Vulnerability Analysis of the Financial Network

Since the financial crisis in 2007-2008, the vulnerability of a financial system has become a major concern in financial engineering. In this paper, we analyze the vulnerability of a financial network based on the linear optimization model introduced by Eisenberg and Noe (2001), where the right hand side of the constraints is subject to market shock and only limited information regarding the liability matrix is exposed. We develop a new extended sensitivity analysis to characterize the conditions under which a bank is solvent, default or bankrupted, and estimate the probability of insolvency and the probability of bankruptcy under mild conditions on the market shock and the network structure. Particularly, we show that while an increment in the total asset may not able to improve the stability of the financial system, a larger asset inequality in the system will reduce its stability. Moreover, under certain assumption on the market shock and the network structure, we show that the least stable network can be attained at some network with a monopoly node, which also has the highest probability of insolvency. The probability of bankruptcy in the network when all the nodes receive shocks is estimated. We also estimate the impact of bankruptcy of the monopoly node in a well-balanced network and explore the domino effect of bankruptcy when the network has a tridiagonal structure. Numerical experiments are presented to verify the theoretical conclusions