arXiv subjects

S. Drozdz

Publications and source records attributed to S. Drozdz.

At least 19 recordsLinked to original sources

arXiv2011

Accuracy analysis of the box-counting algorithm

Accuracy of the box-counting algorithm for numerical computation of the fractal exponents is investigated. To this end several sample mathematical fractal sets are analyzed. It is shown that the standard deviation obtained for the fit of the fractal scaling in the log-log plot strongly underestimates the actual error. The real computational error was found to have power scaling with respect to the number of data points in the sample ($n_{tot}$). For fractals embedded in two-dimensional space the error is larger than for those embedded in one-dimensional space. For fractal functions the error is even larger. Obtained formula can give more realistic estimates for the computed generalized fractal exponents' accuracy.

nlin.AO↗

arXiv2010

Coherent Patterns in Nuclei and in Financial Markets

In the area of traditional physics the atomic nucleus belongs to the most complex systems. It involves essentially all elements that characterize complexity including the most distinctive one whose essence is a permanent coexistence of coherent patterns and of randomness. From a more interdisciplinary perspective, these are the financial markets that represent an extreme complexity. Here, based on the matrix formalism, we set some parallels between several characteristics of complexity in the above two systems. We, in particular, refer to the concept - historically originating from nuclear physics considerations - of the random matrix theory and demonstrate its utility in quantifying characteristics of the coexistence of chaos and collectivity also for the financial markets. In this later case we show examples that illustrate mapping of the matrix formulation into the concepts originating from the graph theory. Finally, attention is drawn to some novel aspects of the financial coherence which opens room for speculation if analogous effects can be detected in the atomic nuclei or in other strongly interacting Fermi systems.

physics.data-an↗

arXiv2009

Complex Systems: From Nuclear Physics to Financial Markets

We compare correlations and coherent structures in nuclei and financial markets. In the nuclear physics part we review giant resonances which can be interpreted as a coherent structure embedded in chaos. With similar methods we investigate the financial empirical correlation matrix of the DAX and Dow Jones. We will show, that if the time-zone delay is properly accounted for, the two distinct markets largely merge into one. This is reflected by the largest eigenvalue that develops a gap relative to the remaining, chaotic eigenvalues. By extending investigations of the specific character of financial collectivity we also discuss the criticality-analog phenomenon of the financial log-periodicity and show specific examples.

q-fin.ST↗

arXiv2008

Scale free effects in world currency exchange network

A large collection of daily time series for 60 world currencies' exchange rates is considered. The correlation matrices are calculated and the corresponding Minimal Spanning Tree (MST) graphs are constructed for each of those currencies used as reference for the remaining ones. It is shown that multiplicity of the MST graphs' nodes to a good approximation develops a power like, scale free distribution with the scaling exponent similar as for several other complex systems studied so far. Furthermore, quantitative arguments in favor of the hierarchical organization of the world currency exchange network are provided by relating the structure of the above MST graphs and their scaling exponents to those that are derived from an exactly solvable hierarchical network model. A special status of the USD during the period considered can be attributed to some departures of the MST features, when this currency (or some other tied to it) is used as reference, from characteristics typical to such a hierarchical clustering of nodes towards those that correspond to the random graphs. Even though in general the basic structure of the MST is robust with respect to changing the reference currency some trace of a systematic transition from somewhat dispersed -- like the USD case -- towards more compact MST topology can be observed when correlations increase.

q-fin.ST↗

arXiv2008

Minimal Spanning Tree graphs and power like scaling in FOREX networks

Correlation matrices of foreign exchange rate time series are investigated for 60 world currencies. Minimal Spanning Tree (MST) graphs for the gold, silver and platinum are presented. Inverse power like scaling is discussed for these graphs as well as for four distinct currency groups (major, liquid, less liquid and non-tradable). The worst scaling has been found for USD and related currencies.

