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At each node of a binomial tree, a claim with next-period values $V_u,V_d$ has valuewhere $q$ is the local risk-neutral probability in a binomial market. For an American claim, replace this continuation value by the maximum of continuation and immediate exercise value.

Under a risk-neutral measure, the discounted stock must be a martingale. If $q$ is the probability of the factor $1+b$, this condition issoThe hypothesis $a<r<b$ ensures $0<q<1$, making this equivalent to the original probability measure. Since the $N$ returns are independent under $Q$, the number of up moves has a binomial distribution, and hence

Now $R=1+r=9/8$. The local risk-neutral up probabilities solve $RS=q{S}_u+(1-q)S_d$. They areThe terminal payoff of the European put option with strike $5$ is $0,0,2$ at stock prices $7,5,3$. Backward option pricing gives time-one valuesand henceThe first-period stock holding in the replicating portfolio in a binomial market isThus the hedge initially shorts $2/9$ of a share.