American Institute of Mathematical Sciences

Advanced
September  2015, 10(3): 543-557. doi: 10.3934/nhm.2015.10.543

Boltzmann-type models for price formation in the presence of behavioral aspects

Carlo Brugna 1, and  Giuseppe Toscani 2,

1. 

Department of Physics, Via Bassi, 6, 27100 Pavia, Italy
2. 

University of Pavia, Department of Mathematics, Via Ferrata 1, 27100 Pavia, Italy

Received  August 2014 Revised  January 2015 Published  July 2015

We introduce and discuss a new kinetic system for a financial market composed by agents that may belong to two different trader populations, whose behavior determines the price dynamic of a certain stock. Our mesoscopic description is based on the microscopic Lux--Marchesi model [16,17], and share analogies with the recent kinetic model by Maldarella and Pareschi [18], from which it differs in various points. In particular, it takes into account price acceleration, as well as a microscopic binary interaction for the exchange between the two populations of agents. Various numerical simulations show that the model can describe realistic situations, like regimes of boom and crashes, as well as the invariance of the large-time behavior with respect to the number of agents of the market.
Keywords: markets and trades, Kinetic models, opinion formation., multi-agent systems.
Mathematics Subject Classification: Primary: 35B40, 91B60; Secondary: 82C4.
Citation: Carlo Brugna, Giuseppe Toscani. Boltzmann-type models for price formation in the presence of behavioral aspects. Networks and Heterogeneous Media, 2015, 10 (3) : 543-557. doi: 10.3934/nhm.2015.10.543
References:
[1]

R. Bapna, W. Jank and G. Shmueli, Price formation and its dynamics in online auctions, Decision Support Systems, 44 (2008), 641-656.

[2]

A. Chakraborti, Distributions of money in models of market economy, Int. J. Modern Phys. C, 13 (2002), 1315-1321. doi: 10.1142/S0129183102003905.

[3]

A. Chakraborti and B. K. Chakrabarti, Statistical mechanics of money: Effects of saving propensity, Eur. Phys. J. B, 17 (2000), 167-170.

[4]

A. Chatterjee, B. K. Chakrabarti and S. S. Manna, Pareto law in a kinetic model of market with random saving propensity, Physica A, 335 (2004), 155-163. doi: 10.1016/j.physa.2003.11.014.

[5]

A. Chatterjee, S. Yarlagadda and B. K. Chakrabarti, Eds., Econophysics of Wealth Distributions, New Economic Window Series, Springer-Verlag, Milan, 2005.

[6]

A. Chatterjee, B. K. Chakrabarti and R. B. Stinchcombe, Master equation for a kinetic model of trading market and its analytic solution, Phys. Rev. E, 72 (2005), 026126. doi: 10.1103/PhysRevE.72.026126.

[7]

S. Cordier, L. Pareschi and G. Toscani, On a kinetic model for a simple market economy, J. Stat. Phys., 120 (2005), 253-277. doi: 10.1007/s10955-005-5456-0.

[8]

M. Cristelli, L. Pietronero and A. Zaccaria, Critical overview of agent-based models for economics, in Proceedings of the School of Physics E. Fermi, course CLXXVI, Varenna, 2010. E-Print: arXiv:1101.1847.

[9]

A. Drăgulescu and V. M. Yakovenko, Statistical mechanics of money, Eur. Phys. Jour. B, 17 (2000), 723-729.

[10]

B. Düring, D. Matthes and G. Toscani, Kinetic equations modelling wealth redistribution: A comparison of approaches, Phys. Rev. E, 78 (2008), 056103. doi: 10.1103/PhysRevE.78.056103.

[11]

B. Düring, D. Matthes and G. Toscani, A Boltzmann type approach to the formation of wealth distribution curves, Riv. Mat. Univ. Parma, 1 (2009), 199-261.

[12]

D. Kahneman and A. Tversky, Prospect theory: An analysis of decision under risk, Econometrica, 47 (1979), 183-214. doi: 10.1017/CBO9780511609220.014.

[13]

D. Kahneman and A. Tversky, Choices, values, and frames, American Psychologist, 39 (1984), 341-350. doi: 10.1037/0003-066X.39.4.341.

