Components of multifractality in high-frequency stock returns

We analyzed multifractal properties of 5-min stock returns from a period of over two years for 100 highly capitalized American companies. The two sources: fat-tailed probability distributions and non-linear temporal correlations, vitally contribute to the observed multifractal dynamics of the returns. For majority of the companies the temporal correlations constitute a much more significant related factor, however.     

Alternation of different fluctuation regimes in the stock market dynamics

Based on the tick-by-tick stock prices from the German and American stock markets, we study the statistical properties of the distribution of the individual stocks and the index returns in highly collective and noisy intervals of trading, separately. We show that periods characterized by the strong inter-stock couplings can be associated with the distributions of index fluctuations which reveal more pronounced tails than in the case of weaker couplings in the market. During periods of strong correlations in the German market these distributions can even reveal an apparent Lévy-stable component.     

Modelling spikes in electricity markets using excitable dynamics

We present a general methodology to model spikes in deregulated electricity markets using excitable dynamics in a multi-regime switching approach. In particular, we propose a two-regime switching model and a three-regime switching model in which the spikes phenomenon is described by a FitzHugh-Nagumo excitable dynamics. Both models seems to be interesting candidates for describing the main characteristics of electricity price dynamics as the occurrence of stable periods in which prices fluctuate around some long-run mean, and turbulent periods in which prices experience jumps and spikes of very large magnitude. In agreement with market data, both models can produce probability distributions of price returns with positive skewness and very high values of kurtosis.     

Inefficiency in Latin-American market indices

We explore the deviations from efficiency in the returns and volatility returns of Latin-American market indices. Two different approaches are considered. The dynamics of the Hurst exponent is obtained via a wavelet rolling sample approach, quantifying the degree of long memory exhibited by the stock market indices under analysis. On the other hand, the Tsallis q entropic index is measured in order to take into account the deviations from the Gaussian hypothesis. Different dynamic rankings of inefficieny are obtained, each of them contemplates a different source of inefficiency. Comparing with the results obtained for a developed country (US), we confirm a similar degree of long-range dependence for our emerging markets. Moreover, we show that the inefficiency in the Latin-American countries comes principally from the non-Gaussian form of the probability distributions.     

Dynamics of electricity market correlations

Electricity market participants rely on demand and price forecasts to decide their bidding strategies, allocate assets, negotiate bilateral contracts, hedge risks, and plan facility investments. However, forecasting is hampered by the non-linear and stochastic nature of price time series. Diverse modeling strategies, from neural networks to traditional transfer functions, have been explored. These approaches are based on the assumption that price series contain correlations that can be exploited for model-based prediction purposes. While many works have been devoted to the demand and price modeling, a limited number of reports on the nature and dynamics of electricity market correlations are available. This paper uses detrended fluctuation analysis to study correlations in the demand and price time series and takes the Australian market as a case study. The results show the existence of correlations in both demand and prices over three orders of magnitude in time ranging from hours to months. However, the Hurst exponent is not constant over time, and its time evolution was computed over a subsample moving window of 250 observations. The computations, also made for two Canadian markets, show that the correlations present important fluctuations over a seasonal one-year cycle. Interestingly, non-linearities (measured in terms of a multifractality index) and reduced price predictability are found for the June-July periods, while the converse behavior is displayed during the December-January period. In terms of forecasting models, our results suggest that non-linear recursive models should be considered for accurate day-ahead price estimation. On the other hand, linear models seem to suffice for demand forecasting purposes.     

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.     

A wavelet based investigation of long memory in stock returns

Using a wavelet-based maximum likelihood fractional integration estimator, we test long memory (return predictability) in the returns at the market, industry and firm level. In an analysis of emerging market daily returns over the full sample period, we find that long-memory is not present and in approximately twenty percent of 175 stocks there is evidence of long memory. The absence of long memory in the market returns may be a consequence of contemporaneous aggregation of stock returns. However, when the analysis is carried out with rolling windows evidence of long memory is observed in certain time frames. These results are largely consistent with that of detrended fluctuation analysis. A test of firm-level information in explaining stock return predictability using a logistic regression model reveal that returns of large firms are more likely to possess long memory feature than in the returns of small firms. There is no evidence to suggest that turnover, earnings per share, book-to-market ratio, systematic risk and abnormal return with respect to the market model is associated with return predictability. However, degree of long-range dependence appears to be associated positively with earnings per share, systematic risk and abnormal return and negatively with book-to-market ratio.     

