Introduction

What constitutes an obligor (corporate or sovereign entity) default? How do we model and quantify it? These are two simple questions. The first question can be answered (at least from a quantitative analyst's perspective) by stipulating what constitutes a default from a legal perspective (such as failure to meet a debt obligation, legal bankruptcy or restructuring of a company) and enshrining this in legally enforceable terms and conditions of a contract (specifically the types of events which will trigger contingent cashflows). The second question, which is the subject of this chapter, is more difficult to answer.

In this chapter we introduce the standard market models for characterising obligor default. The fundamental modelling assumption is that default events happen randomly and at unknown times which cannot be predicted deterministically. Default events must therefore be modelled using the powerful machinery of probability theory. For modelling the default of single obligors, the relevant quantity is the default time. The statistical properties of this random variable are postulated to be governed by a Poisson process. Poisson processes occur throughout the mathematical and physical sciences and are used to model probabilistically events which are rare and discrete. Modelling default times makes sense since credit derivatives have payoffs that are a function of the timing of defaults of the underlying reference obligors. Unfortunately, introducing discontinuous jumps in variables (such as credit spreads) renders a lot of the machinery of traditional no-arbitrage financial mathematics ineffective.