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 & 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.   Google Scholar

[2]

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

[3]

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

[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.  doi: 10.1016/j.physa.2003.11.014.  Google Scholar

[5]

A. Chatterjee, S. Yarlagadda and B. K. Chakrabarti, Eds., Econophysics of Wealth Distributions,, New Economic Window Series, (2005).   Google Scholar

[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).  doi: 10.1103/PhysRevE.72.026126.  Google Scholar

[7]

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

[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, (2010).   Google Scholar

[9]

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

[10]

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

[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.   Google Scholar

[12]

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

[13]

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

[14]

M. Levy, H. Levy and S. Solomon, Microscopic Simulation of Financial Markets: From Investor Behaviour to Market Phoenomena,, Academic Press, (2000).   Google Scholar

[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.  doi: 10.1016/S0167-2681(97)00088-7.  Google Scholar

[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.  doi: 10.1142/S0219024900000826.  Google Scholar

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

J. Voit, The Statistical Mechanics of Financial Markets,, Springer Verlag, (2005).   Google Scholar

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.   Google Scholar

[2]

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

[3]

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

[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.  doi: 10.1016/j.physa.2003.11.014.  Google Scholar

[5]

A. Chatterjee, S. Yarlagadda and B. K. Chakrabarti, Eds., Econophysics of Wealth Distributions,, New Economic Window Series, (2005).   Google Scholar

[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).  doi: 10.1103/PhysRevE.72.026126.  Google Scholar

[7]

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

[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, (2010).   Google Scholar

[9]

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

[10]

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

[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.   Google Scholar

[12]

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

[13]

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

[14]

M. Levy, H. Levy and S. Solomon, Microscopic Simulation of Financial Markets: From Investor Behaviour to Market Phoenomena,, Academic Press, (2000).   Google Scholar

[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.  doi: 10.1016/S0167-2681(97)00088-7.  Google Scholar

[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.  doi: 10.1142/S0219024900000826.  Google Scholar

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

J. Voit, The Statistical Mechanics of Financial Markets,, Springer Verlag, (2005).   Google Scholar

[1]

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

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

Zhongkui Li, Zhisheng Duan, Guanrong Chen. Consensus of discrete-time linear multi-agent systems with observer-type protocols. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 489-505. doi: 10.3934/dcdsb.2011.16.489
[4]

Yibo Zhang, Jinfeng Gao, Jia Ren, Huijiao Wang. A type of new consensus protocol for two-dimension multi-agent systems. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 345-357. doi: 10.3934/naco.2017022
[5]

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

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 & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2020111
[7]

Hongru Ren, Shubo Li, Changxin Lu. Event-triggered adaptive fault-tolerant control for multi-agent systems with unknown disturbances. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020379
[8]

Seung-Yeal Ha, Dohyun Kim, Jaeseung Lee, Se Eun Noh. Emergent dynamics of an orientation flocking model for multi-agent system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (4) : 2037-2060. doi: 10.3934/dcds.2020105
[9]

Brendan Pass. Multi-marginal optimal transport and multi-agent matching problems: Uniqueness and structure of solutions. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1623-1639. doi: 10.3934/dcds.2014.34.1623
[10]

Tyrone E. Duncan. Some partially observed multi-agent linear exponential quadratic stochastic differential games. Evolution Equations & Control Theory, 2018, 7 (4) : 587-597. doi: 10.3934/eect.2018028
[11]

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

Marco Caponigro, Anna Chiara Lai, Benedetto Piccoli. A nonlinear model of opinion formation on the sphere. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4241-4268. doi: 10.3934/dcds.2015.35.4241
[13]

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

Dieter Armbruster, Christian Ringhofer, Andrea Thatcher. A kinetic model for an agent based market simulation. Networks & Heterogeneous Media, 2015, 10 (3) : 527-542. doi: 10.3934/nhm.2015.10.527
[15]

Giacomo Albi, Lorenzo Pareschi, Mattia Zanella. Opinion dynamics over complex networks: Kinetic modelling and numerical methods. Kinetic & Related Models, 2017, 10 (1) : 1-32. doi: 10.3934/krm.2017001
[16]

N. Bellomo, A. Bellouquid. From a class of kinetic models to the macroscopic equations for multicellular systems in biology. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 59-80. doi: 10.3934/dcdsb.2004.4.59
[17]

Pierre Carcaud, Pierre-Henri Chavanis, Mohammed Lemou, Florian Méhats. Evaporation law in kinetic gravitational systems described by simplified Landau models. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 907-934. doi: 10.3934/dcdsb.2010.14.907
[18]

Martin Parisot, Mirosław Lachowicz. A kinetic model for the formation of swarms with nonlinear interactions. Kinetic & Related Models, 2016, 9 (1) : 131-164. doi: 10.3934/krm.2016.9.131
[19]

Domenica Borra, Tommaso Lorenzi. Asymptotic analysis of continuous opinion dynamics models under bounded confidence. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1487-1499. doi: 10.3934/cpaa.2013.12.1487
[20]

Pierre Degond, Hailiang Liu. Kinetic models for polymers with inertial effects. Networks & Heterogeneous Media, 2009, 4 (4) : 625-647. doi: 10.3934/nhm.2009.4.625

2018 Impact Factor: 0.871

Tools

Metrics

  • PDF downloads (28)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences