×

New collectivity measures for financial covariances and correlations. (English) Zbl 07582070

Summary: Complex systems are usually non-stationary and their dynamics is often dominated by collective effects. Collectivity, defined as coherent motion of the whole system or of some of its parts, manifests itself in the time-dependent structures of covariance and correlation matrices. The largest eigenvalue corresponds to the collective motion of the system as a whole, while the other large, isolated, eigenvalues indicate collectivity in parts of the system. In the case of finance, these are industrial sectors. By removing the collective motion of the system as a whole, the latter effects are much better revealed. We measure a remaining collectivity to which we refer as average sector collectivity. We identify collective signals around the Lehman Brothers crash and after the dot-com bubble burst. For the Lehman Brothers crash, we find a potential precursor. We analyze 213 US stocks over a period of more than 30 years from 1990 to 2021. We plot the average sector collectivity versus the collectivity corresponding to the largest eigenvalue to study the whole market trajectory in a two dimensional space spanned by both collectivities. Therefore, we capture the average sector collectivity in a much more precise way. Additionally, we observe that larger values in the average sector collectivity are often accompanied by trend shifts in the mean covariances and mean correlations. As of 2015/2016 the collectivity in the US stock markets changed fundamentally.

MSC:

82-XX Statistical mechanics, structure of matter
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Mandelbrot, B. B., The variation of certain speculative prices, (Fractals and Scaling in Finance: Discontinuity, Concentration, Risk. Selecta Volume E (1997), Springer: Springer New York), 371-418 · Zbl 1005.91001
[2] Schwert, G. W., Why does stock market volatility change over time?, J. Finance, 44, 5, 1115-1153 (1989)
[3] Longin, F.; Solnik, B., Is the correlation in international equity returns constant: 1960-1990?, J. Int. Money Finance, 14, 1, 3-26 (1995)
[4] Mantegna, R. N.; Stanley, H. E., Introduction to Econophysics: Correlations and Complexity in Finance (1999), Cambridge University Press: Cambridge University Press Cambridge
[5] Bouchaud, J.-P.; Potters, M., Theory of Financial Risk and Derivative Pricing: From Statistical Physics to Risk Management (2003), Cambridge University Press: Cambridge University Press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo · Zbl 1194.91008
[6] Kwapień, J.; Drożdż, S., Physical approach to complex systems, Phys. Rep., 515, 3, 115-226 (2012)
[7] Kutner, R.; Ausloos, M.; Grech, D.; Di Matteo, T.; Schinckus, C.; Eugene Stanley, H., Econophysics and sociophysics: Their milestones & challenges, Physica A, 516, 240-253 (2019)
[8] Marčenko, V. A.; Pastur, L. A., Distribution of eigenvalues for some sets of random matrices, Math. USSR-Sbornik, 1, 4, 457-483 (1967) · Zbl 0162.22501
[9] Kwapień, J.; Drożdż, S.; ecimka, P. O., The bulk of the stock market correlation matrix is not pure noise, Physica A, 359, 589-606 (2006)
[10] Laloux, L.; Cizeau, P.; Bouchaud, J.-P.; Potters, M., Noise dressing of financial correlation matrices, Phys. Rev. Lett., 83, 1467-1470 (1999)
[11] Noh, J. D., Model for correlations in stock markets, Phys. Rev. E, 61, 5981-5982 (2000)
[12] Gopikrishnan, P.; Rosenow, B.; Plerou, V.; Stanley, H. E., Quantifying and interpreting collective behavior in financial markets, Phys. Rev. E, 64, Article 035106 pp. (2001)
[13] Plerou, V.; Gopikrishnan, P.; Rosenow, B.; Amaral, L. A.N.; Guhr, T.; Stanley, H. E., Random matrix approach to cross correlations in financial data, Phys. Rev. E, 65, Article 066126 pp. (2002)
[14] Song, D.-M.; Tumminello, M.; Zhou, W.-X.; Mantegna, R. N., Evolution of worldwide stock markets, correlation structure, and correlation-based graphs, Phys. Rev. E, 84, Article 026108 pp. (2011)
