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| In [[financial economics]], a '''state-price security''', also called an '''Arrow-Debreu security''' (from its origins in the [[Arrow-Debreu model]]), is a contract that agrees to pay one unit of a [[numeraire]] (a currency or a commodity) if a particular state occurs at a particular time in the future and pays zero numeraire in all the other states. The price of this security is the '''state price''' of this particular state of the world, which may be represented by a vector. The state price vector is the vector of state prices for all states.<ref>[http://economics.about.com/od/economicsglossary/g/statepricev.htm economics.about.com] Accessed June 18, 2008</ref>
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| As such, any derivatives contract whose settlement value is a function of an underlying whose value is uncertain at contract date can be decomposed as a linear combination of its Arrow-Debreu securities, and thus as a weighted sum of its state prices.
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| The [[Arrow-Debreu model]] (also referred to as the Arrow-Debreu-McKenzie model or ADM model) is the central model in the [[General equilibrium|General Equilibrium Theory]] and uses state prices in the process of proving the existence of a unique general equilibrium.
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| ==Example==
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| Imagine a world where two states are possible tomorrow: peace (P) and war (W). Denote the random variable which represents the state as ω; denote tomorrow's random variable as ω<sub>1</sub>. Thus, ω<sub>1</sub> can take two values: ω<sub>1</sub>=P and ω<sub>1</sub>=W.
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| Let's imagine that:
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| * There is a security that pays off £1 if tomorrow's state is "P" and nothing if the state is "W". The price of this security is q<sub>P</sub>
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| * There is a security that pays off £1 if tomorrow's state is "W" and nothing if the state is "P". The price of this security is q<sub>W</sub>
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| The prices q<sub>P</sub> and q<sub>W</sub> are the state prices.
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| The factors that affect these state prices are:
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| * The ''probabilities'' of ω<sub>1</sub>=P and ω<sub>1</sub>=W. The more likely a move to W is, the higher the price q<sub>W</sub> gets, since q<sub>W</sub> insures the agent against the occurrence of state W. The seller of this insurance would demand a higher premium (if the economy is efficient).
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| * The ''preferences'' of the agent. Suppose the agent has a standard [[Concave function|concave]] [[utility]] function which depends on the state of the world. Assume that the agent loses an equal amount if the state is "W" as he would gain if the state was "P". Now, even if you assume that the above-mentioned probabilities ω<sub>1</sub>=P and ω<sub>1</sub>=W are equal, the changes in utility for the agent are not: Due to his decreasing marginal utility, the utility gain from a "peace dividend" tomorrow would be lower than the utility lost from the "war" state. If our agent were [[rational expectations|rational]], he would pay more to insure against the down state than his net gain from the up state would be.
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| ==Application to financial assets==
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| If the agent buys both q<sub>P</sub> and q<sub>W</sub>, he has secured £1 for tomorrow. He has purchased a riskless bond. The price of the bond is b<sub>0</sub> = q<sub>P</sub> + q<sub>W</sub>.
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| Now consider a security with state-dependent payouts (e.g. an equity security, an option, a risky bond etc.). It pays c<sub>k</sub> if ω<sub>1</sub>=k -- i.e. it pays c<sub>P</sub> in peacetime and c<sub>W</sub> in wartime). The price of this security is c<sub>0</sub> = q<sub>P</sub>c<sub>P</sub> + q<sub>W</sub>c<sub>W</sub>.
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| Generally, the usefulness of state prices arises from their linearity: Any security can be valued as the sum over all possible states of state price times payoff in that state: <math>c_0 = \sum_k q_k\times c_k</math>.<ref>http://www.theponytail.net/DOL/DOLnode38.htm</ref>
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| Analogously, for a [[continuous random variable]] indicating a continuum of possible states, the value is found by [[integration (mathematics)|integrating]] over the [[state price density]].
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| ==See also==
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| *[[Complete market]]
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| *[[Incomplete markets]]
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| ==References==
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| {{reflist}}
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| {{economics}}
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| [[Category:Economics of uncertainty]]
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