Abstract
Filiz et al. (in arXiv:0809.1393 (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 G=(V,E), where the vertices V correspond to the firms and the edges E 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 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 {±1}-valued spin is +1. We ask whether there is a suitable continuous-time Markov chain (X t ) t≥0 taking values in the subsets of V such that X 0=∅, X r ⊆X s for r≤s (that is, once a firm defaults, it stays in default), the distribution of X T for some fixed time T is the one given by the default model, and the distribution of X t for other times t is described by a probability distribution in the same family as the default model. In terms of the equivalent Ising model, this corresponds to asking if it is possible to begin at time 0 with a configuration in which every spin is −1 and then flip spins one at a time from −1 to +1 according to Markovian dynamics so that the configuration of spins at each time is described by some Ising model and at time T the configuration is distributed according to the prescribed Ising 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.
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Evans, S.N., Hening, A. Nonexistence of Markovian time dynamics for graphical models of correlated default. Queueing Syst 69, 293–312 (2011). https://doi.org/10.1007/s11134-011-9261-y
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DOI: https://doi.org/10.1007/s11134-011-9261-y