A New Look at Calendar Anomalies: Multifractality and Day-of-the-Week Effect

Stock markets can become inefficient due to calendar anomalies known as the day-of-the-week effect. Calendar anomalies are well known in the financial literature, but the phenomena remain to be explored in econophysics. This paper uses multifractal analysis to evaluate if the temporal dynamics of market returns also exhibit calendar anomalies such as day-of-the-week effects. We apply multifractal detrended fluctuation analysis (MF-DFA) to the daily returns of market indices worldwide for each day of the week. Our results indicate that distinct multifractal properties characterize individual days of the week. Monday returns tend to exhibit more persistent behavior and richer multifractal structures than other day-resolved returns. Shuffling the series reveals that multifractality arises from a broad probability density function and long-term correlations. The time-dependent multifractal analysis shows that the Monday returns’ multifractal spectra are much wider than those of other days. This behavior is especially persistent during financial crises. The presence of day-of-the-week effects in multifractal dynamics of market returns motivates further research on calendar anomalies for distinct market regimes.     

Multifractal analysis of stock exchange crashes

We analyze the complexity of rare events of the DJIA Index. We reveal that the returns of the time series exhibit strong multifractal properties meaning that temporal correlations play a substantial role. The effect of major stock market crashes can be best illustrated by the comparison of the multifractal spectra of the time series before and after the crash. Aftershock periods compared to foreshock periods exhibit richer and more complex dynamics. Compared to an average crash, calculated by taking into account the larger 5 crashes of the DJIA Index, the 1929 event exhibits significantly more increase in multifractality than the 1987 crisis.     

Multifractal detrended fluctuation analysis of the Chinese stock index futures market

Based on the multifractal detrended fluctuation analysis (MF-DFA) and multifractal spectrum analysis, this paper empirically studies the multifractal properties of the Chinese stock index futures market. Using a total of 2942 ten-minute closing prices, we find that the Chinese stock index futures returns exhibit long-range correlations and multifractality, making the single-scale index insufficient to describe the futures price fluctuations. Further, by comparing the original time series with the transformed time series through shuffling procedure and phase randomization procedure, we show the existence of two different sources of the multifractality for the Chinese stock index futures market. Our results suggest that the multifractality is mainly due to long-range correlations, although the fat-tailed probability distributions also contribute to such multifractal behaviour.     

Joint multifractal analysis based on wavelet leaders

Mutually interacting components form complex systems and these components usually have long-range cross-correlated outputs. Using wavelet leaders, we propose a method for characterizing the joint multifractal nature of these long-range cross correlations; we call this method joint multifractal analysis based on wavelet leaders (MF-X-WL). We test the validity of the MF-X-WL method by performing extensive numerical experiments on dual binomial measures with multifractal cross correlations and bivariate fractional Brownian motions (bFBMs) with monofractal cross correlations. Both experiments indicate that MF-X-WL is capable of detecting cross correlations in synthetic data with acceptable estimating errors. We also apply the MF-X-WL method to pairs of series from financial markets (returns and volatilities) and online worlds (online numbers of different genders and different societies) and determine intriguing joint multifractal behavior.     

The Role of Information in Managing Interactions from a Multifractal Perspective

In the framework of the multifractal hydrodynamic model, the correlations informational entropy–cross-entropy manages attractive and repulsive interactions through a multifractal specific potential. The classical dynamics associated with them imply Hubble-type effects, Galilei-type effects, and dependences of interaction constants with multifractal degrees at various scale resolutions, while the insertion of the relativistic amendments in the same dynamics imply multifractal transformations of a generalized Lorentz-type, multifractal metrics invariant to these transformations, and an estimation of the dimension of the multifractal Universe. In such a context, some correspondences with standard cosmologies are analyzed. Since the same types of interactions can also be obtained as harmonics mapping between the usual space and the hyperbolic plane, two measures with uniform and non-uniform temporal flows become functional, temporal measures analogous with Milne’s temporal measures in a more general manner. This work furthers the analysis published recently by our group in “Towards Interactions through Information in a Multifractal Paradigm”.     

