= Perpetual American call option
{title2=$S_*={pK}/{(p-1)}$}
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 $\sigma^2p(p-1)/2+(r-y)p-r=0$. Its value is $(S_*-K)(S/S_*)^p$ below $S_*=pK/(p-1)$ 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 $S_*$. In the exercise region its discounted-generator residual is $rK-yS\leq0$: the root equation gives $yS_*=K[r+\sigma^2p/2]>rK$. 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 $S_*$ attains it; on paths not yet hitting, the bounded discounted continuation value tends to zero. This supplies the <optimal stopping> verification. As $y\downarrow0$, $p\downarrow1$, the threshold diverges and the value tends to $S$.
\Image[/perpetual-call-exercise-threshold.png]
{title=Early and late exercise thresholds both lose value relative to the optimal perpetual-call threshold}
{description=An ideal Black-Scholes example with strike one, interest rate 0.05, volatility 0.2 and dividend yield 0.045 gives exponent two and optimal threshold two. A late fixed-policy value may lie below the immediate exercise payoff; it cannot be the price of an unrestricted American option.}
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