احتمال حرکت های بزرگ در بازارهای مالی

Based on empirical financial time series, we show that the “silence-breaking” probability follows a super-universal power law: the probability of observing a large movement is inversely proportional to the length of the on-going low-variability period. Such a scaling law has been previously predicted theoretically [R. Kitt, J. Kalda, Physica A 353 (2005) 480], assuming that the length-distribution of the low-variability periods follows a multi-scaling power law.

The power laws and scaling behaviour are present in numerous aspects of human societies and in the nature. One of the first promoters of the concept of the power law was Vilfredo Pareto (cf. Ref. [1]), who studied the wealth distribution in different societies. Further, Harvard linguistics professor George Zipf, again a representative of social sciences, observed that only few words in English language are used very often, and most of the words are used rarely. (cf. Ref. [2]). Nowadays, physicists are very used to the power laws, which, however, are sometimes somewhat counter-intuitive. Indeed, the presence of a power law means that there are some representatives of a population, which are very different from the typical members of that population. For example, as the result of a social evolution, the wealth of a single individual can qualitatively change the wealth of a large community. This is completely different from the biological evolution: e.g. the weight of a single living creature makes always only a tiny contribution to the net biomass of the corresponding (non-small) population. The presence of a wide spectrum of power laws in finances can be ascribed to the fact that socially interacting humans form large complex systems, which are characterized by self-organized criticality [3]. This makes finances (alongside with the turbulence) a fruitful polygon for studying various aspects of scale-invariance. Indeed, various power laws have been observed in financial time series since 1960s, by B. Mandelbrot (cf. Ref. [4] and [5] and the references therein). In 1999, Ausloos and Ivanova also reported the multi-fractality in financial time series (cf. Ref. [6]). Around 1990s, the studies of scale-invariance in finances became more extensive, cf. Ref. [7], [8], [9] and [10], effectively creating a new branch of statistical physics — the econophysics. A recent overview of the progress in understanding the scaling and its universality in finances can be found in Ref. [11]. The aim of the current study is to contribute to the understanding of the origins of universality. Mathematically, our basic idea is very simple, nearly trivial. However, when dealing with the sources of universality, mathematically simple and robust models have better chances of describing reality, than complex and elaborate constructions; cf. the Occam’s razor. The attempt to successfully and systematically predict the direction of future movements of asset prices can be compared to the attempts of inventing the perpetum mobile. However, the attempt to characterize and predict the risk (or volatility) may offer significantly better results and therefore the volatility is one of the most-studied phenomena in Econophysics. One of the most challenging features of the volatility dynamics are the intermittently appearing extreme price movements, which are often accompanied by overall increase of volatility over a certain time window. Traditionally, such a behaviour has been described by multi-fractal spectra. The multi-fractal analysis is undoubtedly a powerful tool; however, due to the involved mathematical methods, it is not well suited for practical applications of the prediction and optimisation of risks. This observation motivated us to introduce a complementary method of the length-distribution analysis of the low-variability periods [12], [13] and [14]. The low-variability periods are defined as consequent time periods, during which the price changes of the observed asset (as compared to the local average over a sliding window of width ww) remains under a pre-set threshold δδ. The illustration of the method is shown in Fig. 1. This method (with certain modifications) was developed independently within different contexts, and put under extensive tests by several research groups. So, the effects of the long-term memory and clustering of extreme events in various time series (cf. Ref. [15], [16], [17], [18] and [19]), are, in fact, closely related to (and in a certain sense covered by) the low-variability period analysis. The same applies to the studies of the time intervals ττ between volatilities which are above a threshold qq [20], [21], [22], [23], [24] and [25]). Full-size image (81 K) Fig. 1. Variability ΔΔ of the DAX index and the respective low-variability periods for 4 years (2004–2007), using View the MathML sourcew=1day (a) and View the MathML sourcew=10days (b). For a pre-fixed threshold level δ=2σδ=2σ (where σσ is the standard deviation of the signal), low-variability intervals of duration τiτi are formed as the intervals corresponding to such graph segments, which lay entirely inside the gray area. The small graphs in top share the time axis with the bottom graph, and illustrate the fragmentation of