A mathematical treatment of bank monitoring incentives

TL;DR

This paper develops a mathematical framework for analyzing optimal contracts in a principal/agent model with moral hazard, bank monitoring, and contagion, using stochastic control and Hamilton-Jacobi-Bellman equations.

Contribution

It provides a comprehensive mathematical formulation and explicit solutions for optimal contracts in a complex bank monitoring model, avoiding more complicated stochastic differential equations.

Findings

Explicit value function and optimal contract derived

Recursive HJB equations simplified the analysis

Limit case of non-impatient bank studied

Abstract

In this paper, we take up the analysis of a principal/agent model with moral hazard introduced in [17], with optimal contracting between competitive investors and an impatient bank monitoring a pool of long-term loans subject to Markovian contagion. We provide here a comprehensive mathematical formulation of the model and show using martingale arguments in the spirit of Sannikov [18] how the maximization problem with implicit constraints faced by investors can be reduced to a classical stochastic control problem. The approach has the advantage of avoiding the more general techniques based on forward-backward stochastic differential equations described in [6] and leads to a simple recursive system of Hamilton-Jacobi-Bellman equations. We provide a solution to our problem by a verification argument and give an explicit description of both the value function and the optimal contract. Finally, we study the limit case where the bank is no longer impatient.