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"Slimming" of power law tails by increasing market returns

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  • D. Sornette

    (Univ. Nice/CNRS and UCLA)

Abstract

We introduce a simple generalization of rational bubble models which removes the fundamental problem discovered by [Lux and Sornette, 1999] that the distribution of returns is a power law with exponent less than 1, in contradiction with empirical data. The idea is that the price fluctuations associated with bubbles must on average grow with the mean market return r. When r is larger than the discount rate r_delta, the distribution of returns of the observable price, sum of the bubble component and of the fundamental price, exhibits an intermediate tail with an exponent which can be larger than 1. This regime r>r_delta corresponds to a generalization of the rational bubble model in which the fundamental price is no more given by the discounted value of future dividends. We explain how this is possible. Our model predicts that, the higher is the market remuneration r above the discount rate, the larger is the power law exponent and thus the thinner is the tail of the distribution of price returns.

Suggested Citation

  • D. Sornette, 2000. ""Slimming" of power law tails by increasing market returns," Papers cond-mat/0010112, arXiv.org, revised Sep 2001.
  • Handle: RePEc:arx:papers:cond-mat/0010112
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    References listed on IDEAS

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    1. Sornette, D & Malevergne, Y, 2001. "From rational bubbles to crashes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 299(1), pages 40-59.
    2. Sornette, Didier, 2000. "Stock market speculation: Spontaneous symmetry breaking of economic valuation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 284(1), pages 355-375.
    3. Richard B. Olsen & Ulrich A. Müller & Michel M. Dacorogna & Olivier V. Pictet & Rakhal R. Davé & Dominique M. Guillaume, 1997. "From the bird's eye to the microscope: A survey of new stylized facts of the intra-daily foreign exchange markets (*)," Finance and Stochastics, Springer, vol. 1(2), pages 95-129.
    4. Sornette, Didier, 1998. "Linear stochastic dynamics with nonlinear fractal properties," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 250(1), pages 295-314.
    5. Behzad T. Diba & Herschel I. Grossman, 1987. "On the Inception of Rational Bubbles," The Quarterly Journal of Economics, Oxford University Press, vol. 102(3), pages 697-700.
    6. De Vries, C.G. & Leuven, K.U., 1994. "Stylized Facts of Nominal Exchange Rate Returns," Papers 94-002, Purdue University, Krannert School of Management - Center for International Business Education and Research (CIBER).
    7. Lux, T. & M. Marchesi, "undated". "Scaling and Criticality in a Stochastic Multi-Agent Model of a Financial Market," Discussion Paper Serie B 438, University of Bonn, Germany, revised Jul 1998.
    8. Mehra, Rajnish & Prescott, Edward C., 1985. "The equity premium: A puzzle," Journal of Monetary Economics, Elsevier, vol. 15(2), pages 145-161, March.
    9. D. Sornette, 2000. "Stock Market Speculation: Spontaneous Symmetry Breaking of Economic Valuation," Papers cond-mat/0004001, arXiv.org.
    10. Pagan, Adrian, 1996. "The econometrics of financial markets," Journal of Empirical Finance, Elsevier, vol. 3(1), pages 15-102, May.
    11. Y. Malevergne & D. Sornette, 2001. "Multi-dimensional rational bubbles and fat tails," Quantitative Finance, Taylor & Francis Journals, vol. 1(5), pages 533-541.
    12. Adam, M C & Szafarz, A, 1992. "Speculative Bubbles and Financial Markets," Oxford Economic Papers, Oxford University Press, vol. 44(4), pages 626-640, October.
    13. Mehra, Rajnish & Prescott, Edward C., 1988. "The equity risk premium: A solution?," Journal of Monetary Economics, Elsevier, vol. 22(1), pages 133-136, July.
    14. Rietz, Thomas A., 1988. "The equity risk premium a solution," Journal of Monetary Economics, Elsevier, vol. 22(1), pages 117-131, July.
    15. Camerer, Colin, 1989. " Bubbles and Fads in Asset Prices," Journal of Economic Surveys, Wiley Blackwell, vol. 3(1), pages 3-41.
    16. Burke, Jonathan L., 2000. "General Equilibrium When Economic Growth Exceeds Discounting," Journal of Economic Theory, Elsevier, vol. 94(2), pages 141-162, October.
    17. P. Gopikrishnan & M. Meyer & L.A.N. Amaral & H.E. Stanley, 1998. "Inverse cubic law for the distribution of stock price variations," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 3(2), pages 139-140, July.
    18. Blanchard, Olivier Jean, 1979. "Speculative bubbles, crashes and rational expectations," Economics Letters, Elsevier, vol. 3(4), pages 387-389.
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    Cited by:

    1. Sornette, D & Malevergne, Y, 2001. "From rational bubbles to crashes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 299(1), pages 40-59.
    2. Y. Malevergne & D. Sornette, 2001. "Multi-dimensional Rational Bubbles and fat tails: application of stochastic regression equations to financial speculation," Papers cond-mat/0101371, arXiv.org.

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