them into low-variability intervals by plotting the interval index ii versus time tt. Figure options Our previous studies [14] and [26] have shown that • financial time series are typically characterized by multi-scaling behaviour of the low-variability periods (power law can be observed for a certain range of the parameters δδ and ww, and the scaling exponent depends on these parameters); • theoretically, a multi-fractal time series follow also a multi-scaling behaviour of the low-variability periods; the scaling exponent can be expressed via the multi-fractal exponents; • as compared to the multi-fractal analysis, the analysis of the low-variability periods is easier to implement, has higher resolution of time-scales, and the results of the analysis can be interpreted more straightforwardly. These findings agree well with the above cited independent studies. The market fluctuations have been also modelled as Lévy flights and continuum time random walks (CTRW); therefore it is also of interest to mention that in the case of uncorrelated Lévy flights, there is no power law for the length-distribution of the low-variability periods: length-distribution decays exponentially; the same applies to Gaussian time series, e.g. (non)persistent random walks. Meanwhile, in the case of CTRW, there is a mono-scaling behaviour of the low-variability periods: the scaling exponent does not depend on the parameters δδ and ww, and is defined by the exponent of the waiting-time distribution [27]. In addition to the above listed results, we have shown [26] that the very presence of a power law for the probability distribution function of the low-variability segment lengths (even if observed for a narrow range of the parameters δδ and ww) bears interesting consequence: the probability pp of observing a large movement (exceeding the threshold parameter δδ) in the time series during the next period of duration ww, is inversely proportional to the length of the on-going low-variability period (i.e. to the time elapsed since the most recent large movement; note that by definition, p<1p<1, and p=1p=1 would imply that for the given parameters, the next price movement exceeds definitely the threshold δδ). This super-universal scaling law has been derived independently by Bogachev, et al. [17] and [18]. Here we repeat the derivation briefly. First, we assume that the length of the periods is measured in the units of the window length ww. Then, the probability of a large movement during the unit time is given by the ratio of (a) the number of those low-variability periods, the length of which is exactly nn, Na=R(n)−R(n+1)Na=R(n)−R(n+1), and (b) the number of those low-variability periods, which have length m≥nm≥n, Nb=R(n)Nb=R(n). Here, R(n)R(n) denotes the cumulative length-distribution function of the low-variability periods with length equal to nn [i.e. R(n)R(n) represents the number of low-variability periods with length greater than or equal to nn time units]. So, we can express the probability of “silence-breaking” (i.e. the probability that the next movement exceeding the threshold δδ) as a function of the on-going “silent” period nn: equation(1) p(n)=[R(n)−R(n+1)]/R(n).p(n)=[R(n)−R(n+1)]/R(n). Turn MathJax on If nn is large, the difference R(n)−R(n+1)R(n)−R(n+1) can be calculated approximately as View the MathML source−dRdn. Upon applying the power law R(n)=R0n−α(δ,w)R(n)=R0n−α(δ,w), we arrive at Na≈αR0n−α−1Na≈αR0n−α−1 and Nb≈R0n−αNb≈R0n−α. Bearing in mind that p(n)=Na/Nbp(n)=Na/Nb, the final result is written as equation(2) p(n)≈αn−1.p(n)≈αn−1. Turn MathJax on Since the scaling exponent of this law is independent of the parameters δδ and ww (as well as of the scaling exponent αα), it can be called super-universal. The Eq. (2) can be also denoted as the law of the silence-breaking probability, because it describes the probability of a large movement after a longer “silent” period (characterized by small movements). Recently, similar application of the risk measurement was proposed by Wang, et al. [21]. As demonstrated above, the mathematics behind this super-universal law is extremely simple. However, we do believe that the consequences of it are profound, and it allows us to shed light into the origin of universality in the dynamics of market fluctuations. In fact, a very similar behaviour (with a similar mathematical origin) has been detected in the context of direct avalanches in self-organized critical systems (cf. Ref. [28]), and proven to be a useful tool in understanding the universality in burst dynamics. These arguments motivated us to put Eq. (2) under several tests, using both the data of real market fluctuations, as well as the surrogate data. The aim is to clarify, how well is the super-universal law followed by the real-world time series, for which the scaling laws are far from being perfect (and is for certain applications better described, e.g., by stretched exponentials [15]), and for which the finite-size effects can (possibly) mask the theoretical scaling behaviour. In order to show the presence of the 1/n1/n-law in relatively short time series (with unavoidably poor statistics of long low-variability periods), an appropriate data analysis technique has been developed (see Section 2.1)