A Refined Laplace-Carson Transform Approach to Valuing Convertible Bonds

Abstract

This paper deals with a refinement of the Laplace-Carson transform (LCT) approach to option pricing, with a special emphasis on valuing defaultable and non-callable convertible bonds (CBs), but not limited to it. What we are actually aiming at is refining the plain LCT approach to meet possibly general American derivatives. The setup is a standard 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 begin with the plain LCT approach that generates a complex solution with little prospect of further analysis. To improve this solution, we introduce the notion of premium decomposition, which separates the CB value into the associated European CB value and an early conversion premium. By the LCT approach combined with the premium decomposition, we obtain a much simpler and closed-form solution for the CB value and an optimal conversion boundary. By virtue of the simplified solution, we can easily characterize asymptotic properties of the early conversion boundary. Finally, we show that our refined LCT approach is broadly applicable to a more general class of claims with optimal stopping structure.