mρ hb (mρ bb > mρ bh) among the m agents inspected are bullish (bearish). The condition (1) (resp. (2)) corresponds to imitative (resp. antagonistic) behavior. In the limit where the number N of agents is infinite, the dynamics of the fraction of bullish agents is deterministic and exhibits chaotic behavior in a significant domain of the parameter space {ρ hb , ρ bh , ρ hh , ρ bb , m}. A typical chaotic trajectory is characterized by intermittent phases of chaos, quasi-periodic behavior and super-exponentially growing bubbles followed by crashes. A typical bubble starts initially by growing at an exponential rate and then crosses over to a nonlinear power law growth rate leading to a finitetime singularity. The reinjection mechanism provided by the contrarian behavior introduces a finite-size effect, rounding off these singularities and leads to chaos. We document the main stylized facts of this model in the symmetric and asymmetric cases. This model is one of the rare agent-based models that give rise to interesting non-periodic complex dynamics in the "thermodynamic" limit (of an infinite number N of agents). We also discuss the case of a finite number of agents, which introduces an endogenous source of noise superimposed on the chaotic dynamics. "Human behavior is a main factor in how markets act. Indeed, sometimes markets act quickly, violently with little warning. [.. .] Ultimately, history tells us that there will be a correction of some significant dimension. I have no doubt that, human nature being what it is, that it is going to happen again and again." Alan Greenspan, Chairman of the Federal Reserve of the USA, before the Committee on Banking and Financial Services, U.S. House of Representatives, July 24, 1998."> mρ hb (mρ bb > mρ bh) among the m agents inspected are bullish (bearish). The condition (1) (resp. (2)) corresponds to imitative (resp. antagonistic) behavior. In the limit where the number N of agents is infinite, the dynamics of the fraction of bullish agents is deterministic and exhibits chaotic behavior in a significant domain of the parameter space {ρ hb , ρ bh , ρ hh , ρ bb , m}. A typical chaotic trajectory is characterized by intermittent phases of chaos, quasi-periodic behavior and super-exponentially growing bubbles followed by crashes. A typical bubble starts initially by growing at an exponential rate and then crosses over to a nonlinear power law growth rate leading to a finitetime singularity. The reinjection mechanism provided by the contrarian behavior introduces a finite-size effect, rounding off these singularities and leads to chaos. We document the main stylized facts of this model in the symmetric and asymmetric cases. This model is one of the rare agent-based models that give rise to interesting non-periodic complex dynamics in the "thermodynamic" limit (of an infinite number N of agents). We also discuss the case of a finite number of agents, which introduces an endogenous source of noise superimposed on the chaotic dynamics. "Human behavior is a main factor in how markets act. Indeed, sometimes markets act quickly, violently with little warning. [.. .] Ultimately, history tells us that there will be a correction of some significant dimension. I have no doubt that, human nature being what it is, that it is going to happen again and again." Alan Greenspan, Chairman of the Federal Reserve of the USA, before the Committee on Banking and Financial Services, U.S. House of Representatives, July 24, 1998.">
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Imitation and contrarian behavior: hyperbolic bubbles, crashes and chaos

Author

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  • Anne Corcos

    (CURAPP-ESS - Centre universitaire de recherches sur l'action publique et le politique. Epistémologie et Sciences sociales - UMR CNRS UPJV 7319 - UPJV - Université de Picardie Jules Verne - CNRS - Centre National de la Recherche Scientifique)

  • Jean-Pierre Eckmann

    (Département de Physique Théorique - UNIGE - Université de Genève = University of Geneva, Section de mathématiques [Genève] - UNIGE - Université de Genève = University of Geneva)

  • A. Malaspinas
  • Yannick Malevergne

    (LPMC - Laboratoire de physique de la matière condensée - UNS - Université Nice Sophia Antipolis (1965 - 2019) - CNRS - Centre National de la Recherche Scientifique, SAF - Laboratoire de Sciences Actuarielle et Financière - UCBL - Université Claude Bernard Lyon 1 - Université de Lyon)

  • Didier Sornette

    (IGPP - Institute of Geophysics and Planetary Physics [Los Angeles] - UCLA - University of California [Los Angeles] - UC - University of California, EPSS - Department of Earth, Planetary and Space Sciences [Los Angeles] - UCLA - University of California [Los Angeles] - UC - University of California, LPMC - Laboratoire de physique de la matière condensée - UNS - Université Nice Sophia Antipolis (1965 - 2019) - CNRS - Centre National de la Recherche Scientifique)

