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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?

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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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  • $\begingroup$ How can I do that if I only have the matrix X? Aren't I missing p? Or is there another way to calculate it? $\endgroup$
    – user2034
    Commented Feb 25, 2015 at 13:45

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