Page "Complete market" Paragraph 2

The bet on a coin toss is a simplistic example but illustrates widely applicable concepts, especially in finance.

If markets are complete, it is possible to arrange a portfolio with any conceivable payoff vector.

That is, the state claims available for purchase, represented as payoff vectors, span the payoff space.

A pure security or simple contingent claim is a state claim that pays off in only one state.

Any state-contingent claim can be regarded as a collection of pure securities.

A system of markets is complete if and only if the number of attainable pure securities equals the number of possible states.

Formally, a market is complete with respect to a trading strategy,, if there exists a self-financing trading strategy, such that at any time, the returns of the two strategies, and are equal.

This is equivalent to stating that for a complete market, all cash flows for a trading strategy can be replicated using a similar synthetic trading strategy.

Because a trading strategy can be simplified into a set of simple contingent claims ( strategies paying 1 in one state and 0 in every other state ), a complete market can be generalized as the ability to replicate cash flows of all simple contingent claims.