Marcin Forczek

Bio: Marcin Forczek is an academic researcher from Polish Academy of Sciences. The author has contributed to research in topics: Multifractal system & Hurst exponent. The author has an hindex of 3, co-authored 4 publications receiving 246 citations.

Detrended cross-correlation analysis consistently extended to multifractality.

Abstract: We propose an algorithm, multifractal cross-correlation analysis (MFCCA), which constitutes a consistent extension of the detrended cross-correlation analysis and is able to properly identify and quantify subtle characteristics of multifractal cross-correlations between two time series. Our motivation for introducing this algorithm is that the already existing methods, like multifractal extension, have at best serious limitations for most of the signals describing complex natural processes and often indicate multifractal cross-correlations when there are none. The principal component of the present extension is proper incorporation of the sign of fluctuations to their generalized moments. Furthermore, we present a broad analysis of the model fractal stochastic processes as well as of the real-world signals and show that MFCCA is a robust and selective tool at the same time and therefore allows for a reliable quantification of the cross-correlative structure of analyzed processes. In particular, it allows one to identify the boundaries of the multifractal scaling and to analyze a relation between the generalized Hurst exponent and the multifractal scaling parameter ${\ensuremath{\lambda}}_{q}$. This relation provides information about the character of potential multifractality in cross-correlations and thus enables a deeper insight into dynamics of the analyzed processes than allowed by any other related method available so far. By using examples of time series from the stock market, we show that financial fluctuations typically cross-correlate multifractally only for relatively large fluctuations, whereas small fluctuations remain mutually independent even at maximum of such cross-correlations. Finally, we indicate possible utility of MFCCA to study effects of the time-lagged cross-correlations.

Stock market return distributions: From past to present

Abstract: 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 α > 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.

Minimum spanning tree filtering of correlations for varying time scales and size of fluctuations.

Abstract: Based on a recently proposed $q$-dependent detrended cross-correlation coefficient, ${\ensuremath{\rho}}_{q}$ [J Kwapie\ifmmode \acute{n}\else \'{n}\fi{}, P O\ifmmode \acute{s}\else \'{s}\fi{}wi\ifmmode \mbox{\k{e}}\else \k{e}\fi{}cimka, and S Dro\ifmmode \dot{z}\else \{z}\fi{}d\ifmmode \dot{z}\else \{z}\fi{}, Phys Rev E 92, 052815 (2015)], we generalize the concept of the minimum spanning tree (MST) by introducing a family of $q$-dependent minimum spanning trees ($q\mathrm{MST}\mathrm{s}$) that are selective to cross-correlations between different fluctuation amplitudes and different time scales of multivariate data They inherit this ability directly from the coefficients ${\ensuremath{\rho}}_{q}$, which are processed here to construct a distance matrix being the input to the MST-constructing Kruskal's algorithm The conventional MST with detrending corresponds in this context to $q=2$ In order to illustrate their performance, we apply the $q\mathrm{MSTs}$ to sample empirical data from the American stock market and discuss the results We show that the $q\mathrm{MST}$ graphs can complement ${\ensuremath{\rho}}_{q}$ in disentangling ``hidden'' correlations that cannot be observed in the MST graphs based on ${\ensuremath{\rho}}_{\mathrm{DCCA}}$, and therefore, they can be useful in many areas where the multivariate cross-correlations are of interest As an example, we apply this method to empirical data from the stock market and show that by constructing the $q\mathrm{MSTs}$ for a spectrum of $q$ values we obtain more information about the correlation structure of the data than by using $q=2$ only More specifically, we show that two sets of signals that differ from each other statistically can give comparable trees for $q=2$, while only by using the trees for $q\ensuremath{ e}2$ do we become able to distinguish between these sets We also show that a family of $q\mathrm{MSTs}$ for a range of $q$ expresses the diversity of correlations in a manner resembling the multifractal analysis, where one computes a spectrum of the generalized fractal dimensions, the generalized Hurst exponents, or the multifractal singularity spectra: the more diverse the correlations are, the more variable the tree topology is for different $q$'s As regards the correlation structure of the stock market, our analysis exhibits that the stocks belonging to the same or similar industrial sectors are correlated via the fluctuations of moderate amplitudes, while the largest fluctuations often happen to synchronize in those stocks that do not necessarily belong to the same industry

