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While theoretical options prices are derived from models, such as Black-Scholes, IV and IV skew reminds us that options prices are ultimately based on supply and demand. My question is the following: how can we claim that implied volatility measures the expected volatility of the underlying during the life of the option, when the IV varies by strike? If we are specifically talking about the ATM volatility, which I assume, how do we interpret the IVs of other strikes — especially as we move further into or away from the money? Finally, how do we interpret IV in an illiquid market, in which option prices don’t change for, say, weeks at a time? Do we say “there has been no change in the volatility expectations” — other than theoretical changes due to DvegaDtime — or do we just deem the data stale and irrelevant?

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Nice question. My interpretation is via the concept of a risk premium (i.e. risk adversity of market participants).

Let me introduce the concept of a risk premium first via US corporate bonds: one can observe that the credit spread of these bonds increases as the credit quality decreases. However when looking at actual historical realized defaults of corporate companies in the US across the various credit-quality buckets, one can see that the realized default frequencies are lower than the credit premiums charged: in other words, the expected returns (under the real world probability measure) increase as you move down the credit quality. That is because investors need that extra premium to invest in these lower credit-quality bonds, to be compensated for the increase in risk, specifically tail-risks related to defaults of poor quality bonds during stressed events.

With options, the option writers demand a similar risk premium for writing options that contain a "tail risk": that's why you'd typically observe a higher IV on OTM puts on equities (because there is a tail risk related to stress events such as Covid, for which the put option writer wants to be compensated). You'd observe a similar increase in IV on OTM Call FX options written on USD vs. emerging market currencies (i.e. USDTRY): because here, it would be the OTM calls that are exposed to tail-risk events.

Last but not least, even ITM options would contain some risk premium: imagine that the IV priced into the written option would only reflect the actual expected future realized volatility; then the option writer would make zero money by delta-hedging the option. The option writers demand a premium in general for writing the options & managing the related risks via hedging. That's why I'd always argue that IV is a not a good measure of market's expectation of future volatility: because of risk premiums, the IV always overstates the expected future volatility.

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  • $\begingroup$ This is an excellent answer! Thank you! So how would you estimate the market’s expectation of future volatility? My employer is hell-bent on using options volatility. They don’t seem interested in arguments from historical volatility and homoskedasticity / stationarity. $\endgroup$
    – CasusBelli
    Oct 17, 2020 at 16:03
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    $\begingroup$ @CasusBelli: If you want to convince your employer, I'd suggest running a simple comparison: compute the realized 1-year rolling volatility on something like the SPX500 index over the past 30 years, and compare it to the rolling 1-year IV of ITM calls on the SPX500. You should clearly see that the IV is consistently higher than the realized (perhaps with the exception of tail-risk events, such as 2008 or Covid). Do the same for non-overlapping 1-year horizons: would make up for an interesting analysis. Ps: I'll let someone else comment on GARCH modeling of vol, that is not my expertise. $\endgroup$ Oct 17, 2020 at 16:12
  • $\begingroup$ @Jan: I did some research on your suggestion a while back. I think there is no premium for ITM calls. (OTM puts). The VAR premium is completely driven by OTM puts. But maybe I misinterpreted your argument, though. $\endgroup$ Oct 17, 2020 at 16:45
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    $\begingroup$ Yes, I think that’s what I want to point out. The “IV difference” is usually only - on average (of course) - found in OTM puts. Upside potential is - statistically speaking - not really commanding any IV premium, at least for major stock market indices. Here’s a little summary on that: papers.ssrn.com/sol3/papers.cfm?abstract_id=3189480. $\endgroup$ Oct 17, 2020 at 17:01
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    $\begingroup$ Interesting discussion. I had a quick look into the cited paper. Indeed, it has interesting experiments but at first glance, I cannot see how the paper adresses the specific issue raised by Jan regarding ITM Calls. It seems that the authors only consider OTM with respect to the ATMF and some shifted version of it. Please correct me, if I'm wrong $\endgroup$
    – SI7
    Oct 19, 2020 at 18:02
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great question. Sorry I am quite late to answer this, and if it is still relevant for you. I think the use of "market implied" IVs is to extrapolate a pdf curve for the underlying. As black Scholes assumes constant IV and log normal distribution. When you trace back the IVs to create a probability distribution of the underlying, and then you can do all sorts of speculative analysis of the underlying moves.

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You don't claim that implied vol measures the expected realized vol. This is completely theoretical and hinges on the ability to delta hedge continuously. In reality there is a PnL leak due to discrete hedging and vol autocorrelation. In order to make up for this residual PnL variance, one might as well charge an extra premium.

This is why implied vol value contains a "gotta make up for less than perfect hedging" premium. The exact value depends on strike (because the PnL variance depends on gamma which depends on strike).

Potentially one could do a PCA of the implied vols and that factor should actually track the realized vols better in my opinion.

Movement of implied vol of the same contract is an admission that asset dynamics are not lognormal and just that. The real quantity, $Vol(dS/S)$ is naturally non constant and the expectations reflect that.

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