Large Banks and Systemic Risk: Insights from a Mean-Field Game Model

TL;DR

The paper addresses how a single large bank influences stability and contagion in interbank markets by formulating a dynamic mean-field game with one major agent and many minor agents. It derives explicit best-response trading strategies via convex analysis, establishing an $\bepsilon$-Nash equilibrium for the finite population and a fixed-point mean-field solution, then evaluates default and systemic-risk metrics through Monte Carlo simulations. Results indicate that a moderately large major bank can enhance system stability by improving coordination among small banks, but its default can induce substantial tail-risk amplification, with effects magnified by the major bank’s size and trading frequency. These findings highlight a fundamental trade-off between stabilization through market coordination and heightened systemic risk from potential large-bank failures, informing policy on size, capital requirements, and trading-speed regulation in interbank markets.

Abstract

This paper presents a dynamic game framework to analyze the role of large banks in interbank markets. By extending existing models, we incorporate a large bank as a dynamic decision-maker interacting with multiple small banks. Using the mean-field game methodology and convex analysis, best-response trading strategies are derived, leading to an approximate equilibrium for the interbank market. We investigate the influence of the large bank on the market stability by examining individual default probabilities and systemic risk, through the use of Monte Carlo simulations. Our findings reveal that, when the size of the major bank is not excessively large, it can positively contribute to market stability. However, there is also the potential for negative spillover effects in the event of default, leading to an increase in systemic risk. The magnitude of this impact is further influenced by the size and trading rate of the major bank. Overall, this study provides valuable insights into the management of systemic risk in interbank markets.

Figures (7)

• Figure 1: Loss distribution ($a=5$): (a) total loss distribution for minor banks, (b) loss distribution for minor banks conditional on the major bank not default, and (c) loss distribution for minor banks conditional on the major bank default.
• Figure 2: Convergence of the average state and the market state in the finite population to corresponding quantities in the infinite population.
• Figure 3: Simulated trajectories for 10 small banks, the major bank and the market state in the finite-population case with ($a=5$): (a) $G=0.1$ and a non-defaulting major bank, (b) $G=0.9$ and a non-defaulting major bank, (c) $G=0.1$ and a defaulting major bank, and (d) $G=0.9$ and a defaulting major bank. In all cases.
• Figure 4: Simulated Trajectories for 10 small banks, the major bank and the market state in the finite-population case with ($G=0.5, F=0.5$): (a) $a=1$ and a non-defaulting major bank, (b) $a=10$ and a non-defaulting major bank, (c) $a=1$ and a defaulting major bank, and (d) $a=10$ and a defaulting major bank.
• Figure 5: Simulation Results (from left to right): (a) solution of the ODE system, and (b) the mean-reversion level of different banks after adding the optimal control ($a=5,a_0=2.5,F=0.5,G=0.5,q=q_0=1$).

Theorems & Definitions (4)

• Definition 2.1: Nash Equilibrium
• Definition 2.2: $\epsilon$-Nash property
• Theorem 4.1: Best-Response Transactions
• Theorem 4.2: Interbank Equilibria