q-fin.ST↗

arXiv2008

Different fractal properties of positive and negative returns

We perform an analysis of fractal properties of the positive and the negative changes of the German DAX30 index separately using Multifractal Detrended Fluctuation Analysis (MFDFA). By calculating the singularity spectra $f(α)$ we show that returns of both signs reveal multiscaling. Curiously, these spectra display a significant difference in the scaling properties of returns with opposite sign. The negative price changes are ruled by stronger temporal correlations than the positive ones, what is manifested by larger values of the corresponding Hölder exponents. As regards the properties of dominant trends, a bear market is more persistent than the bull market irrespective of the sign of fluctuations.

q-fin.ST↗

arXiv2008

Cross-correlations in Warsaw Stock Exchange

We study the inter-stock correlations for the largest companies listed on Warsaw Stock Exchange and included in the WIG20 index. Our results from the correlation matrix analysis indicate that the Polish stock market can be well described by a one factor model. We also show that the stock-stock correlations tend to increase with the time scale of returns and they approach a saturation level for the time scales of at least 200 min, i.e. an order of magnitude longer than in the case of some developed markets. We also show that the strength of correlations among the stocks crucially depends on their capitalization. These results combined with our earlier findings together suggest that now the Polish stock market situates itself somewhere between an emerging market phase and a mature market phase.

q-fin.ST↗

arXiv2007

Empirics versus RMT in financial cross-correlations

In order to pursue the issue of the relation between the financial cross-correlations and the conventional Random Matrix Theory we analyse several characteristics of the stock market correlation matrices like the distribution of eigenvalues, the cross-correlations among signs of the returns, the volatility cross-correlations, and the multifractal characteristics of the principal values. The results indicate that the stock market dynamics is not simply decomposable into 'market', 'sectors', and the Wishart random bulk. This clearly is seen when the time series used to construct the correlation matrices are sufficiently long and thus the measurement noise suppressed. Instead, a hierarchically convoluted and highly nonlinear organization of the market emerges and indicates that the relevant information about the whole market is encoded already in its constituents.

q-fin.ST↗

arXiv2007

World currency exchange rate cross-correlations

World currency network constitutes one of the most complex structures that is associated with the contemporary civilization. On a way towards quantifying its characteristics we study the cross correlations in changes of the daily foreign exchange rates within the basket of 60 currencies in the period December 1998 -- May 2005. Such a dynamics turns out to predominantly involve one outstanding eigenvalue of the correlation matrix. The magnitude of this eigenvalue depends however crucially on which currency is used as a base currency for the remaining ones. Most prominent it looks from the perspective of a peripheral currency. This largest eigenvalue is seen to systematically decrease and thus the structure of correlations becomes more heterogeneous, when more significant currencies are used as reference. An extreme case in this later respect is the USD in the period considered. Besides providing further insight into subtle nature of complexity, these observations point to a formal procedure that in general can be used for practical purposes of measuring the relative currencies significance on various time horizons.

q-fin.ST↗

arXiv2007

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.

q-fin.ST↗

arXiv2006

Correlation matrix decomposition of WIG20 intraday fluctuations

Using the correlation matrix formalism we study the temporal aspects of the Warsaw Stock Market evolution as represented by the WIG20 index. The high frequency (1 min) WIG20 recordings over the time period between January 2001 and October 2005 are used. The entries of the correlation matrix considered here connect different distinct periods of the stock market dynamics, like days or weeks. Such a methodology allows to decompose the price fluctuations into the orthogonal eigensignals that quantify different modes of the underlying dynamics. The magnitudes of the corresponding eigenvalues reflect the strengths of such modes. One observation made in this paper is that strength of the daily trend in the WIG20 dynamics systematically decreases when going from 2001 to 2005. Another is that large events in the return fluctuations are primarily associated with a few most collective eigensignals.

physics.data-an↗

arXiv2006

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.

