Impact of stock market structure on intertrade time and price dynamics

Evolution of temporal fluctuation scaling exponent in nonstationary time series using supersymmetric theory of stochastic dynamics

Temporal fluctuation scaling (TFS) is an emergent property of complex systems that relates the variance (Ξ_{2}) and the mean (M_{1}) from an empirical data set in the form Ξ_{2}∼M_{1}^{α_{TFS}}, where the dispersion (fluctuation) of the data has been described in terms of Ξ_{2}. At present, it has been shown that this law of complex systems has different multidisciplinary applications such as characterizing the market rate based on its exponent, explaining the spatial spread of a pandemic or measuring dispersion in a counting process, among others, if it is known how the average value M_{1} of a representative quantity in a system changes. Then, using the path integral formalism and Parisi-Sourlas method, we propose an extension of path integral formalism to understand the origin of the temporal fluctuation scaling and the evolution of its exponent over time in nonstationary time series. To this end, we first show how the probability of transition between two states of a stochastic variable x(t) can be expressed once it is known its cumulant generating function. Also, we introduce a nonlinear term in a cumulant generating function of the form H^{(n)}(p,t;γ)∼p^{n} to obtain a model where the nth moment of the probability distribution evolves arbitrarily. Subsequently, in order to reproduce the temporal fluctuation scaling, a linear combination of H^{(n)}(p,t;γ) with n∈{1,2} is used. Therefore this allows describing how the mean M_{1}(t) and the variance Ξ_{2}(t) of empirical time series evolve. Thence, an analytical expression is deduced for the evolution of the temporal evolution of the temporal fluctuation scaling exponent α_{TFS}(t). Likewise, the validity of the expression found for α_{TFS}(t) is verified with a toy model based on white noise. Finally, this approach is verified in two stock indices (Dow Jones and Sao Paulo stock index) and two currencies (GBP-USD and EUR-USD) with daily data. It is found that this approach accurately captures the evolution of the mean and variance of these four financial derivatives after contrasting the results with a coefficient of determination that depends on H^{(n)}(p,t;γ). Also, it is shown that the temporal fluctuation scaling exponent is a measure of uncertainty or volatility in financial time series.

Temporal Theil scaling in diffusive trajectory time series

Temporal fluctuation scaling (TFS) is a power-law relation between the variance (Ξ) and the mean (Υ) which is present in cumulative time series. Taking into account that Theil index (T) can be assumed as a measure of dispersion and considering diffusive trajectory time series, we find a power-law relation between T and Υ of the form T∼(1-cΥ)^{β}, which we call temporal Theil scaling (TTS). Specifically, by analyzing data of volatility and absolute log-return for 24 nonstationary time series of financial markets, meteorology, and COVID-19 spread, we find that TTS is present in diffusive trajectory time series, while TFS is not present. Furthermore, we show that the power-law relation of TTS has a form that is similar to the relation between order parameter and temperature, which is found in the Ginzburg-Landau theory when the nontrivial critical points of an energy functional F_{η,δ} containing arbitrary powers η and δ of the order parameter are calculated.

Temporal fluctuation scaling in nonstationary time series using the path integral formalism

We model the time evolution of the mean and the variance of nonstationary time series using the path integral formalism with the purpose to obtain the temporal fluctuation scaling presents in complex systems. To this end, we first show how the probability of change between two times of a stochastic variable can be written in terms of a Feynman kernel, where the cumulant generating function of statistical moments is identified as the Hamiltonian of the system. Thus, by including the effects of a stochastic drift and a temporal logarithmic term in the cumulant generating function, we find analytical expressions describing the temporal evolutions of the mean and the variance in terms of cumulants. Starting from these expressions, we obtain the temporal fluctuation scaling written as a general analytical relation between the variance and the mean, in such a way that this relation satisfies a power law, with the exponent being a function on time. Additionally, we study several financial time series associated with changes of prices for some stock indexes and currencies. For this financial time series, we find that the temporal evolution of the mean and the variance, the temporal fluctuation scaling, and the temporal evolution of the exponent which are obtained from this path integral approach are in agreement with those obtained using the empirical data.

