A multifractal approach for stock market inefficiency

In this paper, the multifractality degree in a collection of developed and emerging stock market indices is evaluated. Empirical results suggest that the multifractality degree can be used as a quantifier to characterize the stage of market development of world stock indices. We develop a model to test the relationship between the stage of market development and the multifractality degree and find robust evidence that the relationship is negative, i.e., higher multifractality is associated with a less developed market. Thus, an inefficiency ranking can be derived from multifractal analysis. Finally, a link with previous volatility time series results is established.     

Scaling and memory in the return intervals of realized volatility

We perform return interval analysis of 1-min realized volatility defined by the sum of absolute high-frequency intraday returns for the Shanghai Stock Exchange Composite Index (SSEC) and 22 constituent stocks of SSEC. The scaling behavior and memory effect of the return intervals between successive realized volatilities above a certain threshold q are carefully investigated. In comparison with the volatility defined by the closest tick prices to the minute marks, the return interval distribution for the realized volatility shows a better scaling behavior since 20 stocks (out of 22 stocks) and the SSEC pass the Kolmogorov-Smirnov (KS) test and exhibit scaling behaviors, among which the scaling function for 8 stocks could be approximated well by a stretched exponential distribution revealed by the KS goodness-of-fit test under the significance level of 5%. The improved scaling behavior is further confirmed by the relation between the fitted exponent γ and the threshold q. In addition, the similarity of the return interval distributions for different stocks is also observed for the realized volatility. The investigation of the conditional probability distribution and the detrended fluctuation analysis (DFA) show that both short-term and long-term memory exists in the return intervals of realized volatility.     

The 2006-2008 oil bubble: Evidence of speculation, and prediction

We present an analysis of oil prices in USD and in other major currencies that diagnoses unsustainable faster-than-exponential behavior. This supports the hypothesis that the recent oil price run-up was amplified by speculative behavior of the type found during a bubble-like expansion. We also attempt to unravel the information hidden in the oil supply-demand data reported by two leading agencies, the US Energy Information Administration (EIA) and the International Energy Agency (IEA). We suggest that the found increasing discrepancy between the EIA and IEA figures provides a measure of the estimation errors. Rather than a clear transition to a supply restricted regime, we interpret the discrepancy between the IEA and EIA as a signature of uncertainty, and there is no better fuel than uncertainty to promote speculation! Our post-crash analysis confirms that the oil peak in July 2008 occurred within the expected 80% confidence interval predicted with data available in our pre-crash analysis.     

Statistical properties of volatility return intervals of Chinese stocks

The statistical properties of the return intervals τ_q between successive 1-min volatilities of 30 liquid Chinese stocks exceeding a certain threshold q are carefully studied. The Kolmogorov-Smirnov (KS) test shows that 12 stocks exhibit scaling behaviors in the distributions of τ_q for different thresholds q. Furthermore, the KS test and weighted KS test show that the scaled return interval distributions of 6 stocks (out of the 12 stocks) can be nicely fitted by a stretched exponential function with γ≈0.31 under the significance level of 5%, where is the mean return interval. The investigation of the conditional probability distribution P_q(τ|τ₀) and the mean conditional return interval 〈τ|τ₀〉 demonstrates the existence of short-term correlation between successive return interval intervals. We further study the mean return interval 〈τ|τ₀〉 after a cluster of n intervals and the fluctuation F(l) using detrended fluctuation analysis, and find that long-term memory also exists in the volatility return intervals.     

Statistical properties of trading volume of Chinese stocks

The cumulative distribution of trading volume is investigated for Chinese stocks. Different from the power-law scaling of mature markets, the distribution is well fitted by a stretched exponential function . With the autocorrelation function and the detrended fluctuation analysis, the long-range autocorrelation of trading volume is revealed. The conditional dependence of volume on volatility and the volume-volatility cross-correlation are studied, and a positive long-range correlation between volume and volatility is observed.     

