Why Two Assets Cannot Determine State Prices Across Three States

Summary

State prices are the current values of claims that each pay one unit in exactly one state and nothing in the others. To infer them from traded assets, each state-specific payoff must be replicable by a portfolio of those assets. The example has a bond paying one in all states and a stock with different payoffs across three states. The answer describes testing replication by solving a linear system for each state-specific payoff vector.

In this setup, only two instruments are available for three states, so the payoff vectors cannot span all three independent state claims. The system therefore has no solution for the missing state claims, and the state price vector cannot be uniquely determined from these securities alone. This illustrates market incompleteness: the issue is the number of independent payoff directions, not the choice of an arbitrary target payoff. The answer is conceptual and does not calculate alternative bounds or state prices under additional assumptions. The question’s stock payoff notation repeats one state label, but the answer proceeds using payoffs of 80, 100, and 120 across the three states.

Key ideas

• A state price is the value of a claim paying only in one specified state.
• State prices can be recovered by replicating each state-specific payoff with traded assets.
• Two assets cannot span three independent state payoffs in the stated example.
• When state claims cannot be replicated, the available securities do not determine a complete state price vector.

Tags

# Determine state price vectors?


# Determine state price vectors?












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 by gt6989b (score 2, accepted)

https://quant.stackexchange.com/a/9307

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.