AM  Vol.5 No.19 , November 2014
A Trading Execution Model Based on Mean Field Games and Optimal Control
Abstract: We present a trading execution model that describes the behaviour of a big trader and of a multitude of retail traders operating on the shares of a risky asset. The retail traders are modeled as a population of “conservative” investors that: 1) behave in a similar way, 2) try to avoid abrupt changes in their trading strategies, 3) want to limit the risk due to the fact of having open positions on the asset shares, 4) in the long run want to have a given position on the asset shares. The big trader wants to maximize the revenue resulting from the action of buying or selling a (large) block of asset shares in a given time interval. The behaviour of the retail traders and of the big trader is modeled using respectively a mean field game model and an optimal control problem. These models are coupled by the asset share price dynamic equation. The trading execution strategy adopted by the retail traders is obtained solving the mean field game model. This strategy is used to formulate the optimal control problem that determines the behaviour of the big trader. The previous mathematical models are solved using the dynamic programming principle. In some special cases explicit solutions of the previous models are found. An extensive numerical study of the trading execution model proposed is presented. The interested reader is referred to the website: to find material including animations, an interactive application and an app that helps the understanding of the paper. A general reference to the work of the authors and of their coauthors in mathematical finance is the website:
Cite this paper: Fatone, L. , Mariani, F. , Recchioni, M. and Zirilli, F. (2014) A Trading Execution Model Based on Mean Field Games and Optimal Control. Applied Mathematics, 5, 3091-3116. doi: 10.4236/am.2014.519294.

[1]   Bertsimas, D. and Lo, A.W. (1998) Optimal Control of Liquidation Costs. Journal of Financial Markets, 1, 1-50.

[2]   Almgren, R. and Chriss, N. (2000) Optimal Execution of Portfolio Transactions. Journal of Risk, 3, 5-39.

[3]   Almgren, R. (2003) Optimal Execution with Nonlinear Impact Functions and Trading Enhanced Risk. Applied Mathematical Finance, 10, 1-18.

[4]   Almgren, R. (2012) Optimal Trading with Stochastic Liquidity and Volatility. SIAM Journal of Financial Mathematics, 3, 163-181.

[5]   Gatheral, J. and Schied, A. (2011) Optimal Trade Execution under Geometric Brownian Motion in the Almgren and Chriss Framework. International Journal of Theoretical and Applied Finance, 14, 353-368.

[6]   Schied, A. (2013) Robust Strategies for Optimal Order Execution in the Almgren-Chriss Framework. Applied Mathematical Finance, 20, 264-286.

[7]   Ankirchner, S., Blanchet-Scalliet, C. and Eyraud-Loisel, A. (2012) Optimal Liquidation with Directional Views and Additional Information. Working Paper:

[8]   Guéant, O. (2013) Execution and Block Trade Pricing with Optimal Constant Rate of Participation.

[9]   Guéant, O. and Lehalle, C.A. (2013) General Intensity Shapes in Optimal Liquidation. Mathematical Finance, Published Online.

[10]   Lasry, J.M. and Lions, P.L. (2007) Mean Field Games. Japanese Journal of Mathematics, 2, 239-260.

[11]   Lachapelle, A. and Wolfram, M.T. (2011) On a Mean Field Game Approach Modeling Congestion and Aversion in Pedestrian Crowds. Transportation Research Part B: Methodological, 45, 1572-1589.

[12]   Guéant, O., Lasry, J.M. and Lions, P.L. (2010) Mean Field Games and Oil Production. In: Lasry, J.M., Lautier, D. and Fessler, D., Eds., The Economics of Sustainable Development, Editions Economica, Paris, 139-162.

[13]   Lachapelle, A., Salomon, J. and Turinici, G. (2010) Computation of Mean Field Equilibria in Economics. Mathematical Models and Methods in Applied Sciences, 20, 567-588.

[14]   Shen, M. and Turinici, G. (2012) Liquidity Generated by Heterogeneous Beliefs and Costly Estimation. Networks and Heterogeneous Media, 7, 349-361.

[15]   Couillet, R., Perlaza, S.M., Tembine, H. and Debbah, M. (2012) Electric Vehicles in the Smart Grid: A Mean Field Game Analysis. IEEE Journal on Selected Areas in Communications: Smart Grid Communications Series, 30, 1086-1096.

[16]   Guéant, O., Lasry, J.M. and Lions, P.L. (2011) Mean Field Games and Applications. In: Cousin, A., Crépey, S., Guéant, O., Hobson, D., Jeanblanc, M., Lasry, J.M., et al., Eds., Paris-Princeton Lectures on Mathematical Finance 2010, Lecture Notes in Mathematics, Springer, Berlin, 205-266.

[17]   Kalman, R.E. (1960) A New Approach to Linear Filtering and Prediction Problems. Journal of Basic Engineering, 82, 35-45.

[18]   Guéant, O. (2009) A Reference Case for Mean Field Games Models. Journal de Mathématiques Pures et Appliqués, 92, 276-294.

[19]   Stoer, J. and Bulirsch, R. (1980) Introduction to Numerical Analysis. Springer-Verlag, New York.