2. (Perpetual puto) Consider a perpetual American put option (with 7 = co) For small stock prices it will be advantageous to exercise the put. Let G be the largest such stock price. The time-independent Black-Scholes equation becomes SP" (S)+SP(S)-1 P(S) = 0 for GSo. The appropriate boundary conditions are P (oo) = 0 and P(G)=K-G. G should be chosen to maximize the value of the option

(a) Show that P(S) has the form where y = 21/0 P(S) = aS + aS

(b) Use the two boundary conditions to show that P(S)=(KG)(S/G)"

(c) Finally, choose G to maximize P(S) to conclude that K P(S)= 1+ y (1+y)S YK -Y