We study financial networks with debt contracts and credit default swaps between specific pairs of banks. Given such a financial system, we want to decide which of the banks are in default, and how much of their liabilities can these defaulting banks pay. There can easily be multiple different solutions to this problem, leading to a situation of default ambiguity, and a range of possible solutions to implement for a financial authority. In this paper, we study the properties of the solution space of such financial systems, and analyze a wide range of reasonable objective functions for selecting from the set of solutions. Examples of such objective functions include minimizing the number of defaulting banks, minimizing the amount of unpaid debt, maximizing the number of satisfied banks, and many others. We show that for all of these objectives, it is NP-hard to approximate the optimal solution to an $n^{1-\epsilon}$ factor for any $\epsilon>0$, with $n$ denoting the number of banks. Furthermore, we show that this situation is rather difficult to avoid from a financial regulator's perspective: the same hardness results also hold if we apply strong restrictions on the weights of the debts, the structure of the network, or the amount of funds that banks must possess. However, if we restrict both the network structure and the amount of funds simultaneously, then the solution becomes unique, and it can be found efficiently.
翻译:我们研究的是债务合同和金融网络以及特定银行之间的信用违约互换。 有了这样的金融体系,我们想决定哪些银行违约,以及这些违约银行能够支付多少债务。 这个问题容易有多种不同的解决方案,导致违约模棱两可的局面,以及一系列可能的解决办法来落实金融当局。 在本文件中,我们研究了这些金融体系解决方案空间的特性,分析了从一套解决方案中选择的一系列合理客观功能。 此类客观功能的例子包括:尽量减少违约银行的数量,尽量减少未偿债务的数额,最大限度地增加满意银行的数量,以及许多其他功能。 我们表明,对于所有这些目标,很难找到最佳解决方案的方方面面,即以1-\\\\ epsilon+$为单位,以美元计为一美元,用美元记分辨银行数目。 此外,我们从金融监管者的角度表明,这种情况相当难以避免:如果我们对债务的重量实施严格的限制,那么,那么,如果我们既能同时拥有独特的债务网络,又能同时拥有独特的资金数额,那么,我们很难同时拥有这个网络的金额。