= Perpetual put option
{title2=$S^*=\frac{\beta_-}{\beta_--1}X$}
For risk-neutral <stock> dynamics $dS=(r-\delta)Sdt+\sigma S dW$, with $r>0$ and $\sigma>0$, let $\beta_-<0$ solve $\sigma^2\beta(\beta-1)/2+(r-\delta)\beta-r=0$. A fixed downward trigger $L\in(0,X)$ gives continuation value $(X-L)(S/L)^{\beta_-}$ for $S>L$. Optimizing over the trigger, or imposing <smooth fit>, gives the displayed boundary. The optimal value equals $X-S$ below $S^*$ and the continuation expression above. A verification argument uses its payoff majorization and nonpositive discounted generator. At $r=0$ and $\delta\ge0$, the perpetual value is $X$ for positive spot, approached by triggers tending to zero rather than attained at a positive finite trigger.
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