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.     

Multifractal analysis of the Korean agricultural market

We have studied the long-term memory effects of the Korean agricultural market using the detrended fluctuation analysis (DFA) method. In general, the return time series of various financial data, including stock indices, foreign exchange rates, and commodity prices, are uncorrelated in time, while the volatility time series are strongly correlated. However, we found that the return time series of Korean agricultural commodity prices are anti-correlated in time, while the volatility time series are correlated. The $n$-point correlations of time series were also examined, and it was found that a multifractal structure exists in Korean agricultural market prices.     

The effects of multifractality on the statistics of return intervals

We study the statistics of the return intervals in multifractal data sets with and without linear correlations. In the absence of linear correlations, we find that the nonlinear correlations inherent in multifractal data yield (i) a power-law decay of the autocorrelation function of the return intervals, (ii) a power-law increase of the conditional return period as function of the previous return interval, and (iii) a power-law decay of the probability density function of the return intervals. These features remain unchanged in the presence of linear long-term correlations. Deviations observed in the asymptotic behaviour are probably due to finite size effects. We compare our results with those obtained for uncorrelated and for monofractal long-term correlated data, and demonstrate significant differences. Applications can be found in studying the dynamics of several processes characterised by multifractality, such as turbulence, climate dynamics, heartbeat dynamics, stock market dynamics, and tele-traffic in large networks.     

Effect of nonlinear correlations on the statistics of return intervals in multifractal data sets

We study the statistics of return intervals between events above a certain threshold in multifractal data sets without linear correlations. We find that nonlinear correlations in the record lead to a power-law (i) decay of the autocorrelation function of the return intervals, (ii) increase in the conditional return period, and (iii) decay in the probability density function of the return intervals. We show explicitly that all the observed quantities depend both on the threshold value and system size, and hence there is no simple scaling observed. We also demonstrate that this type of behavior can be observed in real economic records and can be used to improve considerably risk estimation.     

Impact of the tick-size on financial returns and correlations

We demonstrate that the lowest possible price change (tick-size) has a large impact on the structure of financial return distributions. It induces a microstructure as well as possibly altering the tail behavior. On small return intervals, the tick-size can distort the calculation of correlations. This especially occurs on small return intervals and thus contributes to the decay of the correlation coefficient towards smaller return intervals (Epps effect). We study this behavior within a model and identify the effect in market data. Furthermore, we present a method to compensate this purely statistical error.     

Statistical properties of daily ensemble variables in the Chinese stock markets

We study dynamical behavior of the Chinese stock markets by investigating the statistical properties of daily ensemble return and variety defined, respectively, as the mean and the standard deviation of the ensemble daily price return of a portfolio of stocks traded in China's stock markets on a given day. The distribution of the daily ensemble return has an exponential form in the center and power-law tails, while the variety distribution is lognormal in the bulk followed by a power-law tail for large variety. Based on detrended fluctuation analysis, R/S analysis and modified R/S analysis, we find evidence of long memory in the ensemble return and strong evidence of long memory in the evolution of variety.     

Probability distribution function and multiscaling properties in the Korean stock market

We consider the probability distribution function (pdf) and the multiscaling properties of the index and the traded volume in the Korean stock market. We observed the power law of the pdf at the fat tail region for the return, volatility, the traded volume, and changes of the traded volume. We also investigate the multifractality in the Korean stock market. We consider the multifractality by the detrended fluctuation analysis (MFDFA). We observed the multiscaling behaviors for index, return, traded volume, and the changes of the traded volume. We apply MFDFA method for the randomly shuffled time series to observe the effects of the autocorrelations. The multifractality is strongly originated from the long time correlations of the time series.     

Long memory properties in return and volatility: Evidence from the Korean stock market

In this paper, we study the dual long memory property of the Korean stock market. For this purpose, the ARFIMA-FIGARCH model is applied to two daily Korean stock price indices (KOSPI and KOSDAQ). Our empirical results indicate that long memory dynamics in the returns and volatility can be adequately estimated by the joint ARFIMA-FIGARCH model. We also found that the assumption of a skewed Student-t distribution is better for incorporating the tendency of asymmetric leptokurtosis in a return distribution.     

