Power law distribution in high frequency financial data? An econometric analysis

Power law distributions are very common in natural sciences. We analyze high frequency financial data from XETRA and the NYSE using maximum likelihood estimation and the Kolmogorov–Smirnov statistic to test whether the power law hypothesis holds also for these data. We find that the universality and scale invariance properties of the power law are violated. Furthermore, the returns of Daimler Chrysler and SAP traded simultaneously on both exchanges follow a power law at one exchange, but not at the other. These results raise some questions about the no-arbitrage condition. Finally, we find that an exponential function provides a better fit for the tails of the sample distributions than a power law function.     

Semi-Markov regime switching interest rate models and minimal entropy measure

In this paper, we present a discrete time regime switching binomial-like model of the term structure where the regime switches are governed by a discrete time semi-Markov process. We model the evolution of the prices of zero-coupon when given an initial term structure as in the model by Ho and Lee that we aim to extend. We discuss and derive conditions for the model to be arbitrage free and relate this to the notion of martingale measure. We explicitly show that due to the extra source of uncertainty coming from the underlying semi-Markov process, there are an infinite number of equivalent martingale measures. The notion of path independence is also studied in some detail, especially in the presence of regime switches. We deal with the market incompleteness by giving an explicit characterization of the minimal entropy martingale measure. We give an application to the pricing of a European bond option both in a Markov and semi-Markov framework. Finally, we draw some conclusions.     

First and second order semi-Markov chains for wind speed modeling

The increasing interest in renewable energy, particularly in wind, has given rise to the necessity of accurate models for the generation of good synthetic wind speed data. Markov chains are often used for this purpose but better models are needed to reproduce the statistical properties of wind speed data. We downloaded a database, freely available from the web, in which are included wind speed data taken from L.S.I. -Lastem station (Italy) and sampled every 10 min. With the aim of reproducing the statistical properties of this data we propose the use of three semi-Markov models. We generate synthetic time series for wind speed by means of Monte Carlo simulations. The time lagged autocorrelation is then used to compare statistical properties of the proposed models with those of real data and also with a synthetic time series generated through a simple Markov chain.     

Spot foreign exchange market and time series

We investigate high frequency price dynamics in foreign exchange market using data from Reuters information system (the dataset has been provided to us by Olsen and Associates). In our analysis we show that a naïve approach to the definition of price (for example using the spot mid price) may lead to wrong conclusions on price behavior as for example the presence of short term correlations for returns. For this purpose we introduce an algorithm which only uses the non arbitrage principle to estimate real prices from the spot ones. The new definition leads to returns which are not affected by spurious correlations. Furthermore, any apparent information (defined by using Shannon entropy) contained in the data disappears.Received: 12 June 2003, Published online: 9 September 2003PACS: 89.65.Gh Economics; econophysics, financial markets, business and management - 65.40.Gr Entropy and other thermodynamical quantities     

Hitting time distributions in financial markets

We analyze the hitting time distributions of stock price returns in different time windows, characterized by different levels of noise present in the market. The study has been performed on two sets of data from US markets. The first one is composed by daily price of 1071 stocks trade for the 12-year period 1987-1998, the second one is composed by high frequency data for 100 stocks for the 4-year period 1995-1998. We compare the probability distribution obtained by our empirical analysis with those obtained from different models for stock market evolution. Specifically by focusing on the statistical properties of the hitting times to reach a barrier or a given threshold, we compare the probability density function (PDF) of three models, namely the geometric Brownian motion, the GARCH model and the Heston model with that obtained from real market data. We will present also some results of a generalized Heston model.     

Activity spectrum from waiting-time distribution

In high frequency financial data not only returns but also waiting times between trades are random variables. In this work, we analyze the spectra of the waiting-time processes for tick-by-tick trades. The numerical problem, strictly related with the real inversion of Laplace transforms, is analyzed by using Tikhonov's regularization method. We also analyze these spectra by a rough method using a comb of Dirac's delta functions.     

Algorithms for generating surrogate data for sparsely quantized time series

The method of surrogate data is frequently used for a statistical examination of nonlinear properties underlying original data. If surrogate data sets are generated by a null hypothesis that the data are derived by a linear process, a rejection of the hypothesis means that the original data have more complex properties. However, we found that if an algorithm for generating surrogate data, for example, amplitude adjusted Fourier transformed, is applied to sparsely quantized data, there are large discrepancies between their power spectrum and that of the original data in lower frequency regions. We performed some simulations to confirm that these errors often lead to false rejections.In this paper, in order to prevent such drawbacks, we advance an extended hypothesis, and propose two improved algorithms for generating surrogate data that reduce the discrepancies of the power spectra. We also confirm the validity of the two improved algorithms with numerical simulations by showing that the extended null hypothesis can be rejected if the time series is produced from chaotic dynamical systems. Finally, we applied these algorithms for analyzing financial tick data as a real example; then we showed that the extended null hypothesis cannot be rejected because the nonlinear statistics or nonlinear prediction errors exhibited are the same as those of the original financial tick time series.     

