Perpetual reciprocal-payoff American option ($b=\alpha /(1-\alpha )$)

For payoff $g(s)=(1+s{)}^{-1}$ and $\alpha =2r/{\sigma }^2\in (0,1)$, the exercise boundary is $b=\alpha /(1-\alpha )$. The value equals $g$ below $b$ and $(1+b{)}^{-1}(s/b{)}^{-\alpha }$ above it. Value matching, smooth fit, and the obstacle problem verify the first down-crossing policy.