Submitted by Salil Mehta of Statistical Ideas
Volatility In Motion
Views on volatility are often only within recent context, causing sometimes influential advisers and investment banks to misguidedly extrapolate correction or premature rebound calls with far greater frequency then they actually occur (excessive false positives). In this article we see everything through a pertinent wider lens. We focus on how and when volatility moves from one level, to another.
The implied volatility index is currently trading in the upper-teens (about average); a natural question to ask is how long will we might stay there, before breaking out of this range level. Days, weeks, or months? Without fail, all of these choices have been seen through history, though we will show what the probability distribution is on these choices. We show that volatility tends to change quickly within this middling environment, though not so quickly as we often hear suggested in the media. Also the duration of time at any given volatility decile tends to be serially uncorrelated across time. The other interdependent question a trader generally has is about how volatility will change from here. Will the transition profile remain similar to last year’s, or from these volatility levels does it tend to either collapse or rise? Again a true and refined answer sometimes takes into account a joint combination of these questions. It is more a matter of unpredictable economic cycles . Not econometric risk patterns. You'll earn through nine exotic illustrations into volatility, a solid anchor for understanding the changes in volatility distribution that would address the questions just now posed. While we recommend neither trading nor listening to those who tell you to, this reference would be great for anyone curious to understand the transitory behavior of markets.
Before we begin though, we should bring up some preliminary reading covering some basic risk analysis, so that we can appreciate how the vantage of today's article here is different.
volatility, without considering duration
seeing low volatility : today’s article explores volatility by deciles, as opposed to quartiles of volatility treemaps
tail risk
tail risk curves and
tail concordance : today’s article here focuses on transitions in tail and non-tail equity risk, as opposed to focusing on action videos of tail risk events and copulas across asset classes
our autumn of discontent : today’s article focuses on annual time units, as opposed to seasonal or daily tail risk patterns
market convolution
guassian and non-guassian distribution
abnormal risks (shared by a Board member of Lending Club): today’s article looks at historical, non-parametric data, as opposed to advanced probability modeling with guassian (normal) distributions
Volatility by volatility decile
The chart below shows the overall daily distribution of the VIX, over its entire multi-decade history through YTD. We color segregated the distribution by deciles. While the median volatility has been ~18%, the mode we can identify as ~13% (we see it is at 8% of days, or averaging semi-monthly). Because there are not many experiences where the volatility has exceeded 18% (even though in those cases we can see the rise can be phenomenally explosive), the average is not that much greater than the median. The overall average is just ~20%.
In this chart below we present again this volatility, but spread across time. We use the same volatility decile-coloring scheme as before, as well as state the range within each VIX decile. One should notice some interesting characteristics from this chart. An outstanding ~½ of 2009 was spent in the 10th volatility decile, and that also happens to be continuously within that year (so no transitioning breaks in and out of that decile). So much continuous time gives some professionals an erroneous sense that this volatility can occur often, even though it was one of the outer deciles!
And so by definition each year should have an equal 10% of the time spent in the 10th decile (or any other decile). We see payback then for the 10th volatility decile being prominent for many years through 2011, since with 2012 onwards, none of the time has been spent in this decile (and only <1% in the 9th volatility decile!) It’s impossible to tell when in the proximate future we might break out of the nearly 14%-15% typical volatility we’ve seen in recent years, and elevate to the top decile volatility values.
We'll perform traditional non-parametric tests on a similar distribution type below. But it should also be noted that if we used machine learning tools on the data pattern above, then the 9-12 and the 29-81
volatility clusters (a well-known phenomena) would jointly be the first contour to be shaped out from the rest of the distribution. We can see in the chart above that every year would be impacted right away. So only brief in-between periods where the outer deciles consume very little of the year; and we could be nearing one of those times soon. The next non-linear carve-out would be the 12-13 and the 25-29, again jointly. Now these are the 2nd and 9th decile, yet they are subsets of the outer 1st and 10th decile years shown above! See the art by Jeff Koons at the bottom of
this article for a similar theme on support vector machines. The lesson here is that, despite how financial professionals portray their logic in the news, one can't predict the onset of an extreme decile, by the sudden appearance of the next most extreme decile.
Now for the transition matrix, we show the probability distribution of which decile the VIX goes to, based on which decile the VIX is currently in. The entire Markov matrix format we show sums to 1 (100%). As a trader, the take-away from this section is that volatility (approximate levels that is, not movements within) tends to be serially correlated across time (years of continuously low volatility, years of continuously high volatility). Financial news and investors however tend to think there is path dependency however, which hasn't been the case. Right now we have been in a low volatility period for a time; during which time, ongoing calls for 10% corrections and the like don’t make sense.
Next we show the amount of typical time spent in a VIX decile, once there. Obviously each decile, by definition, should only exist for an equal 1/10 amount of time. So what we show below is more of a sense of continuous streak duration, something we explore in greater depth in the Duration by volatility decile section immediately below. We can see more clearly here, where fewer -though lengthier- streaks lie.
