A kinetic equation for economic value estimation with irrationality and herding

Bertram Düring; Ansgar Jüngel; Lara Trussardi

doi:10.3934/krm.2017010

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March 2017, 10(1): 239-261. doi: 10.3934/krm.2017010

A kinetic equation for economic value estimation with irrationality and herding

Bertram Düring ^1, , Ansgar Jüngel ^2, and Lara Trussardi ^2,

1.	Department of Mathematics, University of Sussex, Pevensey Ⅱ, Brighton BN1 9QH, United Kingdom
2.	Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8-10,1040 Wien, Austria

Received January 2016 Revised May 2016 Published November 2016

Fund Project: The first author is supported by the Leverhulme Trust research project grant "Novel discretisations for higher-order nonlinear PDE" (RPG-2015-69). The second and third authors acknowledge partial support from the European Union in the FP7-PEOPLE-2012-ITN Program under Grant Agreement Number 304617 and from the Austrian Science Fund (FWF), grants P22108, P24304, and W1245.

A kinetic inhomogeneous Boltzmann-type equation is proposed to model the dynamics of the number of agents in a large market depending on the estimated value of an asset and the rationality of the agents. The interaction rules take into account the interplay of the agents with sources of public information, herding phenomena, and irrationality of the individuals. In the formal grazing collision limit, a nonlinear nonlocal Fokker-Planck equation with anisotropic (or incomplete) diffusion is derived. The existence of global-in-time weak solutions to the Fokker-Planck initial-boundary-value problem is proved. Numerical experiments for the Boltzmann equation highlight the importance of the reliability of public information in the formation of bubbles and crashes. The use of Bollinger bands in the simulations shows how herding may lead to strong trends with low volatility of the asset prices, but eventually also to abrupt corrections.

Keywords: Inhomogeneous Boltzmann equation, public information, herding, Fokker-Planck equation, existence of solutions, sociophysics.

Mathematics Subject Classification: 35Q20, 35Q84, 91G80, 35K65.

Citation: Bertram Düring, Ansgar Jüngel, Lara Trussardi. A kinetic equation for economic value estimation with irrationality and herding. Kinetic and Related Models, 2017, 10 (1) : 239-261. doi: 10.3934/krm.2017010

References:

