Analysis of Free Boundaries for Convertible Bonds, with a call feature

TL;DR

This paper investigates the behavior and regularity of free boundaries in convertible bonds with call features, especially near points where the free boundary touches the fixed boundary, using robust methods applicable to nonlinear equations.

Contribution

It provides new insights into the free boundary behavior near contact points with fixed boundaries and establishes regularity results for solutions and boundaries in nonlinear settings.

Findings

Free boundary may reach the fixed boundary in bonds with call features.

Regularity of solutions and free boundaries is established up to the fixed boundary.

Methods are applicable to fully nonlinear equations and yield uniform regularity results.

Abstract

Convertible bonds give rise to the so-called free boundary; i.e., an unknown boundary between continuation and conversion regions of the bond. The characteristic feature of such a bond, with an extra call feature, is that the free boundary may reach all the way to the fixed boundary. Our intention in this paper is to study the behavior of the free boundary in the vicinity of a touching point with the fixed boundary. Along the lines of our analysis we will also produce some results on regularity of solutions (value of the bond) up to the fixed boundary. Our methods are robust and of general nature, and can be applied to fully nonlinear equations. In particular, we shall obtain uniform results for the regularity of both solutions and their free boundaries.