Research on the relationship between the multifractality and long memory of realized volatility in the SSECI

We examine the multifractal properties of the realized volatility (RV) and realized bipower variation (RBV) series in the Shanghai Stock Exchange Composite Index (SSECI) by using the multifractal detrended fluctuation analysis (MF-DFA) method. We find that there exist distinct multifractal characteristics in the volatility series. The contributions of two different types of source of multifractality, namely, fat-tailed probability distributions and nonlinear temporal correlations, are studied. By using the unit root test, we also find the strength of the multifractality of the volatility time series is insensitive to the sampling frequency but that the long memory of these series is sensitive.     

A multifractal detrended fluctuation analysis (MDFA) of the Chinese growth enterprise market (GEM)

A multifractal, detrended fluctuation approach is used to analyze the growth enterprise market (GEM) in China involving a range of correlations in fluctuations of share prices (fat tail), persistent and anti-persistent states. Our analysis exhibits company-specific multifractal characteristics, which vary among the companies listed in the same industry, e.g., the power-law cross-correlations between computer and electronics sectors. These results may help reduce the risk in complex financial markets.     

Multifractal properties of the Indian financial market

We investigate the multifractal properties of the logarithmic returns of the Indian financial indices (BSE & NSE) by applying the multifractal detrended fluctuation analysis. The results are compared with that of the US S&P 500 index. Numerically we find that qth-order generalized Hurst exponents h(q) and τ(q) change with the moments q. The nonlinear dependence of these scaling exponents and the singularity spectrum f(α) show that the returns possess multifractality. By comparing the MF-DFA results of the original series to those for the shuffled series, we find that the multifractality is due to the contributions of long-range correlations as well as the broad probability density function. The financial markets studied here are compared with the Binomial Multifractal Model (BMFM) and have a smaller multifractal strength than the BMFM.     

A multifractal detrended fluctuation analysis of trading behavior of individual and institutional traders in Tehran stock market

Employing the multifractal detrended fluctuation analysis (MF-DFA), the multifractal properties of trading behavior of individual and institutional traders in the Tehran Stock Exchange (TSE) are numerically investigated. Using daily trading volume time series of these two categories of traders, the scaling exponents, generalized Hurst exponents, generalized fractal dimensions and singularity spectrum are derived. Furthermore, two main sources of multifractality, i.e. temporal correlations and fat-tailed probability distributions are also examined. We also compare our results with data of S&P 500. Results of this paper suggest that for both classes of investors in TSE, multifractality is mainly due to long-range correlation while for S&P 500, the fat-tailed probability distribution is the main source of multifractality.     

Price–volume cross-correlation analysis of CSI300 index futures

We investigate the cross-correlation between price returns and trading volumes for the China Securities Index 300 (CSI300) index futures, which are the only stock index futures traded on the China Financial Futures Exchange (CFFEX). The basic statistics suggest that distributions of these two time series are not normal but exhibit fat tails. Based on the detrended cross-correlation analysis (DCCA), we obtain that returns and trading volumes are long-range cross-correlated. The existence of multifractality in the cross-correlation between returns and trading volumes has been proven with the multifractal detrended cross-correlation analysis (MFDCCA) algorithm. The multifractal analysis also confirms that returns and trading volumes have different degrees of multifractality. We further perform a cross-correlation statistic to verify whether the cross-correlation significantly exists between returns and trading volumes for CSI300 index futures. In addition, results of the test for lead-lag effect demonstrate that contemporaneous cross-correlation of return and trading volume series is stronger than cross-correlations of leaded or lagged series.     

Are developed and emerging agricultural futures markets multifractal? A comparative perspective

