Dynamical model and nonextensive statistical mechanics of a market index on large time windows

Tsallis Entropy for Cross-Shareholding Network Configurations

In this work, we develop the Tsallis entropy approach for examining the cross-shareholding network of companies traded on the Italian stock market. In such a network, the nodes represent the companies, and the links represent the ownership. Within this context, we introduce the out-degree of the nodes-which represents the diversification-and the in-degree of them-capturing the integration. Diversification and integration allow a clear description of the industrial structure that were formed by the considered companies. The stochastic dependence of diversification and integration is modeled through copulas. We argue that copulas are well suited for modelling the joint distribution. The analysis of the stochastic dependence between integration and diversification by means of the Tsallis entropy gives a crucial information on the reaction of the market structure to the external shocks-on the basis of some relevant cases of dependence between the considered variables. In this respect, the considered entropy framework provides insights on the relationship between in-degree and out-degree dependence structure and market polarisation or fairness. Moreover, the interpretation of the results in the light of the Tsallis entropy parameter gives relevant suggestions for policymakers who aim at shaping the industrial context for having high polarisation or fair joint distribution of diversification and integration. Furthermore, a discussion of possible parametrisations of the in-degree and out-degree marginal distribution-by means of power laws or exponential functions- is also carried out. An empirical experiment on a large dataset of Italian companies validates the theoretical framework.

Efficiency of the Brazilian Bitcoin: A DFA Approach

Bitcoin’s evolution has attracted the attention of investors and researchers looking for a better understanding of the efficiency of cryptocurrency markets, considering their prices and volatility. The purpose of this paper is to contribute to this understanding by studying the degree of persistence of the Bitcoin measured by the Hurst exponent, considering prices from the Brazilian market, and comparing with Bitcoin in USD as a benchmark. We applied Detrended Fluctuation Analysis (DFA), for the period from 9 April 2017 to 30 June 2018, using daily closing prices, with a total of 429 observations. We focused on two prices of Bitcoins resulting from negotiations made by two different Brazilian financial institutions: Foxbit and Mercado. The results indicate that Mercado and Foxbit returns tend to follow Bitcoin dynamics and all of them show persistent behavior, although the persistence in slightly higher for the Brazilian Bitcoin. However, this evidence does not necessarily mean opportunities for abnormal profits, as aspects such as liquidity or transaction costs could be impediments to this occurrence.

An Intrinsic Entropy Model for Exchange-Traded Securities

This paper introduces an intrinsic entropy model which can be employed as an indicator for gauging investors’ interest in a given exchange-traded security, along with the state of the overall market corroborated by individual security trading data. Although the syntagma of intrinsic entropy might sound somehow pleonastic, since entropy itself characterizes the fundamentals of a system, we would like to make a clear distinction between entropy models based on the values that a random variable may take, and the model that we propose, which employs actual stock exchange trading data. The model that we propose for the intrinsic entropy does not include any exogenous factor that could influence the level of entropy. The intrinsic entropy signals if the market is either inclined to buy the security or rather to sell it. We further explore the usage of the intrinsic entropy model for algorithmic trading, in order to demonstrate the value of our model in assisting investors’ intraday stock portfolio selection, along with timely generated signals for supporting the buy/sell decision-making process. The test results provide empirical evidence that the proposed intrinsic entropy model can be used as an indicator for evaluating the direction and the intensity of intraday trading activity of an exchange-traded security. The data employed for testing consisted of historical intraday transactions executed on The Bucharest Stock Exchange (BVB).

