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I'm trying to justify the lower bound for the price of a contingent claim (a European one) which is not marketable in an arbitrage free market. I would like to have your advice on my way to do it:

First, I start with the set $\{V_0(\theta) : V_T(\theta)\leq H\; and\;\theta\; admissible\}$ where $V_0(\theta)$ is the initial value of a portfolio with strategy $\theta$. I would like to show that $C_{-}:=\sup{\{V_0(\theta) : V_T(\theta)\leq H\; and\;\theta\; admissible\}}$ is a lower bound for the pricing of the contingent claim. Indeed, consider a price $V_0(\theta)<C_{-}$, then the buyer can short sell $C_{-}$, buy $V_0(\theta)$ and invest $(C_{-}-V_0(\theta))$ in a risk free asset such that at the maturity date $T$ the buyer can face the short selling by using $H$ and he receives $(C_{-}-V_0(\theta))(1+r)^T$ which is strictly positive.

However I am wondering if my arbitrage strategy is correct since why one will buy $C_{-}$ if there exists $V_0(\theta)<C_{-}$ ?

I think I have misunderstood something Thank you a lot

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