[1]	E. Altshuler, O. Ramos, Y. Núñez, J. Fernández, A. Batista-Leyva and C. Noda, Symmetry breaking in escaping ants, Aner. Natur, 166 (2005), 643-649. doi: 10.1086/498139.
[2]	A. Amadori and R. Natalini, Entropy solutions to a strongly degenerate anisotropic convection-diffusion equation with application to utility theory, J. Math. Anal. Appl., 284 (2003), 511-531. doi: 10.1016/S0022-247X(03)00339-1.
[3]	C. Avery and P. Zemsky, Multidimensional uncertainty and herd behavior in financial markets, Amer. Econ. Rev., 88 (1998), 724-748.
[4]	A. Banerjee, A simple model of herd behavior, Quart. J. Econ., 107 (1992), 797-817. doi: 10.2307/2118364.
[5]	S. Bickhchandani, D. Hirshleifter and I. Welch, A theory of fads, fashion, custim, and cultural change as informational cascades, J. Polit. Econ., 100 (1992), 992-1026.
[6]	G. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows Oxford University Press, Oxford, 1995.
[7]	L. Boudin and F. Salvarani, A kinetic approach to the study of opinion formation, ESAIM Math. Mod. Anal. Num., 43 (2009), 507-522. doi: 10.1051/m2an/2009004.
[8]	M. Brunnermeier, Asset Pricing Under Asymmetric Information: Bubbles, Crashes, Technical Analysis, and Herding Oxford University Press, Oxford, 2001. doi: 10.1093/0198296983.001.0001.
[9]	M. Burger, P. Markowich and J.-F. Pietschmann, Continuous limit of a crowd motion and herding model: Analysis and numerical simulations, Kinetic Related Models, 4 (2011), 1025-1047. doi: 10.3934/krm.2011.4.1025.
[10]	V. Comincioli, L. Della Croce and G. Toscani, A Boltzmann-like equation for choice formation, Kinet. Relat. Models, 2 (2009), 135-149. doi: 10.3934/krm.2009.2.135.
[11]	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.
[12]	P. Degond, C. Appert-Rolland, M. Moussaid, J. Pettré and G. Theraulaz, A hierarchy of heuristic-based models of crowd dynamics, J. Stat. Phys., 152 (2013), 1033-1068. doi: 10.1007/s10955-013-0805-x.
[13]	M. Delitala and T. Lorenzi, A mathematical model for value estimation with public information and herding, Kinet. Relat. Models, 7 (2014), 29-44. doi: 10.3934/krm.2014.7.29.
[14]	A. Devenow and I. Welch, Rational herding in financial economics, Europ. Econ. Rev., 40 (1996), 603-615. doi: 10.1016/0014-2921(95)00073-9.
[15]	G. Dimarco and L. Pareschi, High order asymptotic-preserving schemes for the Boltzmann equation, Comptes Rendus Methématique, 350 (2012), 481-486. doi: 10.1016/j.crma.2012.05.010.
[16]	B. Düring, P. Markowich, J.-F. Pietschmann and M.-T. Wolfram, Boltzmann and Fokker-Planck equations modelling opinion formation in the presence of strong leaders, Proc. Roy. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 465 (2009), 3687-3708. doi: 10.1098/rspa.2009.0239.
[17]	B. Düring and M. -T. Wolfram, Opinion dynamics: Inhomogeneous Boltzmann-type equations modelling opinion leadership and political segregation Proc. Roy. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 471 (2015), 20150345, 21pp. doi: 10.1098/rspa.2015.0345.
[18]	J. Dyer, A. Johansson, D. Helbing, I. Couzin and J. Krause, Leadership, consensus decision making and collective behaviour in humans, Phil. Trans. Roy. Soc. B: Biol. Sci., 364 (2009), 781-789. doi: 10.1098/rstb.2008.0233.
[19]	M. Escobedo, J.-L. Vázquez and E. Zuazua, Entropy solutions for diffusion-convection equations with partial diffusivity, Trans. Amer. Math. Soc., 343 (1994), 829-842. doi: 10.1090/S0002-9947-1994-1225573-2.
[20]	S. Fong, J. Tai and Y.W. Si, Trend following algorithms for technical trading in stock markets, J. Emerging Tech. Web Intell., 3 (2011), 136-145. doi: 10.4304/jetwi.3.2.136-145.
[21]	W. Hamilton, Geometry for the selfish herd, J. Theor. Biol., 31 (1971), 295-311. doi: 10.1016/0022-5193(71)90189-5.
[22]	T. Hillen, K. Painter and M. Winkler, Anisotropic diffusion in oriented environments can lead to singularity formation, Europ. J. Appl. Math., 24 (2013), 371-413. doi: 10.1017/S0956792512000447.
[23]	D. Hirshleifer, Investor psychology and asset pricing, J. Finance, 56 (2001), 1533-1597.
[24]	P. Jahangiri, A. Nejat, J. Samadi and A. Aboutalebi, A high-order Monte Carlo algorithm for the direct simulation of Boltzmann equation, J. Comput. Phys., 231 (2012), 4578-4596. doi: 10.1016/j.jcp.2012.02.029.
[25]	T. Lux and M. Marchesi, Scaling and criticality in a stochastic multi-agent model of a financial market, Nature, 397 (1999), 498-500.
[26]	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.
[27]	L. Pareschi and G. Toscani, Interacting Multiagent Systems, Kinetic Equations and Monte Carlo methods, Oxford University Press, Oxford, 2014.
[28]	R. Raafat, N. Chater and C. Frith, Herding in humans, Trends Cognitive Sci., 13 (2009), 420-428. doi: 10.1016/j.tics.2009.08.002.
[29]	L. Rook, An economic psychological approach to herd behavior, J. Econ., 40 (2006), 75-95. doi: 10.1080/00213624.2006.11506883.
[30]	R. Shiller, Irrational Exuberance Princeton University Press, Princeton, 2015. doi: 10.1515/9781400865536.
[31]	G. Toscani, Kinetic models of opinion formation, Commun. Math. Sci., 4 (2006), 481-496. doi: 10.4310/CMS.2006.v4.n3.a1.
[32]	E. Zeidler, Nonlinear Functional Analysis and Applications Vol. ⅡA. Springer, New York, 1990. doi: 10.1007/978-1-4612-0985-0.

