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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), 641-656. Google Scholar

[2]

A. Chakraborti, Distributions of money in models of market economy, Int. J. Modern Phys. C, 13 (2002), 1315-1321. 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), 167-170. 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), 155-163. 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, Springer-Verlag, Milan, 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), 026126. 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), 253-277. 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, course CLXXVI, Varenna, 2010. E-Print: arXiv:1101.1847. Google Scholar

[9]

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

[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.  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), 199-261.  Google Scholar

[12]

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

[13]

D. Kahneman and A. Tversky, Choices, values, and frames, American Psychologist, 39 (1984), 341-350. 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, San Diego, 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), 143-165. 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), 675-702. 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), 498-500. Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[26]

J. Voit, The Statistical Mechanics of Financial Markets, Springer Verlag, Berlin, 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), 641-656. Google Scholar

[2]

A. Chakraborti, Distributions of money in models of market economy, Int. J. Modern Phys. C, 13 (2002), 1315-1321. 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), 167-170. 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), 155-163. 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, Springer-Verlag, Milan, 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), 026126. 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), 253-277. 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, course CLXXVI, Varenna, 2010. E-Print: arXiv:1101.1847. Google Scholar

[9]

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

[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.  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), 199-261.  Google Scholar

[12]

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

[13]

D. Kahneman and A. Tversky, Choices, values, and frames, American Psychologist, 39 (1984), 341-350. 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, San Diego, 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), 143-165. 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), 675-702. 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), 498-500. Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[26]

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

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