[14]

M. Levy, H. Levy and S. Solomon, Microscopic Simulation of Financial Markets: From Investor Behaviour to Market Phoenomena, Academic Press, San Diego, 2000.

[15]

T. Lux, The socio-economic dynamics of speculative markets: Interacting agents, chaos, and the fat tails of return distributions, Journal of Economic Behavior & Organization, 33 (1998), 143-165. doi: 10.1016/S0167-2681(97)00088-7.

[16]

T. Lux and M. Marchesi, Volatility clustering in financial markets: A microscopic simulation of interacting agents, International Journal of Theoretical and Applied Finance, 3 (2000), 675-702. doi: 10.1142/S0219024900000826.

[17]

T. Lux and M. Marchesi, Scaling and criticality in a stochastic multi-agent model of a financial market, Nature, 397 (1999), 498-500.

[18]

D. Maldarella and L. Pareschi, Kinetic models for socio-economic dynamics of speculative markets, Physica A, 391 (2012), 715-730. doi: 10.1016/j.physa.2011.08.013.

[19]

R. N. Mantegna and H. E. Stanley, An Introduction to Econophysics Correlations and Complexity in Finance, Cambridge University Press, Cambridge, 2007.

[20]

D. Matthes and G. Toscani, On steady distributions of kinetic models of conservative economies, J. Stat. Phys., 130 (2008), 1087-1117. doi: 10.1007/s10955-007-9462-2.

[21]

G. Naldi, L. Pareschi and G. Toscani, Eds., Mathematical Modelling of Collective Behavior in Socio-economic and Life Sciences, Birkhäuser, Boston, 2010. doi: 10.1007/978-0-8176-4946-3.

[22]

L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods, Oxford University Press, Oxford, 2014.

[23]

L. Pareschi and G. Toscani, Wealth distribution and collective knowledge. A Boltzmann approach, Phil. Trans. R. Soc. A, 372 (2014), 20130396, 15pp. doi: 10.1098/rsta.2013.0396.

[24]

F. Slanina, Inelastically scattering particles and wealth distribution in an open economy, Phys. Rev. E, 69 (2004), 046102. doi: 10.1103/PhysRevE.69.046102.

[25]

G. Toscani, Kinetic models of opinion formation, Comm. Math. Scie., 4 (2006), 481-496. doi: 10.4310/CMS.2006.v4.n3.a1.

[26]

J. Voit, The Statistical Mechanics of Financial Markets, Springer Verlag, Berlin, 2005.

show all references

References:
[1]

R. Bapna, W. Jank and G. Shmueli, Price formation and its dynamics in online auctions, Decision Support Systems, 44 (2008), 641-656.

[2]

A. Chakraborti, Distributions of money in models of market economy, Int. J. Modern Phys. C, 13 (2002), 1315-1321. doi: 10.1142/S0129183102003905.

[3]

A. Chakraborti and B. K. Chakrabarti, Statistical mechanics of money: Effects of saving propensity, Eur. Phys. J. B, 17 (2000), 167-170.

[4]

A. Chatterjee, B. K. Chakrabarti and S. S. Manna, Pareto law in a kinetic model of market with random saving propensity, Physica A, 335 (2004), 155-163. doi: 10.1016/j.physa.2003.11.014.

[5]

A. Chatterjee, S. Yarlagadda and B. K. Chakrabarti, Eds., Econophysics of Wealth Distributions, New Economic Window Series, Springer-Verlag, Milan, 2005.

[6]

A. Chatterjee, B. K. Chakrabarti and R. B. Stinchcombe, Master equation for a kinetic model of trading market and its analytic solution, Phys. Rev. E, 72 (2005), 026126. doi: 10.1103/PhysRevE.72.026126.

[7]

S. Cordier, L. Pareschi and G. Toscani, On a kinetic model for a simple market economy, J. Stat. Phys., 120 (2005), 253-277. doi: 10.1007/s10955-005-5456-0.

[8]

M. Cristelli, L. Pietronero and A. Zaccaria, Critical overview of agent-based models for economics, in Proceedings of the School of Physics E. Fermi, course CLXXVI, Varenna, 2010. E-Print: arXiv:1101.1847.

[9]

A. Drăgulescu and V. M. Yakovenko, Statistical mechanics of money, Eur. Phys. Jour. B, 17 (2000), 723-729.

[10]

B. Düring, D. Matthes and G. Toscani, Kinetic equations modelling wealth redistribution: A comparison of approaches, Phys. Rev. E, 78 (2008), 056103. doi: 10.1103/PhysRevE.78.056103.

[11]

B. Düring, D. Matthes and G. Toscani, A Boltzmann type approach to the formation of wealth distribution curves, Riv. Mat. Univ. Parma, 1 (2009), 199-261.

[12]

D. Kahneman and A. Tversky, Prospect theory: An analysis of decision under risk, Econometrica, 47 (1979), 183-214. doi: 10.1017/CBO9780511609220.014.

[13]

D. Kahneman and A. Tversky, Choices, values, and frames, American Psychologist, 39 (1984), 341-350. doi: 10.1037/0003-066X.39.4.341.

[14]

M. Levy, H. Levy and S. Solomon, Microscopic Simulation of Financial Markets: From Investor Behaviour to Market Phoenomena, Academic Press, San Diego, 2000.

[15]

T. Lux, The socio-economic dynamics of speculative markets: Interacting agents, chaos, and the fat tails of return distributions, Journal of Economic Behavior & Organization, 33 (1998), 143-165. doi: 10.1016/S0167-2681(97)00088-7.

[16]

T. Lux and M. Marchesi, Volatility clustering in financial markets: A microscopic simulation of interacting agents, International Journal of Theoretical and Applied Finance, 3 (2000), 675-702. doi: 10.1142/S0219024900000826.

[17]

T. Lux and M. Marchesi, Scaling and criticality in a stochastic multi-agent model of a financial market, Nature, 397 (1999), 498-500.

[18]

D. Maldarella and L. Pareschi, Kinetic models for socio-economic dynamics of speculative markets, Physica A, 391 (2012), 715-730. doi: 10.1016/j.physa.2011.08.013.

[19]

R. N. Mantegna and H. E. Stanley, An Introduction to Econophysics Correlations and Complexity in Finance, Cambridge University Press, Cambridge, 2007.

[20]

D. Matthes and G. Toscani, On steady distributions of kinetic models of conservative economies, J. Stat. Phys., 130 (2008), 1087-1117. doi: 10.1007/s10955-007-9462-2.

[21]

G. Naldi, L. Pareschi and G. Toscani, Eds., Mathematical Modelling of Collective Behavior in Socio-economic and Life Sciences, Birkhäuser, Boston, 2010. doi: 10.1007/978-0-8176-4946-3.

[22]

L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods, Oxford University Press, Oxford, 2014.

[23]

L. Pareschi and G. Toscani, Wealth distribution and collective knowledge. A Boltzmann approach, Phil. Trans. R. Soc. A, 372 (2014), 20130396, 15pp. doi: 10.1098/rsta.2013.0396.

[24]

F. Slanina, Inelastically scattering particles and wealth distribution in an open economy, Phys. Rev. E, 69 (2004), 046102. doi: 10.1103/PhysRevE.69.046102.

[25]

G. Toscani, Kinetic models of opinion formation, Comm. Math. Scie., 4 (2006), 481-496. doi: 10.4310/CMS.2006.v4.n3.a1.

[26]

J. Voit, The Statistical Mechanics of Financial Markets, Springer Verlag, Berlin, 2005.

[1]

Richard Carney, Monique Chyba, Chris Gray, George Wilkens, Corey Shanbrom. Multi-agent systems for quadcopters. Journal of Geometric Mechanics, 2022, 14 (1) : 1-28. doi: 10.3934/jgm.2021005
[2]

Nadia Loy, Andrea Tosin. Boltzmann-type equations for multi-agent systems with label switching. Kinetic and Related Models, 2021, 14 (5) : 867-894. doi: 10.3934/krm.2021027
[3]

Mei Luo, Jinrong Wang, Yumei Liao. Bounded consensus of double-integrator stochastic multi-agent systems. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022088
[4]

Giulia Cavagnari, Antonio Marigonda, Benedetto Piccoli. Optimal synchronization problem for a multi-agent system. Networks and Heterogeneous Media, 2017, 12 (2) : 277-295. doi: 10.3934/nhm.2017012
[5]

Zhiyong Sun, Toshiharu Sugie. Identification of Hessian matrix in distributed gradient-based multi-agent coordination control systems. Numerical Algebra, Control and Optimization, 2019, 9 (3) : 297-318. doi: 10.3934/naco.2019020
[6]

Rui Li, Yingjing Shi. Finite-time optimal consensus control for second-order multi-agent systems. Journal of Industrial and Management Optimization, 2014, 10 (3) : 929-943. doi: 10.3934/jimo.2014.10.929
[7]

Xi Zhu, Meixia Li, Chunfa Li. Consensus in discrete-time multi-agent systems with uncertain topologies and random delays governed by a Markov chain. Discrete and Continuous Dynamical Systems - B, 2020, 25 (12) : 4535-4551. doi: 10.3934/dcdsb.2020111
[8]

Zhongkui Li, Zhisheng Duan, Guanrong Chen. Consensus of discrete-time linear multi-agent systems with observer-type protocols. Discrete and Continuous Dynamical Systems - B, 2011, 16 (2) : 489-505. doi: 10.3934/dcdsb.2011.16.489
[9]

Yibo Zhang, Jinfeng Gao, Jia Ren, Huijiao Wang. A type of new consensus protocol for two-dimension multi-agent systems. Numerical Algebra, Control and Optimization, 2017, 7 (3) : 345-357. doi: 10.3934/naco.2017022
[10]

Hongru Ren, Shubo Li, Changxin Lu. Event-triggered adaptive fault-tolerant control for multi-agent systems with unknown disturbances. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1395-1414. doi: 10.3934/dcdss.2020379
[11]

Ke Yang, Wencheng Zou, Zhengrong Xiang, Ronghao Wang. Fully distributed consensus for higher-order nonlinear multi-agent systems with unmatched disturbances. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1535-1551. doi: 10.3934/dcdss.2020396
[12]

Xiaojin Huang, Hongfu Yang, Jianhua Huang. Consensus stability analysis for stochastic multi-agent systems with multiplicative measurement noises and Markovian switching topologies. Numerical Algebra, Control and Optimization, 2021  doi: 10.3934/naco.2021024
[13]

Zhongqiang Wu, Zongkui Xie. A multi-objective lion swarm optimization based on multi-agent. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022001
[14]

Simone Fagioli, Emanuela Radici. Opinion formation systems via deterministic particles approximation. Kinetic and Related Models, 2021, 14 (1) : 45-76. doi: 10.3934/krm.2020048
[15]

Seung-Yeal Ha, Dohyun Kim, Jaeseung Lee, Se Eun Noh. Emergent dynamics of an orientation flocking model for multi-agent system. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 2037-2060. doi: 10.3934/dcds.2020105
[16]

Brendan Pass. Multi-marginal optimal transport and multi-agent matching problems: Uniqueness and structure of solutions. Discrete and Continuous Dynamical Systems, 2014, 34 (4) : 1623-1639. doi: 10.3934/dcds.2014.34.1623
[17]

Tyrone E. Duncan. Some partially observed multi-agent linear exponential quadratic stochastic differential games. Evolution Equations and Control Theory, 2018, 7 (4) : 587-597. doi: 10.3934/eect.2018028
[18]

Hong Man, Yibin Yu, Yuebang He, Hui Huang. Design of one type of linear network prediction controller for multi-agent system. Discrete and Continuous Dynamical Systems - S, 2019, 12 (4&5) : 727-734. doi: 10.3934/dcdss.2019047
[19]

Marco Caponigro, Anna Chiara Lai, Benedetto Piccoli. A nonlinear model of opinion formation on the sphere. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4241-4268. doi: 10.3934/dcds.2015.35.4241
[20]

Sergei Yu. Pilyugin, M. C. Campi. Opinion formation in voting processes under bounded confidence. Networks and Heterogeneous Media, 2019, 14 (3) : 617-632. doi: 10.3934/nhm.2019024

2020 Impact Factor: 1.213

Tools

Metrics

  • PDF downloads (142)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy