In
financial economics
Financial economics, also known as finance, is the branch of economics characterized by a "concentration on monetary activities", in which "money of one type or another is likely to appear on ''both sides'' of a trade".William F. Sharpe"Financial ...
, a state-price security, also called an Arrow–Debreu security (from its origins in the
Arrow–Debreu model
In mathematical economics, the Arrow–Debreu model suggests that under certain economic assumptions (convex preferences, perfect competition, and demand independence) there must be a set of prices such that aggregate supplies will equal aggreg ...
), a pure security, or a primitive security 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. The state price
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
is the vector of state prices for all states.
See .
The
Arrow–Debreu model
In mathematical economics, the Arrow–Debreu model suggests that under certain economic assumptions (convex preferences, perfect competition, and demand independence) there must be a set of prices such that aggregate supplies will equal aggreg ...
(also referred to as the Arrow–Debreu–McKenzie model or ADM model) is the central model in
general equilibrium theory
In economics, general equilibrium theory attempts to explain the behavior of supply, demand, and prices in a whole economy with several or many interacting markets, by seeking to prove that the interaction of demand and supply will result in an ov ...
and uses state prices in the process of proving the existence of a unique general equilibrium.
State prices may relatedly be applied in
derivatives
The derivative of a function is the rate of change of the function's output relative to its input value.
Derivative may also refer to:
In mathematics and economics
* Brzozowski derivative in the theory of formal languages
* Formal derivative, an ...
pricing and hedging: a contract whose settlement value is a function of an underlying asset 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;
see
Contingent claim analysis In finance, a contingent claim is a derivative whose future payoff depends on the value of another “underlying” asset,Dale F. Gray, Robert C. Merton and Zvi Bodie. (2007). Contingent Claims Approach to Measuring and Managing Sovereign Credit Ri ...
.
Breeden and
Litzenberger's work in 1978 established the latter, more general use of state prices in finance.
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 ω
1. Thus, ω
1 can take two values: ω
1=P and ω
1=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 q
P
* 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
W
The prices q
P and q
W are the state prices.
The factors that affect these state prices are:
* "Time preferences for consumption and the productivity of capital".
That is to say that the
time value of money
The time value of money is the widely accepted conjecture that there is greater benefit to receiving a sum of money now rather than an identical sum later. It may be seen as an implication of the later-developed concept of time preference.
The t ...
affects the state prices.
* The ''probabilities'' of ω
1=P and ω
1=W. The more likely a move to W is, the higher the price q
W gets, since q
W 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
concave
Concave or concavity may refer to:
Science and technology
* Concave lens
* Concave mirror
Mathematics
* Concave function, the negative of a convex function
* Concave polygon, a polygon which is not convex
* Concave set
* The concavity
In ca ...
utility
As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosopher ...
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
1=P and ω
1=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
Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abili ...
, he would pay more to insure against the down state than his net gain from the up state would be.
Application to financial assets
If the agent buys both q
P and q
W, he has secured £1 for tomorrow. He has purchased a riskless bond. The price of the bond is b
0 = q
P + q
W.
Now consider a security with state-dependent payouts (e.g. an equity security, an option, a risky bond etc.). It pays c
k if ω
1=k ,k=p or w.-- i.e. it pays c
P in peacetime and c
W in wartime). The price of this security is c
0 = q
Pc
P + q
Wc
W.
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
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
indicating a continuum of possible states, the value is found by
integrating over the
state price density
State may refer to:
Arts, entertainment, and media Literature
* '' State Magazine'', a monthly magazine published by the U.S. Department of State
* ''The State'' (newspaper), a daily newspaper in Columbia, South Carolina, United States
* ''Our ...
.
See also
*
Asset pricing
In financial economics, asset pricing refers to a formal treatment and development of two main Price, pricing principles, outlined below, together with the resultant models.
There have been many models developed for different situations, but cor ...
*
Complete market
In economics, a complete market (aka Arrow-Debreu market or complete system of markets) is a market with two conditions:
# Negligible transaction costs and therefore also perfect information,
# there is a price for every asset in every possible st ...
*
Contingent claim analysis In finance, a contingent claim is a derivative whose future payoff depends on the value of another “underlying” asset,Dale F. Gray, Robert C. Merton and Zvi Bodie. (2007). Contingent Claims Approach to Measuring and Managing Sovereign Credit Ri ...
*
Incomplete markets In economics, incomplete markets are markets in which there does not exist an Arrow–Debreu security for every possible state of nature. In contrast with complete markets, this shortage of securities will likely restrict individuals from transfer ...
*
Stochastic discount factor The concept of the stochastic discount factor (SDF) is used in financial economics and mathematical finance. The name derives from the price of an asset being computable by "discounting" the future cash flow \tilde_i by the stochastic factor \tilde, ...
*
List of asset pricing articles
*
References
{{economics
Financial risk
Financial economics
Pricing