An Outlook on Correlations in Stock Prices

We present an outlook of the studies on correlations in the price timeseries of stocks, discussing the construction and applications of "asset tree". The topic discussed here should illustrate how the complex economic system (financial market) enrichens the list of existing dynamical systems that physicists have been studying for long.Comment: 6 pages, RevTeX format. To appear in the Conference Proceedings of ECONOPHYS-KOLKATA II: International Workshop on Econophysics of Stock Markets and Minority Games", February 14-17, 2006, SINP, Kolkata, as a book chapter in Eds. A. Chatterjee and B.K. Chakrabarti, Econophysics of Stock and other Markets, (Springer-Verlag (Italia), Milan, 2006

Linear and nonlinear market correlations: characterizing financial crises and portfolio optimization

Pearson correlation and mutual information based complex networks of the day-to-day returns of US S&P500 stocks between 1985 and 2015 have been constructed in order to investigate the mutual dependencies of the stocks and their nature. We show that both networks detect qualitative differences especially during (recent) turbulent market periods thus indicating strongly fluctuating interconnections between the stocks of different companies in changing economic environments. A measure for the strength of nonlinear dependencies is derived using surrogate data and leads to interesting observations during periods of financial market crises. In contrast to the expectation that dependencies reduce mainly to linear correlations during crises we show that (at least in the 2008 crisis) nonlinear effects are significantly increasing. It turns out that the concept of centrality within a network could potentially be used as some kind of an early warning indicator for abnormal market behavior as we demonstrate with the example of the 2008 subprime mortgage crisis. Finally, we apply a Markowitz mean variance portfolio optimization and integrate the measure of nonlinear dependencies to scale the investment exposure. This leads to significant outperformance as compared to a fully invested portfolio.Comment: 12 pages, 11 figures, Phys. Rev. E, accepte

Relation between Financial Market Structure and the Real Economy: Comparison between Clustering Methods

We quantify the amount of information filtered by different hierarchical clustering methods on correlations between stock returns comparing it with the underlying industrial activity structure. Specifically, we apply, for the first time to financial data, a novel hierarchical clustering approach, the Directed Bubble Hierarchical Tree and we compare it with other methods including the Linkage and k-medoids. In particular, by taking the industrial sector classification of stocks as a benchmark partition, we evaluate how the different methods retrieve this classification. The results show that the Directed Bubble Hierarchical Tree can outperform other methods, being able to retrieve more information with fewer clusters. Moreover, we show that the economic information is hidden at different levels of the hierarchical structures depending on the clustering method. The dynamical analysis on a rolling window also reveals that the different methods show different degrees of sensitivity to events affecting financial markets, like crises. These results can be of interest for all the applications of clustering methods to portfolio optimization and risk hedging.Comment: 31 pages, 17 figure

Structural and topological phase transitions on the German Stock Exchange

We find numerical and empirical evidence for dynamical, structural and topological phase transitions on the (German) Frankfurt Stock Exchange (FSE) in the temporal vicinity of the worldwide financial crash. Using the Minimal Spanning Tree (MST) technique, a particularly useful canonical tool of the graph theory, two transitions of the topology of a complex network representing FSE were found. First transition is from a hierarchical scale-free MST representing the stock market before the recent worldwide financial crash, to a superstar-like MST decorated by a scale-free hierarchy of trees representing the market's state for the period containing the crash. Subsequently, a transition is observed from this transient, (meta)stable state of the crash, to a hierarchical scale-free MST decorated by several star-like trees after the worldwide financial crash. The phase transitions observed are analogous to the ones we obtained earlier for the Warsaw Stock Exchange and more pronounced than those found by Onnela-Chakraborti-Kaski-Kert\'esz for S&P 500 index in the vicinity of Black Monday (October 19, 1987) and also in the vicinity of January 1, 1998. Our results provide an empirical foundation for the future theory of dynamical, structural and topological phase transitions on financial markets