Portfolio credit risk with Archimedean copulas: asymptotic analysis and efficient simulation

TL;DR

This paper develops asymptotic analysis and efficient simulation algorithms for large credit portfolio losses using Archimedean copulas, capturing extremal dependence and significantly improving Monte Carlo methods.

Contribution

It introduces a new model with Archimedean copulas for credit dependence and derives asymptotics and variance reduction algorithms for risk estimation.

Findings

Asymptotic tail probability estimates for portfolio losses

Two variance reduction algorithms with logarithmic efficiency

Variance reduction by orders of magnitude in simulations

Abstract

In this paper, we study large losses arising from defaults of a credit portfolio. We assume that the portfolio dependence structure is modelled by the Archimedean copula family as opposed to the widely used Gaussian copula. The resulting model is new, and it has the capability of capturing extremal dependence among obligors. We first derive sharp asymptotics for the tail probability of portfolio losses and the expected shortfall. Then we demonstrate how to utilize these asymptotic results to produce two variance reduction algorithms that significantly enhance the classical Monte Carlo methods. Moreover, we show that the estimator based on the proposed two-step importance sampling method is logarithmically efficient while the estimator based on the conditional Monte Carlo method has bounded relative error as the number of obligors tends to infinity. Extensive simulation studies are conducted to highlight the efficiency of our proposed algorithms for estimating portfolio credit risk. In particular, the variance reduction achieved by the proposed conditional Monte Carlo method, relative to the crude Monte Carlo method, is in the order of millions.