Free boundary problem concerning pricing convertible bond

In this paper, we consider some behaviors of the optimal conversion boundaries (i.e. free boundaries) of American‐style convertible bond with finite horizon in some case. The bond's holder may convert it into the stock of its issued firm at any time before maturity, and the firm may call it at any time before maturity. Its pricing model is a parabolic variational inequality, in which the fundamental variables are time and the stock price of the bond's issuer. We achieve some properties of the free boundary, besides the existence and uniqueness of the solution of the variational inequality, such as: the monotonicity, the boundedness, smoothness and its starting point. Moreover, we analyze the relationship between the free boundary and the parameters in the problem, as well as, obtain the critical condition where the free boundary is a constant independent of time. Copyright © 2011 John Wiley & Sons, Ltd.