An SPDE model for systemic risk with endogenous contagion

By Ben M. Hambly, Andreas Søjmark. Finance and Stochastics, 2018

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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.

Subfields: Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/cea4666d6b7111ddde03f62b2159b59895c4a6d6