Consider a perpetual American put option (with \(T=\infty\) ). 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

for \(G \leq S \leq \infty\). The appropriate boundary conditions are \(P(\infty)=0\) 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 \(\gamma=2 r / \sigma^{2}\).

(b) Use the two boundary conditions to show that

(c) Finally, choose \(G\) to maximize \(P(S)\) to conclude that

Transcribed Image Text:

o sp" (S)+rSP' (S) rP (S) = 0