# State prices

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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), 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 is the vector of state prices for all states. [1] [2] [3] As such, any derivatives 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.

Financial economics 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". Its concern is thus the interrelation of financial variables, such as prices, interest rates and shares, as opposed to those concerning the real economy. It has two main areas of focus: asset pricing and corporate finance; the first being the perspective of providers of capital, i.e. investors, and the second of users of capital.

A vector space is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called axioms, listed below.

In finance, a derivative is a contract that derives its value from the performance of an underlying entity. This underlying entity can be an asset, index, or interest rate, and is often simply called the "underlying." Derivatives can be used for a number of purposes, including insuring against price movements (hedging), increasing exposure to price movements for speculation or getting access to otherwise hard-to-trade assets or markets. Some of the more common derivatives include forwards, futures, options, swaps, and variations of these such as synthetic collateralized debt obligations and credit default swaps. Most derivatives are traded over-the-counter (off-exchange) or on an exchange such as the New York Stock Exchange, while most insurance contracts have developed into a separate industry. In the United States, after the financial crisis of 2007–2009, there has been increased pressure to move derivatives to trade on exchanges. Derivatives are one of the three main categories of financial instruments, the other two being stocks and debt. The oldest example of a derivative in history, attested to by Aristotle, is thought to be a contract transaction of olives, entered into by ancient Greek philosopher Thales, who made a profit in the exchange. Bucket shops, outlawed a century ago, are a more recent historical example.

## Contents

The Arrow-Debreu model (also referred to as the Arrow-Debreu-McKenzie model or ADM model) is the central model in 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 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 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 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 rational, he would pay more to insure against the down state than his net gain from the up state would be.

In 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.

Within economics the concept of utility is used to model worth or value, but its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or satisfaction within the theory of utilitarianism by moral philosophers such as Jeremy Bentham and John Stuart Mill. But the term has been adapted and reapplied within neoclassical economics, which dominates modern economic theory, as a utility function that represents a consumer's preference ordering over a choice set. As such, it is devoid of its original interpretation as a measurement of the pleasure or satisfaction obtained by the consumer from that choice.

In economics, "rational expectations" are model-consistent expectations, in that agents inside the model are assumed to "know the model" and on average take the model's predictions as valid. Rational expectations ensure internal consistency in models involving uncertainty. To obtain consistency within a model, the predictions of future values of economically relevant variables from the model are assumed to be the same as that of the decision-makers in the model, given their information set, the nature of the random processes involved, and model structure. The rational expectations assumption is used especially in many contemporary macroeconomic models.

## 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 ,k=p or w.-- 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: ${\displaystyle c_{0}=\sum _{k}q_{k}\times c_{k}}$.

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

In economics, a complete market is a market with two conditions:

1. Negligible transaction costs and therefore also perfect information,
2. there is a price for every asset in every possible state of the world

In economics, incomplete markets are markets in which the number of Arrow–Debreu securities is less than the number of states of nature. In contrast with complete markets, this shortage of securities will likely restrict individuals from transferring the desired level of wealth among states.

The stochastic discount factor (SDF) is a concept in financial economics and mathematical finance.

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In probability theory, the expected value of a random variable, intuitively, is the long-run average value of repetitions of the same experiment it represents. For example, the expected value in rolling a six-sided die is 3.5, because the average of all the numbers that come up is 3.5 as the number of rolls approaches infinity. In other words, the law of large numbers states that the arithmetic mean of the values almost surely converges to the expected value as the number of repetitions approaches infinity. The expected value is also known as the expectation, mathematical expectation, EV, average, mean value, mean, or first moment.

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 overall general equilibrium. General equilibrium theory contrasts to the theory of partial equilibrium, which only analyzes single markets.

The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing derivative investment instruments. From the partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that the option has a unique price regardless of the risk of the security and its expected return. The formula led to a boom in options trading and provided mathematical legitimacy to the activities of the Chicago Board Options Exchange and other options markets around the world. It is widely used, although often with adjustments and corrections, by options market participants.

In mathematical finance, a risk-neutral measure is a probability measure such that each share price is exactly equal to the discounted expectation of the share price under this measure. This is heavily used in the pricing of financial derivatives due to the fundamental theorem of asset pricing, which implies that in a complete market a derivative's price is the discounted expected value of the future payoff under the unique risk-neutral measure. Such a measure exists if and only if the market is arbitrage-free.

