Abstract
We study the optimal interventions of a regulator (a central bank or government) on the illiquidity default contagion process in a large, heterogeneous, unsecured interbank lending market. The regulator has only partial information on the interbank connections and aims to minimize the fraction of final defaults with minimal interventions. We derive the analytical results of the asymptotic optimal intervention policy and the asymptotic magnitude of default contagion in terms of the network characteristics. We extend the results of Amini, Cont and Minca’s work to incorporate interventions and adopt the dynamics of Amini, Minca and Sulem’s model to build heterogeneous networks with degree sequences and initial equity levels drawn from arbitrary distributions. Our results generate insights that the optimal intervention policy is “monotonic” in terms of the intervention cost, the closeness to invulnerability and connectivity. The regulator should prioritize interventions on banks that are systematically important or close to invulnerability. Moreover, the regulator should keep intervening on a bank once having intervened on it. Our simulation results show a good agreement with the theoretical results.
Highlights
The systemic risk in a financial network has been drawing more and more interest of regulators and researchers, especially after the Asian financial crisis in the late 1990’s and the more recent economic recession during 2007-2009
We present in Appendix A: Proofs all the proofs and in Appendix B: Wormald’s theorem and Appendix C: Extended pontryagin maximum principle the two theorems used in the proofs as well as a list of notations in Appendix D: Preliminary list of notations
Based on the dynamics of the contagion process under interventions described in the previous section, we will show that the state variable Sk, the accumulative interventions Rk, the number of defaults Dk and the number of unrevealed out-links from the default set DÀk after being scaled by n all converge to a deterministic process which depends on the solution of the system of ordinary differential equations (ODEs) we will present
Summary
The systemic risk in a financial network has been drawing more and more interest of regulators and researchers, especially after the Asian financial crisis in the late 1990’s and the more recent economic recession during 2007-2009. Based on the dynamics of the contagion process under interventions described, we will show that the state variable Sk, the accumulative interventions Rk, the number of defaults Dk and the number of unrevealed out-links from the default set DÀk (defined later) after being scaled by n all converge to a deterministic process which depends on the solution of the system of ODEs we will present now. To summarize the results we have so far, we have shown in proposition 3 and proposition 4 that the state variable Sk, the accumulative interventions Rk, the number of defaults Dk and the number of unrevealed out-links from the default set DÀk after being scaled by n all converge to a deterministic process which depends on the solution of ODEs in definition 4. In theorem 3 the first case indicates that the network is highly vulnerable and interventions are costly, the regulator rather lets the whole network default without implementing any interventions, while in the second case interventions are less expensive or the contagion effect is not as high, it is better for the regulator to implement interventions to counteract the contagion process
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