Overview

In this chapter, we deal with the problem of finding an optimal positive-definite (PD) approximation for a given correlation matrix. Consider a non-positive-definite (NPD) correlation matrix ( ?) ^NPD. We call NPD matrices "illegal" and positive-definite matrices "legal". Correlation matrices that are NPD can arise from various sources. As discussed before, stressed correlation matrices are desirable to probe correlation risk. Such stressed matrices are produced by moving the individual correlation matrix elements ? _?? from their current values ? _?? ^Current by amounts ?? _??.

We did present a way in the last chapter for the ?? _?? to be chosen while preserving PD constraints by using the - sphere geometrical formalism.

However, as also described in the last chapter, we might want to move ?? _?? by hand using (e.g.) historical volatilities, maximum moves, etc. However, the constraints among the various correlations make it difficult (read impossible) to preserve the PD constraint. The stressed matrix is therefore most probably NPD.

One good reason to stress the correlations is in order to use them in Monte-Carlo simulations generating stressed-correlated movements of underlying variables ^[1]. However, if a correlation matrix is NPD, it is useless for simulation. This is because a NPD matrix has negative eigenvalues and so the real square root matrix needed for simulations does not exist ^[2]. Hence, a method is needed to render NPD matrices positive definite in such a way that the stressed character of...