A basic story that is told over and over again about financial markets is:

- Things mean-revert.

- A thing is away from its mean.

- Therefore that thing will mean-revert.

- Soon and horribly.
^{1}

This is I think roughly the way to read the notion, which my Bloomberg View colleague Mohamed El-Erian examined today, that low readings on the VIX -- an index of implied volatility in short-dated S&P 500 index options -- mean that the market is "complacent." So:

- Equity volatility is basically a mean-reverting thing.
^{2}

- The volatility index is below its long-run average.

- Therefore volatility will go higher.

- Soon and horribly.

- Run, you fools!
- Why are you being so complacent?

There are some simple problems with that story. One is that "things mean-revert" does not actually prove that "this thing will mean-revert soon and horribly": Sure, things will eventually get more volatile, but the fact that things are especially calm now doesn't prove that they'll be much more volatile soon.^{3}
There is an obvious analogue to this problem in stock prices: Stock valuations seem, by some measures, to be high right now, and valuation multiples are roughly speaking mean-reverting, So maybe that means that future returns will be low. But that is not necessarily a good reason to sell stocks.^{4}

And there is a further problem. A share of stock is a perpetual thing, more or less; you can talk sensibly about stock valuations reflecting people's expectations for the coming, like, decades. Uber is not valued at $17 billion based on this year's GAAP net income. The VIX, on the other hand, is a necessarily short-dated thing: It's implied volatility for the next month.^{5}
If you expect 2018 to be a great year for Uber, go ahead and bid up Uber's stock. If you expect 2018 to be volatile, there is absolutely no reason to buy options on the S&P 500 index that expire in July 2014. Those options will do nothing for you in 2018. Stock prices discount the entire future, somehow or other. The VIX discounts a month.^{6}

How is the VIX at predicting even that month, by the way? So-so. El-Erian notes that some people "believe the VIX is a better reflection of the past than the future." These people are correct. Here, I made you a chart:

The light blue line is the VIX over the last five years or so; the dark blue line is trailing 30-day realized volatility of the S&P 500. They match up nicely: The higher volatility has been for the last 30 days, the higher the market expects it to be over the next 30 days. The gray line shows what volatility will *actually *be over the next 30 days; it's just the dark blue line shifted in time. It looks ... like the dark blue line shifted in time. The VIX matches up well with the recent past (90 percent correlation); it matches up pretty meh with the near future (74 percent correlation).^{7}

The other thing you might notice about that chart is that the VIX is usually higher than the future realized volatility. In fact, it's higher about 88 percent of the time.^{8}
This makes sense. You can simplistically think of options on the S&P 500 as an insurance-like product, used to hedge the risk of volatility increasing. Traders should overpay for options, versus actual realized volatility, because they're buying insurance, and insurance costs money.^{9}

A simplistic way to think about how much traders are paying for insurance is that it is the difference between

- the "implied volatility" of the options they're buying,
^{10}and

- their actual expectations of future volatility.

That is, VIX is made up of (1) expected future volatility plus (2) insurance premium against volatility being higher than expected. I don't have a good measure of actual expected future volatility, but one bad measure is convenient to hand, which is, actual realized past volatility. Remember that the VIX correlates very well with trailing volatility; a simple guess of what traders expect is that they expect the next 30 days to be exactly like the last 30 days.

So then you can just subtract out realized volatility from the VIX, and what's left is insurance premium:

That is not an *especially *significant-looking chart.^{11}
But there is sort of a normal amount by which VIX exceeds 30-day realized volatility: About 4 points, or about 32 percent of realized volatility. And after spending some time below that average amount, the "insurance premium" component of the VIX has ticked up; on Friday, VIX was about 3.3 points (44 percent) above the last 30 days' realized volatility of 7.4 percent. The VIX reflects volatility expectations, and those expectations reflect the fact that volatility has been low, it is summer, central banks are supportive, and there's no obvious reason for it to go higher. But the VIX also reflects how much traders want to pay for options, in excess of what they expect from future volatility -- that is, how risky they think their forecast of future volatility is. And that number is up a bit. Perhaps all the worrying about complacency is making people less complacent.

