= Solution
{c}
Shouting at zero locks in $S_0-K=-1$, producing terminal payoff $\max(S_2,100)-101$. Its pathwise values are 20 on $uu$ and $-1$ on each other path. Waiting and applying part (c) instead gives values 20 on $uu$, 9 on $ud$, and zero on $du,dd$. Therefore waiting dominates shouting now, strictly on events of positive probability.
Under the optimal <binomial shout option with signed lock-in>, discount the latter terminal payoffs using independent risk-neutral transitions:
$$
V_0=\frac{q^2\,20+q(1-q)\,9}{1.05^2}
=\frac{(9/16)20+(3/16)9}{(21/20)^2}
=\boxed{\frac{575}{49}\approx11.7347.}
$$
The time-one option value is $115/7$ after an up move and zero after a down move. These values also reproduce the price by one-step backward induction.
\Image[/past-exam-of-the-martingales-and-financial-mathematics-course-of-the-university-of-oxford/2008-shout-tree.png]
{title=Optimal shout decisions and the four path-dependent terminal payoffs}
The recombining stock price 99 has two different claim payoffs, because $ud$ has an earlier lock-in while $du$ does not. Keeping the path history is essential.
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