Asymmetry of price returns-Analysis and perspectives from a non-extensive statistical physics point of view

We study how the approach grounded on non-extensive statistical physics can be applied to describe and distinguish different stages of the stock and money market development. A particular attention is given to asymmetric behavior of fat tailed distributions of positive and negative returns. A new method to measure this asymmetry is proposed. It is based on the value of the non-extensive Tsallis parameter q. The new quantifier of the relative asymmetry level between tails in terms of the Tsallis parameters q^± is provided to analyze the effect of memory in data caused by nonlinear autocorrelations. The presented analysis takes into account data of separate stocks from the main developing stock market in Europe, i.e., the Warsaw Stock Exchange (WSE) in Poland and—for comparison—data from the most mature money market (Forex). It is argued that the proposed new quantifier is able to describe the stage of market development and its robustness to speculation. The main strength is put on a description and interpretation of the asymmetry between statistical properties of positive and negative returns for various stocks and for diversified time-lags Δt of data counting. The particular caution in this context is addressed to the difference between intraday and interday returns. Our search is extended to study memory effects and their dependence on the quotation frequency for similar large companies—owners of food-industrial retail supermarkets acting on both Polish and European markets (Eurocash, Jeronimo-Martins, Carrefour, Tesco)—but traded on various European stock markets of diversified economical maturity (respectively in Warsaw, Lisbon, Paris and London). The latter analysis seems to indicate quantitatively that stocks from the same economic sector traded on different markets within European Union (EU) may be a target of diversified level of speculations involved in trading independently on the true economic situation of the company. Our work thus gives indications that the statement:” where you are is more important than who you are” is true on trading markets.

Extended q-Gaussian and q-exponential distributions from gamma random variables

The family of q-Gaussian and q-exponential probability densities fit the statistical behavior of diverse complex self-similar nonequilibrium systems. These distributions, independently of the underlying dynamics, can rigorously be obtained by maximizing Tsallis "nonextensive" entropy under appropriate constraints, as well as from superstatistical models. In this paper we provide an alternative and complementary scheme for deriving these objects. We show that q-Gaussian and q-exponential random variables can always be expressed as a function of two statistically independent gamma random variables with the same scale parameter. Their shape index determines the complexity q parameter. This result also allows us to define an extended family of asymmetric q-Gaussian and modified q-exponential densities, which reduce to the standard ones when the shape parameters are the same. Furthermore, we demonstrate that a simple change of variables always allows relating any of these distributions with a beta stochastic variable. The extended distributions are applied in the statistical description of different complex dynamics such as log-return signals in financial markets and motion of point defects in a fluid flow.

Parameter estimation and forecasting for multiplicative log-normal cascades

We study the well-known multiplicative log-normal cascade process in which the multiplication of Gaussian and log normally distributed random variables yields time series with intermittent bursts of activity. Due to the nonstationarity of this process and the combinatorial nature of such a formalism, its parameters have been estimated mostly by fitting the numerical approximation of the associated non-Gaussian probability density function to empirical data, cf. Castaing et al. [Physica D 46, 177 (1990)]. More recently, alternative estimators based upon various moments have been proposed by Beck [Physica D 193, 195 (2004)] and Kiyono et al. [Phys. Rev. E 76, 041113 (2007)]. In this paper, we pursue this moment-based approach further and develop a more rigorous generalized method of moments (GMM) estimation procedure to cope with the documented difficulties of previous methodologies. We show that even under uncertainty about the actual number of cascade steps, our methodology yields very reliable results for the estimated intermittency parameter. Employing the Levinson-Durbin algorithm for best linear forecasts, we also show that estimated parameters can be used for forecasting the evolution of the turbulent flow. We compare forecasting results from the GMM and Kiyono et al.'s procedure via Monte Carlo simulations. We finally test the applicability of our approach by estimating the intermittency parameter and forecasting of volatility for a sample of financial data from stock and foreign exchange markets.

Analysis of molecular dynamics of colloidal particles in transported dilute samples by self-mixing laser Doppler velocimetry

Colloidal particles in a liquid medium are transported with constant velocity, and dynamic light scattering experiments are performed on the samples by self-mixing laser Doppler velocimetry. The power spectrum of the modulated wave induced by the motion of the colloidal particles cannot be described by the well-known formula for flowing Brownian motion systems, i.e., a combination of Doppler shift, diffusion, and translation. Rather, the power spectrum was found to be described by the q-Gaussian distribution function. The molecular mechanism resulting in this anomalous line shape of the power spectrum is attributed to the anomalous molecular dynamics of colloidal particles in transported dilute samples, which satisfy a nonlinear Langevin equation.

