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Multi-dimensional Rational Bubbles and fat tails: application of stochastic regression equations to financial speculation

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  • Y. Malevergne

    (Univ. Nice/CNRS)

  • D. Sornette

    (Univ. Nice/CNRS and UCLA)

Abstract

We extend the model of rational bubbles of Blanchard and of Blanchard and Watson to arbitrary dimensions d: a number d of market time series are made linearly interdependent via d times d stochastic coupling coefficients. We first show that the no-arbitrage condition imposes that the non-diagonal impacts of any asset i on any other asset j different from i has to vanish on average, i.e., must exhibit random alternative regimes of reinforcement and contrarian feedbacks. In contrast, the diagonal terms must be positive and equal on average to the inverse of the discount factor. Applying the results of renewal theory for products of random matrices to stochastic recurrence equations (SRE), we extend the theorem of Lux and Sornette (cond-mat/9910141) and demonstrate that the tails of the unconditional distributions associated with such d-dimensional bubble processes follow power laws (i.e., exhibit hyperbolic decline), with the same asymptotic tail exponent mu

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Bibliographic Info

Paper provided by arXiv.org in its series Papers with number cond-mat/0101371.

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Date of creation: Jan 2001
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Publication status: Published in Quantitative Finance 1, 533541 (2001)
Handle: RePEc:arx:papers:cond-mat/0101371

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  1. Pagan, Adrian, 1996. "The econometrics of financial markets," Journal of Empirical Finance, Elsevier, vol. 3(1), pages 15-102, May.
  2. Flood, Robert P. & Garber, Peter M. & Scott, Louis O., 1984. "Multi-country tests for price level bubbles," Journal of Economic Dynamics and Control, Elsevier, vol. 8(3), pages 329-340, December.
  3. 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, vol. 3(2), pages 139-140, July.
  4. D. Sornette, 2000. ""Slimming" of power law tails by increasing market returns," Papers cond-mat/0010112, arXiv.org, revised Sep 2001.
  5. Adam, M C & Szafarz, A, 1992. "Speculative Bubbles and Financial Markets," Oxford Economic Papers, Oxford University Press, vol. 44(4), pages 626-40, October.
  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. Rosario N. Mantegna, 1998. "Hierarchical Structure in Financial Markets," Papers cond-mat/9802256, arXiv.org.
  8. Blanchard, Olivier Jean, 1979. "Speculative bubbles, crashes and rational expectations," Economics Letters, Elsevier, vol. 3(4), pages 387-389.
  9. Camerer, Colin, 1989. " Bubbles and Fads in Asset Prices," Journal of Economic Surveys, Wiley Blackwell, vol. 3(1), pages 3-41.
  10. Shleifer, Andrei, 2000. "Inefficient Markets: An Introduction to Behavioral Finance," OUP Catalogue, Oxford University Press, number 9780198292272.
  11. 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.
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