Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 1 / 28I / c

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 $1/9$. HenceThe payoff at the middle stock-price node is path dependent: the up-down path still has strike 864 and pays zero.