Duration by volatility decile
In this chart below we look at the duration the market continuously spends in a volatility decile, before transitioning to another decile. One can notice that the durations are typically brief here (three days), though once volatility is at an extreme level this duration is at times, notably greater.
Next we show the typical duration matrix, where we indicate the typical length of time to move to one VIX decile, from another, whenever such a transition occurs. We notice that with 2012 onwards, the duration within the deciles is more brief (2 days, down from 3 days). Particularly for the lower deciles, the duration of time is still greater than normal even though this has been brought down (e.g., recently 3 days vs. 2 days, while we see in the overall chart above that it is 6 days vs. 3 days). As a trader, the take-away from this section is that one shouldn’t expect to hold their positions for long, with the exception of when volatility is at a severe level (high, or low). Clearly we have been at extreme low volatilities frequently, with 2012 onwards, though it is unclear how much longer that might continue to be the case.
Next we show the amount of time continuously in a VIX decile, based only upon the duration of time spent in the previous decile. If there were a relationship between two, we'd see transition charts similar to those above, where the contour is greatest in a linear fashion as high durations in a volatility decile would typically lead to high durations in another volatility decile. However, as we examine in the Duration by duration decile section immediately below, the high copula style analysis is fairly independent.
Duration by duration decile
The chart below shows the overall duration distribution, similarly segregated by duration decile. What we are showing in the duration distribution (how long volatility continuously remains in a volatility decile before moving to another decile). Other than 1 censored duration at 170 days, the next highest outlier is at ~60 days. We notice that these transitions generally occur immediately within a couple days (65%), with most within 19 days (~99.5%).
So far we know from this article that there are fewer extreme VIX deciles, within the brief duration streaks (those in lower duration decile); and more of those extreme VIX deciles in the lengthier duration streaks.
Conversely, there are fewer middling VIX deciles (such as the period we have often been in, with 2012 onwards) within these lengthier duration streaks.
Also notice that the highest duration is our non-parametric 10th decile; most of these durations are between 7 days, and 20 days. This comports nicely with what we showed earlier in the Duration by volatility decile section.
In this chart above we show the duration deciles per year. We can again notice the pick-up in the low 4th decile duration (i.e., 1 day), with 2012 onwards. We have often seen a similar pattern in the mid-1990s. And this chart may appear more subtle that then volatility chart (2nd chart from the top of this article), though it is statistically significant in the distribution of events per duration (a Chi square of ~0%.) This implies that what we've seen is a lower volatility environment that we’ve recently experienced, as much as it implies a lower duration systematically across volatility deciles.
Lastly, we see that there is a memoryless duration transition, where the duration in the previous volatility decile has no information to provide on the duration in the next volatility decile it moves to. Even if one wishes this to be the case, the market has shown -over multiple decades- to have another scheme. Durations simply are lower then we generally expect, and whenever they burst higher for any reason, they tend to regress expeditiously back down towards one or two days. We show in the article that it is simply the independent, and empirical nature of the duration distribution. Given the more continuous nature of durations given volatility decile, it is simpler to show convolution among the unconditional overall durations where the measurement unit was discrete days. So here is an example of what can expect over a given trading week (five days).
Number and probability of gross volatility decile switches:
0 is ~18%, since the top 2 duration deciles is >5 days
1 is ~26%, since there is a nearly 82% chance of not having 0 duration switches (implying an initial duration of >5), and within each possible initial duration therefore we compute the other remaining 1 possible duration switch such that only 1 occurs within a week
2 or 3 is ~29%, since this is the balance remaining so the column sums to 100%
4 is ~13%, since there are 3 permutations of either immediately seeing 4 1-day durations (43%^4), plus the likelihood of seeing 1 2-day combination in the initial 3 days with 1-day transitions otherwise (3*43%^3*22%)
5 is ~1%, since this is the 1-day duration probability of 43% occurring 5 times
More than ½ of all trading weeks therefore include anywhere between 1 and 3 volatility decile transitions. Also note that this is not the more complex derivation of the number of net volatility switches (since volatility can rise or fall with each transition). Volatility indeed tends to move quite often. But never so fast -with netting- as to expect that if we are in a bottom volatility decile market, to expect a rise to the top decile to occur in a matter of a couple weeks! Good things sometimes seem to take forever, and both financial media and human psychology sometimes can't appreciate that market changes are not spontaneous. Finding one's perspective is paramount; as 20th century singer John Lennon wrote in Beautiful Boy (Darling Boy):
Life is what happens to you while you're busy making other plans.
As a trader, the take-away from this section is again that one shouldn’t expect to hold their positions for long. Certainly not as long as a similar strategy one may have adapted and carried over from during the financial crisis and its initial recovery (all of which concluded years ago).
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