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.     

Empirical scaling laws and the aggregation of non-stationary data

Widely cited evidence for scaling (self-similarity) of the returns of stocks and other securities is inconsistent with virtually all currently-used models for price movements. In particular, state-of-the-art models provide for ubiquitous, irregular, and oftentimes high-frequency fluctuations in volatility (“stochastic volatility”), both intraday and across the days, weeks, and years over which data is aggregated in demonstrations of self-similarity of returns. Stochastic volatility renders these models, which are based on variants and generalizations of random walks, incompatible with self-similarity. We show here that empirical evidence for self-similarity does not actually contradict the analytic lack of self-similarity in these models. The resolution of the mismatch between models and data can be traced to a statistical consequence of aggregating large amounts of non-stationary data.     

Statistical distribution and time correlation of stock returns runs

In this paper, we focus on the statistical features and time correlation of runs which is defined as a sequence of consecutive gain/loss (rise/fall) stock returns. By studying daily data of the Dow Jones industrial average (DJIA), we get the following points: firstly, the distribution of length and magnitude of stock returns runs both follow an exponential law; secondly, runs length do lack significant time correlation, while runs magnitude exhibit a slow decay of time correlation with long persistence up to several months, which implies existence of volatility clustering. We expect the above properties may add new members to the family of stylized facts about stock returns.     

Volatility return intervals analysis of the Japanese market

We investigate scaling and memory effects in return intervals between price volatilities above a certain threshold q for the Japanese stock market using daily and intraday data sets. We find that the distribution of return intervals can be approximated by a scaling function that depends only on the ratio between the return interval τ and its mean 〈τ〉. We also find memory effects such that a large (or small) return interval follows a large (or small) interval by investigating the conditional distribution and mean return interval. The results are similar to previous studies of other markets and indicate that similar statistical features appear in different financial markets. We also compare our results between the period before and after the big crash at the end of 1989. We find that scaling and memory effects of the return intervals show similar features although the statistical properties of the returns are different.     

Multifractality in stock indexes: Fact or Fiction?

Multifractal analysis and extensive statistical tests are performed upon intraday minutely data within individual trading days for four stock market indexes (including HSI, SZSC, S&P 500, and NASDAQ) to check whether the indexes (instead of the returns) possess multifractality. We find that the mass exponent τ(q) is linear and the singularity α(q) is close to 1 for all trading days and all indexes. Furthermore, we find strong evidence showing that the scaling behaviors of the original data sets cannot be distinguished from those of shuffled time series. Hence, the so-called multifractality in the intraday stock market indexes is merely an illusion.     

Comparing the structure of an emerging market with a mature one under global perturbation

In this paper we investigate the Tehran stock exchange (TSE) and Dow Jones Industrial Average (DJIA) in terms of perturbed correlation matrices. To perturb a stock market, there are two methods, namely local and global perturbation. In the local method, we replace a correlation coefficient of the cross-correlation matrix with one calculated from two Gaussian-distributed time series, whereas in the global method, we reconstruct the correlation matrix after replacing the original return series with Gaussian-distributed time series. The local perturbation is just a technical study. We analyze these markets through two statistical approaches, random matrix theory (RMT) and the correlation coefficient distribution. By using RMT, we find that the largest eigenvalue is an influence that is common to all stocks and this eigenvalue has a peak during financial shocks. We find there are a few correlated stocks that make the essential robustness of the stock market but we see that by replacing these return time series with Gaussian-distributed time series, the mean values of correlation coefficients, the largest eigenvalues of the stock markets and the fraction of eigenvalues that deviate from the RMT prediction fall sharply in both markets. By comparing these two markets, we can see that the DJIA is more sensitive to global perturbations. These findings are crucial for risk management and portfolio selection.     

”Direct” causal cascade in the stock market

We use wavelets to decompose the volatility (standard deviation) of intraday (S&P500) return data across scales. We show that when investigating two-point correlation functions of the volatility logarithms across different time scales, one reveals the existence of a causal information cascade from large scales (i.e. small frequencies) to fine scales. We quantify and visualize the information flux across scales. We provide a possible interpretation of our findings in terms of market dynamics. Received: 9 January 1998 / Received in final form and accepted: 13 January 1998     

Hybrid paramagnetic-ferromagnetic quantum computer design based on electron spin arrays and a ferromagnetic nanostripe

Recently the interest of researchers has shifted from the analysis of synchronous relationships of financial instruments to the analysis of more meaningful asynchronous relationships. Both types of analysis are concentrated mostly on Pearson’s correlation coefficient and consequently intraday lead-lag relationships (where one of the variables in a pair is time-lagged) are also associated with them. Under the Efficient-Market Hypothesis such relationships are not possible as all information is embedded in the prices, but in real markets we find such dependencies. In this paper we analyse lead-lag relationships of financial instruments and extend known methodology by using mutual information instead of Pearson’s correlation coefficient. Mutual information is not only a more general measure, sensitive to non-linear dependencies, but also can lead to a simpler procedure of statistical validation of links between financial instruments. We analyse lagged relationships using New York Stock Exchange 100 data not only on an intraday level, but also for daily stock returns, which have usually been ignored.     