[15] MacMahon, M.; Garlaschelli, D., Community detection for correlation matrices, Phys. Rev. X, 5, Article 021006 pp. (2015)
[16] Stepanov, Y.; Rinn, P.; Guhr, T.; Peinke, J.; Schäfer, R., Stability and hierarchy of quasi-stationary states: financial markets as an example, J. Stat. Mech. Theory Exp., 2015, 8, P08011 (2015) · Zbl 1456.62115
[17] Chetalova, D.; Schäfer, R.; Guhr, T., Zooming into market states, J. Stat. Mech. Theory Exp., 2015, 1, P01029 (2015)
[18] Benzaquen, M.; Mastromatteo, I.; Eisler, Z.; Bouchaud, J.-P., Dissecting cross-impact on stock markets: An empirical analysis, J. Stat. Mech. Theory Exp., 2017, 2, Article 023406 pp. (2017)
[19] Potters, M.; Bouchaud, J.-P., A First Course in Random Matrix Theory: For Physicists, Engineers and Data Scientists (2020), Cambridge University Press: Cambridge University Press Cambridge
[20] Allez, R.; Bouchaud, J.-P., Eigenvector dynamics: General theory and some applications, Phys. Rev. E, 86, Article 046202 pp. (2012)
[21] Reigneron, P.-A.; Allez, R.; Bouchaud, J.-P., Principal regression analysis and the index leverage effect, Physica A, 390, 17, 3026-3035 (2011)
[22] Heckens, A. J.; Krause, S. M.; Guhr, T., Uncovering the dynamics of correlation structures relative to the collective market motion, J. Stat. Mech. Theory Exp., 2020, 10, Article 103402 pp. (2020) · Zbl 1459.91223
[23] Heckens, A. J.; Guhr, T., A new attempt to identify long-term precursors for endogenous financial crises in the market correlation structures, J. Stat. Mech. Theory Exp., 2022, 4, Article 043401 pp. (2022) · Zbl 07531149
[24] Sornette, D.; Malevergne, Y.; Muzy, J.-F., What causes crashes?, Risk Mag., 16, 2, 67-72 (2003)
[25] Bouchaud, J.-P., The endogenous dynamics of markets: price impact and feedback loops (2010), arXiv:1009.2928
[26] Danielsson, J.; Shin, H. S.; Zigrand, J.-P., Endogenous extreme events and the dual role of prices, Annu. Rev. Econ., 4, 1, 111-129 (2012)
[27] Bouchaud, J.-P., The endogenous dynamics of markets: A complex system point of view, Procedia Comput. Sci., 7, 22-23 (2011)
[28] Hamilton, J. D., A new approach to the economic analysis of nonstationary time series and the business cycle, Econometrica, 57, 2, 357-384 (1989) · Zbl 0685.62092
[29] Marsili, M., Dissecting financial markets: sectors and states, Quant. Finance, 2, 4, 297-302 (2002) · Zbl 1405.91755
[30] Procacci, P. F.; Aste, T., Forecasting market states, Quant. Finance, 19, 9, 1491-1498 (2019) · Zbl 1420.91553
[31] Uechi, L.; Akutsu, T.; Stanley, H. E.; Marcus, A. J.; Kenett, D. Y., Sector dominance ratio analysis of financial markets, Physica A, 421, 488-509 (2015)
[32] Münnix, M. C.; Shimada, T.; Schäfer, R.; Leyvraz, F.; Seligman, T. H.; Guhr, T.; Stanley, H. E., Identifying states of a financial market, Sci. Rep., 2, 644 (2012)
[33] Rinn, P.; Stepanov, Y.; Peinke, J.; Guhr, T.; Schäfer, R., Dynamics of quasi-stationary systems: Finance as an example, EPL (Europhys. Lett.), 110, 6, 68003 (2015)
[34] Chetalova, D.; Wollschläger, M.; Schäfer, R., Dependence structure of market states, J. Stat. Mech. Theory Exp., 2015, 8, P08012 (2015) · Zbl 1456.91130
[35] Stepanov, Y.; Wellner, E.; Abou-Zeid, T., Multi-asset correlation dynamics with application to trading (2015)
[36] Papenbrock, J.; Schwendner, P., Handling risk-on/risk-off dynamics with correlation regimes and correlation networks, Financial Mark. Portfolio Manag., 29, 2, 125-147 (2015)
[37] Jurczyk, J.; Rehberg, T.; Eckrot, A.; Morgenstern, I., Measuring critical transitions in financial markets, Sci. Rep., 7, 1, 11564 (2017)
[38] Pharasi, H. K.; Sharma, K.; Chatterjee, R.; Chakraborti, A.; Leyvraz, F.; Seligman, T. H., Identifying long-term precursors of financial market crashes using correlation patterns, New J. Phys., 20, 10, Article 103041 pp. (2018)
[39] Qiu, L.; Gu, C.; Xiao, Q.; Yang, H.; Wu, G., State network approach to characteristics of financial crises, Physica A, 492, 1120-1128 (2018)
[40] Pharasi, H. K.; Seligman, E.; Seligman, T. H., Market states: A new understanding (2020), arXiv:2003.07058
