رفتار غیر گسترده شاخص بازار سهام در مقیاس های زمانی میکروسکوپی

This paper presents an empirical investigation of the intraday Brazilian stock market price fluctuations, considering qq-Gaussian distributions that emerge from a non-extensive statistical mechanics. Our results show that, when price returns are measured over intervals less than one hour, the empirical distributions are well fitted by qq-Gaussians with exponential damped tails. Scaling behavior is also observed for these microscopic time intervals. We find that the time evolution of the return distributions is according to a super-diffusive qq-Gaussian stationary process within a nonlinear Fokker–Planck equation. This regime breaks down due to the exponential fall-off of the tails, which in turn, governs the transient dynamics to the long-term macroscopic Gaussian regime. This exponentially damped, non-extensive modeling provides a new framework to investigate the dynamics of other stock markets intraday price fluctuations.

The empirical probability distribution functions (PDFs) of price fluctuations of financial indices for different markets have been reported in the econophysics literature in recent years [1] and [2]. Although there has been a huge progress in the statistical description of these fluctuations, a complete and consistent description of the distributions and its dynamics in the high-frequency regime is still lacking. It is well known that high-frequency financial time series such as foreign exchange rates or stock market returns are long-memory non-Gaussian processes. Such complex behavior calls for advanced theories in the field of statistical physics. Whence, we consider the recently proposed non-extensive statistical physics [3], a theory that generalizes the extensive classical statistical theory for strong dependent variables. A large number of studies [4], [5], [6] and [7] of financial markets have already employed the non-extensive statistics in their analysis. In this paper, we investigate the intraday dynamics of the BOVESPA index, the financial index of the Brazilian stock market, one of the largest emerging markets in the world. We model the distributions of the index price fluctuations by qq-Gaussians, which are a class of stable distributions that emerge from the non-extensive approach, where the parameter qq measures the degree of the non-extensivity of the stochastic process. We find that for time horizons less than one hour, the probability distributions of price fluctuations are well fitted by qq-Gaussians with damped exponential tails. A qq-Gaussian scaling at the center of the distributions with q*=1.75q*=1.75 is observed, due to the presence of strong correlations between price fluctuations. The quasi-stationary q*q*-Gaussian regime breaks down due to the exponentially truncated tails and a new intraday correlated regime emerges at larger time scales, in which the q*q*-Gaussian central part of the distributions is consumed by the exponential tails. The dynamics in the short-time regime is also according to a super-diffusive qq-Gaussian stationary process governed by a non-linear Fokker–Planck equation (FPE). Therefore, a coherent description encompassing the non-extensive statistics is addressed, in which the high-frequency price fluctuations are modeled by a correlated anomalous diffusion process. This paper is organized as follows. In Section 2, we describe the empirical observations for the high-frequency BOVESPA index. In Section 3, we present the non-extensive statistical theory. In Sections 4 and 5, these observations are analyzed according to the non-extensive approach. In Section 6, we present some concluding remarks.