Although there are many reports on the empirical evidence of the existence of multifractality in various financial or commodity markets in current literature, few can be found to compare the multifractal properties of emerging and developed economies, especially for agricultural futures markets in those countries (regions). We therefore chose China as the representative of the transition and emerging economies, and USA as the representative of developed ones. We attempt to find the answers to the following questions: (1) Are all those different markets multifractal? (2) What are the dynamical causes for multifractality in those markets (if any)? (3) Are the multifractality strengths in those markets of the transition and emerging economies weaker (or stronger) than those of the developed ones? To answer these questions, Multifractal Detrended Fluctuation Analysis (MF-DFA) are applied to study some of the representative agricultural futures markets in China and USA, namely, wheat, soy meal, soybean and corn. Our results suggest that all the markets of China and USA exhibit multifractal properties except US soybean market, which is much closer to mono-fractal comparing with China’s soybean market. To investigate the sources of multifractality, shuffling and phase randomization procedures are applied to destroy the temporal correlations and non-Gaussian distributions respectively. We found that multifractality can be mainly attributed to the non-Gaussian probability distribution and secondarily to the nonlinear temporal correlation mechanism for all the markets, except US soybean and soy meal, which derives from some other unknown factors. Furthermore, the average of τ(q) are applied to obtain the multifractal spectra of the two markets as a whole. The results show that the width of the multifractal spectrum of US agricultural futures markets is significantly narrower than that of China’s. Based on our findings, we proposed a hypothesis that the strength of multifractality in developed economies may be weaker than that in emerging and transition ones.     

Correlation of eigenstates in the critical regime of quantum Hall systems

We extend the multifractal analysis of the statistics of critical wave functions in quantum Hall systems by calculating numerically the correlations of local amplitudes corresponding to eigenstates at two different energies. Our results confirm multifractal scaling relations which are different from those occurring in conventional critical phenomena. The critical exponent corresponding to the typical amplitude, [Formula: see text], gives an almost complete characterization of the critical behaviour of eigenstates, including correlations. Our results support the interpretation of the local density of states being an order parameter of the Anderson transition.     

Multifractal analysis on international crude oil markets based on the multifractal detrended fluctuation analysis

The multifractal nature of WTI and Brent crude oil markets is studied employing the multifractal detrended fluctuation analysis. We find that two crude oil markets become more and more efficient for long-term and two Gulf Wars cannot change time scale behavior of crude oil return series. Considering long-term influence caused by Gulf Wars, we find such “turning windows” in generalized Hurst exponents obtained from three periods divided by two Gulf Wars so that WTI and Brent crude oil returns possess different properties above and below the windows respectively. Comparing with the results obtained from three periods we conclude that, before the First Gulf War, international crude oil markets possessed the highest multifractality degree, small-scope fluctuations presented the strongest persistence and large-scope fluctuations presented the strongest anti-persistence. We find that, for two Gulf Wars, the first one made a greater impact on international oil markets; for two markets, Brent was more influenced by Gulf Wars. In addition, we also verified that the multifractal structures of two markets’ indices are not only mainly attributed to the broad fat-tail distributions and persistence, but also affected by some other factors.     

Are crude oil markets multifractal? Evidence from MF-DFA and MF-SSA perspectives

In this article, we investigated the multifractality and its underlying formation mechanisms in international crude oil markets, namely, Brent and WTI, which are the most important oil pricing benchmarks globally. We attempt to find the answers to the following questions: (1) Are those different markets multifractal? (2) What are the dynamical causes for multifractality in those markets (if any)? To answer these questions, we applied both multifractal detrended fluctuation analysis (MF-DFA) and multifractal singular spectrum analysis (MF-SSA) based on the partition function, two widely used multifractality detecting methods. We found that both markets exhibit multifractal properties by means of these methods. Furthermore, in order to identify the underlying formation mechanisms of multifractal features, we destroyed the underlying nonlinear temporal correlation by shuffling the original time series; thus, we identified that the causes of the multifractality are influenced mainly by a nonlinear temporal correlation mechanism instead of a non-Gaussian distribution. At last, by tracking the evolution of left- and right-half multifractal spectra, we found that the dynamics of the large price fluctuations is significantly different from that of the small ones. Our main contribution is that we not only provided empirical evidence of the existence of multifractality in the markets, but also the sources of multifractality and plausible explanations to current literature; furthermore, we investigated the different dynamical price behaviors influenced by large and small price fluctuations.     