Financial correlations at ultra-high frequency: theoretical models and empirical estimation

A detailed analysis of correlation between stock returns at high frequency is compared with simple models of random walks. We focus in particular on the dependence of correlations on time scales – the so-called Epps effect. This provides a characterization of stochastic models of stock price returns which is appropriate at very high frequency.     

Grafting of higher-order correlations of real financial markets into herding models

In this work, we graft the volatility clustering observed in empirical financial time series into the Equiluz and Zimmermann (EZ) model, which was introduced to reproduce the herding behaviors of a financial time series. The original EZ model failed to reproduce the empirically observed power-law exponents of real financial data. The EZ model ordinarily produces a more fat-tailed distribution compared to real data, and a long-range correlation of absolute returns that underlie the volatility clustering. As it is not appropriate to capture the empirically observed correlations in a modified EZ model, we apply a sorting method to incorporate the nonlinear correlation structure of a real financial time series into the generated returns. By doing so, we observe that the slow convergence of distribution of returns is well established for returns generated from the EZ model and its modified version. It is also found that the modified EZ model leads to a less fat-tailed distribution.     

Waiting time distributions of the volatility in the Italian MIB30 index: Clustering or Poisson functions?

We investigate the time behaviour of the Italian MIB30 stock index collected every minute during two months in the period from May 17, 2006, up to July 24, 2006. We find short-range correlations in the price returns and, on the contrary, a long persistent time lag and slow decay in the autocorrelation functions of volatility. Besides, we find that the probability density functions (PDFs) of returns show fat tails, which are well fit by the log-normal model of Castaing [B. Castaing, Y. Gagne, E.J. Hopfinger, Physica D 46 (1990) 177], and a convergence toward a normal distribution for large time scales; we also find that the PDFs of volatility, for short time horizons, fit better with a log-normal distribution than with a Gaussian. Most of these features characterize the indexes and stocks of the largest American, European and Asian markets.We also investigate the distribution of stochastic separation between isolated strong events in the volatility signal. This is interesting because this gives us a deeper understanding about the price formation process. By using a test for the occurrence of local Poisson hypothesis, we show that the process we examined strongly departs from a Poisson statistics, the origin of this failure stemming from the presence of temporal clustering and of a certain amount of memory.     

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.     

Understanding the source of multifractality in financial markets

In this paper, we use the generalized Hurst exponent approach to study the multi-scaling behavior of different financial time series. We show that this approach is robust and powerful in detecting different types of multi-scaling. We observe a puzzling phenomenon where an apparent increase in multifractality is measured in time series generated from shuffled returns, where all time-correlations are destroyed, while the return distributions are conserved. This effect is robust and it is reproduced in several real financial data including stock market indices, exchange rates and interest rates. In order to understand the origin of this effect we investigate different simulated time series by means of the Markov switching multifractal model, autoregressive fractionally integrated moving average processes with stable innovations, fractional Brownian motion and Levy flights. Overall we conclude that the multifractality observed in financial time series is mainly a consequence of the characteristic fat-tailed distribution of the returns and time-correlations have the effect to decrease the measured multifractality.     

Long memory features in the high frequency data of the Korean stock market

This paper examines the long memory property in the high frequency data of KOSPI 200 using the FIAPARCH model. The empirical results indicate that the FIAPARCH model can capture asymmetry and long memory in the volatility of intraday KOSPI 200 returns. Interestingly, the presence of long memory is invariant to the temporally aggregated intraday returns, implying that a long memory phenomenon is an inherent characteristic of the data generating process, not a result of structural breaks.     

Accounting for outliers and calendar effects in surrogate simulations of stock return sequences

Surrogate data analysis (SDA) is a statistical hypothesis testing framework for the determination of weak chaos in time series dynamics. Existing SDA procedures do not account properly for the rich structures observed in stock return sequences, attributed to the presence of heteroscedasticity, seasonal effects and outliers. In this paper we suggest a modification of the SDA framework, based on the robust estimation of location and scale parameters of mean-stationary time series and a probabilistic framework which deals with outliers. A demonstration on the NASDAQ Composite index daily returns shows that the proposed approach produces surrogates that faithfully reproduce the structure of the original series while being manifestations of linear-random dynamics.     

