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Clearing Payments in Dynamic Financial Networks

Giuseppe C. Calafiore, Giulia Fracastoro, Anton V. Proskurnikov

Abstract

This paper proposes a novel dynamical model for determining clearing payments in financial networks. We extend the classical Eisenberg-Noe model of financial contagion to multiple time periods, allowing financial operations to continue after possible initial pseudo defaults, thus permitting nodes to recover and eventually fulfil their liabilities. Optimal clearing payments in our model are computed by solving a suitable linear program, both in the full matrix payments case and in the pro-rata constrained case. We prove that the proposed model obeys the \emph{priority of debt claims} requirement, that is, each node at every step either pays its liabilities in full, or it pays out all its balance. In the pro-rata case, the optimal dynamic clearing payments are unique, and can be determined via a time-decoupled sequential optimization approach.

Clearing Payments in Dynamic Financial Networks

Abstract

This paper proposes a novel dynamical model for determining clearing payments in financial networks. We extend the classical Eisenberg-Noe model of financial contagion to multiple time periods, allowing financial operations to continue after possible initial pseudo defaults, thus permitting nodes to recover and eventually fulfil their liabilities. Optimal clearing payments in our model are computed by solving a suitable linear program, both in the full matrix payments case and in the pro-rata constrained case. We prove that the proposed model obeys the \emph{priority of debt claims} requirement, that is, each node at every step either pays its liabilities in full, or it pays out all its balance. In the pro-rata case, the optimal dynamic clearing payments are unique, and can be determined via a time-decoupled sequential optimization approach.
Paper Structure (15 sections, 8 theorems, 62 equations, 3 figures)

This paper contains 15 sections, 8 theorems, 62 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. The Eisenberg-Noe financial network model
  4. The pro-rata rule
  5. Dynamic financial networks
  6. The absolute priority rule
  7. A sub-optimal sequential approach
  8. Dynamic networks with pro-rated payments
  9. Numerical illustration
  10. Conclusions
  11. Appendix: proofs
  12. Technical preliminaries
  13. Proof of Theorem \ref{['thm.1']}
  14. Proof of Theorem \ref{['thm.prorata']}
  15. A remark on the structure of the cost function

Key Result

Lemma 1

Journal2021 The solution $p^*=p^*[A,c,\bar{p}]$ to eq:clearingopt exists, is unique and does not depend on the choice of $f$, provided that $f$ is decreasing. Additionally,

Figures (3)

  • Figure 1: Strong components of a directed graph: (a) non-isolated; (b) isolated. In (a), $\{4\}$ is a (trivial) single source component, $\{11,\ldots,15\}$ is a single sink component.
  • Figure 2: A four-node liability network.
  • Figure 3: Clearing payments in the example network. Left panel (a) shown the payments under pro-rata rule, Right panel (b) shown the unrestricted clearing payments.

Theorems & Definitions (10)

  • Lemma 1
  • Theorem 1
  • Remark 1
  • Theorem 2
  • Remark 2
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Lemma 2
  • Corollary 1