The St. Petersburg paradox or St. Petersburg lottery is a paradox related to probability and decision theory in economics. It is based on a particular (theoretical) lottery game that leads to a random variable with infinite expected value but nevertheless seems to be worth only a very small amount to the participants. The St. Petersburg paradox is a situation where a naive decision criterion which takes only the expected value into account predicts a course of action that presumably no actual person would be willing to take. Several resolutions are possible.

A risk-free bond is a theoretical bond that repays interest and principal with absolute certainty. The rate of return would be the risk-free interest rate. It is primary security, which pays off 1 unit no matter state of economy is realized at time . So its payoff is the same regardless of what state occurs. Thus, an investor experiences no risk by investing in such an asset.

In finance and economics, systematic risk is vulnerability to events which affect aggregate outcomes such as broad market returns, total economy-wide resource holdings, or aggregate income. In many contexts, events like earthquakes and major weather catastrophes pose aggregate risks that affect not only the distribution but also the total amount of resources. If every possible outcome of a stochastic economic process is characterized by the same aggregate result, the process then has no aggregate risk.

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.

Monte Carlo methods are used in finance and mathematical finance to value and analyze (complex) instruments, portfolios and investments by simulating the various sources of uncertainty affecting their value, and then determining the distribution of their value over the range of resultant outcomes. This is usually done by help of stochastic asset models. The advantage of Monte Carlo methods over other techniques increases as the dimensions of the problem increase.

The numéraire is a basic standard by which value is computed. In mathematical economics it is a tradeable economic entity in terms of whose price the relative prices of all other tradeables are expressed. In a monetary economy, acting as the numéraire is one of the functions of money, to serve as a unit of account: to provide a common benchmark relative to which the worths of various goods and services are measured. Using a numeraire, whether monetary or some consumable good, facilitates value comparisons when only the relative prices are relevant, as in general equilibrium theory. When economic analysis refers to a particular good as the numéraire, one says that all other prices are normalized by the price of that good. For example, if a unit of good g has twice the market value of a unit of the numeraire, then the (relative) price of g is 2. Since the value of one unit of the numeraire relative to one unit of itself is 1, the price of the numeraire is always 1.

In finance, a bond option is an option to buy or sell a bond at a certain price on or before the option expiry date. These instruments are typically traded OTC.

A bond graph is a graphical representation of a physical dynamic system. It allows the conversion of the system into a state-space representation. It is similar to a block diagram or signal-flow graph, with the major difference that the arcs in bond graphs represent bi-directional exchange of physical energy, while those in block diagrams and signal-flow graphs represent uni-directional flow of information. Bond graphs are multi-energy domain and domain neutral. This means a bond graph can incorporate multiple domains seamlessly.

Consumption smoothing is the economic concept used to express the desire of people to have a stable path of consumption. People desire to translate their consumption from periods of high income to periods of low income to obtain more stability and predictability. There exists many states of the world, which means there are many possible outcomes that can occur throughout an individual's life. Therefore, to reduce the uncertainty that occurs, people choose to give up some consumption today to prevent against an adverse outcome in the future.

In decision theory, economics, and finance, a two-moment decision model is a model that describes or prescribes the process of making decisions in a context in which the decision-maker is faced with random variables whose realizations cannot be known in advance, and in which choices are made based on knowledge of two moments of those random variables. The two moments are almost always the mean—that is, the expected value, which is the first moment about zero—and the variance, which is the second moment about the mean.

The Shapley–Folkman lemma is a result in convex geometry with applications in mathematical economics that describes the Minkowski addition of sets in a vector space. Minkowski addition is defined as the addition of the sets' members: for example, adding the set consisting of the integers zero and one to itself yields the set consisting of zero, one, and two:

In the mathematical theory of probability, the drift-plus-penalty method is used for optimization of queueing networks and other stochastic systems.

In theoretical economics, an abstract economy is a model that generalizes both the standard model of a exchange economy in microeconomics, and the standard model of a game in game theory. An equilibrium in an abstract economy generalizes both a competitive equilibrium in microeconomics, and a Nash equilibrium in game-theory.

## References

1. economics.about.com Accessed June 18, 2008
2. Rebonato, Riccardo (8 July 2005). Volatility and Correlation: The Perfect Hedger and the Fox. John Wiley & Sons. pp. 323–. ISBN   978-0-470-09140-1.
3. Dempster; Pliska; Bruno Dupire (13 October 1997). Mathematics of Derivative Securities, ch. "Pricing and Hedging With Smiles". Cambridge University Press. pp. 103–. ISBN   978-0-521-58424-1.