Self-similar additive processes in default modelling

Description

Classically, in reduced form default models the instantaneous default intensity is the modeling object and
survival probabilities are given by the Laplace transform of cumulative hazard process, its integral. Instead,
recent literature has shown a tendency towards specifying the cumulative hazard process directly.

Within this framework we present a new model class where the cumulative hazard is described by a
time-inhomogeneous L\'{e}vy process.

In particular we analyze cumulative hazards given by self-similar additive processes, also known as Sato
processes, as well as specifications obtained via a simple deterministic time-change of a homogeneous L\'{e}vy
process. While the cumulative hazard processes in these two subclasses share the same average behavior over
time, the associated intensities exhibit very different properties.

Concrete models are calibrated to data on the single names included in the iTraxx Europe index and compared with
two Ornstein-Uhlenbeck type intensity models. It is shown how the time-inhomogeneous L\'{e}vy models achieve
similar calibration errors with fever parameters, and with more stable parameter estimates in time. However the
calibration performances of the Sato processes and the time-changes specifications are practically
indistinguishable.