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Financial networks raise a significant computational challenge in identifyinginsolvent firms and evaluating their exposure to systemic risk. This task,known as the clearing problem, is computationally tractable when dealing withsimple debt contracts. However under the presence of certain derivatives calledcredit default swaps (CDSes) the clearing problem is $\textsf{FIXP}$-complete.Existing techniques only show $\textsf{PPAD}$-hardness for finding an$\epsilon$-solution for the clearing problem with CDSes within an unspecifiedsmall range for $\epsilon$.  We present significant progress in both facets of the clearing problem: (i)intractability of approximate solutions; (ii) algorithms and heuristics forcomputable solutions. Leveraging $\textsf{Pure-Circuit}$ (FOCS'22), we providethe first explicit inapproximability bound for the clearing problem involvingCDSes. Our primal contribution is a reduction from $\textsf{Pure-Circuit}$which establishes that finding approximate solutions is $\textsf{PPAD}$-hardwithin a range of roughly 5%.  To alleviate the complexity of the clearing problem, we identify twomeaningful restrictions of the class of financial networks motivated byregulations: (i) the presence of a central clearing authority; and (ii) therestriction to covered CDSes. We provide the following results: (i.) The$\textsf{PPAD}$-hardness of approximation persists when central clearingauthorities are introduced; (ii.) An optimisation-based method for solving theclearing problem with central clearing authorities; (iii.) A polynomial-timealgorithm when the two restrictions hold simultaneously.

Clearing Financial Networks with Derivatives: From Intractability to Algorithms