Improved Hardness Results for the Clearing Problem in Financial Networks with Credit Default Swaps

TL;DR

Two natural decision problems in financial networks of banks connected by debt contracts and credit default swaps are shown to be $\exists\mathbb{R}$-complete, complementing previous $\mathrm{NP}$-hardness results for the approximate setting.

Abstract

We study computational problems in financial networks of banks connected by debt contracts and credit default swaps (CDSs). A main problem is to determine \emph{clearing} payments, for instance right after some banks have been exposed to a financial shock. Previous works have shown the $\varepsilon$-approximate version of the problem to be $\mathrm{PPAD}$-complete and the exact problem $\mathrm{FIXP}$-complete. We show that $\mathrm{PPAD}$-hardness hold when $\varepsilon \approx 0.101$, improving the previously best bound significantly. Due to the fact that the clearing problem typically does not have a unique solution, or that it may not have a solution at all in the presence of default costs, several natural decision problems are also of great interest. We show two such problems to be $\exists\mathbb{R}$-complete, complementing previous $\mathrm{NP}$-hardness results for the approximate setting.

Figures (15)

• Figure 1: Example diagram of financial network.
• Figure 2: Financial network gadget with input bank $u$ and output bank $v$.
• Figure 3: $\mathsf{NAND}$ gadget
• Figure 4: $\mathsf{PURIFY}$ gadget
• Figure 5: Input gadget.

Theorems & Definitions (32)

• Proposition 1
• Corollary 1
• proof
• Definition 1: Clearing recovery rate vector
• Definition 2: Non-degeneracy
• Definition 3: $\varepsilon$-approximate clearing recovery rate vector
• Theorem 1
• Definition 4
• Definition 5
• Theorem 2: Deligkas et al. DeligkasFHM22-Pure-Circuit