 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 $m$ giveThe ratio payoff is therefore independent of ${\mathcal{F}}_m$ under $Q$, and its time-zero price isAt any date $s\le m$, 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.