تناوب از نظام های نوسانات متفاوت در پویایی بازار سهام

Based on the tick-by-tick stock prices from the German and American stock markets, we study the statistical properties of the distribution of the individual stocks and the index returns in highly collective and noisy intervals of trading, separately. We show that periods characterized by the strong inter-stock couplings can be associated with the distributions of index fluctuations which reveal more pronounced tails than in the case of weaker couplings in the market. During periods of strong correlations in the German market these distributions can even reveal an apparent Lévy-stable component.

A series of papers devoted to the analysis of financial data fluctuations disclosed that the corresponding distributions can be characterized by the Paretian scaling [1], [2], [3], [4] and [5]. These studies, based on the large data sets of historical stock prices and on the index values, showed that both the distributions of stock price fluctuations and the distributions of index returns reveal scaling over a broad range of time scales from minutes to days (although a more recent investigation found that scaling is restricted to rather short time scales [6]). A remarkable related issue is that the stocks and indices exhibit similar value of the scaling exponent α≃3.0 [4] and [5]. In accordance with the Central Limit Theorem, the distribution of a random variable being a sum of a number of iid random variables with a finite second moment, has to converge to a normal distribution. From this point of view, the similarity of the distributions for the stocks and the corresponding indices requires that the financial data violate the assumptions of the theorem. And as these data have indeed finite variance, a plausible cause for the problems with the convergence can be related to the correlations among the data. This claim seems to be supported by findings that an artificial S&P index constructed from randomly reshuffled stock returns, presents a much better convergence to a Gaussian than the original index [4]. An appropriate measure of correlations among elements of a system is the spectrum of the correlation matrix eigenvalues, which can be easily compared with the universal properties of random matrices [7]. A few recent works have shown that the financial market can be described by at least one repelled eigenvalue with a magnitude exceeding the likely range of values allowed for a random matrix. This one or more deviating eigenvalues indicate that there are relations between various components of the market [8], [9], [10] and [11]. The main purpose of the present work is a quantitative description of the possible relation between the stock price movements and the properties of the distribution of the corresponding index fluctuations. We showed in a previous analyses which were focused on daily patterns of the German DAX index fluctuations that certain characteristic time intervals of a trading day with high index volatility are associated with fluctuation distributions with properties different from more silent intervals of trading [12] and [13]. Since high volatility is connected with stronger correlations between the stocks [14] and [15] we expect that strong and weak inter-stock correlations are reflected in different properties of the index fluctuations. By choosing a few distinct time scales (1–View the MathML source) we are able to test the stability of the results.

To summarize, the results of our analysis show that the time intervals characterized by strongly collective behaviour of stocks are associated with the distributions of the index returns, whose properties differ from the ones for the intervals dominated by noise. Strongly correlated market can be related to the phenomenon of fat tails of the returns distribution, while faster convergence of such a distribution to normal can be attributed to a decorrelated trading. This might be considered as an empirical argument supporting the hypothesis stating that the important factor responsible for the fat tails of the distributions of the index returns is the inter-stock correlations [5]. Such an effect is observed in both the German and the American market for time scales of at least View the MathML source and in the German market for even View the MathML source time scale. This does not exclude, however, possible influence of other factors which can either amplify the effect of inter-stock correlations or be a distinct source of the non-Gaussian tails (see Ref. [23]). For the DAX market, which is in principle more collective than the Dow Jones one on short time scales [10] and [24], the strong inter-stock couplings which occur both repeatedly in specific periods of a trading day and uniquely at random moments, lead to the occurrence of the Lévy-stable region in the distributions of the index returns. This region, however, comprises returns of moderate size only and its existence does not affect the distributions’ tails. The results of our study indicate that removing spurious overnight returns in the S-windows leads to the distributions which, on short time scales, closely resemble those for the truncated Lévy processes. The existence of and switching between different fluctuation regimes in index evolution during periods of correlated and decorrelated trading resembles the phenomenon of two-phase behaviour of the demand for stocks where the equilibrium and the out-of-equilibrium phase interweave [25]. We do not observe, however, any sudden change of properties of the fluctuations for any of the values of the control parameter λ1, but rather a continuous transition from one type of behaviour to another type, which is best visible for 1 and View the MathML source returns of DAX.