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Now, the proof I have read goes like this:

Take assets 1 and 2, entirely identical. By assumption there is a pricing vector, i.e. $\sum_s\pi_sd^1_s=q^1$ and $\sum_s\pi_sd^2_s=q^2$ where $d^i_j$ is the payoff of asset $i$ in state $j$ and $\pi_i$ is the $i$-th element of the pricing vector $\vec{\pi}$. Since $d^1_s=d^2_s$ for all $s$ (identical assets), we must have $q^1=q^2$.


My question about this proof is that the assumption does not say only one $\vec{\pi}$ exists. It seems possible to me that there can be another price vector and we would have got a different price for the same asset. For example, price vector $(1,2,3)$ and $(2,3,4)$.

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By pricing vector I assume you refer to the state price vector, whose definition can be found here.

According to the above definition, the price of a particular state specifically refers to the the price of a contract paying exactly one unit of numeraire if that state does occur and zero unit otherwise. And state price vector is thus the vector of state prices for all states.

So it seems to me that this state price vector is kind of normalized, only one specific state price vector exists for a given measure. Thus if two assets are identical (providing exactly same cash flows under any state), their price should be equal to each other no matter what. I think this is the point of the law of one price.

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