No-Arbitrage and State Price Vectors in an Incomplete Market

Summary

The document presents a one-period market with three possible states and two assets: a risk-free asset and a risky asset whose returns differ by state. It asks how to prove that no arbitrage exists and how to describe the family of state price vectors when the payoff matrix has fewer assets than states. The stated state probabilities and asset returns provide a concrete example, while the question notes that the price-payoff system cannot be inverted as a square matrix.

The response points out that no-arbitrage does not require a unique state price vector. With an incomplete market, the equations linking asset prices and state prices may admit multiple solutions; existence of a suitable vector is the relevant question. The document recommends solving the linear system to find the family, but does not show the calculation or establish its sign constraints. Thus it introduces the core idea while leaving the requested proof and full characterization incomplete.

Key ideas

• The example has three states but only two traded assets, so the state price vector need not be unique.
• No-arbitrage can be assessed by checking whether a suitable state price vector exists.
• The asset pricing equations form an underdetermined linear system in this market.
• The document suggests row reduction but does not carry out the proof or characterize the solution family.

Tags

# Proving there exists no arbitrage opportunities given 3 states and 2 assets


# Proving there exists no arbitrage opportunities given 3 states and 2 assets












Assume there are 3 states of the world: w1, w2, and w3. Assume there are two assets: a risk-free asset returning Rf in each state, and a risky asset with Return R1 in state w1, R2 in state w2, and R3 in state W3. Assume the probabilities are 1/4 for state w1, 1/2 for state w2, and 1/4 for state w3. Assume Rf=1.0 and R1= 1.1, R2=1.0 and R3= 0.9.

(a) Prove that there are no arbitrage opportunities. (b) Describe the one-dimensional family of state price vectors (q1,q2,q3)>

For (a), I believe this is equivalent to showing there exists a state price vector.

I know p=Xq, but since we are only given two assets X doesn't have an inverse so I don't know how to compute q. Further, we are not given p. How do I show a state price vector exists?

## Answer by user9403 (score 1)

https://quant.stackexchange.com/a/16736

A unique state price vector does not have to exist for there to be no arbitrage. It sounds like the state price vector in question has infinitely many solutions. Try to reduce the price matrix to row echelon form and show that at least one state price vector exists.

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