Optimal Stopping When the Absorbing Boundary is Following After

TL;DR

This paper introduces a novel optimal stopping problem with a moving absorbing boundary linked to the process's maximum, providing explicit solutions and applications to bank capital management.

Contribution

It reduces a complex two-dimensional problem to an infinite set of one-dimensional problems and offers explicit solutions for the moving boundary optimal stopping scenario.

Findings

Explicit solutions for the moving boundary problem.

Reduction of a 2D problem to 1D problems.

Application to bank leverage and capital constraints.

Abstract

We consider a new type of optimal stopping problems where the absorbing boundary moves as the state process X attains new maxima S. More specifically, we set the absorbing boundary as S-b where b is a certain constant. This problem is naturally connected with excursions from zero of the reflected process S-X. We examine this constrained optimization with the state variable X as a spectrally negative Levy process. The problem is in nature a two-dimensional one. The threshold strategy given by the path of X is not in fact optimal. It turns out, however, that we can reduce the original problem to an infinite number of one-dimensional optimal stopping problems, and we find explicit solutions. This work is motivated by the bank's profit maximization with the constraint that it maintain a certain level of leverage ratio. When the bank's asset value severely deteriorates, the bank's required capital requirement shall be violated. This situation corresponds to X<S-b in our setting. This model may well describe a real-life situation where even a big bank can fail because the absorbing boundary is keeping up with the size of the bank.