Non-existence of Markovian time dynamics for graphical models of correlated default

TL;DR

This paper proves that for certain correlated default models based on graphical structures, there is no continuous-time Markov process that can replicate the default distribution dynamics unless the defaults are independent.

Contribution

The paper demonstrates the non-existence of Markovian time dynamics for correlated default models represented by graphical Ising models, except in the trivial independent case.

Findings

No Markov process exists for correlated defaults in the studied cases.

Markovian dynamics only possible when defaults are independent.

Results apply to three natural special cases of the model.

Abstract

Filiz et al. (2008) proposed a model for the pattern of defaults seen among a group of firms at the end of a given time period. The ingredients in the model are a graph, where the vertices correspond to the firms and the edges describe the network of interdependencies between the firms, a parameter for each vertex that captures the individual propensity of that firm to default, and a parameter for each edge that captures the joint propensity of the two connected firms to default. The correlated default model can be re-rewritten as a standard Ising model on the graph by identifying the set of defaulting firms in the default model with the set of sites in the Ising model for which the spin is +1. We ask whether there is a suitable continuous time Markov chain taking values in the subsets of the vertex set such that the initial state of the chain is the empty set, each jump of the chain involves the inclusion of a single extra vertex, the distribution of the chain at some fixed time horizon time is the one given by the default model, and the distribution of the chain for other times is described by a probability distribution in the same family as the default model. We show for three simple but financially natural special cases that this is not possible outside of the trivial case where there is complete independence between the firms.