A Hierarchical Model of Tail Dependent Asset Returns for Assessing Portfolio Credit Risk

Abstract

This paper introduces a multivariate pure-jump Lévy process which allows for skewness and excess kurtosis of single asset returns and for asymptotic tail dependence in the multivariate setting. It is termed Variance Compound Gamma (VCG). The novelty of my approach is that, by applying a two-stage stochastic time change to Brownian motions, I derive a hierarchical structure with different properties of inter-and intra-sector dependence. I investigate the properties of the implied static copula families and come to the conclusion that they are ordered with respect to their parameters and that the lower-tail dependence of the intra-sector copula is increasing in the absolute values of skewness parameters. Furthermore, I show that the joint characteristic function of the VCG asset returns can be explicitly given as a nested Archimedean copula of their marginal characteristic functions. Applied to credit portfolio modelling, the framework introduced results in a more conservative tail risk assessment than a Gaussian framework with the same linear correlation structure, as I show in a simulation study. To foster the simulation efficiency, I provide an Importance Sampling algorithm for the VCG portfolio setting. Non-technical summaryIn this paper, I introduce a novel model of tail-dependent asset returns which can be used for the purposes of structural credit risk modelling. Similar to the copula approach proposed recently by Puzanova (2011), the Variance Compound Gamma (VCG) model presented here implies a hierarchical dependence structure with stronger dependence within pre-specified sectors than between them. The magnitude of sector-specific dependence parameters governing the tail dependence property can vary from one sector to another, allowing the model to cope with concentration risk. An advantage of the VCG framework over the aforementioned copula approach is its more general applicability, which is not limited to a static, one-period consideration of portfolio credit risk. In fact, the fundamental VCG model of asset returns can be utilised for financial modelling (pricing of financial derivatives etc.) whenever using multivariate jump-driven Lévy processes with a hierarchical dependence structure is deemed appropriate.Allowing jumps in the sample paths of the asset returns, the VCG model overcomes the shortcomings of a Gaussian/Brownian framework, such as anticipated default time, symmetric and mesokurtic probability distribution of the underlying and linear dependence structure. The jumps occur simultaneously for asset returns that are evaluated at a common business time. A business time common to all assets in an appropriately specified sector is a stochastic process which represents the irregular flow of information that is only relevant for that particular sector. The sector-specific business times themselves are evaluated at another common random time, which represents the flow of information that is relevant for the whole market, such as changes in the overall macroeconomic conditions. This two-s...