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This paper studies the valuation and optimal strategy of convertible bonds asa Dynkin game by using the reflected backward stochastic differential equationmethod and the variational inequality method. We first reduce such a Dynkingame to an optimal stopping time problem with state constraint, and then in aMarkovian setting, we investigate the optimal strategy by analyzing theproperties of the corresponding free boundary, including its position,asymptotics, monotonicity and regularity. We identify situations when callprecedes conversion, and vice versa. Moreover, we show that the irregularpayoff results in the possibly non-monotonic conversion boundary. Surprisingly,the price of the convertible bond is not necessarily monotonic in time: it mayeven increase when time approaches maturity.

Dynkin Game of Convertible Bonds and Their Optimal Strategy