Structural Credit Risk Models with Subordinated Processes

We discuss structural models based on Merton's framework. First, we observe that the classical assumptions of the Merton model are generally rejected. Secondly, we implement a structural credit risk model based on stable non-Gaussian processes as a representative of subordinated models in order to overcome some drawbacks of the Merton one. Finally, following the KMV-Merton estimation methodology, we propose an empirical comparison between the results obtained from the classical KMV-Merton model and the stable Paretian one. In particular, we suggest alternative parameter estimation for subordinated processes, and we optimize the performance for the stable Paretian model.

Multivariate subordination, self-decomposability and stability

Multivariate subordinators are multivariate Lévy processes that are increasing in each component. Various examples of multivariate subordinators, of interest for applications, are given. Subordination of Lévy processes with independent components by multivariate subordinators is defined. Multiparameter Lévy processes and their subordination are introduced so that the subordinated processes are multivariate Lévy processes. The relations between the characteristic triplets involved are established. It is shown that operator self-decomposability and the operator version of the class Lm property are inherited from the multivariate subordinator to the subordinated process under the condition of operator stability of the subordinand.

Mixing for stationary processes with finite-order multiple Wiener-Itô integral representation

abstractNecessary and sufficient analytical conditions are given for homogeneous multiple Wiener-Itô integral processes (MWIs) to be mixing, and sufficient conditions are given for mixing of general square-integrable Gaussian-subordinated processes. It is shown that every finite or infinite sum Y of MWIs (i.e. every real square-integrable stationary polynomial form in the variables of an underlying weakly mixing Gaussian process) is mixing if the process defined separately by each homogeneous-order term is mixing, and that this condition is necessary for a large class of Gaussian-subordinated processes. Moreover, for homogeneous MWIs Y1, for sums of MWIs of order ≤ 3, and for a large class of square-integrable infinite sums Y1, of MWIs, mixing holds if and only if Y2 has correlation-function decaying to zero for large lags. Several examples of the criteria for mixing are given, including a second-order homogeneous MWI, i.e. a degree two polynomial form, orthogonal to all linear forms, which has auto-correlations tending to zero for large lags but is not mixing.