Interbank lending with benchmark rates: Pareto optima for a class of singular control games

Abstract

We analyze a class of stochastic differential games of singular control, motivated by the study of a dynamic model of interbank lending with benchmark rates. We describe Pareto optima for this game and show how they may be achieved through the intervention of a regulator, whose policy is a solution to a singular stochastic control problem. Pareto optima are characterized in terms of the solutions to a new class of Skorokhod problems with piecewise‐continuous free boundary. Pareto optimal policies are shown to correspond to the enforcement of endogenous bounds on interbank lending rates. Analytical comparison between Pareto optima and Nash equilibria provides insight into the impact of regulatory intervention on the stability of interbank rates.

Dynamics of market making algorithms in dealer markets: Learning and tacit collusion

“…Motivated by Cont et al. (2021), we show that collusion corresponds to a Pareto optimum: Definition $$\vec{\bm{\delta }}^p=((\bm{\delta }^1)^p, \ldots ,(\bm{\delta }^N)^p)\in \mathcal {S}$$ is a Pareto‐optimal policy if and only if there does not exist $$\vec{\bm{\delta }}\in \mathcal {S}$$, such that for all $$\bm{q}\in \prod \limits _{j=1}^N\mathcal {Q}_j$$, $$$$\begin{eqnarray} \forall i\in \lbrace 1,\ldots ,N\rbrace , J_i(\bm{\delta }^i, \bm{\delta }^{-i}; q_i) \ge J_i((\bm{\delta }^i)^p, (\bm{\delta }^{-i})^p; q_i) \nonumber \\ \exists j\in \lbrace 1,\ldots ,N\rbrace , J_j(\bm{\delta }^j, \bm{\delta }^{-j}; q_j) &gt; J_j((\bm{\delta }^j)^p, (\bm{\delta }^{-j})^p; q_i) \end{eqnarray}$$<...$$…”

“…In most of these models, the market maker faces a random environment represented by an order flow represented as a point process, so a natural mathematical modeling framework for the problem is that of intensity control of point processes (Bremaud, 1981). Competition of market makers may be modeled in this setting as a stochastic differential game (Cont et al, 2021;Guo & Xu, 2019;Luo & Zheng, 2021). Cont et al (2021) model inter-bank lending as a stochastic differential game of singular control and study Pareto optimal strategies.…”

“…Competition of market makers may be modeled in this setting as a stochastic differential game (Cont et al, 2021;Guo & Xu, 2019;Luo & Zheng, 2021). Cont et al (2021) model inter-bank lending as a stochastic differential game of singular control and study Pareto optimal strategies. Competition among market makers is studied by Luo and Zheng (2021).…”

“…Cont et al. (2021) model inter‐bank lending as a stochastic differential game of singular control and study Pareto optimal strategies. Competition among market makers is studied by Luo and Zheng (2021).…”

Dynamics of Market Making Algorithms in Dealer Markets: Learning and Tacit Collusion

“…In most of these models, the market maker faces a random environment represented by an order flow represented as a point process, so a natural mathematical modeling framework for the problem is that of intensity control of point processes (Bremaud, 1981). Competition of market makers may be modeled in this setting as a stochastic differential game (Cont et al, 2021;Guo & Xu, 2019;Luo & Zheng, 2021). Cont et al (2021) model inter-bank lending as a stochastic differential game of singular control and study Pareto optimal strategies.…”