A valuation model for perpetual convertible bonds with Markov regime-switching models

This paper develops a valuation model for a perpetual convertible bond when the price dynamics of the underlying share are governed by continuous-time Markovian regime-switching models. We suppose that the appreciation rate and the volatility of the underlying share are modulated by a continuous-time, finite-state, observable Markov chain. The states of this chain are interpreted as the states of an economy. Here the valuation problem of the perpetual convertible bond can be viewed as that of valuing a perpetual stock loan, or a perpetual American option with time-dependent strike price. With the presence of the regime-switching effect, the market in the model is, in general, incomplete. To provide a convenient method to determine a price kernel for valuation, we employ the regime-switching Esscher transform introduced in Elliott, Chan and Siu (2005) [4]. We then adopt the differential equation approach in Guo and Zhang (2004) [7] to solve the optimal stopping problem associated with the valuation of the perpetual convertible bond. Numerical examples are presented to illustrate the practical implementation of the proposed model.