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Mean field game of controls and an application to trade crowding

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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 facing 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 also 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.

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Notes

  1. Large asset managers, like Blackrock, Amundi, Fidelity or Allianz, delegate the implementation of their investment decisions to a dedicated (internal) team: their dealing desk. This team is in charge of trading to drive the real portfolios to their targets. They are implementing on a day to day basis what this paper is modelling.

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Acknowledgements

Authors thank Marc Abeille for his careful reading of Sect. 3.2.1. The first author was partially supported by the ANR (Agence Nationale de la Recherche) Projects ANR-14-ACHN-0030-01 and ANR-16-CE40-0015-01.

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Authors and Affiliations

  1. CEREMADE, Université Paris-Dauphine, PSL University, Place du Marchal de Lattre de Tassigny, 75775, Paris Cedex 16, France

    Pierre Cardaliaguet

  2. Capital Fund Management (CFM), Paris, France

    Charles-Albert Lehalle

  3. Imperial College, London, UK

    Charles-Albert Lehalle

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  1. Pierre Cardaliaguet
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  2. Charles-Albert Lehalle
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Correspondence to Charles-Albert Lehalle.

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Cardaliaguet, P., Lehalle, CA. Mean field game of controls and an application to trade crowding. Math Finan Econ 12, 335–363 (2018). https://doi.org/10.1007/s11579-017-0206-z

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  • DOI: https://doi.org/10.1007/s11579-017-0206-z

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