Written in 1987; printed in Derivatives Pricing: The Classic Collection, P. Carr (ed.), London: Risk Books, 2004.
Consider a portfolio consisting of n loans in equal dollar amounts. Let the probability of default on any one loan be p, and assume that the values of the borrowing companies' assets are correlated with a coefficient ρ for any two companies. We wish to calculate the probability distribution of the percentage gross loss L on the portfolio, that is,
Let be the value of the i-th company's assets, described by a logarithmic Wiener process
where are Wiener processes with
The company defaults on its loan if the value of its assets drops below the contractual value of its obligations Di payable at time T. We thus have