= Solution
Interpret $t=m$ and $T=N$ as trading dates in the discrete model; a payoff at nondates would require an interpolation convention absent from the question. Put $n=N-m$. Under $Q$, the independent stock returns after $m$ give
$$
\frac{S_N}{S_m}=u^Jd^{n-J},\qquad J\sim\operatorname{Binomial}(n,q).
$$
The ratio payoff is therefore independent of $\mathcal F_m$ under $Q$, and its time-zero price is
$$
\boxed{V_0=R^{-N}\sum_{j=0}^n\binom njq^j(1-q)^{n-j}(u^jd^{n-j}-K)^+.}
$$
At any date $s\leq m$, its price is the same expectation discounted only for the remaining $N-s$ periods. The formula uses the ratio payoff actually specified; multiplying it by $S_m$ would define a different, conventional monetary forward-start call.
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