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Imitation and contrarian behaviour: hyperbolic bubbles, crashes and chaos

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  • A. Corcos
  • J-P Eckmann
  • A. Malaspinas
  • Y. Malevergne
  • D. Sornette

Abstract

Imitative and contrarian behaviours 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 (i) at least mρhb (mρbh) among the m agents inspected are bearish (bullish) or (ii) at least mρhh>mρhb (mρbb>mρbh) among the m agents inspected are bullish (bearish). The condition (i) ((ii)) corresponds to imitative (antagonistic) behaviour. In the limit where the number N of agents is infinite, the dynamics of the fraction of bullish agents is deterministic and exhibits chaotic behaviour in a significant domain of the parameter space {ρhb,ρbh,ρhh,ρbb,m}. A typical chaotic trajectory is characterized by intermittent phases of chaos, quasi-periodic behaviour 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 finite-time singularity. The reinjection mechanism provided by the contrarian behaviour 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.

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  • A. Corcos & J-P Eckmann & A. Malaspinas & Y. Malevergne & D. Sornette, 2002. "Imitation and contrarian behaviour: hyperbolic bubbles, crashes and chaos," Quantitative Finance, Taylor & Francis Journals, vol. 2(4), pages 264-281.
  • Handle: RePEc:taf:quantf:v:2:y:2002:i:4:p:264-281
    DOI: 10.1088/1469-7688/2/4/303
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    1. 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.
    2. Lleo, Sébastien & Ziemba, William T., 2015. "Some historical perspectives on the Bond-Stock Earnings Yield Model for crash prediction around the world," International Journal of Forecasting, Elsevier, vol. 31(2), pages 399-425.
    3. 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.
    4. Deng, Lei & Liu, Yun & Xiong, Fei, 2013. "An opinion diffusion model with clustered early adopters," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(17), pages 3546-3554.
    5. Li Lin & Didier Sornette, 2009. "Diagnostics of Rational Expectation Financial Bubbles with Stochastic Mean-Reverting Termination Times," Papers 0911.1921, arXiv.org.
    6. Hongler, Max-Olivier & Gallay, Olivier & Hülsmann, Michael & Cordes, Philip & Colmorn, Richard, 2010. "Centralized versus decentralized control—A solvable stylized model in transportation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(19), pages 4162-4171.
    7. Patrick Bayer & Kyle Mangum & James W. Roberts, 2016. "Speculative Fever: Investor Contagion in the Housing Bubble," NBER Working Papers 22065, National Bureau of Economic Research, Inc.
    8. Carmen Pellicer-Lostao & Ricardo Lopez-Ruiz, 2010. "Transition from Exponential to Power Law Distributions in a Chaotic Market," Papers 1011.5187, arXiv.org.
    9. Omurtag, Ahmet & Sirovich, Lawrence, 2006. "Modeling a large population of traders: Mimesis and stability," Journal of Economic Behavior & Organization, Elsevier, vol. 61(4), pages 562-576, December.
    10. BRATIAN Vasile & BUCUR Amelia, 2017. "The Development And The Current Status Of The Capital Market Hypotheses: A Few Benchmarks," Revista Economica, Lucian Blaga University of Sibiu, Faculty of Economic Sciences, vol. 69(1), pages 22-28, April.
    11. Егорова Людмила Геннадьевна, 2014. "Эффективность Торговых Стратегий Мелких Трейдеров," Проблемы управления, CyberLeninka;Общество с ограниченной ответственностью "СенСиДат-Контрол", issue 5, pages 34-41.
    12. Shiryaev, Albert N. & Zhitlukhin, Mikhail N. & Ziemba, Bill & 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.
    13. Jean-Philippe Bouchaud, 2002. "An introduction to statistical finance," Science & Finance (CFM) working paper archive 313238, Science & Finance, Capital Fund Management.
    14. 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.
    15. 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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