June  2012, 7(2): 349-361. doi: 10.3934/nhm.2012.7.349

Liquidity generated by heterogeneous beliefs and costly estimations

1. 

CEREMADE, Universite Paris Dauphine, Place du Marechal de Lattre de Tassigny, 75016 Paris, France

2. 

CEREMADE, Université Paris Dauphine, Place du Marechal de Lattre de Tassigny, 75016 Paris, France

Received  November 2011 Revised  March 2012 Published  June 2012

We study the liquidity, defined as the size of the trading volume, in a situation where an infinite number of agents with heterogeneous beliefs reach a trade-off between the cost of a precise estimation (variable depending on the agent) and the expected wealth from trading. The "true" asset price is not known and the market price is set at a level that clears the market. We show that, under some technical assumptions, the model has natural properties such as monotony of supply and demand functions with respect to the price, existence of an equilibrium and monotony with respect to the marginal cost of information. We also situate our approach within the Mean Field Games (MFG) framework of Lions and Lasry which allows to obtain an interpretation as a limit of Nash equilibrium for an infinite number of agents.
Citation: Min Shen, Gabriel Turinici. Liquidity generated by heterogeneous beliefs and costly estimations. Networks and Heterogeneous Media, 2012, 7 (2) : 349-361. doi: 10.3934/nhm.2012.7.349
References:
[1]

Yves Achdou, Fabio Camilli and Italo Capuzzo-Dolcetta, Mean field games: Numerical methods for the planning problem, SIAM Journal on Control and Optimization, 50 (2012), 77-109. doi: 10.1137/100790069.

[2]

Yves Achdou and Italo Capuzzo-Dolcetta, Mean field games: Numerical methods, SIAM J. Numer. Anal., 48 (2010), 1136-1162. doi: 10.1137/090758477.

[3]

Marco Avellaneda and Sasha Stoikov, High-frequency trading in a limit order book, Quantitative Finance, 8 (2008), 217-224.

[4]

Agnes Bialecki, Eleonore Haguet and Gabriel Turinici, Trading volume as equilibrium induced by heterogeneous uncertain estimations of a continuum of agents, in preparation, 2012.

[5]

Michael Gallmeyer and Burton Hollifield, An examination of heterogeneous beliefs with a short-sale constraint in a dynamic economy, Review of Finance, 12 (2008), 323-364. doi: 10.1093/rof/rfm036.

[6]

Diogo A. Gomes, Joana Mohr and Rafael Rigao Souza, Discrete time, finite state space mean field games, Journal de Mathématiques Pures et Appliquées (9), 93 (2010), 308-328.

[7]

Olivier Guéant, A reference case for mean field games models, Journal de Mathématiques Pures et Appliquées (9), 92 (2009), 276-294.

[8]

Roger Guesnerie, An exploration of the eductive justifications of the rational-expectations hypothesis, The American Economic Review, 82 (1992).

[9]

Alexandra Hachmeister, "Informed Traders as Liquidity Providers," DUV, 2007.

[10]

E. Jouini and C. Napp, Aggregation of heterogeneous beliefs, Journal of Mathematical Economics, 42 (2006), 752-770. doi: 10.1016/j.jmateco.2006.02.001.

[11]

Elyès Jouini and Clotilde Napp, Heterogeneous beliefs and asset pricing in discrete time: An analysis of pessimism and doubt, Journal of Economic Dynamics and Control, 30 (2006), 1233-1260. doi: 10.1016/j.jedc.2005.05.008.

[12]

Aime Lachapelle, Julien Salomon and Gabriel Turinici, Computation of mean field equilibria in economics, Math. Models Methods Appl. Sci., 20 (2010), 567-588. doi: 10.1142/S0218202510004349.

[13]

Aimé Lachapelle and Marie-Therese Wolfram, On a mean field game approach modeling congestion and aversion in pedestrian crowds, Transportation Research Part B: Methodological, 45 (2011), 1572-1589. doi: 10.1016/j.trb.2011.07.011.

[14]

Jean-Michel Lasry and Pierre-Louis Lions, Mean field games. I. The stationary case, Comptes Rendus Mathematique Acad. Sci. Paris, 343 (2006), 619-625. doi: 10.1016/j.crma.2006.09.019.

[15]

Jean-Michel Lasry and Pierre-Louis Lions, Mean field games. II. Finite horizon and optimal control, Comptes Rendus Mathematique Acad. Sci. Paris, 343 (2006), 679-684. doi: 10.1016/j.crma.2006.09.018.