Network analysis of a financial market based on genuine correlation and threshold method

A financial market is an example of an adaptive complex network consisting of many interacting units. This network reflects market’s behavior. In this paper, we use Random Matrix Theory (RMT) notion for specifying the largest eigenvector of correlation matrix as the market mode of stock network. For a better risk management, we clean the correlation matrix by removing the market mode from data and then construct this matrix based on the residuals. We show that this technique has an important effect on correlation coefficient distribution by applying it for Dow Jones Industrial Average (DJIA). To study the topological structure of a network we apply the removing market mode technique and the threshold method to Tehran Stock Exchange (TSE) as an example. We show that this network follows a power-law model in certain intervals. We also show the behavior of clustering coefficients and component numbers of this network for different thresholds. These outputs are useful for both theoretical and practical purposes such as asset allocation and risk management.     

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.     

Forecasting volatility of fuel oil futures in China: GARCH-type,SV or realized volatility models?

In most previous works on forecasting oil market volatility, squared daily returns were taken as the proxy of unobserved actual volatility. However, as demonstrated by Andersen and Bollerslev (1998) [22], this proxy with too high measurement noise could be perfectly outperformed by a so-called realized volatility (RV) measure calculated by the cumulative sum of squared intraday returns. With this motivation, we further extend earlier works by employing intraday high-frequency data to compare the performance of three typical volatility models in the daily out-of-sample volatility forecasting of fuel oil futures on the Shanghai Futures Exchange (SHFE): the GARCH-type, stochastic volatility (SV) and realized volatility models. By taking RV as the proxy of actual daily volatility and then computing forecasting errors, we find that the realized volatility model based on intraday high-frequency data produces significantly more accurate volatility forecasts than the GARCH-type and SV models based on daily returns. Furthermore, the SV model outperforms many linear and nonlinear GARCH-type models that capture long-memory volatility and/or the asymmetric leverage effect in volatility. These results also prove that abundant volatility information is available in intraday high-frequency data, and can be used to construct more accurate oil volatility forecasting models.     

Statistical analysis of the autoregressive modeling of reverberant speech

Hands-free speech input is required in many modern telecommunication applications that employ autoregressive (AR) techniques such as linear predictive coding. When the hands-free input is obtained in enclosed reverberant spaces such as typical office rooms, the speech signal is distorted by the room transfer function. This paper utilizes theoretical results from statistical room acoustics to analyze the AR modeling of speech under these reverberant conditions. Three cases are considered: (i) AR coefficients calculated from a single observation; (ii) AR coefficients calculated jointly from an M-channel observation (M > 1); and (iii) AR coefficients calculated from the output of a delay-and sum beamformer. The statistical analysis, with supporting simulations, shows that the spatial expectation of the AR coefficients for cases (i) and (ii) are approximately equal to those from the original speech, while for case (iii) there is a discrepancy due to spatial correlation between the microphones which can be significant. It is subsequently demonstrated that at each individual source-microphone position (without spatial expectation), the M-channel AR coefficients from case (ii) provide the best approximation to the clean speech coefficients when microphones are closely spaced (<0.3m).     

Estimating time-varying conditional correlations between stock and foreign exchange markets

This study explores the dynamic interaction between stock market returns and changes in nominal exchange rates. Many financial variables are known to exhibit fat tails and autoregressive variance structure. It is well-known that unconditional covariance and correlation coefficients also vary significantly over time and multivariate generalized autoregressive model (MGARCH) is able to capture the time-varying variance-covariance matrix for stock market returns and changes in exchange rates. The model is applied to daily Euro-Dollar exchange rates and two stock market indexes from the US economy: Dow-Jones Industrial Average Index and S&P500 Index. The news impact surfaces are also drawn based on the model estimates to see the effects of idiosyncratic shocks in respective markets.     

Salient features of dependence in daily US stock market indices

This paper deals with the analysis of long range dependence in the US stock market. We focus first on the log-values of the Dow Jones Industrial Average, Standard and Poors 500 and Nasdaq indices, daily from February, 1971 to February, 2007. The volatility processes are examined based on the squared and the absolute values of the returns series, and the stability of the parameters across time is also investigated in both the level and the volatility processes. A method that permits us to estimate fractional differencing parameters in the context of structural breaks is conducted in this paper. Finally, the “day of the week” effect is examined by looking at the order of integration for each day of the week, providing also a new modeling approach to describe the dependence in this context.     

Wavelet multiscale analysis for Hedge Funds: Scaling and strategies

The wide acceptance of Hedge Funds by Institutional Investors and Pension Funds has led to an explosive growth in assets under management. These investors are drawn to Hedge Funds due to the seemingly low correlation with traditional investments and the attractive returns. The correlations and market risk (the Beta in the Capital Asset Pricing Model) of Hedge Funds are generally calculated using monthly returns data, which may produce misleading results as Hedge Funds often hold illiquid exchange-traded securities or difficult to price over-the-counter securities. In this paper, the Maximum Overlap Discrete Wavelet Transform (MODWT) is applied to measure the scaling properties of Hedge Fund correlation and market risk with respect to the S&P 500. It is found that the level of correlation and market risk varies greatly according to the strategy studied and the time scale examined. Finally, the effects of scaling properties on the risk profile of a portfolio made up of Hedge Funds is studied using correlation matrices calculated over different time horizons.     