Trading Imbalance in Chinese Stock Market-A High-Frequency View

Although an imbalance of buying and selling profoundly affects the formation of market trends, a fine-granularity investigation of this perplexity of trading behavior is still missing. Instead of using existing entropy measures, this paper proposed a new indicator based on transaction dataset that enables us to inspect both the direction and the magnitude of this imbalance at high frequency, which we call “polarity”. The polarity aims to measure the unevenness of the very essence trading desire based on the most micro decision making units. We investigate the relationship between the polarity and the return at both market-level and stock-level and find that the autocorrelated polarities cause a positive relation between lagged polarities and returns, while the current polarity is the opposite. It is also revealed that these associations shift according to the market conditions. In fact, when aggregating the one-minute polarities into daily signals, we find not only significant correlations disclosed by the market polarity and market emotion, but also the reliability of these signals in terms of reflecting the transitions of market-level behavior. These results imply that our presented polarity can reflect the market sentiment and condition in real time. Indeed, the trading polarity provides a new indicator from a high-frequency perspective to understand and foresee the market’s behavior in a data-driven manner.

Transforming Gaussian correlations. Applications to generating long-range power-law correlated time series with arbitrary distribution

The observable outputs of many complex dynamical systems consist of time series exhibiting autocorrelation functions of great diversity of behaviors, including long-range power-law autocorrelation functions, as a signature of interactions operating at many temporal or spatial scales. Often, numerical algorithms able to generate correlated noises reproducing the properties of real time series are used to study and characterize such systems. Typically, many of those algorithms produce a Gaussian time series. However, the real, experimentally observed time series are often non-Gaussian and may follow distributions with a diversity of behaviors concerning the support, the symmetry, or the tail properties. It is always possible to transform a correlated Gaussian time series into a time series with a different marginal distribution, but the question is how this transformation affects the behavior of the autocorrelation function. Here, we study analytically and numerically how the Pearson's correlation of two Gaussian variables changes when the variables are transformed to follow a different destination distribution. Specifically, we consider bounded and unbounded distributions, symmetric and non-symmetric distributions, and distributions with different tail properties from decays faster than exponential to heavy-tail cases including power laws, and we find how these properties affect the correlation of the final variables. We extend these results to a Gaussian time series, which are transformed to have a different marginal distribution, and show how the autocorrelation function of the final non-Gaussian time series depends on the Gaussian correlations and on the final marginal distribution. As an application of our results, we propose how to generalize standard algorithms producing a Gaussian power-law correlated time series in order to create a synthetic time series with an arbitrary distribution and controlled power-law correlations. Finally, we show a practical example of this algorithm by generating time series mimicking the marginal distribution and the power-law tail of the autocorrelation function of real time series: the absolute returns of stock prices.

Chaotic Dynamics of the Fractional-Love Model with an External Environment

Based on the fractional order of nonlinear system for love model with a periodic function as an external environment, we analyze the characteristics of the chaotic dynamic. We analyze the relationship between the chaotic dynamic of the fractional order love model with an external environment and the value of fractional order (α, β) when the parameters are fixed. Meanwhile, we also study the relationship between the chaotic dynamic of the fractional order love model with an external environment and the parameters (a, b, c, d) when the fractional order of the system is fixed. When the parameters of fractional order love model are fixed, the fractional order (α, β) of fractional order love model system exhibit segmented chaotic states with the different fractional orders of the system. When the fractional order (α = β) of the system is fixed, the system shows the periodic state and the chaotic state as the parameter is changing as a result.

Dual-induced multifractality in online viewing activity

Although recent studies have found that the long-term correlations relating to the fat-tailed distribution of inter-event times exist in human activity and that these correlations indicate the presence of fractality, the property of fractality and its origin have not been analyzed. We use both detrended fluctuation analysis and multifractal detrended fluctuation analysis to analyze the time series in online viewing activity separating from Movielens and Netflix. We find long-term correlations at both the individual and communal levels and that the extent of correlation at the individual level is determined by the activity level. These long-term correlations also indicate that there is fractality in the pattern of online viewing. We first find a multifractality that results from the combined effect of the fat-tailed distribution of inter-event times (i.e., the times between successive viewing actions of individuals) and the long-term correlations in online viewing activity and verify this finding using three synthesized series. Therefore, it can be concluded that the multifractality in online viewing activity is caused by both the fat-tailed distribution of inter-event times and the long-term correlations and that this enlarges the generic property of human activity to include not just physical space but also cyberspace.

