Power laws and market crashes

In this paper, we quantitatively investigate the statistical properties of a statistical ensemble of stock prices. We selected 1200 stocks traded on the Tokyo Stock Exchange, and formed a statistical ensemble of daily stock prices for 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 in Japn, and its bursting in the Japanese stock market. We found that the tail of the complementary cumulative distribution function of the ensemble of stock prices in the high value of the price is well described by a power-law distribution, $ P(S>x) \sim x^{-\alpha} $, with an exponent that moves in the range of $ 1.09 < \alpha < 1.27 $. Furthermore, we found that as the power-law exponents $ \alpha $ approached unity, the bubbles collapsed. This suggests that Zipf's law for stock prices is a sign that bubbles are going to burst. PACS: 89.65.Gh