Power Laws and Market Crashes

In this paper, we quantitatively investigate the statistical properties of astatistical ensemble of stock prices. We selected 1200 stocks traded on theTokyo Stock Exchange, and formed a statistical ensemble of daily stock pricesfor each trading day in the 3-year period from January 4, 1999 to December 28,2001, corresponding to the period of the forming of the internet bubble inJapn, and its bursting in the Japanese stock market. We found that the tail ofthe complementary cumulative distribution function of the ensemble of stockprices in the high value of the price is well described by a power-lawdistribution, $ P(S>x) \sim x^{-\alpha} $, with an exponent that moves in therange of $ 1.09 < \alpha < 1.27 $. Furthermore, we found that as the power-lawexponents $ \alpha $ approached unity, the bubbles collapsed. This suggeststhat Zipf's law for stock prices is a sign that bubbles are going to burst. PACS: 89.65.GhComment: 8 pages, 4 figures, forthcoming into Prog. Theor. Phys. Suppl. (2006