Dynamic Defaultable Term Structure Modeling Beyond the Intensity Paradigm

Abstract

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.

From the decompositions of a stopping time to risk premium decompositions

“…The compensator proposed is a special form of our expression (4.7), where the economic shocks (T i ), i ≥ 1 are predictable times and the idiosyncratic time, which is totally inaccessible stopping time T 0 , admits an intensity. As a consequence, the default adjusted probability measure that we define in Definition 5.5 coincides when applied to the model in [19] with the risk neutral measure.…”

“…In [19], the authors introduce a framework for pricing zero recovery defaultable claims. They model directly the compensator of the default time, without relying on the Azéma supermartingale.…”

A Generalized Intensity-Based Framework for Single-Name Credit Risk

“…For the convenience of the reader we state the following proposition giving sufficient conditions for absence of arbitrage in an affine generalized intensity-based setting. It extends [12] where only finitely many risky times were treated. …”

“…In this section we closely follow [12] and shortly state the appropriate affine models which fit the generalized intensity framework. For proofs, we refer the reader to this paper.…”

“…We propose a suitable extension with an additional term allowing for discontinuities in the term structure at certain random times and derive precise drift conditions for an appropriate no-arbitrage condition. The related article [12] only allows for the special case of finitely many risky times, an assumption which is dropped in this article.…”