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Mean Field Game of Controls and An Application To Trade Crowding

Author

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  • Pierre Cardaliaguet

    (CEREMADE)

  • Charles-Albert Lehalle

Abstract

In this paper we formulate the now classical problem of optimal liquidation (or optimal trading) inside a Mean Field Game (MFG). This is a noticeable change since usually mathematical frameworks focus on one large trader in front of a "background noise" (or "mean field"). In standard frameworks, the interactions between the large trader and the price are a temporary and a permanent market impact terms, the latter influencing the public price. In this paper the trader faces the uncertainty of fair price changes too but not only. He has to deal with price changes generated by other similar market participants, impacting the prices permanently too, and acting strategically. Our MFG formulation of this problem belongs to the class of "extended MFG", we hence provide generic results to address these "MFG of controls", before solving the one generated by the cost function of optimal trading. We provide a closed form formula of its solution, and address the case of "heterogenous preferences" (when each participant has a different risk aversion). Last but not least we give conditions under which participants do not need to instantaneously know the state of the whole system, but can "learn" it day after day, observing others' behaviors.

Suggested Citation

  • Pierre Cardaliaguet & Charles-Albert Lehalle, 2016. "Mean Field Game of Controls and An Application To Trade Crowding," Papers 1610.09904, arXiv.org, revised Sep 2017.
  • Handle: RePEc:arx:papers:1610.09904
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    File URL: http://arxiv.org/pdf/1610.09904
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    References listed on IDEAS

    as
    1. Charles-Albert Lehalle & Sophie Laruelle (ed.), 2013. "Market Microstructure in Practice," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 8967.
    2. Olivier Guéant & Charles-Albert Lehalle, 2015. "General Intensity Shapes In Optimal Liquidation," Mathematical Finance, Wiley Blackwell, vol. 25(3), pages 457-495, July.
    3. Fabien Guilbaud & Huyen Pham, 2011. "Optimal High Frequency Trading with limit and market orders," Working Papers hal-00603385, HAL.
    4. X. Brokmann & E. Serie & J. Kockelkoren & J. -P. Bouchaud, 2014. "Slow decay of impact in equity markets," Papers 1407.3390, arXiv.org.
    5. Fabien Guilbaud & Huyen Pham, 2011. "Optimal High Frequency Trading with limit and market orders," Papers 1106.5040, arXiv.org.
    6. Emmanuel Bacry & Adrian Iuga & Matthieu Lasnier & Charles-Albert Lehalle, 2014. "Market impacts and the life cycle of investors orders," Papers 1412.0217, arXiv.org, revised Dec 2014.
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    Cited by:

    1. Ali Al-Aradi & Adolfo Correia & Danilo Naiff & Gabriel Jardim & Yuri Saporito, 2018. "Solving Nonlinear and High-Dimensional Partial Differential Equations via Deep Learning," Papers 1811.08782, arXiv.org.
    2. Clémence Alasseur & Imen Ben Taher & Anis Matoussi, 2020. "An Extended Mean Field Game for Storage in Smart Grids," Journal of Optimization Theory and Applications, Springer, vol. 184(2), pages 644-670, February.
    3. David Evangelista & Yuri Thamsten, 2020. "On finite population games of optimal trading," Papers 2004.00790, arXiv.org, revised Sep 2020.

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