Generalization of the Beck-Cohen superstatistics

Generalized superstatistics, i.e., a "statistics of superstatistics," is proposed. A generalized superstatistical system comprises a set of superstatistical subsystems and represents a generalized hyperensemble. There exists a random control parameter that determines both the density of energy states and the distribution of the intensive parameter for each superstatistical subsystem, thereby forming the third, upper level of dynamics. Generalized superstatistics can be used for nonstationary nonequilibrium systems. The system in which a supercritical multitype age-dependent branching process takes place is an example of a nonstationary generalized superstatistical system. The theory is applied to pair production in a neutron star magnetosphere.

Tsallis distributions and 1/f noise from nonlinear stochastic differential equations

Probability distributions that emerge from the formalism of nonextensive statistical mechanics have been applied to a variety of problems. In this article we unite modeling of such distributions with the model of widespread 1/f noise. We propose a class of nonlinear stochastic differential equations giving both the q-exponential or q-Gaussian distributions of signal intensity, revealing long-range correlations and 1/f(β) behavior of the power spectral density. The superstatistical framework to get 1/f(β) noise with q-exponential and q-Gaussian distributions of the signal intensity is proposed, as well.

Superstatistics as the statistics of quasiequilibrium states: application to fully developed turbulence

In nonequilibrium states, currents are produced by irreversible processes that take a system toward the equilibrium state, where the current vanishes. We demonstrate, in a general setting, that superstatistics arises when the system relaxes to a (stationary) quasiequilibrium state instead, where only the mean current vanishes because of fluctuations. In particular, we show that a current with gaussian white noise takes the system to a unique class of quasiequilibrium states, where the superstatistics coincides with Tsallis escort q distributions. Considering the fully developed turbulence as an example of such quasiequilibrium states, we analytically deduce the power-law spectrum of the velocity structure functions, yielding a correction to the log-normal model, which removes its shortcomings with regard to the decreasing higher order moments and the Novikov inequality, and obtain exponents that agree well with the experimental data.

Different methods to estimate the Einstein-Markov coherence length in turbulence

We study the Markov property of experimental velocity data of different homogeneous isotropic turbulent flows. In particular, we examine the stochastic "cascade" process of nested velocity increments ξ(r):=u(x+r)-u(x) as a function of scale r for different nesting structures. It was found in previous work that, for a certain nesting structure, the stochastic process of ξ(r) has the Markov property for step sizes larger than the so-called Einstein-Markov coherence length l(EM), which is of the order of magnitude of the Taylor microscale λ [Phys. Lett. A 359, 335 (2006)]. We now show that, if a reasonable definition of the effective step size of the process is applied, this result holds independently of the nesting structure. Furthermore, we analyze the stochastic process of the velocity u as a function of the spatial position x. Although this process does not have the exact Markov property, a characteristic length scale l(u(x))≈l(EM) can be identified on the basis of a statistical test for the Markov property. Using a method based on the matrix of transition probabilities, we examine the significance of the non-Markovian character of the velocity u(x) for the statistical properties of turbulence.

Generalized statistical mechanics for superstatistical systems

Mesoscopic systems in a slowly fluctuating environment are often well described by superstatistical models. We develop a generalized statistical mechanics formalism for superstatistical systems, by mapping the superstatistical complex system onto a system of ordinary statistical mechanics with modified energy levels. We also briefly review recent examples of applications of the superstatistics concept for three very different subject areas, namely train delay statistics, turbulent tracer dynamics and cancer survival statistics.

Model for non-Gaussian intraday stock returns

Stock prices are known to exhibit non-Gaussian dynamics, and there is much interest in understanding the origin of this behavior. Here, we present a model that explains the shape and scaling of the distribution of intraday stock price fluctuations (called intraday returns) and verify the model using a large database for several stocks traded on the London Stock Exchange. We provide evidence that the return distribution for these stocks is non-Gaussian and similar in shape and that the distribution appears stable over intraday time scales. We explain these results by assuming the volatility of returns is constant intraday but varies over longer periods such that its inverse square follows a gamma distribution. This produces returns that are Student distributed for intraday time scales. The predicted results show excellent agreement with the data for all stocks in our study and over all regions of the return distribution.

