 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.