پویایی های بازار سهام

We elucidate on several empirical statistical observations of stock market returns. Moreover, we find that these properties are recurrent and are also present in invariant measures of low-dimensional dynamical systems. Thus, we propose that the returns are modeled by the first Poincaré return time of a low-dimensional chaotic trajectory. This modeling, which captures the recurrent properties of the return fluctuations, is able to predict well the evolution of the observed statistical quantities. In addition, it explains the reason for which stocks present simultaneously dynamical properties and high uncertainties. In our analysis, we use data from the S&P 500 index and the Brazilian stock Telebrás.

It is clear that the return distributions of stock market indexes does not have a Gaussian shape as proposed in Ref. [1], mainly due to the pronounced tail of these distributions. In Ref. [2] it is shown that the distribution of the fluctuations in cotton price is a stable Lévy distribution. However, due to the fact that the stable Lévy distribution has infinite variance, it does not fit well the decay of the distribution tail of the return indexes. The fact is that the asymptotic behavior of the return distribution shows faster decay than the one predicted by a Lévy distribution. Recently, a truncated unstable Lévy distribution, a Lévy distribution in the central part followed by an approximately exponential truncation, was proposed to describe the distribution of the return [3], [4] and [5]. Although a truncated Lévy distribution fits a return distribution, its tails follow a power-law asymptotic behavior, characterized by an exponent α≈3, well outside the condition required for the Levy distribution stability (0<α<2), as reported in Ref. [6]. Therefore, besides the relevant progress achieved in the statistical description of these fluctuations, a complete description of the return distributions has not been given in the previous works. In recent works we showed that the return of the S&P 500 index has a Poisson-like distribution [7] and [8]. In agreement with this findings, in Ref. [9] the distribution of the increments of the British Pound/U.S.$ time series was found to decay as an exponential, a process whose distribution is a Poisson. Moreover, the observed Poisson-like distribution is equivalent to a Poisson-like distribution of the first Poincaré return time of a low-dimensional deterministic system. The first Poincaré return time measures the time a chaotic trajectory takes to return to a given reference interval in phase space. One of the purposes of this work is to show that the distribution of the return of a stock index is the same for the recurrent time, i.e., the time the return takes to return to a specified interval of values. Furthermore, the other statistical properties of these two fluctuations can be simulated by measures of the first Poincaré return time of a low-dimensional trajectory. This equivalence, between the fluctuations of the return and a measure typical of chaotic dynamical systems, suggests that the stock market is dynamically recurrent, that is, there is a dynamical process ruling out the stock oscillations. The use of dynamical tools, as the first Poincaré return time, can explain many empirical observations, described in the next section, for the return of a stock index [6], [9] and [10]. In particular, the reason for the preservation, for long time scales, of the return distribution functional form. The model also describes very well how the average time intervals in which high return values occur (rare events) is related with the width of the return distribution (proportional to what is called volatility). With the proposed procedure we explain properties and scales of the distributions for the return and the recurrent time of the S&P index and the Brazilian stock Telebrás. This paper is organized as follows. In Section 2, we describe the many empirical observation for the index S&P 500 and the Brazilian stock Telebrás. In 3, 4 and 5, we describe that these observations can be well reproduced by using the first Poincaré return time, a dynamical variable, to simulate the returns. In Section 6, we present the conclusions of this work.

We demonstrate that several empirical statistical evidences observed in the stock market are also present in the first Poincaré return time of a low-dimensional trajectory. Therefore, this dynamical measurement can be used as the basis of a model for the stock market return. As important as describing the distribution of the return index Rn properly, there is a need to understand more the variations of the average quantities 〈Rn〉 and 〈Rn+〉 with respect to the observation time τ. In case these variations are almost constant, 〈Rn〉 should scale with τ following a linear law, as described by Eq. (30). We showed that there is an equivalence between the temporal measures, the return Rn, and spacial measures, the recurrent time of the return, Tn. This means information on the recurrent cycles of stocks in the stock market gives information on the volatility of this same stock, and vice versa. This equivalence was proven to exist in dynamical systems that are typically “fully” chaotic (as it is the case for Eq. (13)). For a system to be “fully” chaotic, there must exist a homoclinic or heteroclinic orbit in the surroundings of the chaotic set. Also, by “fully” one can think of a dynamics at the chaos-stochastic border, where the dynamical system behaves either as a chaotic or as a stochastic system. Therefore, if this proposed model describes correctly the stock market dynamics, a stock fluctuation should have some statistical properties similar to those observed in randomic fluctuations. We could relate the average time interval that we expect some return value to be observed with respect to the average return (volatility) of the index. This theoretical observation, also observed to happen in the stock market, guide us on how much time one should keep some stock in order to obtain some desired gain. This equation, in fact, can be seen as a test for the existence of properties in the data as observed in dynamically recurrent oscillation. In [29] and [30] the common characteristics of the stock market oscillations with turbulence are proposed. We also found that the empirical observations (i,ii,iv,vii,viii,ix) described in this work are also present in edge turbulence of tokamak plasma [11]. Therefore, we believe the empirical properties, explained to be common of chaotic dynamical systems, could be seen as a set of characteristics to turbulence. In Ref. [6] it is argued that the scaling behavior observed in the distribution of the returns may be connected to the slow decay of the volatility correlations. We do not disagree with this affirmation, but we prefer to emphasize that the scaling behavior is connected to the variations of the averages of the return (tendency) and the variation of the amplitude of the returns (volatility) with time. One example of this is based on a peculiar observation regarding differences in the scaling laws of the average amplitude of the recurrent time, and the amplitude variation of the return with respect to time. The difference is that the recurrent time has a linear scaling law (very small linear correlation decay) and the return has a power-scaling law. This difference is due to the fact that the recurrent time is not affected by time variations of the averages of the return (tendency) and the variation of the amplitude of the returns (volatility) with time. Any stock that verify the empirical properties described in this work should be a candidate to have its dynamics modeled by the proposed model. So, this model is not limited to a specific stock market of any country.