Advances in Heavy Tailed Risk Modeling: A Handbook of Operational Risk

CHAPTERFIVE

Flexible Heavy-Tailed Severity Models: Tempered Stable and Quantile Transforms

This chapter develops further the flexible α-stable severity models discussed previously by extending them into the classes of tempered stable (TS) models and quantile transform models. In particular, we present advanced discussion for risk modellers relating to flexible distributions that complements the flexible severity distributions typically utilized in OpRisk, such as those discussed in Cruz et al. (2015, chapter 9). In particular, this chapter will provide details on how to model losses in OpRisk settings with heavy-tailed severity models such as the following families:

• the TS family with exponential tilting and its relationship to the generalized inverse Gaussian (GIG) and generalized hyperbolic families discussed in Cruz et al. (2015, chapter 9);
• the generalized tempered stable (GTS) family and the modified tempered stable (MTS) family;
• mixture variants that generalize those presented in the generalized hyperbolic family such as the normal-modified stable family;
• quantile function flexible severity models based on Tukey's transform class.

5.1 Tempered and Generalized Tempered Stable Severity Models

In the following sections, we discuss the consideration of modifications to the α-stable family to offer additional flexibility with regard to the modeling of skewness and kurtosis in a dataset. Primarily, the approaches considered will be based on taking the Lévy–Khintchine representation ...