No Arbitrage

If there is a portfolio with a semi-positive return vector, then some spot-market budget sets are unbounded. Equilibrium requires is that no such portfolio exists. This condition may restrict asset prices.

Let \begin{equation*} M=\begin{bmatrix}-q\\ A\end{bmatrix} \end{equation*} with column space $\langle M\rangle$.

The no arbitrage condition is satisfied if there is no portfolio $z$ such that $Mz > 0$. That is, \begin{equation*} \langle M\rangle\cap\R^{S+1}_+=\{0\}. \end{equation*}