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A number of quantitative finance textbooks mention something along the following lines, without further explanation:

A typical feature of implied volatility from stock index options is that it is higher than the historical/realized volatility of the index.

Here I assume that something like the VIX index or short-term ATM implied volatility is used as a measure of the overall level of implied volatility.

The above phenomenon seems to be referred to as the "volatility risk premium", defined on Wikipedia as

...a measure of the extra amount investors demand in order to hold a volatile security, above what can be computed based on expected returns.

Question 1: Can someone elaborate on this argument? Do seller of options overcharge because of the inherent volatility risk that cannot be hedged away? Is this related to the market price of volatility risk?

Also, this consistent overpricing of options can be taken advantage of, e.g. by taking a short position in an ATM straddle (sell ATM call and put), i.e. sell volatility. This seems to be a profitable strategy, see e.g. http://quantpedia.com/Screener/Details/20.

Question 2: Why don't simple strategies like these eliminate the fact that implied volatility is consistently higher than realized volatility?

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Consider what happens when IV is lower than realised vol. The person long the IV would make money. So there would ideally be no one selling IV if it's lower than realised vol on an average.

Next if IV is equal to RV, then the guy selling the option has no incentive to sell since he won't make money on average. Also he has considerable risk in case RV turns out to be higher.

So IV is generally higher so as to compensate the seller for the risks he is taking. Also not only is the seller short Vega, he is also short gamma which is another risk for which the seller need to be compensated. The reverse argument can be made for the guy buying the option.

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    $\begingroup$ There are people who short vol and make money but every once in a while something blows up and you tend to lose a lot of money on short option positions. Like now, IVs are very low and people have been calling shorting vol picking pennies in front of the steam roller. $\endgroup$ – nimbus3000 Mar 18 '17 at 17:29
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    $\begingroup$ The fact that heavy losses on this strategy are concentrated in economic "bad times" (ex. NBER recessions) does suggest a non-diversifiable economic risk that might be linked to a positive risk-premium even in equilibrium. (Just like stocks return more than bonds in the long run, there is no arbitrage there). $\endgroup$ – Alex C Mar 18 '17 at 18:33
  • $\begingroup$ @AlexC: While very short, I consider your comment a better answer to this question than any other two that are currently given. $\endgroup$ – LocalVolatility Mar 18 '17 at 21:51
  • $\begingroup$ The risk-premium argument is also mentioned by Mr. Euan Sinclair at (quora.com/…) However, can you explain his second argument? I understand what he is saying, but not how it is related to the systematic implied vol > realized vol. $\endgroup$ – arni Mar 19 '17 at 13:18
  • $\begingroup$ @LocalVolatility Do you have a further input? $\endgroup$ – arni Mar 20 '17 at 6:19
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Another theory is that stock indexes do not follow a the lognormal distribution assumption of Black-Scholes. For example U.S. equities measured by the Russell 3000 or S&P 500 have a negative skew and excess kurtosis (above a normal or log-normal distribution). The empirical distribution of stock returns has larger losses than a normal or log-normal model using ex-post mean and standard deviation would predict. In other words the realized volatility understates the risk or potential for loss. Therefore in addition to the other reasons mentioned above the implied volatility the option writer charges ought to be higher than ex-post volatility.

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Because implied vol takes into account very large but rare events, while realized vol will only take into account such events if they have occured in the period over which the realized vol was calculated (unlikely, since large low probability events are, by definition, rare). If not convinced, consider an extreme case of say USDHKD (HKD is pegged to the USD). The realized vol is close to zero, while the implied vol is greater than zero reflecting the (very low) risk that the peg breaks.

Secondly, returning to equities, there is a natural appetite to purchase protection against adverse events (ie buy OTM puts) which is not matched by the appetite for selling protection, possibly because the later can only be sustained to a certain point when a crisis hits (when margins can't be met the position is stopped out).

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The main reason, from what I understand, is that it has to do with behavioral finance. The difference between implied volatility and realized volatility is sort of like a measure of risk aversion. Even if the computed expected return is X, investors may demand a small premium on top of it to compensate for the risk.

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