anothercfainnyc

first of all, perpetual puts and calls are of the american type (obviously, otherwise you have to wait for perpetuity to be over before you can exercise which can take a while). you can't value them with BS at all, which is for european-type of exercise. for a dividend-paying stock, an optimal exercise boundary exist and there is a non-trivial formula (due to Robert Merton) for the value of a perpetual puts/calls which is very different from Black-Scholes. look it up in a textbook.

however, what you are describing doesn't fully make sense. if both the issuer and the holder have the right to force conversion at any time, at the same conversion rate, then either one or the other will do it today. then the value of your security is trivially equal to the stock price times the number of conversion shares.