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Is there any way to make use of the Beta of an underlying and index, and the implied volatility of options on that underlying and the index?

To specify, if we have available the implied volatility of a closely related index to a single-name option but not the implied volatility of the option itself, can we take the implied volatility of the index and use the Beta to scale it in a way that it is more representative of the implied volatility of the option on the stock? Or can we do no better than use implied volatility of the index directly.

If not, is there some other way to get a better estimate of the implied volatility?

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    $\begingroup$ I don’t think IV of an index and the beta with respect to that index characterizes the IV of a single-name option. Just as with actual volatility, the single-name IV can have a large and unknown idiosyncratic component. $\endgroup$ – fesman Oct 21 '20 at 10:42
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Beta is a measure of the historical volatility of a security compared with the volatility of an index which contains many stocks. So its value depends on the historical volatility of both securities.

Implied volatility is an estimate of future volatility and it can change dramatically in days, even hours (consider an earnings announcement or other pending news such as clinical trials) and if it does, beta will likely be relatively unchanged.

Therefore, I think that they are independent of each other and no derivation is possible. However, take my opinion with a grain of salt because I'm just a retail guy who doesn't understand 90% of what I read on this BB :->)

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