Suppose we have two risky assets and one risk-free asset in the market. The market is incomplete in that there are three assets and four states. The price vector at $t_0$ is: $\boldsymbol{p_0}=[p^s_{1},p^s_{2}, 1]^\intercal$ and the payoff at $t_1$ is $\boldsymbol{p_1} = \begin{bmatrix} p_1^1 & p_1^2 &p_1^3 & p_1^4 \\ p_2^1 & p_2^2 &p_2^3 & p_2^4 \\ 1 & 1 & 1 & 1 \end{bmatrix}.$ Given that $p_{n}^{4}<p_{n}^{3}<p_{n}^{2}<p_{n}^1$ for $n=1,2$. For the market to be arbitrage-free, we already have $p_{n}^{4}<p^s_{n}<p_{n}^1$ for $ n = 1,2$. To check if there is arbitrage opportunity, I am using the theorem that the no arbitrage condition is satisfied if there exists positive risk-neutral probabilities under this setup. Are there any additional conditions that is needed to make sure there exists positive risk-neutral probabilities?



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