Random walker in temporally deforming higher-order potential forces observed in a financial crisis

Basic peculiarities of market price fluctuations are known to be well described by a recently developed random-walk model in a temporally deforming quadratic potential force whose center is given by a moving average of past price traces [M. Takayasu, T. Mizuno, and H. Takayasu, Physica A 370, 91 (2006)]. By analyzing high-frequency financial time series of exceptional events, such as bubbles and crashes, we confirm the appearance of higher-order potential force in the markets. We show statistical significance of its existence by applying the information criterion. This time series analysis is expected to be applied widely for detecting a nonstationary symptom in random phenomena.

Superstatistical fluctuations in time series: applications to share-price dynamics and turbulence

We report a general technique to study a given experimental time series with superstatistics. Crucial for the applicability of the superstatistics concept is the existence of a parameter beta that fluctuates on a large time scale as compared to the other time scales of the complex system under consideration. The proposed method extracts the main superstatistical parameters out of a given data set and examines the validity of the superstatistical model assumptions. We test the method thoroughly with surrogate data sets. Then the applicability of the superstatistical approach is illustrated using real experimental data. We study two examples, velocity time series measured in turbulent Taylor-Couette flows and time series of log returns of the closing prices of some stock market indices.

Superstatistical distributions from a maximum entropy principle

We deal with a generalized statistical description of nonequilibrium complex systems based on least biased distributions given some prior information. A maximum entropy principle is introduced that allows for the determination of the distribution of the fluctuating intensive parameter beta of a superstatistical system, given certain constraints on the complex system under consideration. We apply the theory to three examples: the superstatistical quantum-mechanical harmonic oscillator, the superstatistical classical ideal gas, and velocity time series as measured in a turbulent Taylor-Couette flow.

Spectral fluctuations of billiards with mixed dynamics: from time series to superstatistics

A statistical analysis of the eigenfrequencies of two sets of superconducting microwave billiards, one with mushroomlike shape and the other from the family of the Limaçon billiards, is presented. These billiards have mixed regular-chaotic dynamics but different structures in their classical phase spaces. The spectrum of each billiard is represented as a time series where the level order plays the role of time. Two most important findings follow from the time series analysis. First, the spectra can be characterized by two distinct relaxation lengths. This is a prerequisite for the validity of the superstatistical approach, which is based on the folding of two distribution functions. Second, the shape of the resulting probability density function of the so-called superstatistical parameter is reasonably approximated by an inverse chi2 distribution. This distribution is used to compute nearest-neighbor spacing distributions and compare them with those of the resonance frequencies of billiards with mixed dynamics within the framework of superstatistics. The obtained spacing distribution is found to present a good description of the experimental ones and is of the same or even better quality as a number of other spacing distributions, including the one from Berry and Robnik. However, in contrast to other approaches toward a theoretical description of spectral properties of systems with mixed dynamics, superstatistics also provides a description of properties of the eigenfunctions in terms of a superstatistical generalization of the Porter-Thomas distribution. Indeed, the inverse chi2 parameter distribution is found suitable for the analysis of experimental resonance strengths in the Limaçon billiards within the framework of superstatistics.

Universal features in the growth dynamics of religious activities

We quantify and analyze the growth dynamics of a religious group in 140 countries for a 47-year period (1959-2005). We find that (i) the distribution of annual logarithmic growth rates exhibits the same functional form for distinct size scales and (ii) the standard deviation of growth rates scales with size as a power law. Both findings hold for distinct measures of religious activity. These results are in surprising agreement with those found in the study of economic activities and scientific research, suggesting that religious activities are governed by universal growth mechanisms. We also compare the empirical findings on religious activities with the predictions of general models recently proposed in the context of complex organizations. Our findings should provide useful information for a better understanding of the mechanisms governing the growth of religion.

