Consider a perpetual American put option (with (T=infty) ). For small stock prices it will be advantageous
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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
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