"It's the volatility that determines the prices" I thought this is only the case with contracts where the payoff is not linear, i.e. limiting the upside or dow

"It's the volatility that determines the prices" I thought this is only the case with contracts where the payoff is not linear, i.e. limiting the upside or downside. Future contract payoffs are linear f(t) = S(T) − K, so the upside is the same as the downside.