A Default System with Overspilling Contagion

Abstract

Abstract. In classical contagion models, default systems are Markovian conditionally on the observation of their stochastic environment, with interacting intensities. This necessitates that the environment evolves autonomously and is not influenced by the history of the default events. We extend the classical literature and allow a default system to have a contagious impact on its environment. In our framework, contagion can either be contained within the default system (i.e., direct contagion from a counterparty to another) or spill from the default system over its environment (indirect contagion). This type of model is of interest whenever one wants to capture within a model possible impacts of the defaults of a class of debtors on the more global economy and vice versa. Motivation and aimsDefault events tend to cluster in time, a phenomenon that arises from diverse causes. The literature on dynamic modelling of defaults proposed so far two major mechanisms that produce this effect. First, there is the so-called cyclical correlation, i.e., the dependence of the debtors' financial situation on some common factors. One can naturally think of some macroeconomic factors that impact the default probabilities of many debtors at a time, as the interest rates, the prices of some commodities or the business cycle; a purely statistical approach using abstract or unobserved factors is also possible. Secondly, there is the socalled counterparty risk or direct contagion, i.e., the default of one debtor represents itself a destabilising factor impacting the default rates of surviving debtors (the counterparties of a defaulted debtor). In order to have these two mechanisms of contagion operational simultaneously it is necessary to distinguish within the model the default system from its environment. The role of the random environment is to carry the cyclical dependence.The most classical approach so far was to consider that conditionally on a given realisation of the random environment, the vector of default indicator processes of the different debtors is a time inhomogeneous Markov chain.While the Markovian assumption is convenient, it necessitates that the environment evolves autonomously and is not influenced by the history of the default events. Our aim here is precisely to remove this assumption. This equates to introducing a new source of contagion, that we call overspilling (or indirect) contagion: the one that transmits from the default system to its environment, subsequently having a feedback effect on the system itself. The construction, by its nature is not Markovian, the default probabilities depend not only on the current state of the default system, but also on the circumstances of the occurrence of the past defaults, more precisely the knowledge of their impact on the environment. The theory of enlargements of filtrations reveals to be the right tool to deal with this non Markovian framework.