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It is usually stated that the implied volatility is statistically generally --- not always --- greater than the realized volatility. It seems this statement is made with regard to the implied volatility at the money or when the strike is equal to the future of the underlying. Is this statement statistically generally true for all strikes?

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    $\begingroup$ You can find a screenshot of IV vs HV in this answer. Big unexpected events like Brexit make HV spike above what IV was for that period . I am voting to close this question because it's easy to check and off topic in my opinion. $\endgroup$
    – AKdemy
    Nov 10, 2023 at 5:27
  • $\begingroup$ @AKdemy: Obviously the statement is not always true but is said to be so most of the time at least for the ATM implied vol. This is a statistical matter and is thus not so "easy to check", least of all by a Bloomberg screenshot. So this is no reason to vote to close this question. $\endgroup$
    – Hans
    Nov 10, 2023 at 9:23
  • $\begingroup$ I would say that the focus on this question regards how the IV/RV-premium is distrubuted over strikes, so I would not vote to close it. Maybe it could gain from an elaboration that it really is about different strikes and perhaps why one should or should not expect a variation across strikes? $\endgroup$
    – Mats Lind
    Nov 10, 2023 at 9:57
  • $\begingroup$ @MatsLind: I have edited the question to reflect the emphasis over the variation of strikes. $\endgroup$
    – Hans
    Nov 10, 2023 at 15:38
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    $\begingroup$ We had a related (but not the same, AFAIR) question some time back. Statistically and economically, the premium is on downside risks, only; we did a study on that some time ago sciencedirect.com/science/article/abs/pii/S0378426620301412 $\endgroup$ Dec 19, 2023 at 20:54

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Most of the time, but not always. When a trader underwrites an option (selling a call or put), they do not get the choice to exercise - the buyer has the choice. So the buyer pays a "premium" for that choice, similar to the idea of insurance. If you later sold the option and it turns out that the realised vol < implied vol of the contract you bought, then you would lose money. Thus, in the long-run, you lose money when realised < implied vol.

It then follows, if one assumes that realised vol < implied vol is always true, like your question, why ever be long options if you always make money being short? Because it's not always true. During black-swan events, realised vol > implied vol. If you look at the VIX, during times when vol is > 40%, traders who are short options are generally selling their options priced at implied vol << realised vol and more often than not, would be losing money.

I go back to the insurance analogue. The insurance buyer (long) "wins" during low probability events, but is losing on a day-to-day basis. The insurance seller (shorter) "wins" small amounts on a day-to-day basis

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  • $\begingroup$ I meant to ask if the statement is "statistically generally" true. I edited my question to emphasize this point. Moreover, my question is about the varacity of this statement on strikes other than the one at the money or equaling to the underlying price. $\endgroup$
    – Hans
    Nov 10, 2023 at 9:28
  • $\begingroup$ @Hans Yes. In fact, the difference is worse further away from ATM. The underlying has only 1 realised volatility, but as you can observe with the smile/smirk, implied vol is larger further away from ATM. Thus, you pay extra a worse case scenario. If we conclude that generally imp vol from ATM options is > realised vol, then obviously ITM and ATM imp vol > realised vol. There is only 1 realised vol. $\endgroup$ Dec 19, 2023 at 23:43
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Standard realized volatility calculation only corresponds theoretically to ATM strikes not OTMs. Unless you have a different definition of realized volatility you cannot compare it to OTM implied.

However, on the whole vol curve level I believe the statement to be true. For a pure ROC analysis you'll generally make more money shorting OTMs than ATMs.

The difference starts to occur when you look at the 3rd order and 4th order moments instead of just variance (i.e. skewness and kurtosis). That will be much higher for a short OTM position than a short ATM one.

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  • $\begingroup$ How do you use the 3rd and 4th order moments of the historical returns? What do you compare them to? $\endgroup$
    – Hans
    Dec 4, 2023 at 22:08
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if you look at v short dated SOFR options - at the moment, realised far outweighs implieds. the reason is, implieds are also v directional and sentimentally driven.

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