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This is a result of Ledoit et al Lemma proof in Appendix of http://www.ledoit.net/9-98.pdf

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There are tons of market where vol smile doesn't exist - either because no one makes a market on the call/put options (private equity, physical real estate comes to mind) or only the ATM option gets traded infrequently. You can't have volatility smile without a vol market. On the other hand (and maybe more relevant to what you are trying to get at), if ...

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Peter Jaeckel has written various papers on this. "by implication" and "Let's be rational" are the most recent ones. He also provides code on his website www.jaeckel.org. (Note: the question asked for literature.)

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Look on Google for Asymptotic behavior of Implied Volatility Near Infinity you will find results like : $$I(K) \stackrel{K\to\infty}{=} \sqrt{\frac{2}{T}}\left(\sqrt{\ln \frac{K}{C(K)}}-\sqrt{\ln\frac{1}{C(K)}}\right) +\text{O}_{K\to \infty}\left(\frac{\ln\ln\frac{1}{C(K)}}{\sqrt{\ln\frac{1}{C(K)}}}\right)$$

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What you are trying to do is fit a volatility surface for a given underlying. Once you have a volatility surface you can price an option for an arbitrary expiration and strike. There are numerous approaches to do this and the linear interpolation methods mentioned in the other examples are okay but be careful in the following situations where there is: a ...

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You could check at the methodology for VIX. The VIX itself yields one number - but you might instead return a set of numbers for your skew analysis.

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You use a form of interpolation(start with linear) between the 30 day to maturity IV and the 90 to get the 60,

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It's clear that over time $t\rightarrow t+dt$ the (expected) implied variance in the BS world (under the risk-neutral measure $\mathbb{Q}$) is $\sigma^2 dt$. The realized variance of an arbitrary single log stock path (under the physical measure $\mathbb{P}$) is \sum_{i=1}^\infty \sigma^2(W_{i+1}-W_i)^2 \approx \sum_{i=1}^n ...

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I succeeded to have the expected answer, and I was on the right way : The implied volatility is the ATM IV The realised volatility is defined as the absolute value of the log return We know that a gaussian random variable is in the range +- 1*sigma with a probability of 68%. So, the correct answer is approximately 32% Should you have any comment I am ...

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