Higher dimensional probability of default in structural models

This thesis describes the joint probability distribution of defaults in two, three and four dimensions. In particular, default as defined by Merton and Black and Cox using analytical and simulated Monte Carlo approaches. Our analytical approach in a Merton setting, utilizes the multivariate normal to compute the joint probability distribution in any dimension. In a Black-Cox setting, analytical solutions are defined in specific dimensions, therefore we rely on a simulated approach. The precision of our simulated approaches are evaluated using 104, 107 and up to 107.5 paths 1. We use our results to compare the probability of defaults in both settings as well as tail dependence, portfolio value and value at risk. Tail dependence is evaluated in two and three dimensions with ρ=0.3 and ρ=0.9. We define covariance parameters in four dimensions; "normal" and "crisis" market conditions, to evaluate portfolio value in a credit and market portfolio and value at risk.