State prices
Encyclopedia
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 pay zero numeraire in all 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.
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.
The Arrow-Debreu model
(also referred to as the Arrow-Debreu-McKenzie model or ADM model) is the central model in the General Equilibrium Theory
and uses state prices in the process of proving the existence of a unique general equilibrium.
Let's imagine that:
The prices qP and qW are the state prices.
The factors that affect these state prices are:
Now consider a security with state-dependent payouts (e.g. an equity security, an option, a risky bond etc.). It pays ck if ω1=k -- i.e. it pays cP in peacetime and cW in wartime). The price of this security is c0 = qPcP + qWcW.
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:
.
Analogously, for a continuous random variable indicating a continuum of possible states, the value is found by integrating over the state-price density.
Financial economics
Financial Economics is the branch of economics concerned with "the allocation and deployment of economic resources, both spatially and across time, in an uncertain environment"....
, a state-price security, also called an Arrow-Debreu security (from its origins in the Arrow-Debreu model
Arrow-Debreu model
In mathematical economics, the Arrow–Debreu model suggests that under certain economic assumptions there must be a set of prices such that aggregate supplies will equal aggregate demands for every commodity in the economy.The model is central to the theory of...
), is a contract that agrees to pay one unit of a numeraire
Numéraire
Numéraire is a basic standard by which values are measured. Acting as the numéraire is one of the functions of money, to serve as a unit of account: to measure the worth of different goods and services relative to one another, i.e. in same units...
(a currency or a commodity) if a particular state occurs at a particular time in the future and pay zero numeraire in all 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.
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.
The Arrow-Debreu model
Arrow-Debreu model
In mathematical economics, the Arrow–Debreu model suggests that under certain economic assumptions there must be a set of prices such that aggregate supplies will equal aggregate demands for every commodity in the economy.The model is central to the theory of...
(also referred to as the Arrow-Debreu-McKenzie model or ADM model) is the central model in the General Equilibrium Theory
General equilibrium
General equilibrium theory is a branch of theoretical economics. It seeks to explain the behavior of supply, demand and prices in a whole economy with several or many interacting markets, by seeking to prove that a set of prices exists that will result in an overall equilibrium, hence general...
and uses state prices in the process of proving the existence of a unique general equilibrium.
Example
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 ω1. Thus, ω1 can take two values: ω1=P and ω1=W.Let's imagine that:
- 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 qP
- 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 qW
The prices qP and qW are the state prices.
The factors that affect these state prices are:
- The probabilities of ω1=P and ω1=W. The more likely a move to W is, the higher the price qW gets, since qW insures the agent against the occurrence of state W. The seller of this insurance would demand a higher premium (if the economy is efficient).
- The preferences of the agent. Suppose the agent has a standard concaveConcave functionIn mathematics, a concave function is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap or upper convex.-Definition:...
utilityUtilityIn economics, utility is a measure of customer satisfaction, referring to the total satisfaction received by a consumer from consuming a good or service....
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 ω1=P and ω1=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 rationalRational expectationsRational expectations is a hypothesis in economics which states that agents' predictions of the future value of economically relevant variables are not systematically wrong in that all errors are random. An alternative formulation is that rational expectations are model-consistent expectations, in...
, he would pay more to insure against the down state that his net gain from the up state would be.
Application to financial assets
If the agent buys both qP and qW, he has secured £1 for tomorrow. He has purchased a riskless bond. The price of the bond is b0 = qP + qW.Now consider a security with state-dependent payouts (e.g. an equity security, an option, a risky bond etc.). It pays ck if ω1=k -- i.e. it pays cP in peacetime and cW in wartime). The price of this security is c0 = qPcP + qWcW.
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:
.Analogously, for a continuous random variable indicating a continuum of possible states, the value is found by integrating over the state-price density.