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Consider an economy with $J = 2$ assets and $S = 3$ states. The $J\times S$ payoff matrix for the two assets is $$X = \begin{pmatrix} 0 & 3 & 3\\ 1 & 1 & 0\\ \end{pmatrix}$$ and the asset prices $P' = (5/9,2)$.

Determine whether there are arbitrage opportunities in this market. In addition, find the minimum and maximum prices for the risk free asset.

Attempted solution: We have $$P = X q = \begin{pmatrix} 0 & 3 & 3\\ 1 & 1 & 0\\ \end{pmatrix} \begin{pmatrix} q_1\\ q_2\\ q_3 \end{pmatrix} = \begin{pmatrix} 3q_2 + 3 q_3\\ q_1 + q_2 \end{pmatrix}$$ I believe we need to fix $q_2$ but the solution states that $q = (5/9 - q_2,q_2,6/9 - q_2)$. For $q_2\in (0,5/9)$ all $q$'s are positive so there is no arbitrage. $q_f \in (6/9,11/9)$.

I am not sure how we arrive at this solution any suggestions are greatly appreciated.

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There was an error in my professors question he changed $$P = \begin{pmatrix} 2\\ 5/9 \end{pmatrix}$$ Thus when we fix $q_2$ we get $$q = (5/9 - q_2,q_2,6/9 - q_2)$$ Thus for $q_2\in (0,5/9)$ all $q$'s are positive so there is no arbitrage. Then clearly $q_f = (6/9,11/9)$

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