Defaultable term structures driven by semimartingales

TL;DR

This paper extends the modeling of credit risky bond term structures by incorporating general semimartingales, jumps, and flexible recovery schemes, providing comprehensive conditions for arbitrage-free markets.

Contribution

It introduces a generalized Heath-Jarrow-Morton framework driven by semimartingales with jumps and broad recovery assumptions, expanding the theoretical foundation for credit risk modeling.

Findings

Derived generalized drift conditions for no-arbitrage

Extended HJM approach with jump components

Established criteria for local martingale measures

Abstract

We consider a market with a term structure of credit risky bonds in the single-name case. We aim at minimal assumptions extending existing results in this direction: first, the random field of forward rates is driven by a general semimartingale. Second, the Heath-Jarrow-Morton approach is extended with an additional component capturing those future jumps in the term structure which are visible from the current time. Third, the associated recovery scheme is as general as possible, it is only assumed to be non-increasing. In this general setting we derive generalized drift conditions which characterize when a given measure is a local martingale measure, thus yielding no asymptotic free lunch with vanishing risk (NAFLVR), the right notion for this large financial market to be free of arbitrage.