 Solution

If the first move is up, $S_1=110$ and $S_2$ can be 121 or 99. A shout now gives $\max (S_2,110)-101$, hence payoffs 20 and 9. Waiting until maturity or never shouting gives 20 and 0. The first policy is no worse on either branch and strictly better on the down branch, so shout after the first up move.

If the first move is down, $S_1=90$ and the terminal prices are 99 or 81. A shout would give $-2$ or $-11$, while waiting gives zero. Thus do not shout after the first down move. These comparisons use terminal payoffs directly, so no intermediate discounting can change their ordering.