Abstract
Imitative and contrarian behaviors are the two typical opposite attitudes of investors in stock markets. We introduce a simple model to investigate their interplay in a stock market where agents can take only two states, bullish or bearish. Each bullish (bearish) agent polls m "friends" and changes her opinion to bearish (bullish) if (1) at least mρ hb (mρ bh) among the m agents inspected are bearish (bullish) or (2) at least mρ hh > mρ hb (mρ bb > mρ bh) among the m agents inspected are bullish (bearish). The condition (1) (resp. (2)) corresponds to imitative (resp. antagonistic) behavior. In the limit where the number N of agents is infinite, the dynamics of the fraction of bullish agents is deterministic and exhibits chaotic behavior in a significant domain of the parameter space {ρ hb , ρ bh , ρ hh , ρ bb , m}. A typical chaotic trajectory is characterized by intermittent phases of chaos, quasi-periodic behavior and super-exponentially growing bubbles followed by crashes. A typical bubble starts initially by growing at an exponential rate and then crosses over to a nonlinear power law growth rate leading to a finitetime singularity. The reinjection mechanism provided by the contrarian behavior introduces a finite-size effect, rounding off these singularities and leads to chaos. We document the main stylized facts of this model in the symmetric and asymmetric cases. This model is one of the rare agent-based models that give rise to interesting non-periodic complex dynamics in the "thermodynamic" limit (of an infinite number N of agents). We also discuss the case of a finite number of agents, which introduces an endogenous source of noise superimposed on the chaotic dynamics. "Human behavior is a main factor in how markets act. Indeed, sometimes markets act quickly, violently with little warning. [.. .] Ultimately, history tells us that there will be a correction of some significant dimension. I have no doubt that, human nature being what it is, that it is going to happen again and again." Alan Greenspan, Chairman of the Federal Reserve of the USA, before the Committee on Banking and Financial Services, U.S. House of Representatives, July 24, 1998.

Suggested Citation

  • Anne Corcos & Jean-Pierre Eckmann & A. Malaspinas & Yannick Malevergne & Didier Sornette, 2002. "Imitation and contrarian behavior: hyperbolic bubbles, crashes and chaos," Post-Print hal-03833822, HAL.
    • Handle: RePEc:hal:journl:hal-03833822
    DOI: 10.1088/1469-7688/2/4/303
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    2. Kaizoji, Taisei & Leiss, Matthias & Saichev, Alexander & Sornette, Didier, 2015. "Super-exponential endogenous bubbles in an equilibrium model of fundamentalist and chartist traders," Journal of Economic Behavior & Organization, Elsevier, vol. 112(C), pages 289-310.
    3. Li Lin & Didier Sornette, 2009. "Diagnostics of Rational Expectation Financial Bubbles with Stochastic Mean-Reverting Termination Times," Papers 0911.1921, arXiv.org.
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    5. Егорова Людмила Геннадьевна, 2014. "Эффективность Торговых Стратегий Мелких Трейдеров," Проблемы управления, CyberLeninka;Общество с ограниченной ответственностью "СенСиДат-Контрол", issue 5, pages 34-41.
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    11. Shiryaev, Albert N. & Zhitlukhin, Mikhail N. & Ziemba, William T., 2014. "Land and stock bubbles, crashes and exit strategies in Japan circa 1990 and in 2013," LSE Research Online Documents on Economics 59288, London School of Economics and Political Science, LSE Library.
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    13. Westerhoff, Frank H., 2004. "Greed, fear and stock market dynamics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 343(C), pages 635-642.
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    18. T. Kaizoji & M. Leiss & A. Saichev & D. Sornette, 2011. "Super-exponential endogenous bubbles in an equilibrium model of rational and noise traders," Papers 1109.4726, arXiv.org, revised Mar 2014.
    19. D. Sornette & R. Woodard, "undated". "Financial Bubbles, Real Estate bubbles, Derivative Bubbles, and the Financial and Economic Crisis," Working Papers CCSS-09-003, ETH Zurich, Chair of Systems Design.
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    21. Norman Schofield, 2015. "Climate Change, Collapse and Social Choice Theory," Czech Economic Review, Charles University Prague, Faculty of Social Sciences, Institute of Economic Studies, vol. 9(1), pages 007-035, October.
    22. Frederik Herzberg, 2008. "Black-Scholes theory for an underlying with multiple attractors," Quantitative Finance, Taylor & Francis Journals, vol. 8(5), pages 453-457.
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    24. Lorenzo Zanello RIva, 2012. "El efecto día en cinco índices bursátiles de América Latina," Documentos Departamento de Economía 18081, Universidad del Norte.

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