Fast CDO computations in the affine Markov chain model

Abstract: It is shown that collateralized debt obligations (CDOs), which are complex basket credit derivatives depending on a large number M of firms (M ≥ 100 is typical in most contexts), can be priced in a mathematically consistent and computationally efficient manner within the affine Markov chain (AMC) framework for multifirm credit migration introduced in a companion paper [Hurd and Kuznetsov (2006)]. Thus the AMC framework, which has the potential to generate a wide range of realistic dynamic effects for both default spreads and default correlations, is shown to have the flexibility to handle these complex yet important securities. In addition to inheriting the advantages of the AMC framework, our proposed CDO method possesses a number of more specific merits. First, we are able to prove exact formulas for pricing the premium and insurance legs which handle many popular variations of CDOs such as nonhomogeneous hazard rates and unequal notational amounts that industry practitioners need to use. Second, several reliable approximation schemes are available that reduce the exact formulas to low dimensional integrals which can often be computed on a desktop computer in fractions of seconds. Finally, sensitivities and hedge ratios for CDOs are also computable for a modest computational overhead. In the latter sections of the paper, we develop a representative version of the AMC framework and present a number of sample CDO computations which illustrate the power of the method.