Entropy measures, entropy estimators, and their performance in quantifying complex dynamics: Effects of artifacts, nonstationarity, and long-range correlations

Entropy measures are widely applied to quantify the complexity of dynamical systems in diverse fields. However, the practical application of entropy methods is challenging, due to the variety of entropy measures and estimators and the complexity of real-world time series, including nonstationarities and long-range correlations (LRC). We conduct a systematic study on the performance, bias, and limitations of three basic measures (entropy, conditional entropy, information storage) and three traditionally used estimators (linear, kernel, nearest neighbor). We investigate the dependence of entropy measures on estimator- and process-specific parameters, and we show the effects of three types of nonstationarities due to artifacts (trends, spikes, local variance change) in simulations of stochastic autoregressive processes. We also analyze the impact of LRC on the theoretical and estimated values of entropy measures. Finally, we apply entropy methods on heart rate variability data from subjects in different physiological states and clinical conditions. We find that entropy measures can only differentiate changes of specific types in cardiac dynamics and that appropriate preprocessing is vital for correct estimation and interpretation. Demonstrating the limitations of entropy methods and shedding light on how to mitigate bias and provide correct interpretations of results, this work can serve as a comprehensive reference for the application of entropy methods and the evaluation of existing studies.

Model of the Dynamic Construction Process of Texts and Scaling Laws of Words Organization in Language Systems

Scaling laws characterize diverse complex systems in a broad range of fields, including physics, biology, finance, and social science. The human language is another example of a complex system of words organization. Studies on written texts have shown that scaling laws characterize the occurrence frequency of words, words rank, and the growth of distinct words with increasing text length. However, these studies have mainly concentrated on the western linguistic systems, and the laws that govern the lexical organization, structure and dynamics of the Chinese language remain not well understood. Here we study a database of Chinese and English language books. We report that three distinct scaling laws characterize words organization in the Chinese language. We find that these scaling laws have different exponents and crossover behaviors compared to English texts, indicating different words organization and dynamics of words in the process of text growth. We propose a stochastic feedback model of words organization and text growth, which successfully accounts for the empirically observed scaling laws with their corresponding scaling exponents and characteristic crossover regimes. Further, by varying key model parameters, we reproduce differences in the organization and scaling laws of words between the Chinese and English language. We also identify functional relationships between model parameters and the empirically observed scaling exponents, thus providing new insights into the words organization and growth dynamics in the Chinese and English language.

Collective attention and stock prices: evidence from Google Trends data on Standard and Poor's 100

Today´s connected world allows people to gather information in shorter intervals than ever before, widely monitored by massive online data sources. As a dramatic economic event, recent financial crisis increased public interest for large companies considerably. In this paper, we exploit this change in information gathering behavior by utilizing Google query volumes as a "bad news" indicator for each corporation listed in the Standard and Poor´s 100 index. Our results provide not only an investment strategy that gains particularly in times of financial turmoil and extensive losses by other market participants, but reveal new sectoral patterns between mass online behavior and (bearish) stock market movements. Based on collective attention shifts in search queries for individual companies, hence, these findings can help to identify early warning signs of financial systemic risk. However, our disaggregated data also illustrate the need for further efforts to understand the influence of collective attention shifts on financial behavior in times of regular market activities with less tremendous changes in search volumes.

Impact of stock market structure on intertrade time and price dynamics

We analyse times between consecutive transactions for a diverse group of stocks registered on the NYSE and NASDAQ markets, and we relate the dynamical properties of the intertrade times with those of the corresponding price fluctuations. We report that market structure strongly impacts the scale-invariant temporal organisation in the transaction timing of stocks, which we have observed to have long-range power-law correlations. Specifically, we find that, compared to NYSE stocks, stocks registered on the NASDAQ exhibit significantly stronger correlations in their transaction timing on scales within a trading day. Further, we find that companies that transfer from the NASDAQ to the NYSE show a reduction in the correlation strength of transaction timing on scales within a trading day, indicating influences of market structure. We also report a persistent decrease in correlation strength of intertrade times with increasing average intertrade time and with corresponding decrease in companies' market capitalization–a trend which is less pronounced for NASDAQ stocks. Surprisingly, we observe that stronger power-law correlations in intertrade times are coupled with stronger power-law correlations in absolute price returns and higher price volatility, suggesting a strong link between the dynamical properties of intertrade times and the corresponding price fluctuations over a broad range of time scales. Comparing the NYSE and NASDAQ markets, we demonstrate that the stronger correlations we find in intertrade times for NASDAQ stocks are associated with stronger correlations in absolute price returns and with higher volatility, suggesting that market structure may affect price behavior through information contained in transaction timing. These findings do not support the hypothesis of universal scaling behavior in stock dynamics that is independent of company characteristics and stock market structure. Further, our results have implications for utilising transaction timing patterns in price prediction and risk management optimization on different stock markets.