Systemic Risk and Interbank Lending

We propose a simple model of the banking system incorporating a game feature, where the evolution of monetary reserve is modeled as a system of coupled Feller diffusions. The optimization reflects the desire of each bank to borrow from or lend to a central bank through manipulating its lending preference and the intention of each bank to deposit in the central bank in order to control the reserve and the corresponding volatility for cost minimization. The Markov Nash equilibrium for finite many players generated by minimizing the linear quadratic cost subject to Cox-Ingersoll-Ross type processes creates liquidity and deposit rate. The adding liquidity leads to a flocking effect implying stability or systemic risk depending on the level of the growth rate, but the deposit rate diminishes the growth of the total monetary reserve causing a large number of bank defaults. The central bank acts as a central deposit corporation. In addition, the corresponding mean field game in the case of the number of banks N large and the infinite time horizon stochastic game with the discount factor are also discussed.