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A test for arbitrage opportunities with an LP is to minimize the cost of setting up the portfolio, subject to the restriction that the portfolio loses money in no state of the world. (Note that in your formulation you are missing the actual objective; you only list constraints.) If you find a portfolio that has a negative cost (i.e. you get paid for holding ...


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An incomplete market can be free of arbitrage. For an arbitrage, we need a portfolio vector $p\in\mathbb{R}^2$ such that $Ap\geq0_{\mathbb{R}^3}$, in the sense that all rows (states) have a non-negative payoff and that at least one row has a strictly positive payoff. In addition, $\langle p,S\rangle\leq0_{\mathbb{R}}$, i.e. the arbitrage should be costless (...


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