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 :)
