Abstract: In this paper, we prove that there exist two boundaries which are the optimal exercise boundary and the worst exercise boundary by changing the valuation of convertible bonds into the two-free-boundary promblem of the Black-Scholes equation.When the value of a company acrosses increasingly the optimal exercise boundary,the return of its irredeemable bonds is largest if they are converted into its stocks.When the vale of a company acrosses decreasingly the worst exercise boundary,the value of its irredeemable bonds is zero if they are converted into its stocks.

Key words: rredeemable, convertible bonds, Black-Scholes, two-free-boundary, the vale of a company, rredeemable, convertible bonds, Black-Scholes, two-free-boundary