Valuing Convertible Bonds with Coupons: A Premium-Decomposition Refinement (Financial Modeling and Analysis)

Abstract

This paper deals with valuing defaultable and non-callable convertible bonds (CBs) with continuous coupon payments. The setup is the Black-Scholes-Merton framework where the underlying firm value evolves according to a geometric Brownian motion. The valuation of CBs can be formulated as an optimal stopping problem, due to the possibility of voluntary conversion prior to maturity. We focus on the notion of premium decomposition, which separates the CB value into the associated European CB value and an early conversion premium. By the Laplace-Carson transform (LCT) approach combined with the premium decomposition, we obtain closed-form LCT solutions for the CB value and the early conversion boundary. They have much simpler expressions than plain LCT solutions without using the premium decomposition. By virtue of the simplicity, we can easily characterize asymptotic properties of the early conversion boundary close or at infinite time to expiry.