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Futures payoff is indeed $S_t-F_0$, but the $t$ in question is the maturity date of futures. In this derivation $t$ denotes maturity date of the option, which is always before the futures maturity. Therefore, on the day of option maturities, the futures did not expire yet, but the value of the futures position is $F_t-F_0$ (in mark-to-market sense, you can ...

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