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A Better Way to Model the VIX

 
Tuesday, November 28th, 2017 | Vance Harwood
 

Models are useful. They help us understand the world around us and aid us in predicting what will happen next. But it’s important to remember that models don’t necessarily reflect the underlying reality of the thing we’re modeling. The Ptolemaic model of the solar system assumed the Earth was the center of everything but in spite of that spectacular error, it did a good job of predicting the motions of the stars, planets, moon, and sun. It was the best model available for over a thousand years. But new data (e.g., phases of Venus as revealed by Galileo’s telescope) and errors in predicting the motions of the planets demonstrated that the sun-centric Copernican/Kepler models were superior.

There are a lot of models for the Cboe’s VIX. None of them are particularly good at predicting what the VIX will do tomorrow but they can be useful in predicting general behaviors of the VIX. The most popular model for the VIX (although people might not recognize it as a model) is simple mean reversion.

 Simple Mean Reversion

Car gas mileage is a good example of a simple mean reverting process.

Over time your car’s gas mileage will exhibit an average value, e.g., 28 miles per gallon. You don’t expect to get the same mileage with every tank because you know that there are factors that make a difference with your mileage (e.g., city vs highway driving, tire air pressure, and wind direction) but over time you expect your mileage to cluster around that average value. If you get 32 miles per gallon on one tank of gas you reasonably expect that next time you check it will likely be closer to 28. If the values start varying significantly from the average you start wondering if something has changed with the car itself (e.g., needs a tune-up)

A mean-reverting random walk is a relatively simple model and fits some of the basic behaviors of the VIX. Specifically, over time the mean value of the VIX has stayed stable at around 20 and the VIX exhibits range bound behavior—with all-time lows around 9 and all-time highs around 80.

However, there are many aspects of the VIX that aren’t well explained by a simple mean-reverting model. For example, a simple mean reverting process will have its mode value (the most frequently occurring values) close to its mean. This is not the case of the VIX; its mode is around 12.4—a long way away from its mean. The histograms below show that difference visually.

 

Another VIX behavior that departs from a simple mean-reverting process is the abrupt cessation of values below 9—almost a wall. For a normal mean reverting process you would not expect such a sharp cut off at the low end.

The Acid Test:  How good is the model for predicting the future?

Having a good model for a process is useful because it can help us predict at least some aspects of the future. For example, we can use our average gas mileage to decide whether we need to gas up before entering a long stretch of highway without gas stations.

A simple mean-reverting model is not particularly good at predicting the future moves of the VIX. If the VIX is low (e.g., 12 or below) a simple mean-reverting model predicts that since the VIX is far from its mean that will likely increase soon. But history shows this is usually not the case. Often the VIX can be quite content to hang around 12. This leads to news stories quoting various pundits stating “The VIX is broken” –when in reality they are just using an inferior model.

 As I said earlier there are VIX models out there that address some of these deficiencies. Unfortunately, the ones I know of are complex and not very intuitive. I believe the model that I describe below can improve our intuition considerably without adding too much complexity.

A Better VIX Model

A better way to view the VIX is that it behaves like the combination of two interacting processes: a specialized mean reverting process and a “jump” process. The jump process captures the behavior that all VIX watchers know—its propensity to occasionally have large percentage moves up and down. Since 1990 there’ve been over 86 times where the VIX has increased 30% or more in a 10 day period. The occurrence of these spikes is effectively random with a probability of happening on any given day of around 1.28%. It’s like a roulette table with 78 slots, 77 of them black and one red. If the ball lands on a black the normal reigns—if red then things get crazy. The graph below shows a histogram of the number of days between these 30% spikes in the VIX since 1990.

There’s nothing that prevents reds on consecutive spins nor is there some rule that reds become really likely if you haven’t had a red in a while. The roulette ball has no memory of where it landed on previous spins.

VIX jumps are generally not just one-day events; subjectively it looks like it takes around two weeks before the market reverts to more typical behavior. The model assumes that when a jump occurs it essentially drives the behavior of the VIX for 10 trading days.

The other process, the specialized mean reverting process, addresses the non-jump mode of the VIX—which is historically around 85% of the time. One of the key behaviors it needs to address is the slow relaxation in the mean value of the VIX after a big volatility spike rekindles a generally fearful attitude in the market. This decay process continues (unless interrupted by another VIX jump) until the average monthly VIX values drop into the 11-12 range.

The chart below illustrates this relaxation process.

 

This characteristic can be modeled by expecting the short term mean of the VIX (when it’s not jumping) to decay exponentially until it reaches its “quiet” mean of around 11.75. It works well to quantify this decay as having a time constant of 150 days.

With this approach, sans jumps, the difference between the current VIX value and its long-term quiet value will decay by 50% in 104 days. So if the VIX is at 30 the model predicts the mean will decline to 20.75 in 104 days [30- (30-11.5)*0.5=20.75].  If there are no jumps for the next 104 days the VIX’s mean would decline to 16.13. If a jump occurs in the interim the short term mean is reset to the VIX’s value at the end of the jump.

