Determine state price vectors?

Question

I have 3 states with two assets, stocks and bonds.

The bond has a payoff of 1 in every state of the world. And the stock has a current price of $S_0 = 100$ and payoffs of $S_1(w_1)=80$, $S_1(w_3)=100$ and $S_1(w_3)=120$..

I want to compute the state price vectors:

I know that the state price vectors can be computed using $\sum_{k=1}^K \psi (D\theta)_k>0$ or just $W=D\times \theta $ where D is the payoff matrix, $\theta$ is the replication portfolio.

I also know that D is just the matrix of the payoffs therefore: $$\begin{pmatrix} 1 & 80 \\ 1 & 100 \\ 1 & 120 \end{pmatrix} \times \begin{pmatrix} \psi_1 \\ \psi_2 \end{pmatrix}$$

However, i do not know which W to chose?

There I appreciate your answers!

Answer 1

The state price vector are the prices of securities which pay \$1 if and only if that state of the world occurs. This is just a question of being able to replicate the payoffs $$ \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} $$ with payoff vectors $\vec{b} = [1,1,1]^T$ and $\vec{s} = [80, 100, 120]^T$. This is just a matter of Gaussian Elimination.

The problem is, however, that no such solution exists. That means it is not possible to determine state prices in such a scenario.

The high-level problem is, you need as many (independent) instruments as the states of the world, and you have 2 instruments for 3 states.