跳至主要内容

A Trading Execution Model Based on Mean Field Games and Optimal Control

Read full paper at:
http://www.scirp.org/journal/PaperInformation.aspx?PaperID=51277#.VGGkBWfHRK0

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: http://www.econ.univpm.it/recchioni/finance/w19 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: http://www.econ.univpm.it/recchioni/finance.
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.
http://dx.doi.org/10.1016/S1386-4181(97)00012-8
[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.
http://dx.doi.org/10.1137/090763470
[4] Almgren, R. (2012) Optimal Trading with Stochastic Liquidity and Volatility. SIAM Journal of Financial Mathematics, 3, 163-181.
http://dx.doi.org/10.1137/090763470
[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.
http://dx.doi.org/10.1142/S0219024911006577
[6] Schied, A. (2013) Robust Strategies for Optimal Order Execution in the Almgren-Chriss Framework. Applied Mathematical Finance, 20, 264-286.
http://dx.doi.org/10.1080/1350486X.2012.683963
[7] Ankirchner, S., Blanchet-Scalliet, C. and Eyraud-Loisel, A. (2012) Optimal Liquidation with Directional Views and Additional Information. Working Paper:
http://hal.archives-ouvertes.fr/hal-00735298
[8] Guéant, O. (2013) Execution and Block Trade Pricing with Optimal Constant Rate of Participation.
http://arxiv.org/pdf/1210.7608v3.pdf
[9] Guéant, O. and Lehalle, C.A. (2013) General Intensity Shapes in Optimal Liquidation. Mathematical Finance, Published Online.
http://onlinelibrary.wiley.com/doi/10.1111/mafi.12052/pdf
[10] Lasry, J.M. and Lions, P.L. (2007) Mean Field Games. Japanese Journal of Mathematics, 2, 239-260.
http://dx.doi.org/10.1007/s11537-007-0657-8
[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.
http://dx.doi.org/10.1142/S0218202510004349
[14] Shen, M. and Turinici, G. (2012) Liquidity Generated by Heterogeneous Beliefs and Costly Estimation. Networks and Heterogeneous Media, 7, 349-361.
http://dx.doi.org/10.3934/nhm.2012.7.349
[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.
http://dx.doi.org/10.1115/1.3662552
[18] Guéant, O. (2009) A Reference Case for Mean Field Games Models. Journal de Mathématiques Pures et Appliqués, 92, 276-294.
http://dx.doi.org/10.1016/j.matpur.2009.04.008
[19] Stoer, J. and Bulirsch, R. (1980) Introduction to Numerical Analysis. Springer-Verlag, New York.
http://dx.doi.org/10.1007/978-1-4757-5592-3                 eww141111lx
标签：

评论

发表评论

此博客中的热门博文

A Comparison of Methods Used to Determine the Oleic/Linoleic Acid Ratio in Cultivated Peanut (Arachis hypogaea L.)

Cultivated peanut ( Arachis hypogaea L.) is an important oil and food crop. It is also a cheap source of protein, a good source of essential vitamins and minerals, and a component of many food products. The fatty acid composition of peanuts has become increasingly important with the realization that oleic acid content significantly affects the development of rancidity. And oil content of peanuts significantly affects flavor and shelf-life. Early generation screening of breeding lines for high oleic acid content greatly increases the efficiency of developing new peanut varieties. The objective of this study was to compare the accuracy of methods used to classify individual peanut seed as high oleic or not high oleic. Three hundred and seventy-four (374) seeds, spanning twenty-three (23) genotypes varying in oil composition (i.e. high oleic (H) or normal/not high oleic (NH) inclusive of all four peanut market-types (runner, Spanish, Valencia and Virginia), were individually tested ...
发表评论
阅读全文

Location Optimization of a Coal Power Plant to Balance Costs against Plant’s Emission Exposure

Fuel and its delivery cost comprise the biggest expense in coal power plant operations. Delivery of electricity from generation to consumers requires investment in power lines and transmission grids. Placing a coal power plant or multiple power plants near dense population centers can lower transmission costs. If a coalmine is nearby, transportation costs can also be reduced. However, emissions from coal plants play a key role in worsening health crises in many countries. And coal upon combustion produces CO 2 , SO 2 , NO x , CO, Metallic and Particle Matter (PM10 & PM2.5). The presence of these chemical compounds in the atmosphere in close vicinity to humans, livestock, and agriculture carries detrimental health consequences. The goal of the research was to develop a methodology to minimize the public’s exposure to harmful emissions from coal power plants while maintaining minimal operational costs related to electric distribution losses and coal logistics. The objective was...
发表评论
阅读全文

Evaluation of the Safety and Efficacy of Continuous Use of a Home-Use High-Frequency Facial Treatment Appliance

At present, many home-use beauty devices are available in the market. In particular, many products developed for facial treatment use light, e.g., a flash lamp or a light-emitting diode (LED). In this study, the safety of 4 weeks’ continuous use of NEWA TM , a high-frequency facial treatment appliance, every alternate day at home was verified, and its efficacy was evaluated in Japanese individuals with healthy skin aged 30 years or older who complained of sagging of the facial skin.  Transepidermal water loss (TEWL), melanin levels, erythema levels, sebum secretion levels, skin color changes and wrinkle improvement in the facial skin were measured before the appliance began to be used (study baseline), at 2 and 4 weeks after it had begun to be used, and at 2 weeks after completion of the 4-week treatment period (6 weeks from the study baseline). In addition, data obtained by subjective evaluation by the subjects themselves on a visual analog scale (VAS) were also analyzed. Fur...
发表评论
阅读全文