Credit Monte Carlo
Overview
Simulating Defaults
{% d = \Phi^{-1}(pd) %}
{% A_i \leq d %}
{% A_i = wZ + \sqrt{1-w^2} e_i %}
{% Var(A_i) = Var(wZ) + Var(\sqrt{1-w^2} e_i) %}
{% = w^2 + 1-w^2 = 1 %}
covariance of sum
{% cov(A_i, A_j) = w^2 %}
{% \rho_{ij}^{asset} = cor(A_i, A_j) = w^2 %}
{% w = \sqrt{\rho_{ij}^{asset}} %}
correlations
Modeling Correlations
{% p_{ij} = \Phi(d_i,d_j, \rho_{ij}^{asset}) %}