show all references

References:

[1]	E. Altshuler, O. Ramos, Y. Núñez, J. Fernández, A. Batista-Leyva and C. Noda, Symmetry breaking in escaping ants, Aner. Natur, 166 (2005), 643-649. doi: 10.1086/498139.
[2]	A. Amadori and R. Natalini, Entropy solutions to a strongly degenerate anisotropic convection-diffusion equation with application to utility theory, J. Math. Anal. Appl., 284 (2003), 511-531. doi: 10.1016/S0022-247X(03)00339-1.
[3]	C. Avery and P. Zemsky, Multidimensional uncertainty and herd behavior in financial markets, Amer. Econ. Rev., 88 (1998), 724-748.
[4]	A. Banerjee, A simple model of herd behavior, Quart. J. Econ., 107 (1992), 797-817. doi: 10.2307/2118364.
[5]	S. Bickhchandani, D. Hirshleifter and I. Welch, A theory of fads, fashion, custim, and cultural change as informational cascades, J. Polit. Econ., 100 (1992), 992-1026.
[6]	G. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows Oxford University Press, Oxford, 1995.
[7]	L. Boudin and F. Salvarani, A kinetic approach to the study of opinion formation, ESAIM Math. Mod. Anal. Num., 43 (2009), 507-522. doi: 10.1051/m2an/2009004.
[8]	M. Brunnermeier, Asset Pricing Under Asymmetric Information: Bubbles, Crashes, Technical Analysis, and Herding Oxford University Press, Oxford, 2001. doi: 10.1093/0198296983.001.0001.
[9]	M. Burger, P. Markowich and J.-F. Pietschmann, Continuous limit of a crowd motion and herding model: Analysis and numerical simulations, Kinetic Related Models, 4 (2011), 1025-1047. doi: 10.3934/krm.2011.4.1025.
[10]	V. Comincioli, L. Della Croce and G. Toscani, A Boltzmann-like equation for choice formation, Kinet. Relat. Models, 2 (2009), 135-149. doi: 10.3934/krm.2009.2.135.
[11]	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.
[12]	P. Degond, C. Appert-Rolland, M. Moussaid, J. Pettré and G. Theraulaz, A hierarchy of heuristic-based models of crowd dynamics, J. Stat. Phys., 152 (2013), 1033-1068. doi: 10.1007/s10955-013-0805-x.
[13]	M. Delitala and T. Lorenzi, A mathematical model for value estimation with public information and herding, Kinet. Relat. Models, 7 (2014), 29-44. doi: 10.3934/krm.2014.7.29.
[14]	A. Devenow and I. Welch, Rational herding in financial economics, Europ. Econ. Rev., 40 (1996), 603-615. doi: 10.1016/0014-2921(95)00073-9.
[15]	G. Dimarco and L. Pareschi, High order asymptotic-preserving schemes for the Boltzmann equation, Comptes Rendus Methématique, 350 (2012), 481-486. doi: 10.1016/j.crma.2012.05.010.
[16]	B. Düring, P. Markowich, J.-F. Pietschmann and M.-T. Wolfram, Boltzmann and Fokker-Planck equations modelling opinion formation in the presence of strong leaders, Proc. Roy. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 465 (2009), 3687-3708. doi: 10.1098/rspa.2009.0239.
[17]	B. Düring and M. -T. Wolfram, Opinion dynamics: Inhomogeneous Boltzmann-type equations modelling opinion leadership and political segregation Proc. Roy. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 471 (2015), 20150345, 21pp. doi: 10.1098/rspa.2015.0345.
[18]	J. Dyer, A. Johansson, D. Helbing, I. Couzin and J. Krause, Leadership, consensus decision making and collective behaviour in humans, Phil. Trans. Roy. Soc. B: Biol. Sci., 364 (2009), 781-789. doi: 10.1098/rstb.2008.0233.
[19]	M. Escobedo, J.-L. Vázquez and E. Zuazua, Entropy solutions for diffusion-convection equations with partial diffusivity, Trans. Amer. Math. Soc., 343 (1994), 829-842. doi: 10.1090/S0002-9947-1994-1225573-2.
[20]	S. Fong, J. Tai and Y.W. Si, Trend following algorithms for technical trading in stock markets, J. Emerging Tech. Web Intell., 3 (2011), 136-145. doi: 10.4304/jetwi.3.2.136-145.
[21]	W. Hamilton, Geometry for the selfish herd, J. Theor. Biol., 31 (1971), 295-311. doi: 10.1016/0022-5193(71)90189-5.
[22]	T. Hillen, K. Painter and M. Winkler, Anisotropic diffusion in oriented environments can lead to singularity formation, Europ. J. Appl. Math., 24 (2013), 371-413. doi: 10.1017/S0956792512000447.
[23]	D. Hirshleifer, Investor psychology and asset pricing, J. Finance, 56 (2001), 1533-1597.
[24]	P. Jahangiri, A. Nejat, J. Samadi and A. Aboutalebi, A high-order Monte Carlo algorithm for the direct simulation of Boltzmann equation, J. Comput. Phys., 231 (2012), 4578-4596. doi: 10.1016/j.jcp.2012.02.029.
[25]	T. Lux and M. Marchesi, Scaling and criticality in a stochastic multi-agent model of a financial market, Nature, 397 (1999), 498-500.
[26]	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.
[27]	L. Pareschi and G. Toscani, Interacting Multiagent Systems, Kinetic Equations and Monte Carlo methods, Oxford University Press, Oxford, 2014.
[28]	R. Raafat, N. Chater and C. Frith, Herding in humans, Trends Cognitive Sci., 13 (2009), 420-428. doi: 10.1016/j.tics.2009.08.002.
[29]	L. Rook, An economic psychological approach to herd behavior, J. Econ., 40 (2006), 75-95. doi: 10.1080/00213624.2006.11506883.
[30]	R. Shiller, Irrational Exuberance Princeton University Press, Princeton, 2015. doi: 10.1515/9781400865536.
[31]	G. Toscani, Kinetic models of opinion formation, Commun. Math. Sci., 4 (2006), 481-496. doi: 10.4310/CMS.2006.v4.n3.a1.
[32]	E. Zeidler, Nonlinear Functional Analysis and Applications Vol. ⅡA. Springer, New York, 1990. doi: 10.1007/978-1-4612-0985-0.

