Arrow-Debreu Securities: Definition, Pricing, Example

This post is a brief and somewhat incomplete look at Arrow-Debreu pricing theory, based on my lecture notes.

Hope it helps!

What is an Arrow-Debreu Security

Arrow-Debreu securities (or state-contingent claims, or simply state claims) are financial instruments that pay off in different states of the world.

More specifically, an A-D security pays one unit of consumption in one state of the world and pays zero otherwise. In contrast, a complex security is one that pays off in more than one state of nature.

In general, state-contingent claims are financial assets whose value depends on how a specific situation plays out.

For example, a stock call option gives you the right to buy a specific number of shares at a predetermined price, but only if the stock reaches a certain price within a specific time frame. This is a state-contingent claim.

In other words, the claim’s value depends on the market’s behavior and is therefore contingent.

In this context, states of the world are the potential future outcomes. They determine the likelihood of a good outcome materializing at the specified time. As a result, the security is worth more if the likelihood of a good outcome is bigger than the likelihood of a bad outcome.

An Arrow-Debreu security is a theoretical concept in economics that defines a contract for the exchange of a specific quantity of a financial asset for a predetermined price at a future date. This enables the transfer of risk between parties.

It is named after economists Kenneth Arrow and Gerard Debreu, who were awarded the Nobel Prize in Economic Sciences for their contributions to the study of general equilibrium theory.

In reality, Arrow-Debreu securities don’t exist because the concept of states of the world is abstract. It’s simply a proxy for the building blocks of asset pricing.

Arrow-Debreu Pricing

Arrow-Debreu prices represent the prices at which assets would be traded if markets were perfectly competitive.

What does perfectly competitive mean?

A perfectly competitive market is a theoretical concept in economics that describes a market with the following conditions:

• Lots of buyers and sellers.
• All buyers have all information about the market and the goods and services available.
• The goods traded are identical, so no buyer or seller has any advantage over others.
• No single buyer or seller has the power to influence market prices.
• The interactions of all buyers and sellers are what determine prices.
• Everyone can freely enter and exit the market.
• Prices reflect the true value of goods. Thus, the market is efficient and the allocation of goods and services is optimal.

Also, since all Arrow-Debreu prices are strictly positive, there are no arbitrage opportunities in the market.

Risk-neutral pricing is essentially equivalent to A-D pricing, and it is the center of mathematical finance and asset pricing.

Now, here’s how to calculate state prices:

How to Calculate Arrow-Debreu Prices

To calculate Arrow-Debreu prices, you must first identify all the assets traded in the market you’re studying, as well as the potential states of the world.

Arrow-Debreu pricing is given by the following equation:

\[p=Xq\]

Where:

• p is the vector of equilibrium prices.
• X is the payoff matrix.
• q is the unique vector of state prices.

This means that given the securities traded in an economy and their respective equilibrium prices, you can find the corresponding Arrow-Debreu prices:

\[q=X^{-1}p\]

Let’s go through a quick Arrow-Debreu pricing example to put these concepts in motion:

Arrow-Debreu Equilibrium Example

Consider a two-date (today and tomorrow) economy where there are three states of the world tomorrow. Three securities exist in this economy:

• A risk-free security with a gross return of 1.1.
• A risky security that pays 5, 10, and 15 in the three states of tomorrow, and its price (equilibrium price) today is 8.
• A call option on the risky security with an exercise price of 12 and a current price of 1.

Thus, the matrix of states and assets is as follows:

\[X=\begin{bmatrix}1.1 & 1.1 & 1.1 \\5 & 10 & 15 \\ 0 & 0 & 3\end{bmatrix}\]

The equilibrium prices have embedded the preferences of agents in the economy (utility functions) and their attitudes towards risk, which determine supply and demand conditions.

How do you find the Arrow-Debreu prices?

First, you need to check if the following conditions are true to ensure market completeness:

• The number of states is equal to the number of assets, which indicates the market is complete.
• The payoffs of the 3 assets are linearly independent. In other words, you must check if the payoff matrix is not singular and if the determinant of the matrix is different than zero. If it is zero, the market is incomplete. If the payoff of the assets is linearly dependent, you can replicate one of them as a portfolio of the others, which means that security doesn’t really exist.

You’ll find that the determinant of this matrix is 16.5, hence different than zero. Since all other conditions also hold true, you are ready to compute the A-D prices.

\[p=Xq\Leftrightarrow q=X^{-1}p=\begin{bmatrix}1.1 & 1.1 & 1.1 \\5 & 10 & 15 \\ 0 & 0 & 3\end{bmatrix}^{-1}\begin{bmatrix}1 \\8 \\1 \end{bmatrix}=\begin{bmatrix}0.5515 \\0.0242 \\0.3333 \end{bmatrix}\]

Any doubts? Feedback? Questions? Leave them in the comments below!