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I hope anybody can help me. According to Gatheral and Jacquier (https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2033323) no Butterfly Arbitrage can be expressed like this:

Define the function $\displaystyle g(k):= \left( 1- \frac{k \omega'(k)}{2 \omega(k)}\right)^2- \frac{\omega'(k)}{4} \left(\frac{1}{\omega(k)}+\frac{1}{4} \right) + \frac{\omega''(k)}{2}$

A slice is free of Butterfly Arbitrage $\Leftrightarrow$ \ the function $g(k) \geq 0 \ \forall k \in \mathbb{R}$ and $ \lim \limits_{k \rightarrow \infty} d_+(k)=-\infty$. The second condition here is equivalent to call prices converging to 0 as $k \to \infty$ (if I saw this right this is needed to have a density). Now I'm asking is there an arbitrage strategy to "see" the arbitrage if this point isn't fullfilled?

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    $\begingroup$ The condition $g(k)\ge 0$ basically says that the option price is a convex function of the strike. If this condition is violated, you can constructed an arbitrage strategy as in this answer. $\endgroup$ – Gordon Jan 17 at 17:19
  • $\begingroup$ But why do I need $\lim \limits_{k \to \infty}d_+(k)=-\infty$? $\endgroup$ – J.Doe Jan 21 at 11:15

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