On Stock-Price Fluctuations in the Periods of Booms and Stagnations

Abstract

The statistical properties of the fluctuations of financial prices have been widely researched since Mandelbrot [1] and Fama [2] presented an evidence that return distributions can be well described by a symmetric Levy stable law with tail index close to 1.7. Many empirical studies have shown that the tails of the distributions of returns and volatility follow approximately a power law with estimates of the tail index falling in the range 2 to 4 for large value of returns and volatility. (See, for examples, de Vries [3]; Pagan [4]; Longin [5], Lux [6]; Guillaume et al. [7]; Muller et al. [8]; Gopikrishnan et al. [9], Gopikrishnan et al. [10], Plerou et al. [11], Liu et al. [12]). However, there is also evidence against power-law tails. For instance, Barndorff-Nielsen [13], and Eberlein et al. [14] have respectively fitted the distributions of returns using normal inverse Gaussian, and hyperbolic distribution. Laherrere and Sornette [15] have suggested to describe the distributions of returns by the stretched-exponential distribution. Dragulescu and Yakovenko [16] have shown that the distributions of returns have been approximated by exponential distributions. More recently, Malevergne, Pisarenko and Sornette [17] have suggested that the tails ultimately decay slower than any stretched exponential distribution but probably faster than power laws with reasonable exponents as a result from various statistical tests of returns.