Bank monitoring incentives under moral hazard and adverse selection

In this paper, we extend the optimal securitisation model of Pag\`es [50] and Possama\"i and Pag\`es [51] between an investor and a bank to a setting allowing both moral hazard and adverse selection. Following the recent approach to these problems of Cvitani\'c, Wan and Yang [14], we characterise explicitly and rigorously the so-called credible set of the continuation and temptation values of the bank, and obtain the value function of the investor as well as the optimal contracts through a recursive system of first-order variational inequalities with gradient constraints. We provide a detailed discussion of the properties of the optimal menu of contracts.