Suppose we do not have a particular pricing model, we have just a frictionless market with constant interest rate (say $0$), and some traded stock $S$ which does not pay dividends. For any expiry $T$ to price options/contingent claims consistently we need a pricing rule, or equivalently a pricing (risk-neutral) probability measure. In particular, to price European call options we only need a marginal of such probability measure being the distribution of $S_T$.
Standard non-arbitrage arguments imply that the futures price $F(0,T) = \Bbb E^{\Bbb Q}S_T$ must satisfy $F(0,T) = S_T$, otherwise using a static replication strategy we can exploit mispricing in case of inequality. Hence, for the pricing distribution $\Bbb Q$ at least the first moment is fixed externally. What about the rest of the distribution? Say, we only focus on distributions that can be completely recovered from their moments. The first one is fixed, what about others, are we completely free in choosing them given the conditions above?