Correlation, hierarchies, and networks in financial markets

TL;DR

This paper explores methods to analyze correlation matrices in financial markets, focusing on hierarchical structures, filtering procedures, and stability measures to improve understanding of asset relationships.

Contribution

It introduces techniques to construct hierarchical trees and networks from correlation matrices and links these to nested factor models, enhancing analysis of financial data.

Findings

Hierarchical trees effectively represent asset correlations.

Filtering procedures retain stable information despite fluctuations.

Kullback-Leibler distance quantifies filtering stability.

Abstract

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.