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Writefor the product of the last $N-M$ stock factors. The forward-start payoff isUnder the risk-neutral measure the binomial factors are independent with the same probability $q$ at every node. Hence $S_M$ and $R_{M,N}$ are independent, whileThe forward-start call option consequently has priceAn ordinary $(N-M)$-period call with strike $K$ has priceThe required equality therefore holds for