Complete market
Encyclopedia
In economics
, a complete market (or complete system of markets) is one in which the complete set of possible gambles on future states-of-the-world can be constructed with existing asset
s without friction
. Every agent is able to exchange every good, directly or indirectly, with every other agent without transaction costs. Here goods are state-contingent; that is, a good includes the time and state of the world in which it is consumed. So for instance, an umbrella tomorrow if it rains is a distinct good from an umbrella tomorrow if it is clear. The study of complete markets is central to state-preference theory. The theory can be traced to the work of Kenneth Arrow
(1964), Gérard Debreu
(1959), Arrow & Debreu (1954) and Lionel McKenzie(1954). Arrow and Debreu were awarded the Nobel Memorial Prize in Economics (Arrow in 1972, Debreu in 1983), largely for their work in developing the theory of complete markets and applying it to the problem of general equilibrium.
A state of the world is a complete specification of the values of all relevant variables over the relevant time horizon. A state-contingent claim, or state claim, is a contract whose future payoffs depend on future states of the world. For example, suppose you can bet on the outcome of a coin toss. If you guess the outcome correctly, you will win one dollar, and otherwise you will lose one dollar. A bet on heads is a state claim, with payoff of one dollar if heads is the outcome, and payoff of negative one dollar if tails is the outcome. "Heads" and "tails" are the states of the world in this example. A state-contingent claim can be represented as a payoff vector with one element for each state of the world, e.g. (payoff if heads, payoff if tails). So a bet on heads can be represented as ($1, -$1) and a bet on tails can be represented as (-$1, $1). Notice that by placing one bet on heads and one bet on tails, you have a state-contingent claim of ($0, $0); that is, the payoff is the same regardless of which state of the world occurs. By placing combinations of bets, if fractional bets are allowed, any payoff vector in
can be obtained.
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 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
:
A put option can be synthetically created by solving the parity for put:
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 RN.
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
, a complete market (or complete system of markets) is one in which the complete set of possible gambles on future states-of-the-world can be constructed with existing asset
Asset
In financial accounting, assets are economic resources. Anything tangible or intangible that is capable of being owned or controlled to produce value and that is held to have positive economic value is considered an asset...
s without friction
Frictionless market
A Frictionless market is a financial market without transaction costs. Friction is a type of market incompleteness. Every complete market is frictionless, but the converse does not hold. In a frictionless market the solvency cone is the halfspace normal to the unique price vector. The...
. Every agent is able to exchange every good, directly or indirectly, with every other agent without transaction costs. Here goods are state-contingent; that is, a good includes the time and state of the world in which it is consumed. So for instance, an umbrella tomorrow if it rains is a distinct good from an umbrella tomorrow if it is clear. The study of complete markets is central to state-preference theory. The theory can be traced to the work of Kenneth Arrow
Kenneth Arrow
Kenneth Joseph Arrow is an American economist and joint winner of the Nobel Memorial Prize in Economics with John Hicks in 1972. To date, he is the youngest person to have received this award, at 51....
(1964), Gérard Debreu
Gerard Debreu
Gérard Debreu was a French economist and mathematician, who also came to have United States citizenship. Best known as a professor of economics at the University of California, Berkeley, where he began work in 1962, he won the 1983 Nobel Memorial Prize in Economics.-Biography:His father was the...
(1959), Arrow & Debreu (1954) and Lionel McKenzie(1954). Arrow and Debreu were awarded the Nobel Memorial Prize in Economics (Arrow in 1972, Debreu in 1983), largely for their work in developing the theory of complete markets and applying it to the problem of general equilibrium.
A state of the world is a complete specification of the values of all relevant variables over the relevant time horizon. A state-contingent claim, or state claim, is a contract whose future payoffs depend on future states of the world. For example, suppose you can bet on the outcome of a coin toss. If you guess the outcome correctly, you will win one dollar, and otherwise you will lose one dollar. A bet on heads is a state claim, with payoff of one dollar if heads is the outcome, and payoff of negative one dollar if tails is the outcome. "Heads" and "tails" are the states of the world in this example. A state-contingent claim can be represented as a payoff vector with one element for each state of the world, e.g. (payoff if heads, payoff if tails). So a bet on heads can be represented as ($1, -$1) and a bet on tails can be represented as (-$1, $1). Notice that by placing one bet on heads and one bet on tails, you have a state-contingent claim of ($0, $0); that is, the payoff is the same regardless of which state of the world occurs. By placing combinations of bets, if fractional bets are allowed, any payoff vector in
can be obtained.The bet on a coin toss is a simplistic example but illustrates widely applicable concepts, especially in finance
Finance
"Finance" is often defined simply as the management of money or “funds” management Modern finance, however, is a family of business activity that includes the origination, marketing, and management of cash and money surrogates through a variety of capital accounts, instruments, and markets created...
. 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
Linear span
In the mathematical subfield of linear algebra, the linear span of a set of vectors in a vector space is the intersection of all subspaces containing that set...
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
Trading strategy
In finance, a trading strategy is a predefined set of rules for making trading decisions.Traders, investment firms and fund managers use a trading strategy to help make wiser investment decisions and help eliminate the emotional aspect of trading. A trading strategy is governed by a set of rules...
if there exists a self-financing trading strategy such that at any time
, 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
Put–call parity
In financial mathematics, put–call parity defines a relationship between the price of a European call option and European put option in a frictionless market —both with the identical strike price and expiry, and the underlying being a liquid asset. In the absence of liquidity, the existence of a...
:
A put option can be synthetically created by solving the parity for put:
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 RN.
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