Completing correlation matrices

TL;DR

This paper introduces a method for completing incomplete correlation matrices by maximizing entropy, especially useful in financial models with many variables, using graph-based techniques for accurate construction.

Contribution

It presents a novel entropy-maximization approach for correlation matrix completion, leveraging graph theory to ensure valid correlation matrices in complex financial models.

Findings

Method successfully completes correlation matrices in financial applications.

Graph-based construction ensures positive definiteness for chordal graphs.

Extensions to larger multi-currency models are feasible.

Abstract

We describe a way to complete a correlation matrix that is not fully specified. Such matrices often arise in financial applications when the number of stochastic variables becomes large or when several smaller models are combined in a larger model. We argue that the proper completion to consider is the matrix that maximizes the entropy of the distribution described by the matrix. We then give a way to construct this matrix starting from the graph associated with the incomplete matrix. If this graph is chordal our construction will result in a proper correlation matrix. We give a detailed description of the construction for a cross-currency model with six stochastic variables and describe extensions to larger models involving more currencies.