= Completeness and uniqueness of dominated martingale measures
In an arbitrage-free one-period market with finitely many assets and an <equivalent martingale measure>, <market completeness> is equivalent to uniqueness of the <dominated martingale measure>. Completeness prices every <indicator function>, forcing two measures to agree on every event. Conversely, if the attainable payoff span has dimension $k$ but is not the whole payoff space, choose $k+1$ disjoint positive-probability events. Their moment vectors against a basis of attainable payoffs are linearly dependent. A nonzero bounded linear combination $h$ of their indicators consequently has zero pricing moments. Multiplying an existing equivalent density by $1+\varepsilon h$, for sufficiently small positive and negative $\varepsilon$, constructs distinct <equivalent martingale measures>. This proves the converse without assuming that the original space was finite. On a general probability space, completeness here means replication of all bounded claims, or equivalently all square-integrable claims when asset payoffs are square-integrable.
Back to article page
