 Perpetual American call option

An American call option with no expiry. For geometric Brownian motion with drift $r-y$ under a risk-neutral measure, positive dividend yield $y$, $r>0$, and volatility $\sigma >0$, let $p>1$ solve ${\sigma }^{2}p(p-1)/2+(r-y)p-r=0$. Its value is $(S_{\ast }-K)(S/S_{\ast }{)}^p$ below $S_{\ast }=pK/(p-1)$ and $S-K$ above. The power solves the stationary Black-Scholes equation with continuous stock dividends; value matching and smooth fit determine the threshold.

The candidate dominates the positive part $(S-K{)}^{+}$ because its continuation branch is convex and tangent to $S-K$ at $S_{\ast }$. In the exercise region its discounted-generator residual is $rK-yS\le 0$: the root equation gives $y{S}_{\ast }=K[r+{\sigma }^{2}p/2]>rK$. Applying Itô's formula to the discounted value, localized if necessary, gives an upper bound on every exercise policy. Stopping at the first hit of $S_{\ast }$ attains it; on paths not yet hitting, the bounded discounted continuation value tends to zero. This supplies the optimal stopping verification. As $y\downarrow 0$, $p\downarrow 1$, the threshold diverges and the value tends to $S$.