The two main approaches in credit risk are the structural approach pioneered by Merton and the reduced‐form framework proposed by Jarrow and Turnbull and by Artzner and Delbaen. The goal of this paper is to provide a unified view on both approaches. This is achieved by studying reduced‐form approaches under weak assumptions. In particular, we do not assume the global existence of a default intensity and allow default at fixed or predictable times, such as coupon payment dates, with positive probability. In this generalized framework, we study dynamic term structures prone to default risk following the forward‐rate approach proposed by Heath, Jarrow, and Morton. It turns out that previously considered models lead to arbitrage possibilities when default can happen at a predictable time. A suitable generalization of the forward‐rate approach contains an additional stochastic integral with atoms at predictable times and necessary and sufficient conditions for an appropriate no‐arbitrage condition are given. For efficient implementations, we develop a new class of affine models that do not satisfy the standard assumption of stochastic continuity. The chosen approach is intimately related to the theory of enlargement of filtrations, for which we provide an example by means of filtering theory where the Azéma supermartingale contains upward and downward jumps, both at predictable and totally inaccessible stopping times.

The goal of this article is to study in detail the pricing and calibration in market models for credit portfolios. Starting from the framework of market models driven by time-inhomogeneous Lévy processes in a top-down approach proposed in Eberlein, Grbac, and Schmidt (2012) we consider a slightly simplified setup which eases calibration. This leads to a new class of affine models which are highly tractable. Conditions for absence of arbitrage under various types of contagion are given and valuation formulas for single tranche CDOs and options on CDO spreads are obtained. A simple two-factor affine diffusion model is calibrated to iTraxx data using the EM-algorithm together with an extended Kalman filter. The model shows a very good fit to all tranches and all maturities over the full observation period of four years.

The intensity of a default time is obtained by assuming that the default indicator process has an absolutely continuous compensator. Here we drop the assumption of absolute continuity with respect to the Lebesgue measure and only assume that the compensator is absolutely continuous with respect to a general σ-finite measure. This allows for example to incorporate the Merton-model in the generalized intensity-based framework. We propose a class of generalized Merton models and study absence of arbitrage by a suitable modification of the forward rate approach of Heath-Jarrow-Morton (1992). Finally, we study affine term structure models which fit in this class. They exhibit stochastic discontinuities in contrast to the affine models previously studied in the literature.

No abstract

No abstract