The other part of the specialized mean reverting process mimics the day-to-day volatility of the VIX. I used a formal mean reverting diffusion process (Ornstein-Uhlenbeck) to accomplish this. Despite its scary name, you can think of it as a random walk with the thing “walking” being attached to the mean with a spring—similar to walking a dog with a springy leash. The further the dog gets from you the larger the force pulling the dog back to you.

Unlike the simple mean-reverting model often used for the VIX, this process has a much tighter distribution, with the extreme values effectively limited to around +-20% from the mean. When the VIX is quiet this process replicates the firm lower limit on the VIX, a VIX of 9 is -21.74% lower than a quiet mean of 11.75.

Simulating the Model

 To implement/validate this model I estimated the key input variables and then used Excel to simulate 20-year volatility sequences. I then compared these time series to the actual VIX history and tuned the model’s parameters such that the key characteristics (e.g., volatility, mean, mode, decay rates) were similar to the VIX’s historic values.

Resulting histogram of historic VIX values vs the simulated combined process

 

The next chart shows an example 20 year time series of the simulated VIX combined process compared to the historic VIX. The two series aren’t time synchronized; my intent is to show how the simulated VIX time series has the same visual feel as the real VIX.

 

Conclusion

This improved model is not a path to riches. It isn’t any better than other models at predicting when VIX jumps will occur. However, this model does help us understand how the VIX behaves over longer time spans. In particular, during times of sustained low volatility, it predicts that the VIX will tend to stay low until the next significant VIX spike and not trend up like the simple mean-reverting model demands.

 

 

 

Quant Corner

  • The mean-reverting diffusion process used is an Ornstein-Uhlenbeck mean reverting diffusion process using a log-normal distribution. The volatility was set at an annualized 112% and the return to mean strength parameter ETA set to 0.3. The mean of the process is determined by the previous day’s VIX value minus the exponential decay factor that will decay the mean down to 11.75 over time if there are no additional jumps (Tau of 150 days). If the mean has decayed down to 11.75 the process acts very similarly to the VIX’s low volatility regimes (e.g., 2004-2006, 2016-2017) with the “return to mean” factor effectively acting to keep the VIX  higher than 9.0
  • The Jump process used (with a few small tweaks) is a compound Poisson process where the probability of a jump sequence is random with a probability of a jump being 1.28% per day. The jump sequence and its daily amplitudes are determined using a technique borrowed from rappers called sampling. Instead of trying to recreate the decidedly non-Gaussian distribution of VIX jumps I reused historic VIX jumps by randomly selecting, and replaying one of the more than 85 jump sequences since 1990 where the VIX jumped more than 30% in 10 days. Each jump sequence is 10 days long, with the first 2 days being the behavior before the jump.

How Much Should We Expect the VIX to Move?

 
Friday, March 10th, 2017 | Vance Harwood
 

Every couple of months it seems like there’s an uptick in articles about the CBOE’s VIX Index being broken or manipulated.   Generally I expect the percentage moves in the VIX to be around a factor of 4 in the opposite direction of SPX (S&P 500).  But there are significant eccentricities in the VIX that I factor in, for example Fridays tend to be down days, Mondays tend to be up.

The chart below shows the percentage moves at close for VIX (blue bars) and SPX (red line) for the first 12 trading days of November 2012, along with my -4X rule of thumb (green bars).  The black ovals show 5 days where the VIX went opposite the expected direction. In addition, on two days, the 1st and the 16th the VIX moved far more than a -4X factor.


One of these days, the 12th, has at least a partial explanation.  That was the day that the VIX calculation shifted from using November / December SPX options to December / January options.   If you’re interested see Bill Luby’s post for more information on this phenomenon.

I did an analysis of SPX and VIX since 1990 to see the actual historical ratios between their percentage moves.   I excluded daily SPX percentage moves of less than +-0.1% because they often give very high, nonsensical ratio values.


The average ratio value was -4.77, but as you can see there is a wide spread.   About 20% of the time the ratio is positive (data to the right of the red line).

Each of the blue bars in the histogram shows how many days had a VIX% / SPX% ratio in each 0.25 wide bin.  For example, there were 120 days where the ratio was between -3.25 and -3.5.   I also plotted a normal distribution—which shows this distribution is more concentrated and has wider tails than a Gaussian distribution.

While not broken, and probably not manipulated, the VIX as a fear gauge leaves a lot to be desired.   Given its past performance it’s not reasonable to expect it to negatively correlate with the S&P 500 every day.  However, I think it does give us a very good feel for the mood of the SPX options market.  A single SPX option has the leverage of a $200K+ investment in the S&P 500, so it tends to be the domain of professional / institutional investors.  They aren’t always right, but they aren’t dummies, and they’re voting with their wallets.   Last week they were trading as if they thought the market decline was over, and at least for today, it looks like they were right.

