Monte Carlo for large credit portfolios with potentially high correlations

Abstract

In this paper we develop efficient Monte Carlo methods for large credit portfolios. We assume the default indicators admit a Gaussian copula. Therefore, we are able to embed the default correlations into a continuous Gaussian random field, which is capable of incorporating an infinite size portfolio and potentially highly correlated defaults. We are particularly interested in estimating the expectations, such as the expected number of defaults given that there is at least one default and the expected loss given at least one default. All these quantities turn out to be closely related to the geometric structure of the random field. We will heavily employ random field techniques to construct importance sampling based estimators and provide rigorous efficiency analysis.

On the conditional distributions and the efficient simulations of exponential integrals of Gaussian random fields

“…Then, the unit share value of the portfolio is 1 n w i S i = 1 n w(t i )e f (t i ) . See [19,43] for detailed discussions on the random field representations of large portfolios.…”

Some asymptotic results of Gaussian random fields with varying mean functions and the associated processes

“…This change of measure was first proposed by [44] to derive tail asymptotics of supremum of non-Gaussian random fields. This technique substantially simplifies the analysis (though the derivations are still complicated) and may potentially lead to efficient importance sampling algorithms to numerically compute (1.1) and (1.2); see [49,2,3,15] for a connection between the change of measure and efficient computation of tail probabilities. In addition, without too many modifications, one can foresee that the proposed change of measure can be adapted to certain non-Gaussian random fields such as those in [44], which uses a change-of-measure technique to develop the approximations of suprema of non-Gaussian random fields with functional expansions.…”

“…(1.2); see [49,2,3,15] for a connection between the change of measure and efficient computation of tail probabilities. In addition, without too many modifications, one can foresee that the proposed change of measure can be adapted to certain non-Gaussian random fields such as those in [44], which uses a change-of-measure technique to develop the approximations of suprema of non-Gaussian random fields with functional expansions.…”