Influence of the Investor’s Behavior on the Complexity of the Stock Market

One of the pillars of the finance theory is the efficient-market hypothesis, which is used to analyze the stock market. However, in recent years, this hypothesis has been questioned by a number of studies showing evidence of unusual behaviors in the returns of financial assets (“anomalies”) caused by behavioral aspects of the economic agents. Therefore, it is time to initiate a debate about the efficient-market hypothesis and the “behavioral finances.” We here introduce a cellular automaton model to study the stock market complexity, considering different behaviors of the economical agents. From the analysis of the stationary standard of investment observed in the simulations and the Hurst exponents obtained for the term series of stock index, we draw conclusions concerning the complexity of the model compared to real markets. We also investigate which conditions of the investors are able to influence the efficient market hypothesis statements.     

Isolating scattering resonances of an air-filled spherical shell using iterative, single-channel time reversal

Iterative, single-channel time reversal is employed to isolate backscattering resonances of an air-filled spherical shell in a frequency range of 0.5-20 kHz. Numerical simulations of free-field target scattering suggest improved isolation of the dominant target response frequency in the presence of varying levels of stochastic noise, compared to processing returns from a single transmission and also coherent averaging. To test the efficacy of the technique in a realistic littoral environment, monostatic scattering experiments are conducted in the Gulf of Mexico near Panama City, Florida. The time reversal technique is applied to returns from a hollow spherical shell target sitting proud on a sandy bottom in 14 m deep water. Distinct resonances in the scattering response of the target are isolated, depending upon the bandwidth of the sonar system utilized.     

Determining anomalous dynamic patterns in price indexes of the London Metal Exchange by data synchronization

Data synchronization based on the Kuramoto model for collective synchronization and hypothesis testing based on the rank test combined with the random shuffling surrogate method are applied to finding major feature patterns of weekly nonferrous metal returns from the time series of daily spot and futures price indexes in the London Metal Exchange since 1989. Our results suggest the existence of day-of-the-week anomalies in the metal returns. We conjecture that such anomalies are large-scale manifestations of synchronously accumulated risk-aversive actions of individual market players.     

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.     

Wave and place fixed DPOAE maps of the human ear

Human intermodulation distortion product otoacoustic emissions (DPOAE) can be a mixture of low and high latency components. They have different level, phase, and suppression characteristics, which indicate that emissions arise both from the frequency region of the primary tones directly and indirectly via the DP frequency place. Which component dominates the measured DPOAE in the ear canal depends on the stimulus parameters, especially the frequency ratio, f2/f1. Interference between the two emissions adds complexity to measurements of DPOAE. The behavior and even existence of whichever emission route is lower in level often cannot directly be deduced from the raw DPOAE data because the other emission covers it. It is therefore not known whether both emissions are present for all stimulus parameters or whether the trends seen in each emission when they are the dominant emission route continue under stimulus conditions when they are not dominant. In this study, the two DPOAE components are separated by a post-processing method. Previously, maps of raw DPOAE data against f2/f1 and DP frequency have been obtained. To separate the components, sets of data consisting of f2/f1 sweeps were transformed by an inverse Fourier transform into the time domain. The low and high latency components appeared as two distinct peaks because of their different phase gradients. These peaks were separated by windowing in the time domain and two frequency domain maps were reconstructed, representing the low and high latency DPOAEs. It was found that the low latency component of the 2 f1-f2 DP was only emitted strongly with f2/f1 between approximately 1.1 and 1.3. The removal of the high latency component revealed the low ratio edge of this region, at which the level falls sharply. However, the low latency emission has been traced at reduced amplitude over a wide range of stimulus parameters. Although previously only observed at small frequency ratios, the high latency component was found to be present widely in the lower sideband, its level reducing slowly at larger f2/f1. Its phase behavior changes in the lower sideband, being approximately constant with DP frequency at small ratios of f2/f1, but deviating from this at wider ratios. These results support the hypothesis that a DPOAE component which propagates to and is re-emitted from the DP frequency place (place fixed emission) is present across a wide parameter range. However, for all but the close primary condition the lower sideband DPOAE is dominated by direct emission from the region of f2 and f1 wave interaction (wave fixed emission). A simple transmission line model is presented to illustrate how the observed DPOAE maps can arise on the basis of this hypothesis.     

Dynamical mechanism of two-phase phenomena in financial markets

Two-phase behavior of the Korean treasury bond (KTB) futures in the Korean exchange market is investigated in this study. To show that the two-phase phenomena are due to heavy-tailed behavior of distribution of price returns, actual data from the KTB futures market with shuffled data and a generated time series are examined according to the Brownian process. In addition, we study the correlation inherent in the KTB futures and its Brownian walk, describing the extent to which the volatility clustering plays a crucial role in equilibrium and nonequilibrium states of financial markets. It is shown that the two-phase behavior essentially results from heavy-tailed behavior of the distribution of price returns. This two-phase behavior does not appear to be relevant to volatility clustering.