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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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First, you are in fact given $p$. You can think of return as a security that costs 1 in current period and pays off $R$ in the next period. The price vector and the payoff matrix are thus $$ p = \begin{bmatrix} 1 \\ 1 \end{bmatrix}, \ X = \begin{bmatrix} 1.1 & 1.0 & 0.9 \\ 1.0 & 1.0 & 1.0 \end{bmatrix} $$ and we need to find positive vector $q$ of state prices such that $p = Xq$. Because the system is undetermined, there can be many such vectors, but as long as some of them are positive, we can be sure there's no arbitrage. To actually do the computation, one way would be to treat $q_3$ as a parameter, solve for $q_1, q_2$ as functions of $q_3$, then try to find $q_3 > 0$ such that implied values of $q_1,q_2$ are positive. But in this case it's easy to see that $q = (\frac{1}{3}, \frac{1}{3}, \frac{1}{3})$ does the job.

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