# Prove unique arbitrage-free price implies attainable

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.

• 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? – Trevor Hansen Dec 12 '18 at 15:28
• @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. – YellowRiver Dec 12 '18 at 23:13