Long term memory in extreme returns of financial time series

It is well known that while daily price returns of financial markets are uncorrelated, their absolute values (‘volatility’) are long-term correlated. Here we provide evidence that certain subsequences of the returns themselves also exhibit long-term memory. These subsequences consist of maxima (or minima) of returns in consecutive time windows of R days. Our analysis shows that for both stocks and currency exchange rates, long-term correlations are significant for R≥4. We argue that this long-term memory which is similar to that observed in volatility clustering sheds further insight on price dynamics that might be used for risk estimation.     

Ordered and disordered dynamics in monolayers of rolling particles

We consider the ordered and disordered dynamics for monolayers of rolling self-interacting particles modeling water molecules. The rolling constraint represents a simplified model of a strong, but rapidly decaying bond with the surface. We show the existence and nonlinear stability of ordered lattice states, as well as disturbance propagation through and chaotic vibrations of these states. We study the dynamics of disordered gas states and show that there is a surprising and universal linear connection between distributions of angular and linear velocity, allowing definition of temperature.     

Characteristic distributions of finite-time Lyapunov exponents

We study the probability densities of finite-time or local Lyapunov exponents in low-dimensional chaotic systems. While the multifractal formalism describes how these densities behave in the asymptotic or long-time limit, there are significant finite-size corrections, which are coordinate dependent. Depending on the nature of the dynamical state, the distribution of local Lyapunov exponents has a characteristic shape. For intermittent dynamics, and at crises, dynamical correlations lead to distributions with stretched exponential tails, while for fully developed chaos the probability density has a cusp. Exact results are presented for the logistic map, x-->4x(1-x). At intermittency the density is markedly asymmetric, while for "typical" chaos, it is known that the central limit theorem obtains and a Gaussian density results. Local analysis provides information on the variation of predictability on dynamical attractors. These densities, which are used to characterize the nonuniform spatial organization on chaotic attractors, are robust to noise and can, therefore, be measured from experimental data.     

Statistical properties, dynamic conditional correlation and scaling analysis: Evidence from Dow Jones and Nasdaq high-frequency data

This paper investigates statistical properties of high-frequency intraday stock returns across various frequencies. Both time series and panel data are utilized to explore the properties of probability distribution, dynamic conditional correlations, and scaling analysis in Dow Jones Industrial Average (DJIA) and Nasdaq intraday returns across 10-min, 30-min, 60-min, 120-min, and 390-min frequencies. The evidence shows that both returns and volatility (standard deviation) increase with the increasing scaling from 10-min to 390-min intervals. By fitting an AR(1)-GARCH(1,1) model to intraday data, we find that AR(1) coefficients are negative for DJIA returns and positive for Nasdaq, exhibiting a positive and negative feedback strategy in DJIA and Nasdaq, respectively. The evidence also shows that these coefficients are statistically significant for either including or excluding opening returns for the 10-min and 30-min frequencies. By examining the dynamic conditional correlation between the DJIA and the Nasdaq across different frequencies, a positive correlation ranging from 0.6 to 0.8 was found. In addition, the variance of the dynamic correlation coefficients is decreasing and appears to be stable for the 2001-2003 period. Finally, both returns on DJIA and Nasdaq satisfy the stable Lévy distributions, implying that both markets can converge to equilibrium by self-governing mechanism after shocks. Results of this work provide relevant implications for investors and policy makers.     

