= Binomial shout option with signed lock-in
A holder may choose a shout time $\tau$ and receive $\max(S_T-S_\tau,0)+S_\tau-K=\max(S_T,S_\tau)-K$, or choose no shout and receive $(S_T-K)^+$. The lock-in term is signed, so shouting below strike can create a negative payoff. Backward induction must compare this actual contract with waiting. For $S_0=100$, multipliers $1.1,0.9$, bank growth $1.05$, strike 101 and two periods, the optimal policy waits at zero, shouts after the first up move, and never shouts after the first down move. The risk-neutral price is $575/49$. Claim payoffs at a recombined stock node can differ according to earlier shout history.
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