Arrow Debreu Pricing
Overview
IN Arrow Debreu pricing, we focus on one time period, and divide the states of the world one period forward
into a finite number ({% K %}) of states. In addition, we assume there are {% N %} securities.
The define the payoff matrix to be a matrix of the payoffs of each of the {% N %} securities
in each of the {% K %} future states of the world.
{%
D = \begin{bmatrix}
d_{11} & ... & d_{1K} \\
. & ... & .\\
d_{N1} & ... & d_{Nk} \\
\end{bmatrix}
%}
The a portfolio is defined as a vector of holdings of each asset.
{%
\begin{bmatrix}
\theta_1 \\
. \\
\theta_K \\
\end{bmatrix}
%}
If the prices of each asset is given by a column vector {% S %}, then the cost of the portfolio is
{% S^T \vec{\theta} %}
The payoffs of the given portfolio in each state of the world is given by
{% D \vec{\theta} %}
Arbitrage
{% \theta %} is an arbitrage portfolio if the following is satisfied
{% \vec{S}^T \vec{\theta} = 0 %} and {% D \vec{\theta} \geq 0 %}
where in this case {% \geq %} means that each element of the vector is greater than or equal to .
Complete Markets
In a complete market, we have a set of {% K %} assets such that the matrix {% D %} of these asset payoffs is
square and invertible. When this is the case, if given a payment vector {% P %} for the future state of the
world, we can construct a portfolio {% \psi %} that replicates those payments.
{% \psi = D^{-1} \vec{S} %}
Pricing Kernel
When the future states of the world are not discrete, the same logic can apply, but now we must
integrate instead of summing.
{% P = \int \Psi(x) D(x) dx %}
