Complete market

In economics, a complete market is one in which the complete set of possible gambles on future states-of-the-world can be constructed with existing assets.

Formally, a market is complete with respect to a trading strategy if there exists a self-financing trading strategy such that at any time $t$, the returns of the two strategies are equal. This is equal to stating that for a complete market, all cash flows for a trading strategy can be replicated by a similar synthetic trading strategy. Because a trading strategy can be simplified into a set of simple contingent claims (a trading strategy that pays 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.

For example, consider the put–call parity:

${\text{ Call }}+{\text{ Present Value of Strike Price }}={\text{ Put }}+{\text{ Underlying security }}\,$

A put option can be synthetically created by solving the parity for put:

${\text{ Put }}={\text{ Call }}+{\text{ Present Value of Strike Price }}-{\text{ Underlying security }}\,$

A put is synthesized by buying the call, investing the strike at the risk free rate, and shorting the stock. If the calls on the stock are not traded in the market, the market is considered an incomplete market because it does not provide the ability to replicate the returns of a put option.

Often used to describe insurance markets the model of a complete market occurs if agents can buy insurance contracts to protect themselves against any future time and state-of-the-world.

For example, if a market is a finite state market with dimension N, then a complete market would be one where there exist traded assets with payoffs that form a basis for $R^{N}$.

Dynamically Complete Market

In order for a market to be complete, it must be possible to instantaneously enter into any position regarding any future state of the market. In contrast, a market is called dynamically complete if it is possible to construct a self-financing trading strategy that will have the same cash-flow. In other words, a complete market allows you to place all of your bet at once, while a dynamically complete market may require that you execute subsequent trades after making your initial investment. The requirement that the strategy be self-financing means that subsequent trades must be cash-flow neutral (you cannot contribute or withdraw any additional funds). Any complete market is also dynamically complete.

Further Reading

• Mark D. Flood (1991), "An Introduction to Complete Markets", Federal Reserve Bank of St. Louis, Review, March/April 1991

See also

• Incomplete markets