physics.data-an↗

arXiv2006

Multifractal Model of Asset Returns versus real stock market dynamics

There is more and more empirical evidence that multifractality constitutes another and perhaps the most significant financial stylized fact. A realistic model of the financial dynamics should therefore incorporate this effect. The most promising in this respect is the Multifractal Model of Asset Returns (MMAR) introduced by Mandelbrot in which multifractality is carried by time deformation. In our study we focus on the Lux extension to MMAR and empirical data from Warsaw Stock Exchange. We show that this model is able to reproduce relevant aspects of the real stock market dynamics.

physics.soc-ph↗

arXiv2006

Complexity characteristics of currency networks

A large set of daily FOREX time series is analyzed. The corresponding correlation matrices (CM) are constructed for USD, EUR and PLZ used as the base currencies. The triangle rule is interpreted as constraints reducing the number of independent returns. The CM spectrum is computed and compared with the cases of shuffled currencies and a fictitious random currency taken as a base currency. The Minimal Spanning Tree (MST) graphs are calculated and the clustering effects for strong currencies are found. It is shown that for MSTs the node rank has power like, scale free behavior. Finally, the scaling exponents are evaluated and found in the range analogous to those identified recently for various complex networks.

physics.soc-ph↗

arXiv2006

Asymmetric matrices in an analysis of financial correlations

Financial markets are highly correlated systems that reveal both the inter-market dependencies and the correlations among their different components. Standard analyzing techniques include correlation coefficients for pairs of signals and correlation matrices for rich multivariate data. In the latter case one constructs a real symmetric matrix with real non-negative eigenvalues describing the correlation structure of the data. However, if one performs a correlation-function-like analysis of multivariate data, when a stress is put on investigation of delayed dependencies among different types of signals, one can calculate an asymmetric correlation matrix with complex eigenspectrum. From the Random Matrix Theory point of view this kind of matrices is closely related to Ginibre Orthogonal Ensemble (GinOE). We present an example of practical application of such matrices in correlation analyses of empirical data. By introducing the time lag, we are able to identify temporal structure of the inter-market correlations. Our results show that the American and German stock markets evolve almost simultaneously without a significant time lag so that it is hard to find imprints of information transfer between these markets. There is only an extremely subtle indication that the German market advances the American one by a few seconds.

physics.soc-ph↗

arXiv2006

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.

cond-mat.other↗

arXiv2005

The bulk of the stock market correlation matrix is not pure noise

We analyse the structure of the distribution of eigenvalues of the stock market correlation matrix with increasing length of the time series representing the price changes. We use 100 highly-capitalized stocks from the American market and relate result to the corresponding ensemble of Wishart random matrices. It turns out that systematically more eigenvalues stay beyond the borders prescribed by this variant of the Random Matrix Theory (RMT). This may indicate that even the bulk of the spectrum of the stock market correlation matrix carries some sort of correlations that are masked by a measurement noise when the time series used to construct the matrix are short. We also study some other characteristics of the "noisy" eigensignals, like their return distributions, temporal correlations or their multifractal spectra and the results support the above conclusions.

physics.soc-ph↗

arXiv2005

Detecting subtle effects of persistence in the stock market dynamics

The conventional formal tool to detect effects of the financial persistence is in terms of the Hurst exponent. A typical corresponding result is that its value comes out close to 0.5, as characteristic for geometric Brownian motion, with at most small departures from this value in either direction depending on the market and on the time scales involved. We study the high frequency price changes on the American and on the German stock markets. For both corresponding indices, the Dow Jones and the DAX respectively, the Hurst exponent analysis results in values close to 0.5. However, by decomposing the market dynamics into pairs of steps such that an elementary move up (down) is followed by another move up (down) and explicitly counting the resulting conditional probabilities we find values typically close to 60%. This effect of persistence is particularly visible on the short time scales ranging from 1 up to 3 minutes, decreasing gradually to 50% and even significantly below this value on the larger time scales. We also detect some asymmetry in persistence related to the moves up and down, respectively. This indicates a subtle nature of the financial persistence whose characteristics escape detection within the conventional Hurst exponent formalism.

physics.soc-ph↗