Optimal bailout strategies resulting from the drift controlled supercooled Stefan problem

TL;DR

This paper models a central bank's optimal bailout strategy as a drift-controlled supercooled Stefan problem, providing a mathematical framework and numerical methods to determine minimal intervention policies in systemic risk scenarios.

Contribution

It introduces a novel connection between systemic risk control and the supercooled Stefan problem, deriving optimal strategies through mean field control and numerical policy gradient methods.

Findings

Optimal strategies involve subsidizing banks within a specific equity region.

The value function converges as the number of institutions increases.

Numerical simulations demonstrate effective bailout policies.

Abstract

We consider the problem faced by a central bank which bails out distressed financial institutions that pose systemic risk to the banking sector. In a structural default model with mutual obligations, the central agent seeks to inject a minimum amount of cash in order to limit defaults to a given proportion of entities. We prove that the value of the central agent's control problem converges as the number of defaultable institutions goes to infinity, and that it satisfies a drift controlled version of the supercooled Stefan problem. We compute optimal strategies in feedback form by solving numerically a regularized version of the corresponding mean field control problem using a policy gradient method. Our simulations show that the central agent's optimal strategy is to subsidise banks whose equity values lie in a non-trivial time-dependent region.