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I am trying to solve the perpetual American option problem. Currently I'm following this (slide 9). The stock price is modelled as Ito's process.

$dS_t = (\mu-D_0)S_tdt\ +\ \sigma S_tdW_t $

where $D_0$ is the dividend. Basically the problem tries to find optimal stopping time to exercise the option i.e. we try to optimize the value function $V(S)$ w.r.t stopping time $\tau$.

$V(S) = \sup_{\tau \in T}\ \mathbb{E}[e^{-r\tau}\max(S_\tau -K,\ 0)] $

Once we have the Value function $V(S)=AS^\beta$, we use the boundary condition at optimal time to get relation between $A$ and $S^*$.

$V(S^{*})=S^*-K$

To get another relation (slide 12) for solving for both $A$ and $S^*$ we try to maximize $A$ so as to increase the value function as much as possible. I do not see any justification for this.

Contrary to the derivation in the above reference, another reference (equation 6) uses smooth pasting condition i.e.

$V'(S^*) = f'(S*)$ ; $\ \ f(S) = S-K$

The difference in the techniques to obtain the second relation between $S^*$ and $A$ leads to different solutions. I'm not convinced with the argument of maximizing $A$. So if anybody can justify it, it'd be great. Also which solution is actually optimal?

Thanks :)

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