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 the correlation matrix 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.Comment: 37 pages, 9 figures, 3 table

On the interplay between multiscaling and stocks dependence

We find a nonlinear dependence between an indicator of the degree of multiscaling of log-price time series of a stock and the average correlation of the stock with respect to the other stocks traded in the same market. This result is a robust stylized fact holding for different financial markets. We investigate this result conditional on the stocks' capitalization and on the kurtosis of stocks' log-returns in order to search for possible confounding effects. We show that a linear dependence with the logarithm of the capitalization and the logarithm of kurtosis does not explain the observed stylized fact, which we interpret as being originated from a deeper relationship.Comment: 19 pages, 8 figures, 9 table

Spanning Trees and bootstrap reliability estimation in correlation based networks

We introduce a new technique to associate a spanning tree to the average linkage cluster analysis. We term this tree as the Average Linkage Minimum Spanning Tree. We also introduce a technique to associate a value of reliability to links of correlation based graphs by using bootstrap replicas of data. Both techniques are applied to the portfolio of the 300 most capitalized stocks traded at New York Stock Exchange during the time period 2001-2003. We show that the Average Linkage Minimum Spanning Tree recognizes economic sectors and sub-sectors as communities in the network slightly better than the Minimum Spanning Tree does. We also show that the average reliability of links in the Minimum Spanning Tree is slightly greater than the average reliability of links in the Average Linkage Minimum Spanning Tree.Comment: 17 pages, 3 figure

A model for correlations in stock markets

We propose a group model for correlations in stock markets. In the group model the markets are composed of several groups, within which the stock price fluctuations are correlated. The spectral properties of empirical correlation matrices reported in [Phys. Rev. Lett. {\bf 83}, 1467 (1999); Phys. Rev. Lett. {\bf 83}, 1471 (1999.)] are well understood from the model. It provides the connection between the spectral properties of the empirical correlation matrix and the structure of correlations in stock markets.Comment: two pages including one EPS file for a figur

Correlation filtering in financial time series

We apply a method to filter relevant information from the correlation coefficient matrix by extracting a network of relevant interactions. This method succeeds to generate networks with the same hierarchical structure of the Minimum Spanning Tree but containing a larger amount of links resulting in a richer network topology allowing loops and cliques. In Tumminello et al. \cite{TumminielloPNAS05}, we have shown that this method, applied to a financial portfolio of 100 stocks in the USA equity markets, is pretty efficient in filtering relevant information about the clustering of the system and its hierarchical structure both on the whole system and within each cluster. In particular, we have found that triangular loops and 4 element cliques have important and significant relations with the market structure and properties. Here we apply this filtering procedure to the analysis of correlation in two different kind of interest rate time series (16 Eurodollars and 34 US interest rates).Comment: 10 pages 7 figure

Persistence in a Random Bond Ising Model of Socio-Econo Dynamics

We study the persistence phenomenon in a socio-econo dynamics model using computer simulations at a finite temperature on hypercubic lattices in dimensions up to 5. The model includes a ` social\rq local field which contains the magnetization at time $t$. The nearest neighbour quenched interactions are drawn from a binary distribution which is a function of the bond concentration, $p$. The decay of the persistence probability in the model depends on both the spatial dimension and $p$. We find no evidence of ` blocking\rq in this model. We also discuss the implications of our results for possible applications in the social and economic fields. It is suggested that the absence, or otherwise, of blocking could be used as a criterion to decide on the validity of a given model in different scenarios.Comment: 11 pages, 4 figure

Evolution and anti-evolution in a minimal stock market model

We present a novel microscopic stock market model consisting of a large number of random agents modeling traders in a market. Each agent is characterized by a set of parameters that serve to make iterated predictions of two successive returns. The future price is determined according to the offer and the demand of all agents. The system evolves by redistributing the capital among the agents in each trading cycle. Without noise the dynamics of this system is nearly regular and thereby fails to reproduce the stochastic return fluctuations observed in real markets. However, when in each cycle a small amount of noise is introduced we find the typical features of real financial time series like fat-tails of the return distribution and large temporal correlations in the volatility without significant correlations in the price returns. Introducing the noise by an evolutionary process leads to different scalings of the return distributions that depend on the definition of fitness. Because our realistic model has only very few parameters, and the results appear to be robust with respect to the noise level and the number of agents we expect that our framework may serve as new paradigm for modeling self generated return fluctuations in markets.Comment: 13 pages, 5 figure

Bid distributions of competing agents in simple models of auctions

Models of auctions or tendering processes are introduced. In every round of bidding the players select their bid from a probability distribution and whenever a bid is unsuccessful, it is discarded and replaced. For simple models, the probability distributions evolve to a stationary power law with the exponent dependent only on the number of players. For most situations, the system converges towards a state where all players are identical. A number of variations of this model are introduced and the application of these models to the dynamics of market makers is discussed. The effect of price uncertainty on bid distributions is presented. An underlying market structure generates heterogenous agents which do not have power law bid distribution in general

Economic sector identification in a set of stocks traded at the New York Stock Exchange: a comparative analysis

We review some methods recently used in the literature to detect the existence of a certain degree of common behavior of stock returns belonging to the same economic sector. Specifically, we discuss methods based on random matrix theory and hierarchical clustering techniques. We apply these methods to a set of stocks traded at the New York Stock Exchange. The investigated time series are recorded at a daily time horizon. All the considered methods are able to detect economic information and the presence of clusters characterized by the economic sector of stocks. However, different methodologies provide different information about the considered set. Our comparative analysis suggests that the application of just a single method could not be able to extract all the economic information present in the correlation coefficient matrix of a set of stocks.Comment: 13 pages, 8 figures, 2 Table

Scaling theory of temporal correlations and size dependent fluctuations in the traded value of stocks

Records of the traded value f_i(t) of stocks display fluctuation scaling, a proportionality between the standard deviation sigma(i) and the average : sigma(i) ~ f(i)^alpha, with a strong time scale dependence alpha(dt). The non-trivial (i.e., neither 0.5 nor 1) value of alpha may have different origins and provides information about the microscopic dynamics. We present a set of recently discovered stylized facts, and then show their connection to such behavior. The functional form alpha(dt) originates from two aspects of the dynamics: Stocks of larger companies both tend to be traded in larger packages, and also display stronger correlations of traded value.Comment: 8 pages, 7 figures, 1 table, accepted to Phys. Rev.