Correlation, Hierarchies, and Networks in Financial Markets

We discuss some methods to quantitatively investigate the properties of correlation matrices. Correlation matrices play an important role in portfolio optimization and in several other quantitative descriptions of asset price dynamics in financial markets. Specifically, we discuss how to define and obtain hierarchical trees, correlation based trees and networks from a correlation matrix. The hierarchical clustering and other procedures performed on the correlation matrix to detect statistically reliable aspects of it are seen as filtering procedures of the correlation matrix. We also discuss a method to associate a hierarchically nested factor model to a hierarchical tree obtained from a correlation matrix. The information retained in filtering procedures and its stability with respect to statistical fluctuations is quantified by using the Kullback-Leibler distance.

[1]  T. Aste,et al.  Correlation based networks of equity returns sampled at different time horizons , 2007 .

[2]  V. Plerou,et al.  Universal and Nonuniversal Properties of Cross Correlations in Financial Time Series , 1999, cond-mat/9902283.

[3]  J. Bull,et al.  An Empirical Test of Bootstrapping as a Method for Assessing Confidence in Phylogenetic Analysis , 1993 .

[4]  M. Tumminello,et al.  Kullback-Leibler distance as a measure of the information filtered from multivariate data. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  J. Bouchaud,et al.  Noise Dressing of Financial Correlation Matrices , 1998, cond-mat/9810255.

[6]  Dan Tsafrir,et al.  Sorting points into neighborhoods (SPIN): data analysis and visualization by ordering distance matrices , 2005, Bioinform..

[7]  W. Bialek,et al.  Information-based clustering. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[8]  Mantegna,et al.  Taxonomy of stock market indices , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[9]  F. Lillo,et al.  High-frequency cross-correlation in a set of stocks , 2000 .

[10]  B. Swart,et al.  Quantitative Finance , 2006, Metals and Energy Finance.

[11]  M Marsili,et al.  Data clustering and noise undressing of correlation matrices. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  Alexander Kraskov,et al.  Published under the scientific responsability of the EUROPEAN PHYSICAL SOCIETY Incorporating , 2002 .

[13]  Blatt,et al.  Superparamagnetic clustering of data. , 1998, Physical review letters.

[14]  Jean-Philippe Bouchaud,et al.  Theory of Financial Risk and Derivative Pricing: Foreword , 2003 .

[15]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[16]  Huaiyu Zhu.  On Information and Sufficiency , 1997 .

[17]  Fabrizio Lillo,et al.  Sector identification in a set of stock return time series traded at the London Stock Exchange , 2005 .

[18]  P. Anderson.  More is different. , 1972, Science.

[19]  R. Tibshirani,et al.  An Introduction to the Bootstrap , 1995 .

[20]  HERBERT A. SIMON,et al.  The Architecture of Complexity , 1991 .

[21]  R. Mantegna.  Hierarchical structure in financial markets , 1998, cond-mat/9802256.

[22]  Michael R. Anderberg,et al.  Cluster Analysis for Applications , 1973 .

[23]  K. Strimmer,et al.  Statistical Applications in Genetics and Molecular Biology A Shrinkage Approach to Large-Scale Covariance Matrix Estimation and Implications for Functional Genomics , 2011 .

[24]  A Hutt,et al.  Analysis of spatiotemporal signals: a method based on perturbation theory. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[25]  G. Biroli,et al.  The Student ensemble of correlation matrices: eigenvalue spectrum and Kullback-Leibler entropy , 2007, 0710.0802.

[26]  M. Stephanov,et al.  Random Matrices , 2005, hep-ph/0509286.

[27]  S. T. Buckland,et al.  An Introduction to the Bootstrap. , 1994 .

[28]  J. Bouchaud,et al.  Theory Of Financial Risk And Derivative Pricing , 2000 .

[29]  Fabrizio Lillo,et al.  Shrinkage and spectral filtering of correlation matrices: a comparison via the Kullback-Leibler distance , 2007, 0710.0576.

[30]  Proceedings of the American Philosophical Society , 2022 .

[31]  김삼묘,et al.  “Bioinformatics” 특집을 내면서 , 2000 .

[32]  David Hinkley,et al.  Bootstrap Methods: Another Look at the Jackknife , 2008 .

[33]  Fabrizio Lillo,et al.  Cluster analysis for portfolio optimization , 2005, physics/0507006.

[34]  M. Tumminello,et al.  Hierarchically nested factor model from multivariate data , 2007 .

[35]  M Tumminello,et al.  A tool for filtering information in complex systems. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[36]  N. L. Johnson,et al.  Multivariate Analysis , 1958, Nature.

[37]  Olivier Ledoit,et al.  Improved estimation of the covariance matrix of stock returns with an application to portfolio selection , 2003 .

[38]  K. Kaski,et al.  Dynamics of market correlations: taxonomy and portfolio analysis. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[39]  Jean-Philippe Bouchaud,et al.  Financial Applications of Random Matrix Theory: Old Laces and New Pieces , 2005 .

[40]  Bernd Rosenow,et al.  Portfolio optimization and the random magnet problem , 2002 .

[41]  Fabrizio Lillo,et al.  Degree stability of a minimum spanning tree of price return and volatility , 2003 .

[42]  Kathryn Fraughnaugh,et al.  Introduction to graph theory , 1973, Mathematical Gazette.

[43]  Fabrizio Lillo,et al.  Spanning Trees and bootstrap Reliability Estimation in Correlation-Based Networks , 2007, Int. J. Bifurc. Chaos.

[44]  F. Lillo,et al.  Topology of correlation-based minimal spanning trees in real and model markets. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.