Solution
= Solution
Only paths whose first move is down have positive payoffs. On the down-up path the raised strike gives $992-960=32$, and on down-down it gives $992-384=608$. Their <probabilities> are $2/9$ and $1/9$. Hence
$$
\boxed{V_0=R^{-2}\left(\frac29\cdot32+\frac19\cdot608\right)=42.}
$$
The payoff at the middle stock-price node is path dependent: the up-down path still has strike 864 and pays zero.