There are many articles and posts here claiming that the implied volatility is the expectation of future realized volatility. I don't understand. To begin with, isn't implied volatility homogeneous to a volatility and realized volatility homogeneous to a volatility times the square root of a time? How something not in the same dimension as realized volatility be the expectation of it?
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$\begingroup$ no idea what you're talking about. Just google a primer on implied volatility. $\endgroup$– Edward WatsonMay 17, 2021 at 23:04
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$\begingroup$ @EdwardWatson For example here quant.stackexchange.com/questions/32951/… the answer claims so but I don't see how his equation helps him to say that implied volatilities can be regarded as the market’s expectation of future realized volatilities? $\endgroup$– SuttNFGMay 17, 2021 at 23:13
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This is not true.
IVOL does not generally predict future realized vol (it is definitely not an unbiased predictor). If you plot (quoted) IVOL and strike, you see what is called a smile or skew. You can look here for a stylized example. Realized vol is only one number. So ignoring everything else, this statement cannot be true already. In reality IVOL depends on lots of things, including supply and demand. You can google Volatility Risk Premium for papers.
Last time I checked, ATM 1M vol on Apple was ~ 26%. That for 80% moneyness was +71%. Looking at 71% in an isolated manner will tell you close to nothing about the standard deviation (SD) of future returns. For each strike and maturity there is a different implied volatility. This could be interpreted as the market’s expectation of future volatility between today and the maturity date in the scenario implied by the strike. Personally, I argue it is as much demand and supply as it is expectations.
In terms of scale, in a BS world, IVOL also increases with the square-root of time as time increases. Furthermore it is also an annualized percentage. Daily returns are typically computed in logarithmic terms. See here for an explanation of the use of the natural logarithm. In the context of options this means you are consistent with Black-Scholes.
The link you posted is a variance swap. In plain English, this equation means that the fair variance swap value can be shown to equal the integral of weighted prices of out-of-the-money options over all strikes, with weights inversely proportional to strike squared. That is only indirectly related to implied vol (often computed from a vol surface). However, in terms of variance swaps, you definitely have a one for one relation to realized vol (as realized variance is the squared realized vol; which is exactly how payoffs are defined). $$N_{\text{var}} (\sigma^2_{\text{realized}} - \sigma^2_{K})$$
One last comment: since quoted tenors have gaps in between, interpolation is important. It is insufficient to do straight-line interpolation. Simplest example is the weekend effect (saw-tooth effect) which is more pronounced at the short end. In essence an implication of assuming zero variance over the weekend. If you have important events like non-farm payroll release, elections, referendums etc one would usually apply weights to distribute variance on the vol surface. Now this implies that there is some information content for future SD. Ultimately, ATM IVOL is kind of OK as a predictor (with premium).
