We construct a Markovian set-up that specifies the standard market model for convertible bonds (CBs), namely a model with stock-dependent local default intensity and local volatility. Generically, it can be shown that the problem of convertible bond essentially reduces to the study of an associated Dynkin game, which can be studied using the theory of doubly reflected Backward Stochastic Differential Equations (R2BSDE). In the case of the standard market model for CBs, we show that the associated R2BSDE has a unique solution, we characterize the pre-default price of the CB as the unique viscosity solution of the associated variational inequality, and we give conditions ensuring convergence of deterministic approximation schemes.