Perpetual convertible bonds with credit risk

Abstract

A convertible bond is a security that the holder can convert into a specified number of underlying shares. We enrich the standard model by introducing some default risk of the issuer. Once default has occured payments stop immediately. In the context of a reduced form model with infinite time horizon driven by a Brownian motion, analytical formulae for the no-arbitrage price of this American contingent claim are obtained and characterised in terms of solutions of free boundary problems. It turns out that the default risk changes the structure of the optimal stopping strategy essentially. Especially, the continuation region may become a disconnected subset of the state space.

Keywords:

• convertible bonds
• exchangeable bonds
• default risk
• optimal stopping problems
• free-boundary problems
• smooth fit

AMS Subject Classification:

• 60G40
• 60J50
• 60G44
• 91B28

Acknowledgements

The authors would like to thank Andreas Kyprianou for valuable discussions and comments.

Notes

^1. For sets , ∂A denotes the boundary of A in ℝ_>0, i.e. if A = (a,b) with and then . Furthermore, the closure of A in ℝ_>0 is denoted by , i.e. .