Size selected growth of nanodots: analytical prediction for the selected size

We analyze the multifractal spectra of daily foreign exchange rates for Japan, Hong-Kong, Korea, and Thailand with respect to the United States in the period from 1991 until 2005. We find that the return time series show multifractal spectrum features for all four cases. To observe the effect of the Asian currency crisis, we also estimate the multifractal spectra of limited series before and after the crisis. We find that the Korean and Thai foreign exchange markets experienced a significant increase in multifractality compared to Hong-Kong and Japan. We also show that the multifractality is stronger related to the presence of high values of returns in the series.     

Modeling an efficient Brownian heat engine

We discuss various properties of a homogeneous random multifractal process, which are related to the issue of scale correlations. By design, the process has no built-in scale correlations. However, when it comes to observables like breakdown coefficients, which are based on a coarse-graining of the multifractal field, scale correlations do appear. In the log-normal limit of the model process, the conditional distributions and moments of breakdown coefficients reproduce the observations made in fully developed small-scale turbulence. These findings help to understand several puzzling empirical details, which have been extracted from turbulent data already some time ago.     

Asymmetric multifractal scaling behavior in the Chinese stock market: Based on asymmetric MF-DFA

We utilized asymmetric multifractal detrended fluctuation analysis in this study to examine the asymmetric multifractal scaling behavior of Chinese stock markets with uptrends or downtrends. Results show that the multifractality degree of Chinese stock markets with uptrends is stronger than that of Chinese stock markets with downtrends. Correlation asymmetries are more evident in large fluctuations than in small fluctuations. By discussing the source of asymmetric multifractality, we find that multifractality is related to long-range correlations when the market is going up, whereas it is related to fat-tailed distribution when the market is going down. The main source of asymmetric scaling behavior in the Shanghai stock market are long-range correlations, whereas that in the Shenzhen stock market is fat-tailed distribution. An analysis of the time-varying feature of scaling asymmetries shows that the evolution trends of these scaling asymmetries are similar in the two Chinese stock markets. Major financial and economical events may enhance scaling asymmetries.     

Multifractal Analysis of Human Heartbeat in Sleep

We study the dynamical properties of heart rate variability （HRV） in sleep by analysing the scaling behaviour with the multifractal detrended fluctuation analysis method. It is well known that heart rate is regulated by the interaction of two branches of the autonomic nervous system： the parasympathetic and sympathetic nervous systems. By investigating the multifractal properties of light, deep, rapid-eye-movement （REM） sleep and wake stages, we firstly find an increasing multifractal behaviour during REM sleep which may be caused by augmented sympathetic activities relative to non-REM sleep. In addition, the investigation of long-range correlations of HRV in sleep with second order detrended fluctuation analysis presents irregular phenomena. These findings may be helpful to understand the underlying regulating mechanism of heart rate by autonomic nervous system during wake-sleep transitions.     

Multifractal detrended fluctuation analysis of sheep livestock prices in origin

The multifractal detrended fluctuation analysis (MF-DFA) is used to verify whether or not the returns of time series of prices paid to farmers in original markets can be described by the multifractal approach. By way of example, 5 weekly time series of prices of different breeds, slaughter weight and market differentiation from 2000 to 2012 are analyzed. Results obtained from the multifractal parameters and multifractal spectra show that the price series of livestock products are of a multifractal nature. The Hurst exponent shows that these time series are stationary signals, some of which exhibit long memory (Merino milk-fed in Seville and Segureña paschal in Jaen), short memory (Merino paschal in Cordoba and Segureña milk-fed in Jaen) or even are close to an uncorrelated signals (Merino paschal in Seville). MF-DFA is able to discern the different underlying dynamics that play an important role in different types of sheep livestock markets, such as degree and source of multifractality. In addition, the main source of multifractality of these time series is due to the broadness of the probability function, instead of the long-range correlation properties between small and large fluctuations, which play a clearly secondary role.     

Characterizing emerging European stock markets through complex networks: From local properties to self-similar characteristics

We investigate the properties of the returns of the main emerging stock markets from Europe by means of complex networks. We transform the series of daily returns into complex networks, and analyze the local properties of these networks with respect to degree distributions, clustering, or average line length. We further use the clustering coefficients as quantities describing the local structure of the network, and approach them by using multifractal analysis. We find evidence of scale-free networks and multifractality of clustering coefficients.