Endogenous distress contagion in a dynamic interbank model: how possible future losses may spell doom today

TL;DR

The paper addresses systemic risk in interbank networks by introducing a fully dynamic, stochastic model where distress contagion is endogenous and driven by expectations of future defaults. It employs mark-to-market valuation of interbank claims, a forward–backward equilibrium, and a multinomial-tree implementation to derive clearing solutions, including both single- and multi-maturity settings. Key contributions include the endogenous stochastic volatility mechanism, a dynamic programming principle with a lattice of clearing solutions (often non-unique), and three case studies showing inverted interbank term structures and contagion-driven volatility. The framework provides a principled stress-testing tool for regulatory perspectives, capturing how concerns about future solvency can shape current systemic risk and liquidity dynamics, with practical avenues for estimating, calibrating, and using these insights in supervision and crisis anticipation.

Abstract

We introduce a dynamic and stochastic interbank model with an endogenous notion of distress contagion, arising from rational worries about future defaults and ensuing losses. This entails a mark-to-market valuation adjustment for interbank claims, leading to a forward-backward approach to the equilibrium dynamics whereby future default probabilities are needed to determine today's balance sheets. Distinct from earlier models, the resulting distress contagion acts, endogenously, as a stochastic volatility term that exhibits clustering and down-market spikes. Furthermore, by incorporating multiple maturities, we provide a novel framework for constructing systemic interbank term structures, reflecting the intertemporal risk of contagion. We present the analysis in two parts: first, the simpler single maturity setting that extends the classical interbank network literature and, then, the multiple maturity setting for which we can examine how systemic risk materialises in the shape of the resulting term structures.

Figures (9)

• Figure 1: Stylized book and balance sheet for a firm at time $t$ before maturity of interbank claims.
• Figure 2: Visualization of the tree for $\mathbf{x}$ in Example \ref{['ex:running-tree']} constructed in the manner of \ref{['eq:gbm']}.
• Figure 3: Visualization of the maximal clearing solution in Example \ref{['ex:running-nonunique']}
• Figure 4: A sample path of the external assets $\mathbf{x}$, capital $\mathbf{K}$, and probabilities of solvency $\mathbf{P}$ for two banks. Bank 1 is displayed in blue and bank 2 in red. Our model yields the solid lines. Dashed lines indicate historical price accounting (given by the clearing problem in Appendix \ref{['sec:hpa-1']}).
• Figure 5: Section \ref{['sec:1maturity-cs-corr']}: The impact of the correlation $\rho \in (-1,1)$ between the external asset values on health of the financial system as measured by probabilities of solvency/default.

Theorems & Definitions (44)

• Remark 2.1
• Proposition 2.2
• Example 2.3
• Remark 2.4
• Remark 2.5
• Remark 2.6
• Remark 2.7
• Remark 2.8
• Example 2.9
• Theorem 2.10