An SPDE model for systemic risk with endogenous contagion

We propose a dynamic mean-field model for ‘systemic risk’ in large financial systems, derived from a system of interacting diffusions on the positive half-line with an absorbing boundary at the origin. These diffusions represent the distances-to-default of financial institutions, and absorption at zero corresponds to default. As a way of modelling correlated exposures and herd behaviour, we consider a common source of noise and a form of mean-reversion in the drift. Moreover, we introduce an endogenous contagion mechanism whereby the default of one institution causes a drop in the distances-to-default of the other institutions. In this way, we aim to capture key ‘system-wide’ effects on risk. The resulting mean-field limit is characterised uniquely by a nonlinear SPDE on the half-line with a Dirichlet boundary condition. The density of this SPDE gives the conditional law of a non-standard ‘conditional’ McKean–Vlasov diffusion, for which we provide a novel upper Dirichlet heat kernel type estimate. Depending on the realisations of the common noise and the rate of mean-reversion, the SPDE can exhibit rapid accelerations in the loss of mass at the boundary. In other words, the contagion mechanism can give rise to periods of significant systemic default clustering.