Dynamic Defaultable Term Structure Modelling beyond the Intensity Paradigm

TL;DR

This paper unifies structural and reduced-form credit risk models by extending the forward-rate approach to include default at predictable times, providing a more flexible, arbitrage-free framework for dynamic term structure modeling.

Contribution

It introduces a generalized reduced-form framework allowing default at predictable times and develops affine models without stochastic continuity assumptions.

Findings

Models can incorporate default at predictable times without arbitrage.

A new class of affine models is proposed that relaxes stochastic continuity.

Necessary and sufficient no-arbitrage conditions are established.

Abstract

The two main approaches in credit risk are the structural approach pioneered in Merton (1974) and the reduced-form framework proposed in Jarrow & Turnbull (1995) and in Artzner & Delbaen (1995). The goal of this article 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 with positive probability, such as coupon payment dates. In this generalized framework we study dynamic term structures prone to default risk following the forward-rate approach proposed in Heath-Jarrow-Morton (1992). It turns out, that previously considered models lead to arbitrage possibilities when default may happen at a predictable time with positive probability. A suitable generalization of the forward-rate approach contains an additional stochastic integral with atoms at predictable times and necessary and sufficient conditions for a suitable no-arbitrage condition (NAFL) are given. In the view of efficient implementations we develop a new class of affine models which do not satisfy the standard assumption of stochastic continuity. The chosen approach is intimately related to the theory of enlargement of filtrations, to which we provide a small example by means of filtering theory where the Azema supermartingale contains upward and downward jumps, both at predictable and totally inaccessible stopping times.