Nothing to add to the excellent points above, but just a comment on the term "fair value".
The term is usually used to describe the worth of an option or futures contract as determined by a mathematical model, viz using HV. But as we have all discussed, HV looks backward. What the trader wants to know is what will be the realization of volatility going forward.
Therefore fair value in the sense above is a nebulous concept, because we can't see the future. If we pay up for an option at 50% IV because HV was 50%, and the realized volatility drops to 25%, we haven't got fair value, in hindsight, at all.
I like to look at fair value as being in tune with my volatility projection, therefore for me, fair value might not be anywhere close to that value determined by HV. It's a guess, but it's what I think is fair.
Hence "fair value" is really intrinsic to projected IV as per mazza's and cuttlefish's comments above.