages 1-21, December.
    7. Masaaki Fujii & Akihiko Takahashi, 2021. "``Equilibrium Price Formation with a Major Player and its Mean Field Limit''," CIRJE F-Series CIRJE-F-1162, CIRJE, Faculty of Economics, University of Tokyo.
    8. Ludovic Tangpi & Shichun Wang, 2022. "Optimal Bubble Riding: A Mean Field Game with Varying Entry Times," Papers 2209.04001, arXiv.org, revised Jan 2024.
    9. David Evangelista & Yuri Saporito & Yuri Thamsten, 2022. "Price formation in financial markets: a game-theoretic perspective," Papers 2202.11416, arXiv.org.
    10. Arvind Shrivats & Dena Firoozi & Sebastian Jaimungal, 2020. "A Mean-Field Game Approach to Equilibrium Pricing in Solar Renewable Energy Certificate Markets," Papers 2003.04938, arXiv.org, revised Aug 2021.
    11. Guillermo Alonso Alvarez & Sergey Nadtochiy & Kevin Webster, 2022. "Optimal brokerage contracts in Almgren-Chriss model with multiple clients," Papers 2204.05403, arXiv.org.
    12. Pierre Lavigne & Peter Tankov, 2023. "Decarbonization of financial markets: a mean-field game approach," Papers 2301.09163, arXiv.org.
    13. Guanxing Fu & Paul P. Hager & Ulrich Horst, 2023. "Mean-Field Liquidation Games with Market Drop-out," Papers 2303.05783, arXiv.org, revised Sep 2023.
    14. Olivier F'eron & Peter Tankov & Laura Tinsi, 2020. "Price formation and optimal trading in intraday electricity markets," Papers 2009.04786, arXiv.org, revised Jun 2021.
    15. Arvind V. Shrivats & Dena Firoozi & Sebastian Jaimungal, 2022. "A mean‐field game approach to equilibrium pricing in solar renewable energy certificate markets," Mathematical Finance, Wiley Blackwell, vol. 32(3), pages 779-824, July.
    16. Dena Firoozi & Arvind V Shrivats & Sebastian Jaimungal, 2021. "Principal agent mean field games in REC markets," Papers 2112.11963, arXiv.org, revised Jun 2022.
    17. Philippe Bergault & Leandro S'anchez-Betancourt, 2024. "A Mean Field Game between Informed Traders and a Broker," Papers 2401.05257, arXiv.org.
    18. Alessandro Micheli & Johannes Muhle-Karbe & Eyal Neuman, 2021. "Closed-Loop Nash Competition for Liquidity," Papers 2112.02961, arXiv.org, revised Jun 2023.
    19. Masaaki Fujii & Akihiko Takahashi, 2021. "Equilibrium Price Formation with a Major Player and its Mean Field Limit," Papers 2102.10756, arXiv.org, revised Feb 2022.
    20. Masaaki Fujii, 2022. "Equilibrium pricing of securities in the co-presence of cooperative and non-cooperative populations," Papers 2209.12639, arXiv.org, revised Jun 2023.
    21. Olivier F'eron & Peter Tankov & Laura Tinsi, 2020. "Price formation and optimal trading in intraday electricity markets with a major player," Papers 2011.07655, arXiv.org.
    22. Moritz Voß, 2022. "A two-player portfolio tracking game," Mathematics and Financial Economics, Springer, volume 16, number 6, June.
    23. Sebastian Jaimungal, 2022. "Reinforcement learning and stochastic optimisation," Finance and Stochastics, Springer, vol. 26(1), pages 103-129, January.

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