Scaling and memory effect in volatility return interval of the Chinese stock market

We investigate the probability distribution of the volatility return intervals τ for the Chinese stock market. We rescale both the probability distribution P_q(τ) and the volatility return intervals τ as to obtain a uniform scaling curve for different threshold value q. The scaling curve can be well fitted by the stretched exponential function , which suggests memory exists in τ. To demonstrate the memory effect, we investigate the conditional probability distribution P_q(τ|τ₀), the mean conditional interval 〈τ|τ₀〉 and the cumulative probability distribution of the cluster size of τ. The results show clear clustering effect. We further investigate the persistence probability distribution P_±(t) and find that P₋(t) decays by a power law with the exponent far different from the value 0.5 for the random walk, which further confirms long memory exists in τ. The scaling and long memory effect of τ for the Chinese stock market are similar to those obtained from the United States and the Japanese financial markets.     

Pareto law of the expenditure of a person in convenience stores

We study the statistical laws of the expenditure of a person in convenience stores by analyzing around 100 million receipts. The density function of expenditure exhibits a fat tail that follows a power law. Using the Lorenz curve, the Gini coefficient is estimated to be 0.70; this implies that loyal customers contribute significantly to a store’s sales. We observe the Pareto principle where both the top 25% and 2% of the customers account for 80% and 25% of the store’s sales, respectively.     

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.     

City size distributions for India and China

This paper studies the size distributions of urban agglomerations for India and China. We have estimated the scaling exponent for Zipf’s law with the Indian census data for the years of 1981-2001 and the Chinese census data for 1990 and 2000. Along with the biased linear fit estimate, the maximum likelihood estimate for the Pareto and Tsallis q-exponential distribution has been computed. For India, the scaling exponent is in the range of [1.88, 2.06] and for China, it is in the interval [1.82, 2.29]. The goodness-of-fit tests of the estimated distributions are performed using the Kolmogorov-Smirnov statistic.     

Rational expectations, psychology and inductive learning via moving thresholds

This paper modifies a previously introduced class of heterogeneous agent models in a way that allows for the inclusion of different types of agent motivations and behaviours in a consistent manner. The agents operate within a highly simplified environment where they are only able to be long or short one unit of the asset. The price of the asset is influenced by both an external information stream and the demand of the agents. The current strategy of each agent is defined by a pair of moving thresholds straddling the current price. When the price crosses either of the thresholds for a particular agent, that agent switches position and a new pair of thresholds is generated.The threshold dynamics can mimic different sources of investor motivation, running the gamut from purely rational information-processing, through rational (but often undesirable) behaviour induced by perverse incentives and moral hazards, to purely psychological effects. The simplest model of this kind precisely conforms to the Efficient Market Hypothesis (EMH) and this allows causal relationships to be established between actions at the agent level and violations of EMH price statistics at the global level. In particular, the effects of herding behaviour and perverse incentives are examined.     

Production, depreciation and the size distribution of firms

Many empirical researches indicate that firm size distributions in different industries or countries exhibit some similar characters. Among them the fact that many firm size distributions obey power-law especially for the upper end has been mostly discussed. Here we present an agent-based model to describe the evolution of manufacturing firms. Some basic economic behaviors are taken into account, which are production with decreasing marginal returns, preferential allocation of investments, and stochastic depreciation. The model gives a steady size distribution of firms which obey power-law. The effect of parameters on the power exponent is analyzed. The theoretical results are given based on both the Fokker-Planck equation and the Kesten process. They are well consistent with the numerical results.     

Probability of large movements in financial markets

Based on empirical financial time series, we show that the “silence-breaking” probability follows a super-universal power law: the probability of observing a large movement is inversely proportional to the length of the on-going low-variability period. Such a scaling law has been previously predicted theoretically [R. Kitt, J. Kalda, Physica A 353 (2005) 480], assuming that the length-distribution of the low-variability periods follows a multi-scaling power law.     

Fluctuations of trading volume in a stock market

We consider the probability distribution function of the trading volume and the volume changes in the Korean stock market. The probability distribution function of the trading volume shows double peaks and follows a power law, P(V/〈V〉)∼(V/〈V〉)^−α at the tail part of the distribution with α=4.15(4) for the KOSPI (Korea composite Stock Price Index) and α=4.22(2) for the KOSDAQ (Korea Securities Dealers Automated Quotations), where V is the trading volume and 〈V〉 is the monthly average value of the trading volume. The second peaks originate from the increasing trends of the average volume. The probability distribution function of the volume changes also follows a power law, , where V_r=V(t)−V(t−T) and T is a time lag. The exponents β depend on the time lag T. We observe that the exponents β for the KOSDAQ are larger than those for the KOSPI.     