Improved estimation of Fokker-Planck equations through optimization

An improved method for the description of hierarchical complex systems by means of a Fokker-Planck equation is presented. In particular the limited-memory Broyden-Fletcher-Goldfarb-Shanno algorithm for constraint problems is used to minimize the distance between the numerical solutions of the Fokker-Planck equation and the empirical probability density functions and thus to estimate properly the drift and diffusion term of the Fokker-Planck equation. The optimization routine is applied to a time series of velocity measurements obtained from a turbulent helium gas jet in order to demonstrate the benefits and to quantify the improvements of this optimization routine.

Estimator of a non-Gaussian parameter in multiplicative log-normal models

We study non-Gaussian probability density functions (PDF's) of multiplicative log-normal models in which the multiplication of Gaussian and log-normally distributed random variables is considered. To describe the PDF of the velocity difference between two points in fully developed turbulent flows, the non-Gaussian PDF model was originally introduced by Castaing [Physica D 46, 177 (1990)]. In practical applications, an experimental PDF is approximated with Castaing's model by tuning a single non-Gaussian parameter, which corresponds to the logarithmic variance of the log-normally distributed variable in the model. In this paper, we propose an estimator of the non-Gaussian parameter based on the q th order absolute moments. To test the estimator, we introduce two types of stochastic processes within the framework of the multiplicative log-normal model. One is a sequence of independent and identically distributed random variables. The other is a log-normal cascade-type multiplicative process. By analyzing the numerically generated time series, we demonstrate that the estimator can reliably determine the theoretical value of the non-Gaussian parameter. Scale dependence of the non-Gaussian parameter in multiplicative log-normal models is also studied, both analytically and numerically. As an application of the estimator, we demonstrate that non-Gaussian PDF's observed in the S&P500 index fluctuations are well described by the multiplicative log-normal model.

Superstatistics, thermodynamics, and fluctuations

A thermodynamiclike formalism is developed for superstatistical systems based on conditional entropies. This theory takes into account large-scale variations of intensive variables of systems in nonequilibrium stationary states. Ordinary thermodynamics is recovered as a special case of the present theory, and corrections to it can systematically be evaluated. A generalization of Einstein's relation for fluctuations is presented using a maximum entropy condition.

Random matrix theory within superstatistics

We propose a generalization of the random matrix theory following the basic prescription of the recently suggested concept of superstatistics. Spectral characteristics of systems with mixed regular-chaotic dynamics are expressed as weighted averages of the corresponding quantities in the standard theory assuming that the mean level spacing itself is a stochastic variable. We illustrate the method by calculating the level density, the nearest-neighbor-spacing distributions, and the two-level correlation functions for systems in transition from order to chaos. The calculated spacing distribution fits the resonance statistics of random binary networks obtained in a recent numerical experiment.

From time series to superstatistics

Complex nonequilibrium systems are often effectively described by a "statistics of a statistics", in short, a "superstatistics". We describe how to proceed from a given experimental time series to a superstatistical description. We argue that many experimental data fall into three different universality classes: Chi2 superstatistics (Tsallis statistics), Chi2 inverse superstatistics, and log-normal superstatistics. We discuss how to extract the two relevant well separated superstatistical time scales tau and T, the probability density of the superstatistical parameter beta, and the correlation function for beta from the experimental data. We illustrate our approach by applying it to velocity time series measured in turbulent Taylor-Couette flow, which is well described by log-normal superstatistics and exhibits clear time scale separation.

Stochastic opinion formation in scale-free networks

The dynamics of opinion formation in large groups of people is a complex nonlinear phenomenon whose investigation is just beginning. Both collective behavior and personal views play an important role in this mechanism. In the present work we mimic the dynamics of opinion formation of a group of agents, represented by two states +/-1, as a stochastic response of each agent to the opinion of his/her neighbors in the social network and to feedback from the average opinion of the whole. In the light of recent studies, a scale-free Barabási-Albert network has been selected to simulate the topology of the interactions. A turbulent-like dynamics, characterized by an intermittent behavior, is observed for a certain range of the model parameters. The problem of uncertainty in decision taking is also addressed both from a topological point of view, using random and targeted removal of agents from the network, and by implementing a three-state model, where the third state, zero, is related to the information available to each agent. Finally, the results of the model are tested against the best known network of social interactions: the stock market. A time series of daily closures of the Dow-Jones index has been used as an indicator of the possible applicability of our model in the financial context. Good qualitative agreement is found.

