 Solution

For a finite-state, finite-horizon discrete-time frictionless market with a strictly positive numéraire, the fundamental theorem of asset pricing states no arbitrage if and only if there exists an equivalent martingale measure under which all discounted asset prices are martingales. Equivalence means the same null events as the physical measure. In this setting market completeness is equivalent to uniqueness of the equivalent martingale measure, and a replicable payoff $H$ has price $S_t^0{\mathbb{E}}_Q[H/S_T^0\mid {\mathcal{F}}_t]$. Without finite-state hypotheses, stronger integrability and closure conditions are needed; this is the standard multi-period course version.