[16]

Jean-Michel Lasry and Pierre-Louis Lions, Mean field games, Japanese Journal of Mathematics, 2 (2007), 229-260.

[17]

Pierre-Louis Lions, Mean field games course at Collège de France, video files., Available from: \url{http://www.college-de-france.fr/}., (). 

[18]

Maureen O'Hara, "Market Microstructure Theory," Blackwell Business, March, 1997.

[19]

Emilio Osambela, Asset pricing with heterogeneous beliefs and endogenous liquidity constraints, SSRN eLibrary, 2010.

[20]

Min Shen and Gabriel Turinici, "Mean Field Game Theory Applied in Financial Market Liquidity," internal report, CEREMADE, Université Paris Dauphine, 2011.

show all references

References:
[1]

Yves Achdou, Fabio Camilli and Italo Capuzzo-Dolcetta, Mean field games: Numerical methods for the planning problem, SIAM Journal on Control and Optimization, 50 (2012), 77-109. doi: 10.1137/100790069.

[2]

Yves Achdou and Italo Capuzzo-Dolcetta, Mean field games: Numerical methods, SIAM J. Numer. Anal., 48 (2010), 1136-1162. doi: 10.1137/090758477.

[3]

Marco Avellaneda and Sasha Stoikov, High-frequency trading in a limit order book, Quantitative Finance, 8 (2008), 217-224.

[4]

Agnes Bialecki, Eleonore Haguet and Gabriel Turinici, Trading volume as equilibrium induced by heterogeneous uncertain estimations of a continuum of agents, in preparation, 2012.

[5]

Michael Gallmeyer and Burton Hollifield, An examination of heterogeneous beliefs with a short-sale constraint in a dynamic economy, Review of Finance, 12 (2008), 323-364. doi: 10.1093/rof/rfm036.

[6]

Diogo A. Gomes, Joana Mohr and Rafael Rigao Souza, Discrete time, finite state space mean field games, Journal de Mathématiques Pures et Appliquées (9), 93 (2010), 308-328.

[7]

Olivier Guéant, A reference case for mean field games models, Journal de Mathématiques Pures et Appliquées (9), 92 (2009), 276-294.

[8]

Roger Guesnerie, An exploration of the eductive justifications of the rational-expectations hypothesis, The American Economic Review, 82 (1992).

[9]

Alexandra Hachmeister, "Informed Traders as Liquidity Providers," DUV, 2007.

[10]

E. Jouini and C. Napp, Aggregation of heterogeneous beliefs, Journal of Mathematical Economics, 42 (2006), 752-770. doi: 10.1016/j.jmateco.2006.02.001.

[11]

Elyès Jouini and Clotilde Napp, Heterogeneous beliefs and asset pricing in discrete time: An analysis of pessimism and doubt, Journal of Economic Dynamics and Control, 30 (2006), 1233-1260. doi: 10.1016/j.jedc.2005.05.008.

[12]

Aime Lachapelle, Julien Salomon and Gabriel Turinici, Computation of mean field equilibria in economics, Math. Models Methods Appl. Sci., 20 (2010), 567-588. doi: 10.1142/S0218202510004349.

[13]

Aimé Lachapelle and Marie-Therese Wolfram, On a mean field game approach modeling congestion and aversion in pedestrian crowds, Transportation Research Part B: Methodological, 45 (2011), 1572-1589. doi: 10.1016/j.trb.2011.07.011.

[14]

Jean-Michel Lasry and Pierre-Louis Lions, Mean field games. I. The stationary case, Comptes Rendus Mathematique Acad. Sci. Paris, 343 (2006), 619-625. doi: 10.1016/j.crma.2006.09.019.

[15]

Jean-Michel Lasry and Pierre-Louis Lions, Mean field games. II. Finite horizon and optimal control, Comptes Rendus Mathematique Acad. Sci. Paris, 343 (2006), 679-684. doi: 10.1016/j.crma.2006.09.018.

[16]

Jean-Michel Lasry and Pierre-Louis Lions, Mean field games, Japanese Journal of Mathematics, 2 (2007), 229-260.

[17]

Pierre-Louis Lions, Mean field games course at Collège de France, video files., Available from: \url{http://www.college-de-france.fr/}., (). 

[18]

Maureen O'Hara, "Market Microstructure Theory," Blackwell Business, March, 1997.

[19]

Emilio Osambela, Asset pricing with heterogeneous beliefs and endogenous liquidity constraints, SSRN eLibrary, 2010.

[20]

Min Shen and Gabriel Turinici, "Mean Field Game Theory Applied in Financial Market Liquidity," internal report, CEREMADE, Université Paris Dauphine, 2011.

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