Dynamic properties,scaling and related freezing criteria of two- and three-dimensional colloidal dispersions

The static and dynamic properties of 2- and 3-dimensional dispersions of strongly interacting colloidal spheres are examined. Quasi-2-dimensional dispersions of particles interacting by long range electrostatic and dipolar magnetic forces, respectively, are investigated using Brownian dynamics computer simulations with hydrodynamic interactions included. The dynamics of 3-dimensional bulk dispersions of charge-stabilized and neutral colloidal spheres is determined from a fully self-consistent mode-coupling scheme. For systems with long range repulsive interactions the dynamic correlation functions are shown to obey dynamic scaling in terms of a characteristic relaxation time related to the mean particle distance. Hydrodynamic interactions introduce a second characteristic length scale, and they lead to more restricted scaling behaviour with an enhancement of self-diffusion and, for 2-dimensional systems, to the divergence of the short-time collective diffusion coefficient. As a consequence of dynamic scaling, a dynamic criterion for the onset of colloidal freezing related to long-time self-diffusion is shown to be equivalent to a static freezing criterion related to the 2- and 3-dimensional static structure factors. Alternative freezing criteria are given in terms of the long-time and the mean collective diffusion coefficients.     

Detrended fluctuation analysis of intertrade durations

The intraday pattern, long memory, and multifractal nature of the intertrade durations, which are defined as the waiting times between two consecutive transactions, are investigated based upon the limit order book data and order flows of 23 liquid Chinese stocks listed on the Shenzhen Stock Exchange in 2003. An inverse U-shaped intraday pattern in the intertrade durations with an abrupt drop in the first minute of the afternoon trading is observed. Based on a detrended fluctuation analysis, we find a crossover of power-law scaling behaviors for small box sizes (trade numbers in boxes) and large box sizes and strong evidence in favor of long memory in both regimes. In addition, the multifractal nature of intertrade durations in both regimes is confirmed by a multifractal detrended fluctuation analysis for individual stocks with a few exceptions in the small-duration regime. The intraday pattern has little influence on the long memory and multifractality.     

Generic dynamic scaling in kinetic roughening

We study the dynamic scaling hypothesis in invariant surface growth. We show that the existence of power-law scaling of the correlation functions (scale invariance) does not determine a unique dynamic scaling form of the correlation functions, which leads to the different anomalous forms of scaling recently observed in growth models. We derive all the existing forms of anomalous dynamic scaling from a new generic scaling ansatz. The different scaling forms are subclasses of this generic scaling ansatz associated with bounds on the roughness exponent values. The existence of a new class of anomalous dynamic scaling is predicted and compared with simulations.     

Near-Infrared Spectrum (3000-10 000 cm) of 4-Chlorophenol in Solution

The near-infrared spectra (3000-10 000 cm⁻¹) of 4-chlorophenol-OH and 4-chlorophenol-OD are studied at room temperature in carbon tetrachloride. Some data are also reported for 4-bromophenol-OH and 4-bromophenol-OD. The observed absorptions are assigned to overtones or combinations involving the fundamental transitions which have been recently calculated at the B3LYP/6-311++G(df,pd) level (J. Phys. Chem. A104, 11 685 (2000)). The anharmonicities and the harmonic frequencies of the local ν(OH) and ν(CH) modes and the coupling constants for the binary or ternary combinations are obtained. The extinction coefficients of these transitions are discussed. The half-band width of the ν(OH) and ν(CH) vibrations increases with the absorption frequencies. The experimental harmonic frequencies are obtained from the anharmonicity constants. Comparison with the theoretical frequencies indicates that the low scaling factors ranging from 0.950 to 0.960 characterizing the ν(OH) and ν(CH) vibrations can be accounted for by the large anharmonicity of these vibrations. For these vibrations, the ratio of the experimental harmonic frequencies to the computed frequencies is 0.993. This value does not significantly differ from the average scaling factors of 0.987-0.989 obtained for all the other vibrational modes.     

Time-dependent scaling patterns in high frequency financial data

We measure the influence of different time-scales on the intraday dynamics of financial markets. This is obtained by decomposing financial time series into simple oscillations associated with distinct time-scales. We propose two new time-varying measures of complexity: 1) an amplitude scaling exponent and 2) an entropy-like measure. We apply these measures to intraday, 30-second sampled prices of various stock market indices. Our results reveal intraday trends where different time-horizons contribute with variable relative amplitudes over the course of the trading day. Our findings indicate that the time series we analysed have a non-stationary multifractal nature with predominantly persistent behaviour at the middle of the trading session and anti-persistent behaviour at the opening and at the closing of the session. We demonstrate that these patterns are statistically significant, robust, reproducible and characteristic of each stock market. We argue that any modelling, analytics or trading strategy must take into account these non-stationary intraday scaling patterns.