  The recent banking crisis has again emphasized the importance of
understanding and mitigating systemic risk in financial networks. In this
paper, we study a market-driven approach to rescue a bank in distress based on
the idea of claims trading, a notion defined in Chapter 11 of the U.S.
Bankruptcy Code. We formalize the idea in the context of financial networks by
Eisenberg and Noe. For two given banks v and w, we consider the operation that
w takes over some claims of v and in return gives liquidity to v to ultimately
rescue v. We study the structural properties and computational complexity of
decision and optimization problems for several variants of claims trading.
  When trading incoming edges of v, we show that there is no trade in which
both banks v and w strictly improve their assets. We therefore consider
creditor-positive trades, in which v profits strictly and w remains
indifferent. For a given set C of incoming edges of v, we provide an efficient
algorithm to compute payments by w that result in maximal assets of v. When the
set C must also be chosen, the problem becomes weakly NP-hard. Our main result
here is a bicriteria FPTAS to compute an approximate trade. The approximate
trade results in nearly the optimal amount of assets of v in any exact trade.
Our results extend to the case in which banks use general monotone payment
functions and the emerging clearing state can be computed efficiently.
  In contrast, for trading outgoing edges of v, the goal is to maximize the
increase in assets for the creditors of v. Notably, for these results the
characteristics of the payment functions of the banks are essential. For
payments ranking creditors one by one, we show NP-hardness of approximation
within a factor polynomial in the network size, when the set of claims C is
part of the input or not. Instead, for proportional payments, our results
indicate more favorable conditions.
