State prices

State prices

In financial economics, a state-price security, also called an Arrow-Debreau 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. [ [http://economics.about.com/od/economicsglossary/g/statepricev.htm economics.about.com] Accessed June 18, 2008]

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-Debreau securities, and thus as a weighed 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.

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 qWThe 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 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 she would gain if the state was "P". Now, even if you assume that the abve-mentioned probabilities ω1=P and ω1=W are equal, the changes in utility for the agent are not: Due to her 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, she would pay more to insure against the down state that her net gain from the up state would be.

Application to financial assets

If the agent buys both qP and qW, she has secured £1 for tomorrow. She 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: c_0 = sum_k q_k imes c_k.

Analogously, for a continuous random variable indicating a continuum of possible states, the value is found by integrating over the state-price density.

References


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