We propose a dynamic model for systemic risk in a large financial system , which we formulate as a system of interacting diffusions on the positive half-line with an absorbing boundary at zero . These diffusions represent the distances-to-default of the financial institutions . 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 order to have a general model for systemic (or macroscopic) events, we show that the system converges to a unique mean field limit characterized by a nonlinear SPDE on the half-line ( with a Dirichlet boundary condition ) which governs the conditional law of a 'conditionalMcKean-Vlasov' type diffusion . Depending on the realizations of the common noise and the strength of the mean reversion, the SPDE can exhibit rapid accelerations in the loss of mass at the boundary. In other words, sparked by a devaluation of the common exposures, there are events of small probability that, through amplification by herd behaviour, can give rise to systemic default cascades .
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 .
R
model for systemic risk in a large financial system
mean field model for `systemic risk' in large financial systems
R
formulate as
derive from
R
zero
the origin
R
the financial institutions
financial institutions and absorption at zero corresponds to default
R
order to have a general model for systemic (or macroscopic) events, we show that the system converges to a unique
this way, we aim to capture key `system-wide' effects on risk. The resulting
R
characterized
is characterized uniquely
D
(
R
) which governs
. The density of this SPDE gives
R
'conditionalMcKean-Vlasov' type diffusion
non-standard `conditional' McKean--Vlasov diffusion, for which we provide a novel upper Dirichlet heat kernel type estimate that is essential to the proofs
R
strength of the
rate of
R
sparked by a devaluation of the common exposures, there are events of small probability that, through amplification by herd behaviour,
the contagion mechanism
R
systemic default cascades
periods of significant systemic default clustering