Let H denote the barrier of a perpetual American down-and-out call option. The governing equation for the

Question:

Let H denote the barrier of a perpetual American down-and-out call option. The governing equation for the price of the perpetual American barrier option C_∞(S;r,q) is given by

ds - + (r = q) sdCx ds -rC = 0, H where S^∗_∞is the optimal exercise price. Determine S^∗_∞ and find the option price C_∞(S;r,q).

The optimal exercise price is determined by maximizing the solution for the perpetual American call price among all possible exercise prices, that is,

Co (S; r, q) S* - X max - S* HA+S* - - S*+ H^_ where + and are roots of the quadratic equation: ( 1) + (r