Double free boundary problem for defaultable corporate bond with credit rating migration risks and their asymptotic behaviors

TL;DR

This paper develops a mathematical model for pricing defaultable corporate bonds considering credit rating migration risks, formulating it as a complex free boundary problem with rigorous analysis and numerical validation.

Contribution

It introduces a novel free boundary problem with two free boundaries for bond pricing, proving existence, uniqueness, regularity, and asymptotic behavior of solutions.

Findings

Proved existence, uniqueness, and smoothness of solutions.

Established asymptotic convergence to traveling wave solutions.

Presented numerical results validating the model.

Abstract

In this work, a pricing model for a defaultable corporate bond with credit rating migration risk is established. The model turns out to be a free boundary problem with two free boundaries. The latter are the level sets of the solution but of different kinds. One is from the discontinuous second order term, the other from the obstacle. Existence, uniqueness, and regularity of the solution are obtained. We also prove that two free boundaries are $C^\infty$. The asymptotic behavior of the solution is also considered: we show that it converges to a traveling wave solution when time goes to infinity. Moreover, numerical results are presented.