Additive-multiplicative stochastic models of financial mean-reverting processes

We investigate a generalized stochastic model with the property known as mean reversion, that is, the tendency to relax towards a historical reference level. Besides this property, the dynamics is driven by multiplicative and additive Wiener processes. While the former is modulated by the internal behavior of the system, the latter is purely exogenous. We focus on the stochastic dynamics of volatilities, but our model may also be suitable for other financial random variables exhibiting the mean reversion property. The generalized model contains, as particular cases, many early approaches in the literature of volatilities or, more generally, of mean-reverting financial processes. We analyze the long-time probability density function associated to the model defined through an Itô-Langevin equation. We obtain a rich spectrum of shapes for the probability function according to the model parameters. We show that additive-multiplicative processes provide realistic models to describe empirical distributions, for the whole range of data.

Devil's-staircase-like behavior of the range of random time series with record-breaking fluctuations

We propose insight into the analysis of the record-breaking fluctuations in random time series, which permits to distinguish between the self-organized criticality and the record dynamics (RD) scenarios of system evolution, using a finite time series realization. Performed analysis of the time series associated with the historical prices of different commodities has shown that the evolution of commodity markets is controlled by the record-breaking fluctuations as it is outlined by the RD. Furthermore, we found that the sizes of record-breaking fluctuations follow a fat-tailed distribution and the devil's-staircase-like records of price ranges are multiaffine and persistent, nevertheless, the high moments (q> q(C) >2) of their q-order height-height correlation functions behave logarithmically.

Asymptotics of superstatistics

Superstatistics are superpositions of different statistics relevant for driven nonequilibrium systems with spatiotemporal inhomogeneities of an intensive variable (e.g., the inverse temperature). They contain Tsallis statistics as a special case. We develop here a technique that allows us to analyze the large energy asymptotics of the stationary distributions of general superstatistics. A saddle-point approximation is developed which relates this problem to a variational principle. Several examples are worked out in detail.

Increment definitions for scale-dependent analysis of stochastic data

It is common for scale-dependent analysis of stochastic data to use the increment Delta(t,r) =xi(t+r)-xi(t) of a data set xi(t) as a stochastic measure, where r denotes the scale. For joint statistics of Delta(t,r) and Delta(t, r') the question of how to nest the increments on different scales r, r' is investigated. Here we show that in some cases spurious correlations between scales can be introduced by the common left-justified definition. The consequences for a Markov process are discussed. These spurious correlations can be avoided by an appropriate nesting of increments. We demonstrate this effect for different data sets and show how it can be detected and quantified. The problem allows to propose a unique method to distinguish between experimental data generated by a noiselike or a Langevin-like random-walk process, respectively.

Stochastic cellular automata model for stock market dynamics

In the present work we introduce a stochastic cellular automata model in order to simulate the dynamics of the stock market. A direct percolation method is used to create a hierarchy of clusters of active traders on a two-dimensional grid. Active traders are characterized by the decision to buy, sigma(i) (t)=+1, or sell, sigma(i) (t)=-1, a stock at a certain discrete time step. The remaining cells are inactive, sigma(i) (t)=0. The trading dynamics is then determined by the stochastic interaction between traders belonging to the same cluster. Extreme, intermittent events, such as crashes or bubbles, are triggered by a phase transition in the state of the bigger clusters present on the grid, where almost all the active traders come to share the same spin orientation. Most of the stylized aspects of the financial market time series, including multifractal proprieties, are reproduced by the model. A direct comparison is made with the daily closures of the S&P 500 index.