^{1}
Because this story tends to be told more often when things are doing better than average (bubbles, complacency) than when they're doing worse than average.

^{2}
Is it? Why? Izabella Kaminska is perhaps our leading theorist of "the death of volatility," in which price volatility is not mean-reverting but declining over the long run:

Think of it this way. We live in a world in which corporations as far afield as British Airways, Ocado and Google can anticipate your every wish before you’ve even signalled it.

This is not some cornucopian dream, but rather a reflection of how information technology is undermining the need for pricing because the allocation of goods is now so efficient that urgency, crowding and queuing — the very things that drive prices — have in some cases been eliminated entirely.

Should you really be surprised that volatility is dying?

^{3}
So the headline "VIX Complacency Suggest Stocks Fall Further" was published mid-April; since then the S&P is up 120 points and the VIX has fallen by a further 5 points. "S&P 500 Valuation vs VIX Shows Complacency" in September 2013; since then the S&P is up by almost 300 points and the VIX is down by over 4 points. "VIX Hits Almost Six-Year Lows, Complacency Replaces Fear" in March 2013; since then VIX is down a touch and the S&P is up by almost 400 points. Etc.

^{4}
Here is an excellent long post on that topic, specifically on the question of how much you can tell about future returns from current valuations. (Answer: not much.)

^{5}
See this CBOE white paper for the specifics, which are hairier than that. This RBC note says it's 3-month volatility; meh.

Earlier today I asked on Twitter, "If VIX is low, what period does that mean that investors are complacent about?" Jean-Claude Kommer's sensible answer was "next week."

^{6}
Now this is not strictly true. If you think 2018 will be volatile, then maybe everyone will think that in 2017. And if everyone's going to be worried in 2017, then that might creep over into 2016. And so on back to your expiry in July 2014. But this seems like a stretch?

^{7}
If you're interested, the equivalent numbers for 60-day realized volatility are 91 percent (trailing) and 66 percent (future). For one year (260 trading days), it's 71 percent and 40 percent. (Source: Bloomberg data back to January 2009, my calculations.)

^{8}
That is, VIX is higher than future 30-day volatility on about 88 percent of days in this sample, including in mid-April 2014 and September 2013. See footnote 2: While people were identifying the VIX as a sign of complacency, the VIX was actually **less **complacent than it should have been. (In March 2013 it really was too complacent; realized volatility for the 30 days starting March 11, 2013, was 13.4 percent, versus the 11.56 VIX reading that day.

^{9}
And if you look at the couple of places where the gray line is above the blue lines, it's often **way **above the blue lines, as volatility spikes massively and unexpectedly. So you're really glad you bought protection there. The rest of the time, meh, you pay a little for peace of mind.

^{10}
Which is just the mathematical result of applying the Black-Scholes option pricing formula to the price that they're paying for options. In other words, "implied volatility" is not a real thing; the real thing is that you pay, like, $10 for a put option or whatever. You can then apply a formula to calculate the "implied volatility" of that option, but there's no guarantee that that calculated implied volatility is **what you think future volatility will be**. It's just a number. All we know about your expectations is that you thought that option was worth $10.

^{11}
I've made it perhaps look more significant by normalizing to the realized volatility. So if realized volatility is 20 percent and VIX is 26, then that's a 6 point but 30 **percent **difference between VIX and realized. If realized volatility is 10 percent and the VIX is 13, then that's a 3 point difference, but still a 30 percent difference. As realized volatility declines it makes more sense to me to think of the premium in terms of percentage of realized volatility, but if you think of it in points the chart is even less impressive:

To contact the writer of this article: Matt Levine at mlevine51@bloomberg.net.

To contact the editor responsible for this article: Tobin Harshaw at tharshaw@bloomberg.net.