Convertible Bonds in a Defaultable Diffusion Model

In this paper, we study

convertible securities

(CS) in a primary market model consisting of: a savings account, a stock underlying a CS, and an associated CDS contract (or, alternatively to the latter, a

rolling CDS

more realistically used as an hedging instrument). We model the dynamics of these three securities in terms of Markovian diffusion set-up with default. In this model, we show that a doubly reflected Backward Stochastic Differential Equation associated with a CS has a solution, meaning that super-hedging of the arbitrage value of a convertible security is feasible in the present setup for both issuer and holder at the same initial cost, and we provide the related (super-)hedging strategies. Moreover, we characterize the price of a CS in terms of viscosity solutions of associated variational inequalities and we prove the convergence of suitable approximation schemes. We finally specify these results to convertible bonds and their straight bond and game exchange option components, and provide numerical results.