Correlated Random Walks and the Joint Survival Probability

TL;DR

This paper derives a first-order approximation for the joint survival probability of multiple firms in correlated first passage models, highlighting the impact of asset correlations on credit risk assessment.

Contribution

It introduces a linear perturbation approach to quantify how asset correlations influence joint survival probabilities in first passage models.

Findings

Derived an explicit expression for joint survival probability dependence on correlations.

Compared first passage model results with multivariate normal copula for constant correlations.

Applied the model to estimate five-year joint survival probabilities for industrial firms.

Abstract

First passage models, where corporate assets undergo correlated random walks and a company defaults if its assets fall below a threshold provide an attractive framework for modeling the default process. Typical one year default correlations are small, i.e., of order a few percent, but nonetheless including correlations is very important, for managing portfolio credit risk and pricing some credit derivatives (e.g. first to default baskets). In first passage models the exact dependence of the joint survival probability of more than two firms on their asset correlations is not known. We derive an expression for the dependence of the joint survival probability of $n$ firms on their asset correlations using first order perturbation theory in the correlations. It includes all terms that are linear in the correlations but neglects effects of quadratic and higher order. For constant time independent correlations we compare the first passage model expression for the joint survival probability with what a multivariate normal Copula function gives. As a practical application of our results we calculate the dependence of the five year joint survival probability for five basic industrials on their asset correlations.