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A Mean Field Game between Informed Traders and a Broker

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  • Philippe Bergault
  • Leandro S'anchez-Betancourt

Abstract

We find closed-form solutions to the stochastic game between a broker and a mean-field of informed traders. In the finite player game, the informed traders observe a common signal and a private signal. The broker, on the other hand, observes the trading speed of each of his clients and provides liquidity to the informed traders. Each player in the game optimises wealth adjusted by inventory penalties. In the mean field version of the game, using a G\^ateaux derivative approach, we characterise the solution to the game with a system of forward-backward stochastic differential equations that we solve explicitly. We find that the optimal trading strategy of the broker is linear on his own inventory, on the average inventory among informed traders, and on the common signal or the average trading speed of the informed traders. The Nash equilibrium we find helps informed traders decide how to use private information, and helps brokers decide how much of the order flow they should externalise or internalise when facing a large number of clients.

Suggested Citation

  • Philippe Bergault & Leandro S'anchez-Betancourt, 2024. "A Mean Field Game between Informed Traders and a Broker," Papers 2401.05257, arXiv.org.
    • Handle: RePEc:arx:papers:2401.05257
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    References listed on IDEAS

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    1. 'Alvaro Cartea & Sebastian Jaimungal & Tianyi Jia, 2020. "Trading Foreign Exchange Triplets," Papers 2004.12011, arXiv.org.
    2. Andreas Schrimpf & Vladyslav Sushko, 2019. "FX trade execution: complex and highly fragmented," BIS Quarterly Review, Bank for International Settlements, December.
    3. Alexander Barzykin & Philippe Bergault & Olivier Guéant, 2023. "Algorithmic market making in dealer markets with hedging and market impact," Mathematical Finance, Wiley Blackwell, vol. 33(1), pages 41-79, January.
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    5. 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.
    6. Marcel Nutz & Kevin Webster & Long Zhao, 2023. "Unwinding Stochastic Order Flow: When to Warehouse Trades," Papers 2310.14144, arXiv.org.
    7. repec:hal:wpaper:hal-03941548 is not listed on IDEAS
    8. Marco Avellaneda & Sasha Stoikov, 2008. "High-frequency trading in a limit order book," Quantitative Finance, Taylor & Francis Journals, vol. 8(3), pages 217-224.
    9. Xuancheng Huang & Sebastian Jaimungal & Mojtaba Nourian, 2019. "Mean-Field Game Strategies for Optimal Execution," Applied Mathematical Finance, Taylor & Francis Journals, vol. 26(2), pages 153-185, March.
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