Detrending the realized volatility in the global FX market

There has been growing interest in realized volatility (RV) of financial assets that is calculated using intra-day returns. The choice of optimal time grid for these calculations is not trivial and generally requires analysis of RV dependence on the grid spacing (so-called RV signature). Typical RV signatures have a maximum at the finest time grid spacing available, which is attributed to the microstructure effects. This maximum decays into a plateau at lower frequencies, which implies (almost) stationary return variance. We found that the RV signatures in the modern global FX market may have no plateau or even have a maximum at lower frequencies. Simple averaging methods used to address the microstructure effects in equities have no practical effect on the FX RV signatures. We show that local detrending of the high-frequency FX rate samples yields RV signatures with a pronounced plateau. This implies that FX rates can be described with a Brownian motion having non-stationary trend and stationary variance. We point at a role of algorithmic trading as a possible cause of micro-trends in FX rates.     

Globalization — Entropy unification through the Theil index

A time series is remapped onto an entropy concept, based on the Theil index. The Manhattan distance between these surrogate series is calculated, and contrasted to the usual correlation distance measure. The idea is applied to several Gross Domestic Product (relative increments) of rich countries. Such distances are calculated for various time window sizes. The role of time averaging in such finite size windows is discussed. We construct the locally minimum spanning tree (LMST) corresponding to the distance matrix. Another hierarchical network structure (Unidirectional Minimal Length Path) is compared with the LMST for confirming that the mean distance between the most developed countries on different networks actually decreases in time, — which we consider as a proof of economy globalization. It is stressed that this entropy distance measure seems more suitable in detecting some “phase transition” in time series, like a globalization process than the usual correlation based measure.     

An exploration of commonly observed stylized facts with data from experimental asset markets

We analyze data from experimental asset markets with pooled linear regression models to shed some light on the emergence of fat tails and volatility clustering in return distributions. Our data suggest that the arrival of new information is the most important cause for both stylized facts. After new information arrives we see spikes in volatility as this information is digested in the market. We also find that uninformed traders contribute significantly more to fat tails than do informed traders and that the heterogeneity in fundamental information leads to larger returns.     

Interest rates mapping

The present study deals with the analysis and mapping of Swiss franc interest rates. Interest rates depend on time and maturity, defining term structure of the interest rate curves (IRC). In the present study IRC are considered in a two-dimensional feature space-time and maturity. Exploratory data analysis includes a variety of tools widely used in econophysics and geostatistics. Geostatistical models and machine learning algorithms (multilayer perceptron and Support Vector Machines) were applied to produce interest rate maps. IR maps can be used for the visualisation and pattern perception purposes, to develop and to explore economical hypotheses, to produce dynamic asset-liability simulations and for financial risk assessments. The feasibility of an application of interest rates mapping approach for the IRC forecasting is considered as well.     

Correlation measure to detect time series distances, whence economy globalization

An instantaneous time series distance is defined through the equal time correlation coefficient. The idea is applied to the Gross Domestic Product (GDP) yearly increments of 21 rich countries between 1950 and 2005 in order to test the process of economic globalisation. Some data discussion is first presented to decide what (EKS, GK, or derived) GDP series should be studied. Distances are then calculated from the correlation coefficient values between pairs of series. The role of time averaging of the distances over finite size windows is discussed. Three network structures are next constructed based on the hierarchy of distances. It is shown that the mean distance between the most developed countries on several networks actually decreases in time, —which we consider as a proof of globalization. An empirical law is found for the evolution after 1990, similar to that found in flux creep. The optimal observation time window size is found ?15 years.     

Risk evaluation with enhanced covariance matrix

We propose a route for the evaluation of risk based on a transformation of the covariance matrix. The approach uses a ‘potential’ or ‘objective’ function. This allows us to rescale data from different assets (or sources) such that each data set then has similar statistical properties in terms of their probability distributions. The method is tested using historical data from both the New York and Warsaw stock exchanges.     

Relationship between efficiency and predictability in stock price change

In this study, we evaluate the relationship between efficiency and predictability in the stock market. The efficiency, which is the issue addressed by the weak-form efficient market hypothesis, is calculated using the Hurst exponent and the approximate entropy (ApEn). The predictability corresponds to the hit-rate; this is the rate of consistency between the direction of the actual price change and that of the predicted price change, as calculated via the nearest neighbor prediction method. We determine that the Hurst exponent and the ApEn value are negatively correlated. However, predictability is positively correlated with the Hurst exponent.     

Tail distribution of index fluctuations in World markets

We have investigated the tail distribution of the daily fluctuations in 202 different indices in the stock markets of 59 countries for the time span of the last 20 years. Power law, log-normal, Weibull, exponential and power law with exponential cutoff distributions are considered as possible candidates for the tail distribution of the normalized returns. It is found that the power exponent depends strongly on the choice of the tail threshold and a sizeable number of indices can be better fitted by a distribution function other than the power law at the region that has power law exponent of 3. Also, we have found that the power exponent is not an indicator of the maturity of the market.     

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.     

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.     

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.     

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.     

Effects of time dependency and efficiency on information flow in financial markets

We investigated financial market data to determine which factors affect information flow between stocks. Two factors, the time dependency and the degree of efficiency, were considered in the analysis of Korean, the Japanese, the Taiwanese, the Canadian, and US market data. We found that the frequency of the significant information decreases as the time interval increases. However, no significant information flow was observed in the time series from which the temporal time correlation was removed. These results indicated that the information flow between stocks evidences time-dependency properties. Furthermore, we discovered that the difference in the degree of efficiency performs a crucial function in determining the direction of the significant information flow.     

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.     

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.     

Measuring time dependent volatility and cross-sectional correlation in Australian equity returns

In this study we examine the time-dependent nature of volatility and cross-correlation of Australian equity returns data. Volatility and correlation estimates are calculated using methods that allow for non-stationary behaviour. By averaging the estimates across the entire data set we show that the correlation in ASX stock returns displays evidence of significant time-dependent behaviour. We also find that the volatility estimates do not display similar non-stationary patterns.     

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.     

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.