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Digital assets termed cryptocurrencies are creating new paradigms for financial transactions as well as alternative means of capital. Despite their increasing relevance in financial situations, a complete understanding of the entire system remains lacking. Here the cryptocurrency market is treated as a complex system and analyzed using methods from statistical physics. The complexity–entropy causality plane (or CH plane) is employed in order to explore disorder and complexity in the space of cryptocurrencies. They are found to exist on distinct planar locations in the representation space, ranging from structured to stochastic-like behavior. The temporal trajectories of entropy and statistical complexity of prices vary drastically with position along the plane. Lastly cryptocurrencies appear to be characterized by ordinal patterns that represent strictly decreasing or increasing trends in price. The present analysis expands the understanding of and helps to quantify varying degrees of complexity in cryptocurrencies.

We study auto correlations and cross correlations of daily price returns for seven Brazilian market (Bovespa) sectors using multifractal detrended fluctuation analysis (MF-DFA) and multifractal detrended cross correlation analysis (MF-DXA). We discover rather distinct multifractal behavior for different market sectors, which indicates that individual sectors follow different dynamics from the entire market. Our analysis reveals that most sectors are market efficient due to the lack of long term correlations. Shuffling the series suggests that multifractality in the auto correlations arises both from a broad probability density function and from different long term correlations. Comparisons of multifractal cross correlations between Bovespa and market sectors reveals that some sectors are more affected by multifractality of the entire market, while others are more affected by multifractality of the sectors themselves. A multifractal analysis of cross correlations between different market sectors provides a multifractal description of the Brazilian sectors.

The cryptocurrency market has emerged in recent years with profound implications to central financial institutions such as banking. Bitcoin and other popular cryptocurrencies were recently analyzed using methods from statistical physics, but focus was exclusively on behavior of prices. Here we explore multifractal behavior of daily price changes and daily volume changes of fifty cryptocurrencies using the multifractal detrended fluctuation analysis (MF-DFA). Our results indicate that price changes are more complex than volume changes, and that large and small fluctuations dominate multifractal behavior of price and volume changes, respectively. We also find there is an absence of correlations in price changes, while volume changes present anti-persistent long-term correlations. Shuffling the series reveals that multifractality of both price and volume changes arises from a broad probability density function. The multifractal behavior of the cryptocurrency market is surprisingly similar to that of stock markets, but differs from that of regular exchange rates.

A novel heuristic approach is proposed here for time series data analysis, dubbed Generalized weighted permutation entropy, which amalgamates and generalizes beyond their original scope two well established data analysis methods: Permutation entropy and Weighted permutation entropy. The method introduces a scaling parameter to discern the disorder and complexity of ordinal patterns with small and large fluctuations. Using this scaling parameter, the complexity-entropy causality plane is generalized to the complexity-entropy-scale causality box. Simulations conducted on synthetic series generated by stochastic, chaotic, and random processes, as well as real world data, are shown to produce unique signatures in this three dimensional representation.

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.

Digital assets termed cryptocurrencies are correlated. We analyze cross correlations between price changes of different cryptocurrencies using methods of random matrix theory and minimum spanning trees. We find that the cross correlation matrix exhibits non-trivial hierarchical structures and groupings of cryptocurrency pairs, which are not present in the partial cross correlations. In sharp contrast to the predictions for other financial markets, we discover that most of the eigenvalues in the spectrum of the cross correlation matrix do not agree with the universal predictions of random matrix theory, but the few of the largest eigenvalues deviate as expected. The minimum spanning tree of cryptocurrency cross correlations reveals distinct community structures that are surprisingly stable. Collective behaviors that are present in the cryptocurrency market can be useful for the construction of portfolio of cryptocurrencies as well as for future research on the subject.

