Implications of No Arbitrage

An asset is redundant if $A^j=\sum_{k\neq j}\alpha_kA^k$.

Each $q_k=\pi A^k$. So \begin{equation*} q_j=\pi\sum_{k\neq j}\alpha_kA^k=\sum_{j\neq k}\alpha_k\pi A^k =\sum_{k\neq j}\alpha_kq_k. \end{equation*}

An Arrow Security is an asset that pays off 1 in a single state $s$ and 0 otherwise. If $\pi$ is a state price vector, then $q=\pi$ is the price vector of the Arrow securities.

 

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