Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 1 / 28I (Stochastic Financial Models)

The diagram has common multipliers $u=5/3,d=2/3$ on every branch, with gross riskless return $R=4/3$. The martingale condition $pu+(1-p)d=R$ givesSince $d<R<u$, these probabilities lie strictly between zero and one at each node, and the market is arbitrage-free. All valuations below use discounted conditional expectations under this measure.

The terminal European put pays zero at the upper and middle nodes and $864-384=480$ at the lower one. Its probability is $(1-p{)}^2=1/9$. Thus

At the up node, both immediate exercise and continuation are zero. At the down node the exercise value is $864-576=288$, while the discounted continuation value is $R^{-1}(480/3)=120$. Exercise there is optimal. At time zero immediate exercise pays zero, so

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.