A market is incomplete if some contingent claim cannot be replicated by a self-financing strategy. In a one-period finite-state market, attainable payoff vectors form the span of the traded assets' terminal payoff vectors. If this span is a proper subspace of the full payoff space, the market is incomplete. For example, two linearly independent asset-payoff vectors on three states leave at least one payoff direction unattainable. Absence of arbitrage only requires an appropriate positive state-price density; it does not imply that every claim is attainable.
New to topics? Read the docs here!