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Assume for simplicity that the expiration time of an option is $1$ the initial stock price is $1$ and there is no dividend yield and the risk free return is $0$.

How is it possible to show that the following holds for the implied volatility : $$ \sigma_{\operatorname{imp}}^2 = \operatorname{O}(\log K)$$

Where $K$ is the strike price of the option.

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