Modeling Systemic Default Contagion with a Nonlinear SPDE

Summary

This work develops a dynamic mean-field model for systemic risk in a large financial system. Each institution is represented by a diffusion for its distance to default, with zero acting as an absorbing boundary. The setup includes common noise to represent shared exposures or herd effects, mean reversion in the drift, and endogenous contagion: one institution's default can reduce the distances to default of others.

The limiting behavior is characterized by a nonlinear stochastic partial differential equation on the positive half-line with a boundary condition at zero. Its density describes the conditional law of the associated diffusion, and the authors provide a heat-kernel-style upper bound used in their analysis. The model can produce rapid increases in boundary mass loss under some common-noise realizations and mean-reversion rates, corresponding to clusters of defaults. The excerpt describes a mathematical model and theoretical results; it does not report empirical calibration, validation against crisis data, or trading applications.

Key ideas

• Institutional distance to default is modeled as a diffusion absorbed at zero.
• Common noise and mean reversion represent shared risk and drift dynamics.
• Endogenous contagion lets one default lower other institutions' distances to default.
• A nonlinear SPDE characterizes the mean-field limit and its conditional density.
• Some model conditions produce accelerated default clustering through rapid loss of probability mass at the boundary.

Tags

# An SPDE Model for Systemic Risk with Endogenous Contagion


# An SPDE Model for Systemic Risk with Endogenous Contagion









We propose a dynamic mean field model for `systemic risk' in large financial systems, which we derive 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 can cause 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 characterized 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 that is essential to the proofs. Depending on the realizations 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.

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