= Fixed-threshold value of a perpetual call
{title2=$C_B(S)=(B-K)(S/B)^p,\quad 0\leq S\leq B$}
For the <perpetual American call option> model and a fixed <exercise threshold> $B>K$, this is the discounted <expectation> of $B-K$ at its first hit, with zero payoff if the hit never occurs. The stationary <ordinary differential equation> and boundary conditions $C_B(0)=0$, $C_B(B)=B-K$ prove the formula. At fixed $S<B$, differentiation gives $\partial_BC_B=S^pB^{-p-1}[pK-(p-1)B]$, so the unique maximizing threshold is $pK/(p-1)$. If a larger threshold is imposed, $C_B'(B-)>1$ and its value lies below immediate exercise payoff just below $B$. This is allowed for a committed policy, but violates the exercise lower bound if quoted for an unrestricted <American call option>.
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