Cryptocurrencies represent a new type of financial assets that are traded in a decentralized and transparent way. Recently, cryptocurrencies with large market capitalization (mostly Bitcoin) have been studied theoretically, but a deeper understanding of their underlying mechanisms remains elusive. Here we explore the nonextensivity of price changes for 20 cryptocurrency exchanges from 2013 until 2017. We discover nonextensive triplets in the cryptocurrency market, where the three associated values for Bitcoin are remarkably close to those of the logistic map near the edge of chaos. The current findings strongly indicate that the cryptocurrency market represents a system whose physics is properly described by nonextensive statistical mechanics. Our results shed light on the complex and volatile nature of cryptocurrencies, and establish the first formal link with the nonextensive theory.

The regularity of price fluctuations in exchange rates plays a crucial role in FX market dynamics. Distinct variations in regularity arise from economic, social and political events, such as interday trading and financial crisis. This paper applies a multiscale time-dependent entropy method on thirty-three exchange rates to analyze price fluctuations in the FX. Correlation matrices of entropy values, termed entropic correlations, are in turn used to describe global behavior of the market. Empirical results suggest a weakly correlated market with pronounced collective behavior at bi-weekly trends. Correlations arise from cycles of low and high regularity in long-term trends. Eigenvalues of the correlation matrix also indicate a dominant European market, followed by shifting American, Asian, African, and Pacific influences. As a result, we find that entropy is a powerful tool for extracting important information from the FX market.

This paper examines the effects of financial crises on foreign exchange (FX) markets, where entropy evolution is measured for different exchange rates, using the time-dependent block entropy method. Empirical results suggest that financial crises are associated with significant increase of exchange rate entropy, reflecting instability in FX market dynamics. In accordance with phenomenological expectations, it is found that FX markets with large liquidity and large trading volume are more inert — they recover quicker from a crisis than markets with small liquidity and small trading volume. Moreover, our numerical analysis shows that periods of economic uncertainty are preceded by periods of low entropy values, which may serve as a tool for anticipating the onset of financial crises.

Stock market indices are one of the most investigated complex systems in econophysics. Here we extend the existing literature on stock markets in connection with nonextensive statistical mechanics. We explore the nonextensivity of price volatilities for 34 major stock market indices between 2010 and 2019. We discover that stock markets follow nonextensive statistics regarding equilibrium, relaxation and sensitivity. We find nonextensive behavior in stock markets for developed countries, but not for developing countries. Distances between nonextensive triplets suggest that some stock markets might share similar nonextensive dynamics, while others are widely different. The current findings strongly indicate that the stock market represents a system whose physics is properly described by nonextensive statistical mechanics. Our results shed light on the complex nature of stock market indices, and establish another formal link with the nonextensive theory.

We study auto-correlations and cross-correlations of daily price changes and daily volume changes of thirteen global stock market indices, using multifractal detrended fluctuation analysis (MF-DFA) and multifractal detrended cross-correlation analysis (MF-DXA). We find rather distinct multifractal behavior of price and volume changes. Our results indicate that the time series of price changes are more complex than those of volume changes, and that large fluctuations dominate multifractal behavior of price changes, while small fluctuations dominate multifractal behavior of volume changes. We also find that there is an absence of correlations in price changes, there are anti-persistent long-term correlations in volume changes, and there are anti-persistent long-term cross-correlations between price and volume changes. Shuffling the series reveals that multifractality of both price changes and volume changes arises from a broad probability density function.

We investigate multifractal properties of daily price changes in currency rates using the multifractal detrended fluctuation analysis (MF-DFA). We analyze managed and independent floating currency rates in eight countries, and determine the changes in multifractal spectrum when transitioning between the two regimes. We find that after the transition from managed to independent float regime the changes in multifractal spectrum (position of maximum and width) indicate an increase in market efficiency. The observed changes are more pronounced for developed countries that have a well established trading market. After shuffling the series, we find that the multifractality is due to both probability density function and long term correlations for managed float regime, while for independent float regime multifractality is in most cases caused by broad probability density function.