Uniqueness of equilibrium in a payment system with liquidation costs

doi:10.1016/j.orl.2015.10.005

Operations Research Letters

Volume 44, Issue 1, January 2016, Pages 1-5

https://doi.org/10.1016/j.orl.2015.10.005 Get rights and content

Abstract

We study a financial network where forced liquidations of an illiquid asset have a negative impact on its price, thus reinforcing network contagion. We give conditions for uniqueness of the clearing asset price and liability payments. Our main result holds under mild and natural assumptions on the price impact function: monotonicity of the price impact function and strict monotonicity of the proceeds of liquidation in the liquidated quantity.

Introduction

We study a financial network, in which banks hold interbank liabilities, cash, and shares of an illiquid asset. The settlement of interbank liabilities may force banks to liquidate some shares of the illiquid asset. This has a negative impact on the price of the illiquid asset. Marking to market of banks’ balance sheets reinforces network contagion: lower asset prices may force other banks to default on their interbank liability payments. This results in an entanglement of price mediated contagion and network mediated contagion.

We model the price impact by a given inverse demand function. In equilibrium, this leads to a clearing price and liability payments, given as solution of a fixed point equation. Existence of the fixed point follows by Tarski’s fixed point theorem, as shown in [6]. Uniqueness has remained an open problem. In this paper, we prove uniqueness under some mild and natural technical assumptions.

A key assumption is that the cash proceeds from asset liquidations are strictly increasing in the number of shares liquidated. This assumption is economically reasonable, but is not satisfied in the influential paper [6] for all parameter choices for the exponential inverse demand function.

Our uniqueness result carries over to other interbank clearing mechanisms. For illustration we sketch the proof when there are different seniority classes of interbank liabilities.

We also provide an algorithm for computing the fixed point in our baseline model that terminates in at most $m$ iterations, where $m$ denotes the number of banks in the network. This algorithm is instructive as it corresponds to the actual cascade of bank defaults that leads to the equilibrium and thus has a clear economic interpretation.

Our paper is related to the strand of literature on interbank liability clearing where various mechanisms may reinforce network contagion, e.g, [9], [1], [2], [4].

The remainder of the paper is as follows. In Section 2 we introduce the financial network. Section 3 contains our main result on the uniqueness of the clearing price and liability payments, which is proved in Section 4. In Section 5 we extend our uniqueness result to a financial network with different seniority classes. In Section 6 we provide an algorithm for computing the fixed point along with its economic interpretation.

Section snippets

Financial network

We consider the payment network model of [6] which extends the model of [7] to account for the price impact of the liquidation of external assets. The financial network consists of $m$ interlinked financial institutions (“banks”) $i \in [m] = {1, \dots, m}$ . Bank $i$ holds $γ_{i} \geq 0$ units of a liquid asset (cash), and $y_{i} \geq 0$ units of an illiquid asset. Cash has value one. The illiquid asset has a positive fundamental value $P > 0$ . The total illiquid asset holdings of the banks is denoted by $y_{tot} ≔ \sum_{i \in [m]} y_{i}$ .

Nominal

Existence and uniqueness of equilibrium

In equilibrium, the previous characterization of actual cash flows and price impact lead to a clearing price $P^{*}$ and total liability vector $L^{*} = (L_{1}^{*}, \dots, L_{m}^{*})$ , which can be determined as a fixed point, $Φ (P^{*}, L^{*}) = (P^{*}, L^{*})$ , of the non-linear map $Φ$ on $[P_{\min}, P] \times [0, L]$ , with $L = (L_{1}, \dots, L_{m})$ , given by ${\begin{matrix} Φ_{0} (p, ℓ) = f (\sum_{i \in [m]} \frac{{(L_{i} - γ_{i} - \sum_{j \in [m]} ℓ_{j} Π_{j i})}^{+}}{p} \land y_{i}) \\ Φ_{i} (p, ℓ) = L_{i} \land (y_{i} \cdot p + γ_{i} + \sum_{j \in [m]} ℓ_{j} Π_{j i}), i \in [m] . \end{matrix}$

We have the following lemma.

Lemma 1

The mapping $Φ$ is monotone, continuous and bounded.

Proof

First, note that $Φ_{0} (p, ℓ)$ is a non-decreasing

Proof of Theorem 2

The proof of Theorem 2 builds on the following lemma, which has immediate applications to other interbank liability clearing mechanisms.

Lemma 3

Let $f$ be the inverse demand function as above and let $ζ : [P_{\min}, P] \to [0, y_{tot}]$ be a function satisfying

1.
$ζ (p)$ is continuous and non-increasing in $p \in [P_{\min}, P]$ ;
2.
$p ζ (p)$ is non-decreasing in $p \in [P_{\min}, P]$ .

Then there exists a unique solution

p^{*} \in [P_{\min}, P]

to the equation

p = f (ζ (p)) .

Proof

First observe that $f (ζ (p))$ is a continuous function in $p \in [P_{\min}, P]$ with values in $[P_{\min}, P]$ . Hence

Extension to different seniority classes

We consider $S$ different seniority classes $s \in [S] = {1, \dots, S}$ , with $S$ being the least and 1 the most senior class. The nominal interbank liabilities in each seniority class $s \in [S]$ are represented by a matrix $(L_{i j}^{(s)})$ , where $L_{i j}^{(s)} \geq 0$ denotes the cash-amount that bank $i$ owes bank $j$ in seniority class $s$ . The total nominal liabilities of bank $i$ in seniority class $s$ sum up to $L_{i}^{(s)} = \sum_{j \in [m]} L_{i j}^{(s)} .$ The total nominal liabilities of bank $i$ sum up to $L_{i} = \sum_{s \in [S]} L_{i}^{(s)}$ .

Bank $i$ in turn claims a total nominal cash

Constructing the clearing vector

In this last section, we provide an iterative procedure for our baseline model to identify the clearing liability payment vector and asset price and show that it terminates in at most $m$ iterations. As in [7], this iterative procedure allows us to classify banks in terms of their financial health. The banks that default in the initial round are fundamental defaults, the banks that default in the next round are very fragile, and so on. Fragility however in our case also stems from exposure

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Uniqueness of equilibrium in a payment system with liquidation costs

Abstract

Introduction

Section snippets

Financial network

Existence and uniqueness of equilibrium

Proof of Theorem 2

Extension to different seniority classes

Constructing the clearing vector

To fully net or not to net: Adverse effects of partial multilateral netting

Oper. Res.

Control of interbank contagion under partial information

SIAM J. Financ. Math.