Mathematics and Financial Economics

, Volume 12, Issue 3, pp 335–363 | Cite as

Mean field game of controls and an application to trade crowding

  • Pierre Cardaliaguet
  • Charles-Albert Lehalle


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.


Mean field games Market microstructure Crowding Optimal liquidation Optimal trading Optimal stochastic control 

JEL Classification

C61 C73 G14 D53 



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.


  1. 1.
    Alfonsi, A., Blanc, P.: Dynamic optimal execution in a mixed-market-impact Hawkes price model. Finance Stoch. 20(1), 183–218 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Almgren, R.F., Chriss, N.: Optimal execution of portfolio transactions. J. Risk 3(2), 5–39 (2000)CrossRefGoogle Scholar
  3. 3.
    Almgren, R., Thum, C., Hauptmann, E., Li, H.: Direct estimation of equity market impact. Risk 18, 57–62 (2005)Google Scholar
  4. 4.
    Bacry, E., Iuga, A., Lasnier, M., Lehalle, C.-A.: Market impacts and the life cycle of investors orders. Mark. Microstruct. Liq. 1(2), 1550009 (2015)CrossRefGoogle Scholar
  5. 5.
    Barles, G.: A new stability result for viscosity solutions of nonlinear parabolic equations with weak convergence in time. C. R. Math. 343(3), 173–178 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bensoussan, A., Frehse, J., Yam, P.: Mean Field Games and Mean Field Type Control Theory, vol. 101. Springer, Berlin (2013)CrossRefzbMATHGoogle Scholar
  7. 7.
    Bensoussan, A., Sung, K.C.J., Yam, S.C.P., Yung, S.P.: Linear-quadratic mean field games. J. Optim. Theory Appl. 169(2), 496–529 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bernhard, P.: The Robust Control Approach to Option Pricing and Interval Models: An Overview. In: Breton M., Ben-Ameur H. (eds) Numerical Methods in Finance. Springer, Boston, MA (2005)Google Scholar
  9. 9.
    Bouchard, B., Dang, N.-M., Lehalle, C.-A.: Optimal control of trading algorithms: a general impulse control approach. SIAM J. Financ. Math. 2(1), 404–438 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Brokmann, X., Serie, E., Kockelkoren, J., Bouchaud, J.-P.: Slow decay of impact in equity markets. Mark. Microstruct. Liq. 1(02), 1550007 (2015)CrossRefGoogle Scholar
  11. 11.
    Caines, P.E.: Mean field games. In: Baillieul J., Samad T. (eds) Encyclopedia of Systems and Control, pp. 706–712 (2015)Google Scholar
  12. 12.
    Cardaliaguet, P., Hadikhanloo, S.: Learning in mean field games: the fictitious play. ESAIM Control Optim. Calc. Var. 23(2), 569–591 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Carmona, R., Delarue, F.: Probabilistic Theory of Mean Field Games with Applications. Springer, Berlin (2018)Google Scholar
  14. 14.
    Carmona, R., Fouque, J.-P., Sun, L.-H.: Mean field games and systemic risk. Social Science Research Network Working Paper Series (2013)Google Scholar
  15. 15.
    Carmona, R., Lacker, D.: A probabilistic weak formulation of mean field games and applications. Ann. Appl. Probab. 25(3), 1189–1231 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Cartea, A., Donnelly, R., Jaimungal, S.: Algorithmic trading with model uncertainty. SIAM J. Financ. Math. 8(1), 635–671 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Cartea, Á., Jaimungal, S.: Incorporating order-flow into optimal execution. Social Science Research Network Working Paper Series (2015)Google Scholar
  18. 18.
    Cartea, Á., Jaimungal, S., Penalva, J.: Algorithmic and High-Frequency Trading (Mathematics, Finance and Risk), 1st edn. Cambridge University Press, Cambridge (2015)zbMATHGoogle Scholar
  19. 19.
    Fudenberg, D., Levine, D.K.: The Theory of Learning in Games, vol. 2. MIT Press, Cambridge (1998)zbMATHGoogle Scholar
  20. 20.
    Gomes, D.A.: Mean field games models? A brief survey. Dyn. Games Appl. 4(2), 110–154 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Gomes, D.A., Patrizi, S., Voskanyan, V.: On the existence of classical solutions for stationary extended mean field games. Nonlinear Anal. Theory Methods Appl. 99, 49–79 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Gomes, D.A., Voskanyan, V.K.: Extended deterministic mean-field games. SIAM J. Control Optim. 54(2), 1030–1055 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Guéant, O.: The Financial Mathematics of Market Liquidity: From Optimal Execution to Market Making. Chapman and Hall/CRC, London (2016)zbMATHGoogle Scholar
  24. 24.
    Guéant, O., Lasry, J.-M., Lions, P.-L.: Mean field games and applications. In: Paris-Princeton Lectures on Mathematical Finance 2010, pp. 205–266. Springer, Berlin (2011)Google Scholar
  25. 25.
    Guéant, O., Lehalle, C.-A.: General intensity shapes in optimal liquidation. Math. Finance 25(3), 457–495 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Guilbaud, F., Pham, H.: Optimal High Frequency Trading with limit and market orders. Quant. Finance 13(1), 79–94 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Huang, M., Malhamé, R.P., Caines, P.E.: Large population stochastic dynamic games: closed-loop McKean–Vlasov systems and the Nash certainty equivalence principle. Commun. Inf. Syst. 6(3), 221–252 (2006)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Jaimungal, S., Nourian, M., Huang, X.: Mean-Field Game Strategies for a Major–Minor Agent Optimal Execution Problem. Social Science Research Network Working Paper Series (2015)Google Scholar
  29. 29.
    Kizilkale, A.C., Caines, P.E.: Mean field stochastic adaptive control. IEEE Trans. Automat. Contr. 58(4), 905–920 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Laruelle, S., Lehalle, C.-A., Pagès, G.: Optimal posting price of limit orders: learning by trading. Math. Financ. Econ. 7(3), 359–403 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Lasry, J.-M., Lions, P.-L.: Jeux à champ moyen. I - Le cas stationnaire. C. R. Math. 343(9), 619–625 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Lasry, J.-M., Lions, P.-L.: Jeux à champ moyen. II - Horizon fini et contrôle optimal. C. R. Math. 343(10), 679–684 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Lasry, J.-M., Lions, P.-L.: Mean field games. Jpn. J. Math. 2(1), 229–260 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Lehalle, C.-A., Laruelle, S., Burgot, R., Pelin, S., Lasnier, Matthieu: Market Microstructure in Practice. World Scientific Publishing, Singapore (2013)CrossRefGoogle Scholar
  35. 35.
    Nourian, M., Caines, P.E., Malhamé, R.P., Huang, M.: Mean field LQG control in leader-follower stochastic multi-agent systems: likelihood ratio based adaptation. IEEE Trans. Automat. Contr. 57(11), 2801–2816 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Obizhaeva, A., Wang, J.: Optimal Trading Strategy and Supply/Demand Dynamics. Social Science Research Network Working Paper Series (2005)Google Scholar
  37. 37.
    Waelbroeck, H., Gomes, C.: Is Market Impact a Measure of the Information Value of Trades? Market Response to Liquidity vs. Informed Trades. Social Science Research Network Working Paper Series, July (2013)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.CEREMADEUniversité Paris-Dauphine, PSL UniversityParis Cedex 16France
  2. 2.Capital Fund Management (CFM)ParisFrance
  3. 3.Imperial CollegeLondonUK

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