[41] Pharasi, H. K.; Seligman, E.; Sadhukhan, S.; Seligman, T. H., Dynamics of market states and risk assessment (2020), arXiv:2011.05984
[42] Pharasi, H. K.; Sadhukhan, S.; Majari, P.; Chakraborti, A.; Seligman, T. H., Dynamics of the market states in the space of correlation matrices with applications to financial markets (2021), arXiv:2107.05663
[43] Marti, G.; Nielsen, F.; Bińkowski, M.; Donnat, P., A review of two decades of correlations, hierarchies, networks and clustering in financial markets, (Nielsen, F., Progress in Information Geometry: Theory and Applications (2021), Springer International Publishing: Springer International Publishing Cham), 245-274 · Zbl 1471.91547
[44] Rings, T.; Mazarei, M.; Akhshi, A.; Geier, C.; Tabar, M. R.R.; Lehnertz, K., Traceability and dynamical resistance of precursor of extreme events, Sci. Rep., 9, 1, 1744 (2019)
[45] Wang, S.; Gartzke, S.; Schreckenberg, M.; Guhr, T., Quasi-stationary states in temporal correlations for traffic systems: Cologne orbital motorway as an example, J. Stat. Mech. Theory Exp., 2020, 10, Article 103404 pp. (2020) · Zbl 1459.82199
[46] Bette, H. M.; Jungblut, E.; Guhr, T., Non-stationarity in correlation matrices for wind turbine SCADA-data and implications for failure detection (2021), arXiv:2107.13256
[47] Borghesi, C.; Marsili, M.; Miccichè, S., Emergence of time-horizon invariant correlation structure in financial returns by subtraction of the market mode, Phys. Rev. E, 76, Article 026104 pp. (2007)
[48] Kenett, D. Y.; Shapira, Y.; Madi, A.; Bransburg-Zabary, S.; Gur-Gershgoren, G.; Ben-Jacob, E., Index cohesive force analysis reveals that the US market became prone to systemic collapses since 2002, PLoS ONE, 6, 4, Article e19378 pp. (2011)
[49] Shapira, Y.; Kenett, D. Y.; Ben-Jacob, E., The index cohesive effect on stock market correlations, Eur. Phys. J. B, 72, 4, 657-669 (2009) · Zbl 1189.91130
[50] Kenett, D. Y.; Tumminello, M.; Madi, A.; Gur-Gershgoren, G.; Mantegna, R. N.; Ben-Jacob, E., Dominating clasp of the financial sector revealed by partial correlation analysis of the stock market, PloS ONE, 5, 12, Article e15032 pp. (2010)
[51] Kenett, D. Y.; Huang, X.; Vodenska, I.; Havlin, S.; Stanley, H. E., Partial correlation analysis: applications for financial markets, Quant. Finance, 15, 4, 569-578 (2015)
[52] Anderson, T. W., An Introduction to Multivariate Statistical Analysis (2003), John Wiley and Sons: John Wiley and Sons Hoboken, New Jersey · Zbl 1039.62044
[53] Wikipedia contributors, T. W., Partial correlation — Wikipedia, the free encyclopedia (2021), https://en.wikipedia.org/w/index.php?title=Partial_correlation&oldid=1026106715, [Online; accessed 10-October-2021]
[54] Kritzman, M.; Li, Y.; Page, S.; Rigobon, R., Principal components as a measure of systemic risk, J. Portfolio Manag., 37, 4, 112-126 (2011)
[55] Bisias, D.; Flood, M.; Lo, A. W.; Valavanis, S., A survey of systemic risk analytics, Annu. Rev. Financial Econ., 4, 1, 255-296 (2012)
[56] Zheng, Z.; Podobnik, B.; Feng, L.; Li, B., Changes in cross-correlations as an indicator for systemic risk, Sci. Rep., 2, 1, 888 (2012)
[57] Billio, M.; Getmansky, M.; Lo, A. W.; Pelizzon, L., Econometric measures of connectedness and systemic risk in the finance and insurance sectors, J. Financ. Econ., 104, 3, 535-559 (2012)
[58] Begušić, S.; Kostanjčar, Z.; Kovač, D.; Stanley, H. E.; Podobnik, B., Information feedback in temporal networks as a predictor of market crashes, Complexity, 2018, Article 2834680 pp. (2018)
[59] Huang, X.; Vodenska, I.; Havlin, S.; Stanley, H. E., Cascading failures in bi-partite graphs: Model for systemic risk propagation, Sci. Rep., 3, 1, 1219 (2013)
[60] Musmeci, N.; Aste, T.; Di Matteo, T., Interplay between past market correlation structure changes and future volatility outbursts, Sci. Rep., 6, 1, 36320 (2016)
[61] . Refinitiv Workspace for Students (formals EIKON), https://www.refinitiv.com/ (downloaded: 09-August-2021 and 03-April-2022).
