February 10th, 2018, 1:54 am

Not knowing what particular Hull model you have implemented, it's a bit tricky. But if you are looking to use a standard Gaussian default time copula, you have three random variables: the common factor and an idiosyncratic factor for each name. Denoting the default time of name [$]i[$] with [$] \tau_i [$], then [$] \tau_i = F^{-1}_i (\Phi^{-1} (\sqrt{\rho} X_0 + \sqrt{1-\rho^2} X_i)) [$]. Here [$] F_i [$] denotes the risk neutral survival probability function of time for name [$]i[$], and the [$]X_i[$] are iid standard normal random variables. [$]\rho[$] is the pairwise correlation between the two names, given by exposure to a common factor. The easiest way to do the valuation is to perform a Monte Carlo simulation, but one reasonable alternative is a quadrature scheme. The point is that, given the two default times, the cash flows on your contract are deterministic and can be valued by simply adding up discounted cash flows.