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I just read a Corollary in a finance course note:

Suppose the market is arbitrage free and $C$ is a contingent claim. Then $C$ is attainable if and only if it admits a unique arbitrage-free price.

I don't know how to prove the converse. Assuming we are just dealing with one period model. What I got so far: if it admits a unique arbitrage-free price, then price of a contingent claim $C$, $\Pi(C) = E_Q[\frac{C}{1+r}]$ is unique. But I am not sure how to prove $C = h \cdot S_1$ $ \textbf{P}$- $ a.s.$ for some strategies $h \in$ $\textbf{R}^{N+1}$ where there are $N$ stocks and one bond, to conclude that $C$ is attainable.

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  • $\begingroup$ You should prove the contrapositive. Assuming $C$ not to be attainable allows one to construct non-unique but equivalent martingale measures i.e. $C$ does not admit a unique arbitrage-free price. As the statement is a corollary I assume something like this is proved just before the corollary is stated? $\endgroup$ – Trevor Hansen Dec 12 '18 at 15:28
  • $\begingroup$ @TrevorHansen Thank you! This corollary is right after law of one price. It has two parts and the second part is just the contrapositive in your comment. I just didn't realize that I can use proof by contrapositive. $\endgroup$ – YellowRiver Dec 12 '18 at 23:13

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