Figure 1. Left: Percentage of bubbles (left) and crashes (right) depending on the choice of $\alpha$ and $\beta$. The parameters are $R=0.025$, $W=0.5$, $\delta=\kappa=1$

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Figure 2. Left: percentage of bubbles for varying $\alpha$ ($\beta=0.25$) and constant $\alpha$ ($\beta=0.5$ and $\beta=0.05$). Right: probability distribution function for the logarithmic return (in blue) and normal distribution (in magenta) with the same mean and variance as the return

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Figure 3. Mean asset value $m_w(f(t))$ versus time $t$ for $\alpha=0.05$ (left) and $\alpha=0.5$ (right). The function $W(t)$ is represented by the solid line in between the dashed lines which represent the functions $W(t)+R$ and $W(t)-R$. The parameters are $\beta=0.25$, $R=0.025$, $\delta=2$, $\kappa=1$

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Figure 4. Mean asset value $m_w(f(t))$ versus time for $\delta=0.01$ (left) and $\delta=100$ (right) with $\alpha=0.25$, $\beta=0.2$, $R=0.025$, $\kappa=1$

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Figure 5. Mean asset value $m_w(f(t))$ and Bollinger bands $R^\pm(t)$ versus time for $\alpha=0.05$ (left) and $\alpha=0.2$ (right). The parameters are $\beta=0.25$, $W=0.5$, $\delta=1$, $\kappa=1$

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Figure 6. Mean asset value $m_w(f(t))$ (left column) and Bollinger bands $R^\pm(t)$ (right column) versus time. The function $W(t)$ has a jump at $t=0.2$. The parameters are $\alpha=0.05$, $\beta=0.25$, $R=0.025$, $\delta=\kappa=1$. Upper row: $\eta=\pm 0.06$, lower row: $\eta=\pm 0.18$

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