Perfectly Fitting CDO Prices Across Tranches: A Theoretical Framework with Efficient Algorithms

Abstract

This paper addresses a key challenge in CDO modeling: achieving a perfect fit to market prices across all tranches using a single, consistent model. The existence of such a perfect-fit model implies the absence of arbitrage among CDO tranches and is thus essential for unified risk management and the pricing of nonstandard credit derivatives. To address this central challenge, we face three primary difficulties: standard parametric models typically fail to achieve a perfect fit; the calibration of standard parametric models inherently relies on computationally intensive simulation-based optimization; and there is a lack of formal theory to determine when a perfect-fit model exists and, if it exists, how to construct it. We propose a theoretical framework to overcome these difficulties. We first introduce and define two compatibility levels of market prices: weak compatibility and strong compatibility. Specifically, market prices across all tranches are said to be weakly (resp. strongly) compatible if there exists a single model (resp. a single conditionally i.i.d. model) that perfectly fits these market prices. We then derive sufficient and necessary conditions for both levels of compatibility by establishing a relationship between compatibility and LP problems. Furthermore, under either condition, we construct a corresponding concrete copula model that achieves a perfect fit. Notably, our framework not only allows for efficient verification of weak compatibility and strong compatibility through LP problems but also facilitates the construction of the corresponding copula models that achieve a perfect fit, eliminating the need for simulation-based optimization. The practical applications of our framework are demonstrated in risk management and the pricing of nonstandard credit derivatives.