Mathematics and Financial Economics

, Volume 10, Issue 3, pp 223–262 | Cite as

Efficiency of the price formation process in presence of high frequency participants: a mean field game analysis

  • Aimé Lachapelle
  • Jean-Michel Lasry
  • Charles-Albert LehalleEmail author
  • Pierre-Louis Lions


This paper deals with a stochastic order-driven market model with waiting costs, for orderbooks with heterogenous traders. Offer and demand of liquidity drives price formation and traders anticipate future evolutions of the orderbook. The natural framework we use is mean field game theory, a class of stochastic differential games with a continuum of anonymous players. Several sources of heterogeneity are considered including the mean size of orders. Thus we are able to consider the coexistence of Institutional Investors and high frequency traders (HFT). We provide both analytical solutions and numerical experiments. Implications on classical quantities are explored: orderbook size, prices, and effective bid/ask spread. According to the model, in markets with Institutional Investors only we show the existence of inefficient liquidity imbalances in equilibrium, with two symmetrical situations corresponding to what we call liquidity calls for liquidity. During these situations the transaction price significantly moves away from the fair price. However this macro phenomenon disappears in markets with both Institutional Investors and HFT, although a more precise study shows that the benefits of the new situation go to HFT only, leaving Institutional Investors even with higher trading costs.


Orderbook modeling Mean field games Order-driven market Waiting cost Liquidity equilibrium High frequency trading Market microstructure Price formation process 

JEL Classification

C730 (Stochastic and Dynamic Games) G140 (Information and Market Efficiency) 



This work has been partially granted by the Crédit Agricole Cheuvreux Research Initiative in partnership with the Louis Bachelier Institute, the Collège de France and the Europlace Institute of Finance. Authors thank Ioanid Roşu for fruitful discussions about the orderbook model.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.MFG LabsParisFrance
  2. 2.CEREMADEParisFrance
  3. 3.Capital Fund ManagementParisFrance
  4. 4.Imperial CollegeLondonUK
  5. 5.CEREMADE and Collège de FranceParisFrance

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