 Fixed-threshold value of a perpetual call

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 ${\partial }_B{C}_B=S^p{B}^{-p-1}[pK-(p-1)B]$, so the unique maximizing threshold is $pK/(p-1)$. If a larger threshold is imposed, $C_B^{\prime }(B-)>1$ 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.