[62] Wikipedia contributors, N., Global industry classification standard — Wikipedia, the free encyclopedia (2020), https://en.wikipedia.org/w/index.php?title=Global_Industry_Classification_Standard&oldid=996081862, [Online; accessed 10-January-2021]
[63] Wikipedia contributors, N., List of stock market crashes and bear markets — Wikipedia, The free encyclopedia (2020), https://en.wikipedia.org/w/index.php?title=List_of_stock_market_crashes_and_bear_markets&oldid=995044267, [Online; accessed 5-January-2021]
[64] Plerou, V.; Gopikrishnan, P.; Rosenow, B.; Amaral, L. A.N.; Stanley, H. E., Universal and nonuniversal properties of cross correlations in financial time series, Phys. Rev. Lett., 83, 1471-1474 (1999)
[65] Wang, S.; Gartzke, S.; Schreckenberg, M.; Guhr, T., Quasi-stationary states in temporal correlations for traffic systems: Cologne orbital motorway as an example, J. Stat. Mech. Theory Exp., 2020, 10, Article 103404 pp. (2020) · Zbl 1459.82199
[66] Wishart, J., The generalised product moment distribution in samples from a normal multivariate population, Biometrika, 20A, 1/2, 32-52 (1928) · JFM 54.0565.02
[67] Gupta, A. K.; Nagar, D. K., Matrix Variate Distributions (Monographs and Surveys in Pure and Applied Mathematics), Vol. 104 (2000), Chapman Hall/CRC: Chapman Hall/CRC Boca Raton, London, New York, Washington D.C. · Zbl 0935.62064
[68] Sharpe, W. F., A simplified model for portfolio analysis, Manage. Sci., 9, 2, 277-293 (1963)
[69] Ross, S. A., The arbitrage theory of capital asset pricing, J. Econom. Theory, 13, 3, 341-360 (1976)
[70] Guhr, T.; Kälber, B., A new method to estimate the noise in financial correlation matrices, J. Phys. A: Math. Gen., 36, 12, 3009-3032 (2003) · Zbl 1042.91042
[71] Kenett, D. Y.; Shapira, Y.; Ben-Jacob, E., RMT assessments of the market latent information embedded in the stocks’ raw, normalized, and partial correlations, J. Probab. Stat., 2009, Article 249370 pp. (2009)
[72] Ding, Z.; Granger, C. W.; Engle, R. F., A long memory property of stock market returns and a new model, J. Empir. Financ., 1, 1, 83-106 (1993)
[73] Raddant, M.; Wagner, F., Phase transition in the S&P stock market, J. Econ. Interact. Coord., 11, 2, 229-246 (2016)
[74] Raddant, M.; Wagner, F., Transitions in the stock markets of the US, UK and Germany, Quant. Finance, 17, 2, 289-297 (2017) · Zbl 1402.91990
[75] Chakraborti, A.; Hrishidev, F.; Sharma, K.; Pharasi, H. K., Phase separation and scaling in correlation structures of financial markets, J. Phys.: Complex., 2, 1, Article 015002 pp. (2020)
[76] Drożdż, S.; Grümmer, F.; Górski, A.; Ruf, F.; Speth, J., Dynamics of competition between collectivity and noise in the stock market, Physica A, 287, 3, 440-449 (2000)
[77] Izrailev, F. M., Simple models of quantum chaos: Spectrum and eigenfunctions, Phys. Rep., 196, 5, 299-392 (1990)
[78] Izrailev, F., Quantum localization and statistics of quasienergy spectrum in a classically chaotic system, Phys. Lett. A, 134, 1, 13-18 (1988)
[79] Ross, G. J., Dynamic multifactor clustering of financial networks, Phys. Rev. E, 89, Article 022809 pp. (2014)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.