The VIXs of Christmas Past

 
Monday, July 24th, 2017 | Vance Harwood
 

One of the persistent characteristics of the CBOE‘s VIX® index is the Christmas Effect—the tendency for VIX to drop down to relatively low levels during the Christmas holidays.  The CBOE’s VIX volatility December futures predict this drop for months in advance, and it has come to pass again this year.   I am aware of at least three possible explanations for this:

  1. Option market makers and others short options reduce their prices before the holidays so that they don’t get stuck with time decay (theta) during the multiple days off
  2. Traders in general go on vacation the end of December, volume drops, and the market becomes lethargic, reducing volatility
  3. People expect volatility to decrease, trade accordingly, and it becomes a self-fulfilling prophecy

I am skeptical about calendar based trading strategies (e.g., “crash prone” October was +8.5%, -1.8%, +4%  2011 through 2013) but the Christmas effect has been persistent— perhaps because it’s not easy to profit from it.   The VIX index itself is not investable, and the December VIX futures already discount the effect.

I was curious  how the VIX behaved over the last few years in December and January, so I generated the chart below using VIX historical data from the CBOE.

Xmas-VIX


To make the chart more readable I carried over closing values over weekends/holidays and used a 3-day moving average.  I excluded 2008, even though it shows the Christmas effect because the market that year was clearly in an unusual state.

There does seem to be a fairly consistent low around the 23rd of December and the VIX has consistently increased right after that—at least for a few days.   By mid-January things seem to have settled back into their random ways.

I also wondered how VIX futures behaved around the holidays.     I used my VIX futures master spreadsheet  to generate the chart below showing the behavior of the front month VIX futures, the next ones to expire.
Xmas-VIX-Futures

With the VIX futures the December dip comes a few days earlier.

In my experience, the future is often uncooperative in repeating the past, but this VIX Yet to Come, looks like a reasonable bet for a post-Christmas boost.

Volatility contango—from the beginning

 
Tuesday, November 8th, 2011 | Vance Harwood
 

Bill Luby, of VIX and more, recently pointed out that the 1st / 2nd month volatility futures had recently set a record (now 70 days) for continuous time spent in backwardation—where the value of the 1st month is higher than the 2nd month.   Not just a trivia question, this condition has been feathering the pockets of those holding volatility ETNs like VXX / TVIX, and picking the pockets of  those holding inverse volatility ETNs like XIV and SVXY.   Is this backwardation record a harbinger of structural changes in volatility futures, or is it just the normal response to a market correction?

Just visualizing the history of contango, starting when volatility futures started trading in March of 2004  is not an easy task.  It has two dimensions of time: the term structure of the volatility futures on a given date, and the variation of that term structure over time.  Obtaining the raw data itself is not trivial.  The CBOE provides volatilty futures data back to March 2004 on their web site, but it is in the form of 95 (!) different spreadsheets, and it is incomplete because not all months traded for the first several years.  To get a full data set back to 2004 required a considerable amount of interpolation / extrapolation.  If you are interested in obtaining the spreadsheet that consolidates all this data see this post.

The graphs below focus on the front two months of volatility futures.  The first covers from 2004 to the present.  I have quantified the contango as the percentage difference between the 1st and 2nd month, with the 1st month being the reference.  Negative values indicate a contango state, positive indicates backwardation.

VIX and 1-2 month volatility futures compared, with contango, click to enlarge

A couple things jumped out at me when I saw this graph. First of all, at a 10,000 meter level, first month volatilty futures do a good job of tracking the VIX index.  Certainly they don’t track well during the most volatile periods of VIX, but during the quiet times they are within a few points.  Second, the other than few days in Dec 2008/Jan 2009 the 1st and 2nd month futures were in contango for a long time (128 days) during the 2008/2009 bear market.   Not surprisingly, the graph shows that most of the time, volatility futures are in contango

This next graph zooms in on our current situation, starting in July 2011.

Fall 2011 Contango, click to enlarge

In the Fall 2011 correction the futures lagged the VIX, but more recently have caught up.  The  2nd month futures usually lag both the 1st month futures and VIX.

At least for 1st and 2nd month futures the term structure seems to be behaving like it did in the past.     Next I’ll look at the medium term ( 4 to 7 month) futures to see how they have behaved.

Is XIV behaving correctly?

 
Saturday, March 11th, 2017 | Vance Harwood
 

In spite of its name, XIV is not the inverse of the VIX index—it is the daily percentage inverse of an index called SPVXSP, which you can monitor on Bloomberg here.  This index very closely tracks the same index that VXX uses, SPVXSTR.

Last week XIV did not track VXX’s daily moves particularly well.   There has been a lot of speculation about what was causing this disruption—ranging from turmoil in the futures markets, XIV’s daily re-balancing, to the heavy backwardation in the soon-to-expire August volatility futures.

Below I have ploted VXX and XIV against the values they should have based on the index:

VXX & XIV vs SPVXSTR, click to enlarge

Things do not look seriously out of wack.  Most importantly, we aren’t seeing a divergence between the index and the VXX/XIV prices.  Daily errors are being compensated for over time. The next graph shows the daily VXX/XIV divergence from the index in percent.   The interesting thing here is that VXX is having trouble tracking too—it’s just in the positive direction.

VXX and XIV tracking error, click to enlarge.

Looking at these graphs I’m inclined to say that the tracking problems are not specific to XIV, but rather due to the volatility/disruption of the futures market associated with the S&P downgrade.