Time scales involved in market emergence

In addressing the question of the time scales characteristic for the market formation, we analyze high frequency tick-by-tick data from the NYSE and from the German market. By using returns on various time scales ranging from seconds or minutes up to two days, we compare magnitude of the largest eigenvalue of the correlation matrix for the same set of securities but for different time scales. For various sets of stocks of different capitalization (and the average trading frequency), we observe a significant elevation of the largest eigenvalue with increasing time scale. Our results from the correlation matrix study go in parallel with the so-called Epps effect. There is no unique explanation of this effect and it seems that many different factors play a role here. One of such factors is randomness in transaction moments for different stocks. Another interesting conclusion to be drawn from our results is that in the contemporary markets the emergence of significant correlations occurs on time scales much smaller than in the more distant history.Comment: 13 page

Stock market return distributions: from past to present

We show that recent stock market fluctuations are characterized by the cumulative distributions whose tails on short, minute time scales exhibit power scaling with the scaling index alpha > 3 and this index tends to increase quickly with decreasing sampling frequency. Our study is based on high-frequency recordings of the S&P500, DAX and WIG20 indices over the interval May 2004 - May 2006. Our findings suggest that dynamics of the contemporary market may differ from the one observed in the past. This effect indicates a constantly increasing efficiency of world markets.Comment: to appear in Physica

Effect of detrending on multifractal characteristics

Different variants of MFDFA technique are applied in order to investigate various (artificial and real-world) time series. Our analysis shows that the calculated singularity spectra are very sensitive to the order of the detrending polynomial used within the MFDFA method. The relation between the width of the multifractal spectrum (as well as the Hurst exponent) and the order of the polynomial used in calculation is evident. Furthermore, type of this relation itself depends on the kind of analyzed signal. Therefore, such an analysis can give us some extra information about the correlative structure of the time series being studied.Comment: Presented by P. O\'swi\k{e}cimka at FENS2012 conference, 17 pages, 9 figure

Nonextensive statistical features of the Polish stock market fluctuations

The statistics of return distributions on various time scales constitutes one of the most informative characteristics of the financial dynamics. Here we present a systematic study of such characteristics for the Polish stock market index WIG20 over the period 04.01.1999 - 31.10.2005 for the time lags ranging from one minute up to one hour. This market is commonly classified as emerging. Still on the shortest time scales studied we find that the tails of the return distributions are consistent with the inverse cubic power-law, as identified previously for majority of the mature markets. Within the time scales studied a quick and considerable departure from this law towards a Gaussian can however be traced. Interestingly, all the forms of the distributions observed can be comprised by the single $q$-Gaussians which provide a satisfactory and at the same time compact representation of the distribution of return fluctuations over all magnitudes of their variation. The corresponding nonextensivity parameter $q$ is found to systematically decrease when increasing the time scales.Comment: 14 pages. Physica A in prin

Alternation of different fluctuation regimes in the stock market dynamics

Based on the tick-by-tick stock prices from the German and American stock markets, we study the statistical properties of the distribution of the individual stocks and the index returns in highly collective and noisy intervals of trading, separately. We show that periods characterized by the strong inter-stock couplings can be associated with the distributions of index fluctuations which reveal more pronounced tails than in the case of weaker couplings in the market. During periods of strong correlations in the German market these distributions can even reveal an apparent L\'evy-stable component.Comment: 19 page

The foreign exchange market: return distributions, multifractality, anomalous multifractality and Epps effect

We present a systematic study of various statistical characteristics of high-frequency returns from the foreign exchange market. This study is based on six exchange rates forming two triangles: EUR-GBP-USD and GBP-CHF-JPY. It is shown that the exchange rate return fluctuations for all the pairs considered are well described by the nonextensive statistics in terms of q-Gaussians. There exist some small quantitative variations in the nonextensivity q-parameter values for different exchange rates and this can be related to the importance of a given exchange rate in the world's currency trade. Temporal correlations organize the series of returns such that they develop the multifractal characteristics for all the exchange rates with a varying degree of symmetry of the singularity spectrum f(alpha) however. The most symmetric spectrum is identified for the GBP/USD. We also form time series of triangular residual returns and find that the distributions of their fluctuations develop disproportionately heavier tails as compared to small fluctuations which excludes description in terms of q-Gaussians. The multifractal characteristics for these residual returns reveal such anomalous properties like negative singularity exponents and even negative singularity spectra. Such anomalous multifractal measures have so far been considered in the literature in connection with the diffusion limited aggregation and with turbulence. We find that market inefficiency on short time scales leads to the occurrence of the Epps effect on much longer time scales. Although the currency market is much more liquid than the stock markets and it has much larger transaction frequency, the building-up of correlations takes up to several hours - time that does not differ much from what is observed in the stock markets. This may suggest that non-synchronicity of transactions is not the unique source of the observed effect