Detrended Cross-Correlation Analysis Consistently Extended to Multifractality

Abstract: We propose a novel algorithm - Multifractal Cross-Correlation Analysis (MFCCA) - that constitutes a consistent extension of the Detrended Cross-Correlation Analysis (DCCA) and is able to properly identify and quantify subtle characteristics of multifractal cross-correlations between two time series. Our motivation for introducing this algorithm is that the already existing methods like MF-DXA have at best serious limitations for most of the signals describing complex natural processes and often indicate multifractal cross-correlations when there are none. The principal component of the present extension is proper incorporation of the sign of fluctuations to their generalized moments. Furthermore, we present a broad analysis of the model fractal stochastic processes as well as of the real-world signals and show that MFCCA is a robust and selective tool at the same time, and therefore allows for a reliable quantification of the cross-correlative structure of analyzed processes. In particular, it allows one to identify the boundaries of the multifractal scaling and to analyze a relation between the generalized Hurst exponent and the multifractal scaling parameter $\lambda_q$. This relation provides information about character of potential multifractality in cross-correlations and thus enables a deeper insight into dynamics of the analyzed processes than allowed by any other related method available so far. By using examples of time series from stock market, we show that financial fluctuations typically cross-correlate multifractally only for relatively large fluctuations, whereas small fluctuations remain mutually independent even at maximum of such cross-correlations. Finally, we indicate possible utility of MFCCA to study effects of the time-lagged cross-correlations.

Limit Order Books

Abstract: Limit order books (LOBs) match buyers and sellers in more than half of the world’s financial markets. This survey highlights the insights that have emerged from the wealth of empirical and theoretical studies of LOBs. We examine the findings reported by statistical analyses of historical LOB data and discuss how several LOB models provide insight into certain aspects of the mechanism. We also illustrate that many such models poorly resemble real LOBs and that several well-established empirical facts have yet to be reproduced satisfactorily. Finally, we identify several key unresolved questions about LOBs.

Limit Order Books

Abstract: Limit order books (LOBs) match buyers and sellers in more than half of the world's financial markets. This survey highlights the insights that have emerged from the wealth of empirical and theoretical studies of LOBs. We examine the findings reported by statistical analyses of historical LOB data and discuss how several LOB models provide insight into certain aspects of the mechanism. We also illustrate that many such models poorly resemble real LOBs and that several well-established empirical facts have yet to be reproduced satisfactorily. Finally, we identify several key unresolved questions about LOBs.

Multifractal analysis of financial markets: a review.

Abstract: Multifractality is ubiquitously observed in complex natural and socioeconomic systems. Multifractal analysis provides powerful tools to understand the complex nonlinear nature of time series in diverse fields. Inspired by its striking analogy with hydrodynamic turbulence, from which the idea of multifractality originated, multifractal analysis of financial markets has bloomed, forming one of the main directions of econophysics. We review the multifractal analysis methods and multifractal models adopted in or invented for financial time series and their subtle properties, which are applicable to time series in other disciplines. We survey the cumulating evidence for the presence of multifractality in financial time series in different markets and at different time periods and discuss the sources of multifractality. The usefulness of multifractal analysis in quantifying market inefficiency, in supporting risk management and in developing other applications is presented. We finally discuss open problems and further directions of multifractal analysis.

Multifractal analysis of financial markets

Abstract: Multifractality is ubiquitously observed in complex natural and socioeconomic systems. Multifractal analysis provides powerful tools to understand the complex nonlinear nature of time series in diverse fields. Inspired by its striking analogy with hydrodynamic turbulence, from which the idea of multifractality originated, multifractal analysis of financial markets has bloomed, forming one of the main directions of econophysics. We review the multifractal analysis methods and multifractal models adopted in or invented for financial time series and their subtle properties, which are applicable to time series in other disciplines. We survey the cumulating evidence for the presence of multifractality in financial time series in different markets and at different time periods and discuss the sources of multifractality. The usefulness of multifractal analysis in quantifying market inefficiency, in supporting risk management and in developing other applications is presented. We finally discuss open problems and further directions of multifractal analysis.