We consider the problem of clearing a system of interconnected banks . Prior work has shown that when banks can only enter into simple debt contracts with each other, then a clearing vector of payments can be computed in polynomial time. In this work, we show that the computational complexity of the clearing problem drastically increases when banks can also enter into credit default swaps (CDSs), i.e., financial derivative contracts that depend on the default of another bank . We first show that many important decision problems are NP-hard once CDSs are allowed. This includes deciding if a specific bank is at risk of default and deciding if a clearing vector exists in the first place. Second, we show thatcomputing an approximate solution to the clearing problem with sufficiently small constant error is PPAD-complete  . To prove our results, we demonstrate how financial networks with debt and CDSscan encode Boolean and arithmetic operations. Our results have practical importance for network stress tests and they reveal computational complexity as a new concern regarding the stability of the financial system .
The 2008 financial crisis has been attributed to "excessive complexity" of the financial system due to financial innovation. We employ computational complexity theory to make this notion precise. Specifically, we consider the problem of clearing a financial network after a shock . Prior work has shown that when banks can only enter into simple debt contracts with each other, then this problem can be solved in polynomial time. In contrast, if they can also enter into credit default swaps (CDSs), i.e., financial derivative contracts that depend on the default of another bank , a solution may not even exist. In this work, we show that deciding if a solution exists is NP-complete if CDSs are allowed. This remains true if we relax the problem to \varepsilon-approximate solutions, for a constant \varepsilon. We further show that, under sufficient conditions where a solution is guaranteed to exist, the approximate search problem is PPAD-complete for constant \varepsilon. We then try to isolate the "origin" of the complexity. It turns out that already determining which banks default is hard. Further, we show that the complexity is not driven by the dependence of counterparties on each other, but rather hinges on the presence of so-called naked CDSs . If naked CDSs are not present, we receive a simple polynomial-time algorithm. Our results are of practical importance for regulators' stress tests and regulatory policy .
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We
The 2008 financial crisis has been attributed to "excessive complexity" of the financial system due to financial innovation. We employ computational complexity theory to make this notion precise. Specifically, we
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system of interconnected banks
financial network after a shock
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a clearing vector of payments can be computed
this problem can be solved
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this work, we show that the computational complexity of the clearing problem drastically increases when banks
contrast, if they
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. We first show that many important decision problems are NP-hard once
, a solution may not even exist. In this work, we show that deciding if a solution exists is NP-complete if
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includes deciding if a specific bank is at risk of default and deciding if a clearing vector exists in the first place. Second, we show thatcomputing an approximate solution to the clearing problem with sufficiently small constant error
remains true if we relax the problem to \varepsilon-approximate solutions, for a constant \varepsilon. We further show that, under sufficient conditions where a solution is guaranteed to exist, the approximate search problem
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for constant \varepsilon. We then try to isolate the "origin" of the complexity. It turns out that already determining which banks default is hard. Further, we show that the complexity is not driven by the dependence of counterparties on each other, but rather hinges on the presence of so-called naked CDSs
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To prove our results, we demonstrate how financial networks with debt and CDSscan encode Boolean and arithmetic operations. Our results have
If naked CDSs are not present, we receive a simple polynomial-time algorithm. Our results are of
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network
regulators'
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they reveal computational complexity as a new concern regarding the stability of the financial system
regulatory policy