Hamiltonian and Brownian systems with long-range interactions: III. The BBGKY hierarchy for spatially inhomogeneous systems

We study the growth of correlations in systems with weak long-range interactions. Starting from the BBGKY hierarchy, we determine the evolution of the two-body correlation function by using an expansion of the solutions of the hierarchy in powers of 1/N in a proper thermodynamic limit N→+∞, where N is the number of particles. These correlations are responsible for the “collisional” evolution of the system beyond the Vlasov regime due to finite N effects. We obtain a general kinetic equation that can be applied to spatially inhomogeneous systems and that takes into account memory effects. These peculiarities are specific to systems with unshielded long-range interactions. For spatially homogeneous systems with short memory time like plasmas, we recover the classical Landau (or Lenard-Balescu) equations. An interest of our approach is to develop a formalism that remains in physical space (instead of Fourier space) and that can deal with spatially inhomogeneous systems. This enlightens the basic physics and provides novel kinetic equations with a clear physical interpretation. However, unless we restrict ourselves to spatially homogeneous systems, closed kinetic equations can be obtained only if we ignore some collective effects between particles. General exact coupled equations taking into account collective effects are also given. We use this kinetic theory to discuss the processes of violent collisionless relaxation and slow collisional relaxation in systems with weak long-range interactions. In particular, we investigate the dependence of the relaxation time with the system size N and try to provide a coherent discussion of all the numerical results obtained for these systems.     

Aging in the ferromagnetic phase of terbium

We report on aging, rejuvenation and memory effects in the ferromagnetic phase of pure terbium. We have applied an experimental method specifically for investigating slow dynamics of spin glasses, because these effects cannot be interpreted as conventional diffusion after-effects. Results show that relaxation times of the magnetic response are widely distributed, and isothermal aging shifted the distribution towards longer durations. If the sample was heated/cooled after such isothermal aging, the relaxation times shortened as if aging was starting anew; the behavior resembles that in spin glasses. Uniform magnetization experiments indicate that, unlike rejuvenation in spin glasses, ferromagnetic correlations are not returned to disorder by thermal perturbations. In contrast with memory effects in spin glasses, the effects of isothermal aging cannot be recovered once these disappear, even if the system is returned to its initial temperature. The observed results can be explained as collective pinning of the domain walls for which the potential is given by a rugged temperature-sensitive energy landscape.     

Coherent ultrafast dynamics in the quantum Hall effect regime

We review recent investigations of the femtosecond non-linear optical response of the two-dimensional electron gas (2DEG) in the quantum Hall effect regime. We find that the time and frequency profile of the four-wave-mixing non-linear optical spectrum is strongly influenced by Coulomb correlations between the photoexcited electron-hole pairs and the 2DEG collective excitations. We discuss experimental and theoretical results showing non-Markovian memory effects in the polarization dephasing, and an optically induced time-dependent coupling between the two lowest Landau level magnetoexcitons.     

Dynamics of fluctuations in a reactive system of low spatial dimension

We study, using master equation techniques, the time evolution of the average concentration and fluctuations in the two-speciesn-molecule reactionA+(n-1)XnX in one dimension described by a Glauber-type dynamical lattice model for the specific casesn=2 (bimolecular) andn=3 (trimolecular). The evolution is found to be quite different from that described by the Mean-Field equations even for the bimolecular case, where the steady state is meanfield. For the trimolecular process, the values of fluctuation correlations in the nonequilibrium steady state are well predicted by the fixed points of the dynamical equations obtained from the master equation. In addition, three-point fluctuation correlations are found to play an important role in both processes and are accounted for by an extended Bethe-type ansatz. The bimolecular system shows no memory effects of initial conditions, while the trimolecular system is characterized by memory effects in terms of the average concentration, fluctuations as well as the entropy. The spatial decay of fluctuation correlations is found to be short range at the steady state for the trimolecular system.     

