Perpetual reciprocal-payoff American option
= Perpetual reciprocal-payoff American option
{title2=$b=\alpha/(1-\alpha)$}
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.