Vulnerable Exotic Derivatives

Understanding and managing counterparty credit risk exposure in derivatives contracts has become a crucial element of real-world trading as well as theoretical modeling. But existing models are limited in the number of securities involved and in the assumed dynamics of the underlying asset returns processes. In this article, the authors present a framework with two counterparties who enter into a derivatives contract in which either of them may default, and the derivative’s payoff depends on the joint distribution of n different assets. Three specific examples illustrate application of the approach: an option on a security in which both counterparties are subject to default risk; a vulnerable spread option, in which one risky counterparty issues an option tied to the price spread between two other assets; and a defaultable swap with one underlying and two risky counterparties who commit to a series of future cash flows. The resulting pricing formulas are mathematically complicated, but as closed-form solutions (with the caveat that integral expressions are approximated using finite step size and number of terms in an infinite series), they are much more efficient than Monte Carlo simulation in reaching a given level of accuracy.