"Studies in correlation based financial networks"
STUDIES IN CORRELATION BASED FINANCIAL NETWORKS Network theory provides an approach to complex systems with many interacting units, where the details of the interactions are of lesser importance. Recently this approach has proved to be very useful in a broad field of applications ranging from the Internet to microbiology. In the financial market companies certinly interact with one another, creating an evolving complex system. Although the exact nature of these interactions is not known, they are reflected in temporal correlations based on either stock returns or, alternatively, on flow of capital, calculated as a product of price and volume. We construct correlation based financial networks for a subset of NYSE traded stocks. In the resulting graphs the nodes correspond to stocks and the edges to correlation based ultrametric distances between them. As studies with empirical data have shown, a large majority of eigenvalues for empirical correlation matrices fall within the spectrum predicted for random matrices by random matrix theory. Since these matrices are predominantly noise, a central issue is to prune these systems in such a way that the noise if filtered out but the actual information is retained. We offer two approaches to this problem. In the first approach we construct a minimum spanning tree of edges. We have demonstrated that the MST method leads to a scale-free network, where the scaling exponent is fairly stable over time, except for crash periods, which are characterized by a lower exponent . During crash periods a strong reconfiguration takes place, and the tree shrinks both topologically and in terms of its overall length . We have also demonstrated how the stocks of the minimum risk Markowitz portfolio lie practically at all times on the outskirts of the tree . The second approach is based on agglomerative clustering, i.e. we add a variable number of edges in the graph, one edge at a time, based on their rank. This approach better captures the strong clustering present in the market and leads to a more robust structure than the MST approach . We have also compared some other properties for empirical graphs against those of a completely random graph, for which results are well known. It is postulated that deviations from theoretical predictions are indicative of genuine information. At a critical threshold, the random graph undergoes a radical change in topology related to percolation transition and forms a single giant cluster, a phenomenon not observed for the empirical graph. Differences in mean clustering coefficient lead us to conclude that most information is contained roughly within just 10% of all edges . References: (available at http://www.lce.hut.fi/~jonnela)  J.-P. Onnela, A. Chakraborti, K. Kaski, J. Kertesz, and A. Kanto: Dynamics of market correlations: Taxonomy and portfolio analysis, Physical Review E 68, 056110 (2003).  J.-P. Onnela, A. Chakraborti, K. Kaski, and J. Kertesz: Dynamic asset trees and Black Monday, Physica A 324/1-2, 247-252 (2003).  J.-P. Onnela, A. Chakraborti, K. Kaski, and J. Kertesz: Dynamic asset trees and portfolio analysis, European Physical Journal B 30, 285-288 (2002).  J.-P. Onnela, A. Chakraborti, K. Kaski, J. Kertesz, and A. Kanto: Asset trees and asset graphs in financial markets, Physica Scripta T106, 48-54 (2003).  J.-P. Onnela, K. Kaski, and J. Kertesz: Clustering and information in correlation based financial networks, European Physical Journal B, in press (2004).