Comparison between volatility return intervals of the S&;P 500 index and two common models

We analyze the S&P 500 index data for the 13-year period, from January 1, 1984 to December 31, 1996, with one data point every 10 min. For this database, we study the distribution and clustering of volatility return intervals, which are defined as the time intervals between successive volatilities above a certain threshold q. We find that the long memory in the volatility leads to a clustering of above-median as well as below-median return intervals. In addition, it turns out that the short return intervals form larger clusters compared to the long return intervals. When comparing the empirical results to the ARMA-FIGARCH and fBm models for volatility, we find that the fBm model predicts scaling better than the ARMA-FIGARCH model, which is consistent with the argument that both ARMA-FIGARCH and fBm capture the long-term dependence in return intervals to a certain extent, but only fBm accounts for the scaling. We perform the Student's t-test to compare the empirical data with the shuffled records, ARMA-FIGARCH and fBm. We analyze separately the clusters of above-median return intervals and the clusters of below-median return intervals for different thresholds q. We find that the empirical data are statistically different from the shuffled data for all thresholds q. Our results also suggest that the ARMA-FIGARCH model is statistically different from the S&P 500 for intermediate q for both above-median and below-median clusters, while fBm is statistically different from S&P 500 for small and large q for above-median clusters and for small q for below-median clusters. Neither model can fully explain the entire regime of q studied.     

Compensating asynchrony effects in the calculation of financial correlations

We present a method to compensate statistical errors in the calculation of correlations on asynchronous time series. The method is based on the assumption of an underlying time series. We set up a model and apply it to financial data to examine the decrease of calculated correlations towards smaller return intervals (Epps effect). We show that the discovered statistical effect is a major cause of the Epps effect. Hence, we are able to quantify and to compensate it using only trading prices and trading times.     

Counting statistics of non-Markovian quantum stochastic processes

We derive a general expression for the cumulant generating function (CGF) of non-Markovian quantum stochastic transport processes. The long-time limit of the CGF is determined by a single dominating pole of the resolvent of the memory kernel from which we extract the zero-frequency cumulants of the current using a recursive scheme. The finite-frequency noise is expressed not only in terms of the resolvent, but also initial system-environment correlations. As an illustrative example we consider electron transport through a dissipative double quantum dot for which we study the effects of dissipation on the zero-frequency cumulants of high orders and the finite-frequency noise.     

Persistent dynamic correlations in self-organized critical systems away from their critical point

We show that correlated dynamics and long time memory persist in self-organized criticality (SOC) systems even when forced away from the defined critical point that exists at vanishing drive strength. These temporal correlations are found for all levels of external forcing as long as the system is not overdriven. They arise from the same physical mechanism that produces the temporal correlations found at the vanishing drive limit, namely the memory of past events stored in the system profile. The existence of these correlations contradicts the notion that a SOC time series is simply a random superposition of events with sizes distributed as a power law, as has been suggested by previous studies.     

Dynamics of bid–ask spread return and volatility of the Chinese stock market

The bid–ask spread is taken as an important measure of the financial market liquidity. In this article, we study the dynamics of the spread return and the spread volatility of four liquid stocks in the Chinese stock market, including the memory effect and the multifractal nature. By investigating the autocorrelation function and the Detrended Fluctuation Analysis (DFA), we find that the spread return is the lack of long-range memory, while the spread volatility is long-range time correlated. Besides, the spread volatilities of different stocks present long-range cross-correlations. Moreover, by applying the Multifractal Detrended Fluctuation Analysis (MF-DFA), the spread return is observed to possess a strong multifractality, which is similar to the dynamics of a variety of financial quantities. Different from the spread return, the spread volatility exhibits a weak multifractal nature.     

Long-term memory: a natural mechanism for the clustering of extreme events and anomalous residual times in climate records

We study the statistics of the return intervals between extreme events above a certain threshold in long-term persistent records. We find that the long-term memory leads (i) to a stretched exponential distribution of the return intervals, (ii) to a pronounced clustering of extreme events, and (iii) to an anomalous behavior of the mean residual time to the next event that depends on the history and increases with the elapsed time in a counterintuitive way. We present an analytical scaling approach and demonstrate that all these features can be seen in long climate records. The phenomena should also occur in heartbeat records, Internet traffic, and stock market volatility and have to be taken into account for an efficient risk evaluation.