Sato Processes in Default Modelling

Abstract

In reduced form default models, the instantaneous default intensity is the classical modelling object. Survival probabilities are then given by the Laplace transform of the cumulative hazard defined as the integrated intensity process. Instead, recent literature tends to specify the cumulative hazard process directly. Within this framework we present a new model class where cumulative hazards are described by self-similar additive processes, also known as Sato processes. Furthermore, we analyse specifications obtained via a simple deterministic time change of a homogeneous Levy process. While the processes in these two classes share the same average behaviour over time, the associated intensities exhibit very different properties. Concrete specifications are calibrated to data on all the single names included in the iTraxx Europe index. The performances are compared with those of the classical Cox-Ingersoll-Ross intensity and a recently proposed class of intensity models based on Ornstein-Uhlenbeck-type processes. It is shown that the time-inhomogeneous Levy models achieve comparable calibration errors with fewer parameters and with more stable parameter estimates over time. However, the calibration performance of the Sato processes and the time-change specifications are practically indistinguishable.Credit default swap, reduced form model, Sato process, time-changed Levy process, cumulative hazard,

Investigating the Performance of Non-Gaussian Stochastic Intensity Models in the Calibration of Credit Default Swap Spreads

“…As these processes are similar, we consider only the Sato-Gamma assumption in the empirical analysis. As shown in Kokholm and Nicolato (2010), in the Sato-Gamma model it can be demonstrated that …”

“…Furthermore, as observed by Kokholm and Nicolato (2010), the proposed model has a deterministic intensity process given by the equality…”

“…Recently, Kokholm and Nicolato (2010) analyzed a reduced-form approach where the default time is defined by the direct modelling of the process A t . In their paper, the default is driven by an increasing and additive process, that is a process A t with independent but not necessarily stationary increments.…”

“…As noted in Sects. 2, Kokholm and Nicolato (2010) analyzed a reduced-form approach where the default time is defined by the direct modelling of the process A t . In Sect.…”

“…Finally, we analyze a different framework where the default probability is explained via a Sato process, as proposed by Kokholm and Nicolato (2010). We compare three different families of stochastic processes in our empirical study: (1) the Brownian motion driving the CIR process; (2) Ornstein-Uhlenbeck processes based on Lévy processes, that is, non-Gaussian processes with independent and homogeneous increments (gamma, inverse Gaussian and variance gamma); and (3) additive processes (Sato), that is processes with independent but non-necessary homogeneous increments (see Sato 1999).…”