= Solution
Write
$$
R_{M,N}=\frac{S_N}{S_M}
$$
for the product of the last $N-M$ stock factors. The forward-start payoff is
$$
(S_N-\lambda S_M)^+
=S_M(R_{M,N}-\lambda)^+.
$$
Under 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, while
$$
\mathbb E_QS_M=S_0(1+r)^M.
$$
The <forward-start call option> consequently has price
$$
\begin{aligned}
\operatorname{FSC}(M,N,\lambda)
&=(1+r)^{-N}\mathbb E_Q
\left[S_M(R_{M,N}-\lambda)^+\right]\\
&=S_0(1+r)^{-(N-M)}
\mathbb E_Q\left[(R_{M,N}-\lambda)^+\right].
\end{aligned}
$$
An ordinary $(N-M)$-period call with strike $K$ has price
$$
\operatorname{EC}(N-M,K)
=S_0(1+r)^{-(N-M)}
\mathbb E_Q\left[\left(R_{M,N}-\frac K{S_0}\right)^+\right].
$$
The required equality therefore holds for
$$
\boxed{K=\lambda S_0}.
$$
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