Cross-correlation dynamics in financial time series

The dynamics of the equal-time cross-correlation matrix of multivariate financial time series is explored by examination of the eigenvalue spectrum over sliding time windows. Empirical results for the S&P 500 and the Dow Jones Euro Stoxx 50 indices reveal that the dynamics of the small eigenvalues of the cross-correlation matrix, over these time windows, oppose those of the largest eigenvalue. This behaviour is shown to be independent of the size of the time window and the number of stocks examined.A basic one-factor model is then proposed, which captures the main dynamical features of the eigenvalue spectrum of the empirical data. Through the addition of perturbations to the one-factor model, (leading to a ‘market plus sectors’ model), additional sectoral features are added, resulting in an Inverse Participation Ratio comparable to that found for empirical data. By partitioning the eigenvalue time series, we then show that negative index returns, (drawdowns), are associated with periods where the largest eigenvalue is greatest, while positive index returns, (drawups), are associated with periods where the largest eigenvalue is smallest. The study of correlation dynamics provides some insight on the collective behaviour of traders with varying strategies.     

Randomness in denoised stock returns: The case of Moroccan family business companies

In this paper, we scrutinize entropy in family business stocks listed on Casablanca stock exchange and market index to assess randomness in their returns. For this purpose, we adopt a novel approach based on combination of stationary wavelet transform and Tsallis entropy for empirical analysis of the return series. The obtained empirical results show strong evidence that their respective entropy functions are characterized by opposite dynamics. Indeed, the information contents of their respective dynamics are statistically and significantly different. Obviously, information on regular events carried by family business returns is more certain, whilst that carried by market returns is uncertain. Such results are definitively useful to understand the nonlinear dynamics on returns on family business companies and those of the market. Without a doubt, they could be helpful for quantitative portfolio managers and investors.     

The use of dynamical networks to detect the hierarchical organization of financial market sectors

Two kinds of filtered networks: minimum spanning trees (MSTs) and planar maximally filtered graphs (PMFGs) are constructed from dynamical correlations computed over a moving window. We study the evolution over time of both hierarchical and topological properties of these graphs in relation to market fluctuations. We verify that the dynamical PMFG preserves the same hierarchical structure as the dynamical MST, providing in addition a more significant and richer structure, a stronger robustness and dynamical stability. Central and peripheral stocks are differentiated by using a combination of different topological measures. We find stocks well connected and central; stocks well connected but peripheral; stocks poorly connected but central; stocks poorly connected and peripheral. It results that the Financial sector plays a central role in the entire system. The robustness, stability and persistence of these findings are verified by changing the time window and by performing the computations on different time periods. We discuss these results and the economic meaning of this hierarchical positioning.     

Applying Hybrid ARIMA-SGARCH in Algorithmic Investment Strategies on S&P500 Index

This research aims to compare the performance of ARIMA as a linear model with that of the combination of ARIMA and GARCH family models to forecast S&P500 log returns in order to construct algorithmic investment strategies on this index. We used the data collected from Yahoo Finance with daily frequency for the period from 1 January 2000 to 31 December 2019. By using a rolling window approach, we compared ARIMA with the hybrid models to examine whether hybrid ARIMA-SGARCH and ARIMA-EGARCH can really reflect the specific time-series characteristics and have better predictive power than the simple ARIMA model. In order to assess the precision and quality of these models in forecasting, we compared their equity lines, their forecasting error metrics (MAE, MAPE, RMSE, MAPE), and their performance metrics (annualized return compounded, annualized standard deviation, maximum drawdown, information ratio, and adjusted information ratio). The main contribution of this research is to show that the hybrid models outperform ARIMA and the benchmark (Buy&Hold strategy on S&P500 index) over the long term. These results are not sensitive to varying window sizes, the type of distribution, and the type of the GARCH model.     

Analysis of the efficiency of the Shanghai stock market: A volatility perspective

By applying the rolling window method, we investigate the efficiency of the Shanghai stock market through the dynamic changes of local Hurst exponents based on multifractal detrended fluctuation analysis. We decompose the realized volatility into continuous sample paths and jump components and analyze their long-range correlations of decomposing components. Our results reveal that the efficiency of the Shanghai stock market improved greatly based on the time-varying Hurst exponents.     

