Financial institutions are interconnected by holding debt claims against each other. A default bank may cause its creditors to default, and the risk may be further propagated to up-stream institutes (risk contagion). Such interconnection is a key contributing factor to the past worldwide financial crisis. We show that a good mechanism of default liquidation may improve the total wealth of the financial system and therefore may curb the risk contagion. We formulate this problem as a nonlinear optimization problem with constraints and propose an optimal liquidation policy to minimize the system’s loss. We show that the problem resembles a Markov decision problem (MDP) and therefore we can apply the direct-comparison based optimization approach to solve this problem. Higher order directional derivatives and some optimality properties are obtained. Furthermore, we derive an iterative algorithm which combines both the policy iteration and the gradient based approach to find a local optimal policy, and under some conditions, a global optimal policy. Our work provides a new direction in curbing the risk contagion in financial networks; and it illustrates the advantages of the direct-comparison based approach, originated in the field of discrete event dynamic system, in nonlinear optimization problems.

Introduction

This paper is motivated by two recent developments in financial engineering and performance optimization. First, financial institutions are interconnected by borrowing-and-lending activities among themselves or holding marketable securities against each other (Chen, Liu, & Yao, 2014). Such interconnection is a critical influencing factor to the past worldwide financial crisis and the European sovereignty debt crisis, and could potentially threaten the stability of financial networks. For example, a default bank may cause its creditors to default, and the risk may be further propagated to up-stream institutes. During default liquidation, it is extremely significant to curb such risk contagion among financial networks. The goal of this paper is to propose a new liquidation scheme based on performance optimization and to provide computation algorithms that solve the problem. The problem formulated above has a large dimension and many highly nonlinear constraints, and we need to develop an efficient algorithm for an optimal solution. On this side, a direct-comparison based approach has been developed in the past years to the optimization of nonlinear problems and has been successfully applied to many problems, such as optimization of singular controlled diffusion processes (Ni & Fang, 2013), MDP with long-run average criterion (Cao, 2007, 2015), and variance criterion (Xia, 2016), and nonlinear performance with probability distortion (Cao & Wan, 2017). In this paper, we show that the special features of the financial risk contagion problem make it possible to be solved by the direct-comparison based approach, leading to some new insights to the problem. While network connections have a positive effect by diversifying risk (Haldane, 2009; Summer, 2013), it can have a negative effect by adding spreading channels for risks. When a global shortage for liquidity happens, the systemic risk will be transmitted through the risk-sharing mechanism (Allen & Gale, 2000; Leitner, 2005; Rochet & Tirole, 1996). Such risk contagions may result in consecutive consequences for the financial system, such as declines in asset prices, higher price volatility, more bank defaults, and market inefficiency (Allen & Gale, 2004; Brunnermeier & Pedersen, 2009; Holmstrom & Tirole, 2000).