[Paper Review] Dynamic Default Contagion in Heterogeneous Interbank Systems

Zachary Feinstein, Andreas Søjmark|arXiv (Cornell University)|Oct 28, 2020

Banking stability, regulation, efficiency28 references4 citations

TL;DR

This paper introduces a dynamic default contagion model with endogenous early defaults in heterogeneous interbank systems, generalizing the Gai–Kapadia framework by incorporating insolvency-driven defaults and recovery of face value. It formulates the system as a stochastic particle system, derives a limiting mean-field problem, and establishes conditions under which the system evolves continuously, with a novel mean-field cascade condition characterizing potential jumps.

ABSTRACT

In this work we provide a simple setting that connects the structural modelling approach of Gai-Kapadia interbank networks with the mean-field approach to default contagion. To accomplish this we make two key contributions. First, we propose a dynamic default contagion model with endogenous early defaults for a finite set of banks, generalising the Gai-Kapadia framework. Second, we reformulate this system as a stochastic particle system leading to a limiting mean-field problem. We study the existence of these clearing systems and, for the mean-field problem, the continuity of the system response.

Motivation & Objective

• To develop a dynamic, finite-horizon model of default contagion in interbank networks that allows for early defaults due to insolvency, extending the static Gai–Kapadia framework.
• To connect structural, balance-sheet-based models of systemic risk with continuous-time mean-field approaches, bridging a key gap in the literature.
• To establish the existence of a greatest clearing solution in the finite bank model and derive conditions for continuous evolution in the mean-field limit.
• To introduce and characterize a mean-field cascade condition that identifies jump discontinuities in the limiting system, particularly when interaction constraints are violated.

Proposed method

• Proposes a dynamic balance sheet model with recovery of face value (RFV), where banks face insolvency-driven early defaults based on capital shortfalls.
• Introduces a stochastic particle system representation of the finite interbank network, modeling default timing and capital dynamics via jump processes.
• Derives a limiting mean-field SDE system by taking the number of banks to infinity, ensuring continuous evolution under a bounded interaction constraint.
• Establishes a mean-field cascade condition (3.15) as a recursive, ε-perturbed fixed-point equation to characterize jump sizes when continuity fails.
• Uses the McKean–Vlasov framework as a foundation, but grounds it in financial balance sheet mechanics rather than abstract stochastic processes.
• Applies dominated convergence and iterative approximation to prove well-posedness and existence of solutions in the mean-field limit.

Experimental results

Research questions

• RQ1 How can a dynamic, insolvency-driven default contagion model be constructed in a finite interbank system with endogenous early defaults, generalizing the static Gai–Kapadia model?
• RQ2 What conditions ensure the continuous evolution of the system in the mean-field limit, and when do jumps emerge?
• RQ3 How can a mean-field cascade condition be derived to characterize jump discontinuities in the limiting system when interaction constraints are violated?
• RQ4 What is the connection between the finite particle system and the resulting mean-field SDE, and how is this link formally established?
• RQ5 How does the inclusion of recovery of face value (RFV) affect the default propagation mechanism compared to recovery of market value (RMV)?

Key findings

• The finite interbank system admits a greatest clearing solution under the proposed dynamic default contagion model, ensuring existence and uniqueness of the clearing outcome.
• The mean-field limit of the system evolves continuously in time if the interaction strength remains bounded, as formalized in Theorem 3.4.
• When the interaction constraint is violated, jumps may persist in the limit, and the mean-field cascade condition (3.15) provides a recursive characterization of such jump sizes.
• The mean-field cascade condition is well-defined and convergent due to monotonicity and dominated convergence, ensuring a unique solution for jump amplitudes.
• The model establishes a rigorous link between finite, balance-sheet-based network models and continuous-time mean-field SDEs, enriching the theoretical foundation of systemic risk modeling.
• Numerical implementation of the cascade condition is feasible and stable, as demonstrated in Supplemental B and Figure 5.