Properties of a simple bilinear stochastic model: Estimation and predictability

We analyze the properties of arguably the simplest bilinear stochastic multiplicative process, proposed as a model of financial returns and of other complex systems combining both nonlinearity and multiplicative noise. By construction, it has no linear predictability (zero two-point correlation) but a certain nonlinear predictability (non-zero three-point correlation). It can thus be considered as a paradigm for testing the existence of a possible nonlinear predictability in a given time series. We present a rather exhaustive study of the process, including its ability to produce fat-tailed distributions from Gaussian innovations, the unstable characteristics of the inversion of the key nonlinear parameters and of the two initial conditions necessary for the implementation of a prediction scheme and an analysis of the associated super-exponential sensitivity of the inversion of the innovations in the presence of a large impulse. Our study emphasizes the conditions under which a degree of predictability can be achieved and describe a number of different attempts, which overall illuminates the properties of the process. In conclusion, notwithstanding its remarkable simplicity, the bilinear stochastic process exhibits remarkably rich and complex behavior, which makes it a serious candidate for the modeling of financial time series among others.     

Climate networks around the globe are significantly affected by El Niño

The temperatures in different zones in the world do not show significant changes due to El Ni?o except when measured in a restricted area in the Pacific Ocean. We find, in contrast, that the dynamics of a climate network based on the same temperature records in various geographical zones in the world is significantly influenced by El Ni?o. During El Ni?o many links of the network are broken, and the number of surviving links comprises a specific and sensitive measure for El Ni?o events. While during non-El Ni?o periods these links which represent correlations between temperatures in different sites are more stable, fast fluctuations of the correlations observed during El Ni?o periods cause the links to break.     

Multiple returns for some regular and mixing maps

We study the distributions of the number of visits for some noteworthy dynamical systems, considering whether limit laws exist by taking domains that shrink around points of the phase space. It is well known that for highly mixing systems such limit distributions exhibit a Poissonian behavior. We analyze instead a skew integrable map defined on a cylinder that models a shear flow. Since almost all fibers are given by irrational rotations, we at first investigate the distributions of the number of visits for irrational rotations on the circle. In this last case the numerical results strongly suggest the existence of limit laws when the shrinking domain is chosen in a descending chain of renormalization intervals. On the other hand, the numerical analysis performed for the skew map shows that limit distributions exist even if we take domains shrinking in an arbitrary way around a point, and these distributions appear to follow a power law decay of which we propose a theoretical explanation. It is interesting to note that we observe a similar behavior for domains wholly contained in the integrable region of the standard map. We also consider the case of two or more systems coupled together, proving that the distributions of the number of visits for domains intersecting the boundary between different regions are a linear superposition of the distributions characteristic of each region. Using this result we show that the real limit distributions can be hidden by some finite-size effects. In particular, when a chaotic and a regular region are glued together, the limit distributions follow a Poisson-like law, but as long as the measure of the shrinking domain is not zero, the polynomial behavior of the regular component dominates for large times. Such an analysis seems helpful to understand the dynamics in the regions where ergodic and regular motions are intertwined, as it may occur for the standard map. Finally, we study the distributions of the number of visits around generic and periodic points of the dissipative Henon map. Although this map is not uniformly hyperbolic, the distributions computed for generic points show a Poissonian behavior, as usually occurs for systems with highly mixing dynamics, whereas for periodic points the distributions follow a different law that is obtained from the statistics of first return times by assuming that subsequent returns are independent. These results are consistent with a possible rapid decay of the correlations for the Henon map.     

Exponential smoothing weighted correlations

In many practical applications, correlation matrices might be affected by the ??curse of dimensionality?? and by an excessive sensitiveness to outliers and remote observations. These shortcomings can cause problems of statistical robustness especially accentuated when a system of dynamic correlations over a running window is concerned. These drawbacks can be partially mitigated by assigning a structure of weights to observational events. In this paper, we discuss Pearson??s ?? and Kendall??s ?? correlation matrices, weighted with an exponential smoothing, computed on moving windows using a data-set of daily returns for 300 NYSE highly capitalized companies in the period between 2001 and 2003. Criteria for jointly determining optimal weights together with the optimal length of the running window are proposed. We find that the exponential smoothing can provide more robust and reliable dynamic measures and we discuss that a careful choice of the parameters can reduce the autocorrelation of dynamic correlations whilst keeping significance and robustness of the measure. Weighted correlations are found to be smoother and recovering faster from market turbulence than their unweighted counterparts, helping also to discriminate more effectively genuine from spurious correlations.