Optimal bailout strategies resulting from the drift controlled supercooled Stefan problem

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 forward-backward coupled system of PDEs. Our simulations show that the central agent's optimal strategy is to subsidise banks whose asset values lie in a non-trivial time-dependent region. Finally, we study a linear-quadratic version of the model where instead of the terminal losses, the agent optimises a terminal cost function of the equity values. In this case, we are able to give semi-analytic strategies, which we again illustrate numerically.

Implicit and fully discrete approximation of the supercooled Stefan problem in the presence of blow-ups

“…By compactness of M , after passing to a subsequence if necessary, we may assume that Λ hn Ñ Λ for some Λ with α ´1Λ P M . If x is a càdlàg path, we introduce the notation Γ α rxs :" αΓrα ´1xs, such that Λ solves (7) where the last estimate follows from (37), Lemma 3.4 and the third term is estimated as in the proof of Theorem 3.12. Rearranging terms and absorbing some into the constant C, the claim follows.…”

“…We start by establishing existence (and uniqueness) of minimal solutions which are defined as follows. We call a solution pX, τ , Λq to the McKean-Vlasov problem (7) minimal, if for every solution pX, τ, Λq to (7) we have…”

“…In this section, we consider the numerical approximation of a special case of the McKean-Vlasov problem (7) with Z " B, with B a Brownian motion independent of X 0´. The idea is to replace the Brownian motion by its Donsker approximation, i.e., for h ą 0 1 we consider…”

“…We repeat the same reasoning as previously, obtaining the minimal solution of ( 18) through a fixed-point iteration. We call a solution Λ h of (18) minimal, if Λ h ď Λ h for every Λ h that solves (7). The minimal solution will then give rise to the implicit Donsker approximation scheme presented in Section 4.…”

“…There, Λ t models the proportion of banks defaulted at time t, and α the strength of interbank borrowing (see e.g. [20,34,21,7] for details and further references). Equations with similar mean-field effects through thresholding derive from integrate-and-fire models of neurons as considered in [9].…”