= Perpetual power put option
{title2=$b=\frac{2rK}{2r+n\sigma^2}$}
For a dividend-free <Black-Scholes model> with $r>0$, $\sigma>0$, $K>0$ and positive <integer> $n$, the <optimal stopping> value of the <power put option> is
$$
V(S)=\begin{cases}(K-S)^n,&0\leq S\leq b,\\(K-b)^n(S/b)^{-\gamma},&S>b,\end{cases}\qquad \gamma=\frac{2r}{\sigma^2},\quad b=\frac{\gamma K}{\gamma+n}.
$$
In the continuation region the stationary <Black-Scholes equation> has powers $S$ and $S^{-\gamma}$. Boundedness at infinity removes $S$. Value matching gives the coefficient $(K-b)^nb^\gamma$, maximized at the displayed $b$; equivalently this is the <smooth fit> boundary.
For a verification beyond constant-trigger policies, let $g(S)=(K-S)_+^n$. The maximum property of $b^\gamma(K-b)^n$ gives $V\geq g$. For the <infinitesimal generator> $\mathcal L=rS\partial_S+\sigma^2S^2\partial_{SS}/2$, $(\mathcal L-r)V=0$ above $b$. Below $b$, set $y=S/(K-S)\leq\gamma/n$; then
$$
\frac{(\mathcal L-r)g}{g}=\frac12\sigma^2[n(n-1)y^2-\gamma ny-\gamma]\leq-r(1+y)<0.
$$
The candidate is continuously differentiable across $b$, so the generalized <Itô formula> has no derivative-jump <local time of a semimartingale> term. The discounted candidate is a nonnegative <supermartingale>, bounding every stopping reward by <optional stopping>. First hitting of $b$ attains the bound; the remainder at infinite time vanishes since $V\leq K^n$ and $r>0$. For $n=1$ this recovers the <perpetual put option>. At $r=0$ the value is $K^n$ for $S>0$, approached but not attained at a finite exercise time. At $r<0$ the perpetual value is infinite, because exercise at deterministic times tending to infinity has unbounded discounted <expected value>.
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