Introducing and testing the Carr model of default

The Merton (On the pricing of corporate debt: The risk structure of interest rates. J. Finance, 1974, 29, 449–470) model is often considered the simplest structural model of default. Its famous modification developed by KMV is also widely used and well understood by both practitioners and academics. However, as the underlying asset process is driven by a diffusion process, the Merton model suffers from the well-known issue of the vanishing of credit spreads for progressively shorter maturities. To overcome such shortfall, several modifications have been proposed in the literature; however, none of these, despite succeeding at better pricing spreads and explaining default dynamics, are able to retain the simplicity of the Merton model. In this paper, we propose a new structural model of default driven by an additive process whose pricing formulas are as simple as – if not even simpler than – those of the Merton model. We named such model the Carr model of default after Peter Carr, given his recent passing and the fact that the underlying asset distribution is the one introduced in Carr and Torricelli (Additive logistic processes in option pricing. Finance Stoch., 2021, 25, 689–724) and Carr and Maglione (Compound option pricing and the Roll-Geske-Whaley formula under the conjugate-power Dagum distribution. J. Deriv., 2022, 31, 1–32). We first provide pricing formulas for the firm's claims and credit spreads, and then discuss a general theory of change of measure for such processes in order to introduce a distance-to-default version of this newly-introduced model. Finally, we empirically test the Carr versus the Merton model and document the far better ability of the Carr model to reproduce CDS spreads as well as explain their cross-sectional and time-series variation.