4.2.2b Arrow-Debreu Securities

Arrow-Debreu Securities

An Arrow-Debreu security is a contract that pays exactly one unit of currency if a specific future state occurs, and nothing otherwise.

In an arbitrage-free market with $N$ assets and $s$ possible future states, every asset’s price can be expressed as a linear combination of exactly $s$ Arrow-Debreu securities, with no two securities paying in the same state. The price of an Arrow–Debreu security is called the state price $q_i$ for state $i\in \{1,\dots ,N\}$. For a payoff matrix $X\in {\mathbb{R}}^{s\times N}$ ($N\ge s$) and a state price vector $q\in {\mathbb{R}}^s$, the vector of today’s asset prices $p\in {\mathbb{R}}^N$ is expressed as
$p=X^{T}q.$

The existence of a distinct Arrow–Debreu security for every possible state implies that the market is complete. In other words, the market is complete if $\mathrm{Rank}(X)=s$. This means the $s$-dimensional space of all possible state payoffs is fully spanned by the assets, so every contingent claim can be replicated as a linear combination of these Arrow–Debreu securities.

Properties

(State Price Decomposition) $q=\{\begin{array}{ll}(X^T{)}^{-1}p& \text{if }N=s,\\ (X{X}^T{)}^{-1}Xp& \text{if }N\ge s,\\ \varnothing & \text{otherwise.}\end{array}$

(Uniqueness of State Price) If LOOP holds and market is complete, then $q$ exists and is unique.

(Market Completeness) If $\min (q)>0$, the market is complete.

Examples

Asset Pricing With Sate Prices

We can describe a 3-state, 3-asset complete market as follows:

| State/Asset | ${\text{AD}}_1$ | ${\text{AD}}_2$ | ${\text{AD}}_3$ | ${\text{A}}_1$ | ${\text{A}}_2$ |
|:-------:|:-----:|:-----:|:-----:|:-------:|:-------:|
| $S_1$ | 1 | 0 | 0 | 3 | $u_1$ |
| $S_2$ | 0 | 1 | 0 | 2 | $u_2$ |
| $S_3$ | 0 | 0 | 1 | 1 | $u_3$ |
| Price | $q_1$ | $q_2$ | $q_3$ | | |
Then, the price of asset 1 can be expressed as the linear combination $q_{{\text{A}}_1}=3q_1+2q_2+q_3$. Similarly, the price of asset 2 is $q_{{\text{A}}_2}=u_1{q}_1+u_2{q}_2+u_3{q}_3$.

Mortgage-Backed Securities