Forbidden patterns, permutation entropy and stock market inefficiency

In this paper we introduce two new quantifiers for the stock market inefficiency: the number of forbidden patterns and the normalized permutation entropy. They are model-independent measures, thus they have more general applicability. We find robust evidence that degree of market inefficiency is positively correlated with the number of forbidden patterns and negatively correlated with the permutation entropy. Our empirical results suggest that these two physical tools are useful to discriminate the stage of stock market development and can be easily implemented.     

Statistical properties of world investment networks

We have performed a detailed investigation on the world investment networks constructed from the Coordinated Portfolio Investment Survey (CPIS) data of the International Monetary Fund, ranging from 2001 to 2006. The distributions of degrees and node strengths are scale-free. The weight distributions can be well modeled by the Weibull distribution. The maximum flow spanning trees of the world investment networks possess two universal allometric scaling relations, independent of time and the investment type. The topological scaling exponent is 1.17±0.02 and the flow scaling exponent is 1.03±0.01.     

Evidence of Scaling in Chinese Income Distribution

With income data from Chinese household income projects in 1998-2002, we study the functional form of Chinese income distribution. The fitting results suggest a log-normal distribution plus a power-law tail. This distributional form has changed a lot from its appearance in the early stage of China＇s reform and turns out to be consistent with that of some complete market economies. The uncertainty and diversity of income growth rate aroused by marketing reform are the main causes of current Chinese income distribution.     

Note on Two-Phase Phenomena in Financial Markets

The two-phase behaviour in financial markets actually means the bifurcation phenomenon, which represents the change of the conditional probability from an unimodal to a bimodal distribution. We investigate the bifurcation phenomenon in Hang-Sang index. It is observed that the bifurcation phenomenon in financial index is not universal, but specific under certain conditions. For Hang-Sang index and randomly generated time series, the phenomenon just emerges when the power-law exponent of absolute increment distribution is between 1 and 2 with appropriate period. Simulations on a randomly generated time series suggest the bifurcation phenomenon itself is subject to the statistics of absolute increment, thus it may not be able to reflect essential financial behaviours. However, even under the same distribution of absolute increment, the range where bifurcation phenomenon occurs is fax different from real market to artificial data, which may reflect certain market information.     

Difference in nature of correlation between NASDAQ and BSE indices

We apply a recently developed wavelet based approach to characterize the correlation and scaling properties of non-stationary financial time series. This approach is local in nature and it makes use of wavelets from the Daubechies family for detrending purpose. The built-in variable windows in wavelet transform makes this procedure well suited for the non-stationary data. We analyze daily price of NASDAQ composite index for a period of 20 years, and BSE sensex index, over a period of 15 years. It is found that the long-range correlation, as well as fractal behavior for both the stock index values differ from each other significantly. Strong non-statistical long-range correlation is observed in BSE index, whose removal revealed a Gaussian random noise character for the corresponding fluctuation. The NASDAQ index, on the other hand, showed a multifractal behavior with long-range statistical correlation.     

Multiresolution analysis of fluctuations in non-stationary time series through discrete wavelets

We illustrate the efficacy of a discrete wavelet based approach to characterize fluctuations in non-stationary time series. The present approach complements the multifractal detrended fluctuation analysis (MF-DFA) method and is quite accurate for small size data sets. As compared to polynomial fits in the MF-DFA, a single Daubechies wavelet is used here for detrending purposes. The natural, built-in variable window size in wavelet transforms makes this procedure well suited for non-stationary data. We illustrate the working of this method through the analysis of binomial multifractal model. For this model, our results compare well with those calculated analytically and obtained numerically through MF-DFA. To show the efficacy of this approach for finite data sets, we also do the above comparison for Gaussian white noise time series of different sizes. In addition, we analyze time series of three experimental data sets of tokamak plasma and also spin density fluctuations in 2D Ising model.     

Power-Law Tail in the Chinese Wealth Distribution

We analyse the data from the recently published lists of the richest Chinese from the year 2003 to 2005. The results confirm that in these years the wealth is distributed according to a power law with exponents between 1.758 and 2.285 in the high end. The power distribution is found to be quite robust although the persons in the list change drastically and the wealth increases rapidly. The relation between the wealth and the absolute change of wealth rejects the notion that the wealth evolution is a multiplicative stochastic process.