A Precursor of Market Crashes

In this paper, we quantitatively investigate the properties of a statistical ensemble of stock prices. We focus attention on the relative price defined as $X(t) = S(t)/S(0)$, where $S(0)$ is the initial price. We selected approximately 3200 stocks traded on the Japanese Stock Exchange and formed a statistical ensemble of daily relative prices for each trading day in the 3-year period from January 4, 1999 to December 28, 2001, corresponding to the period in which the {\it internet Bubble} formed and {\it crashes} in the Japanese stock market. We found that the upper tail of the complementary cumulative distribution function of the ensemble of the relative 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 over time. Furthermore, we found that as the power-law exponents $\alpha$ approached {\it two}, the bubble burst. It is reasonable to assume that when the power-law exponents approached {\it two}, it indicates the bubble is about to burst. PACS: 89.65.Gh; Keywords: Market crashes, Power law, PrecursorComment: 12 pages, 5 figures, forthcoming into European Physical Journal

Stock volatility in the periods of booms and stagnations

The aim of this paper is to compare statistical properties of stock price indices in periods of booms with those in periods of stagnations. We use the daily data of the four stock price indices in the major stock markets in the world: (i) the Nikkei 225 index (Nikkei 225) from January 4, 1975 to August 18, 2004, of (ii) the Dow Jones Industrial Average (DJIA) from January 2, 1946 to August 18, 2004, of (iii) Standard and Poor’s 500 index (SP500) from November 22, 1982 to August 18, 2004, and of (iii) the Financial Times Stock Exchange 100 index (FT 100) from April 2, 1984 to August 18, 2004. We divide the time series of each of these indices in the two periods: booms and stagnations, and investigate the statistical properties of absolute log returns, which is a typical measure of volatility, for each period. We find that (i) the tail of the distribution of the absolute log-returns is approximated by a power-law function with the exponent close to 3 in the periods of booms while the distribution is described by an exponential function with the scale parameter close to unity in the periods of stagnations.Stock volatility, booms, stagnations, power-law distributions, and exponential distributions.

A mechanism leading bubbles to crashes: the case of Japan's land markets

In this paper we investigate quantitatively statistical properties of ensemble of {\it land prices} in Japan in the period from 1981 to 2002, corresponding to the period of bubbles and crashes. We find that the tail of the distributions of ensembles of the land prices in the high price range is well described by a power law distribution, $P(S>x) \sim x^{-\alpha}$, and furthermore that as the power-law exponents $\alpha$ approached to unity, the crashes of bubbles occurred.Comment: 4 pages, 1 figur