Decomposing the stock market intraday dynamics

The correlation matrix formalism is used to study temporal aspects of the stock market evolution. This formalism allows to decompose the financial dynamics into noise as well as into some coherent repeatable intraday structures. The present study is based on the high-frequency Deutsche Aktienindex (DAX) data over the time period between November 1997 and September 1999, and makes use of both, the corresponding returns as well as volatility variations. One principal conclusion is that a bulk of the stock market dynamics is governed by the uncorrelated noise-like processes. There exists however a small number of components of coherent short term repeatable structures in fluctuations that may generate some memory effects seen in the standard autocorrelation function analysis. Laws that govern fluctuations associated with those various components are different, which indicates an extremely complex character of the financial fluctuations.Comment: 15 pages, 13 PostScript figure

Identifying Complexity by Means of Matrices

Complexity is an interdisciplinary concept which, first of all, addresses the question of how order emerges out of randomness. For many reasons matrices provide a very practical and powerful tool in approaching and quantifying the related characteristics. Based on several natural complex dynamical systems, like the strongly interacting quantum many-body systems, the human brain and the financial markets, by relating empirical observations to the random matrix theory and quantifying deviations in term of a reduced dimensionality, we present arguments in favour of the statement that complexity is a pheomenon at the edge between collectivity and chaos.Comment: Talk given by S. Drozdz at "Horizons in Complex Systems", Messina, December 5-8, 200

Quantifying dynamics of the financial correlations

A novel application of the correlation matrix formalism to study dynamics of the financial evolution is presented. This formalism allows to quantify the memory effects as well as some potential repeatable intradaily structures in the financial time-series. The present study is based on the high-frequency Deutsche Aktienindex (DAX) data over the time-period between November 1997 and December 1999 and demonstrates a power of the method. In this way two significant new aspects of the DAX evolution are identified: (i) the memory effects turn out to be sizably shorter than what the standard autocorrelation function analysis seems to indicate and (ii) there exist short term repeatable structures in fluctuations that are governed by a distinct dynamics. The former of these results may provide an argument in favour of the market efficiency while the later one may indicate origin of the difficulty in reaching a Gaussian limit, expected from the central limit theorem, in the distribution of returns on longer time-horizons.Comment: 10 pages, 7 PostScript figures, talk presented by the first Author at the NATO ARW on Econophysics, Prague, February 8-10, 2001; to be published in proceedings (Physica A

Wavelet versus Detrended Fluctuation Analysis of multifractal structures

We perform a comparative study of applicability of the Multifractal Detrended Fluctuation Analysis (MFDFA) and the Wavelet Transform Modulus Maxima (WTMM) method in proper detecting of mono- and multifractal character of data. We quantify the performance of both methods by using different sorts of artificial signals generated according to a few well-known exactly soluble mathematical models: monofractal fractional Brownian motion, bifractal Levy flights, and different sorts of multifractal binomial cascades. Our results show that in majority of situations in which one does not know a priori the fractal properties of a process, choosing MFDFA should be recommended. In particular, WTMM gives biased outcomes for the fractional Brownian motion with different values of Hurst exponent, indicating spurious multifractality. In some cases WTMM can also give different results if one applies different wavelets. We do not exclude using WTMM in real data analysis, but it occurs that while one may apply MFDFA in a more automatic fashion, WTMM has to be applied with care. In the second part of our work, we perform an analogous analysis on empirical data coming from the American and from the German stock market. For this data both methods detect rich multifractality in terms of broad f(alpha), but MFDFA suggests that this multifractality is poorer than in the case of WTMM.Comment: substantially extended version, to appear in Phys.Rev.