Why do we need a roll anyway?
If we could directly buy the CBOE’s VIX® index none of this would be necessary. Unfortunately no one has figured out a cost effective approach so we are forced to use the next best thing—VIX Futures. Like options, VIX futures have fixed expiration dates so volatility indexes need a process of rotating their inventory of futures in order to have consistent exposure to volatility. This rotation process is evident in the open interest chart below—the next to expire futures being closed out and the next month of futures being opened.
Indexes and Funds—are different things
Before we dive into the details of how this rotation is dealt with, I’d like to address one source of confusion. ETP’s are not obligated to follow the approach detailed in the indexes. They are allowed to use other approaches (e.g., overthecounter swaps) in their efforts to track their indexes. When ETPs are working properly, their prices closely track the index they specify in their prospectus minus their fees that are deducted on a daily basis.
Because indexes are theoretical constructs they can ignore some practical realities. For example they implicitly assume fractional VIX futures contracts exist and that the next day’s position can be put in place at market close—even though calculating that position requires market close information. I’m sure these issues cause headaches for the fund managers, but to their credit the funds usually closely track their index.
The Index Calculation
The details for the index (ticker SPVXSTR) that VXX tracks are detailed in VXX’s prospectus, pages PS21 through PS22. The math is general enough that it covers both the short term index that VXX uses and the midterm index VXZ uses—which adds to its complexity. The equations use Sigma (?) notation, which probably makes it challenging for people that haven’t studied college level mathematics. I will present the math below using high school level algebra.
Except for interest calculations all references to days are trading days, excluding market holidays and weekends.
The volatility indexes used by short term volatility ETPs (list of all USA volatility ETPs) utilize the same roll algorithm—at the end of each trading day they systematically reduce the portion of the overall portfolio allocated to the nearest to expiration contracts (which I call M1) and increase the number of the next month’s contracts (M2).
The mix percentages are set by the number of trading days remaining on the M1 contract and the total number of days it’s the next to expire contract (varies between 16 and 25 days). So if there are 10 days before expiration of the M1 contract out of a total of 21 the mix ratio for M1 will be 10/21 and 11/21 for M2. At close on the Tuesday before the Wednesday morning M1 expiration there’s no mix because 100% of the portfolio is invested in M2 contracts.
It’s important to understand that the mix is managed as a portfolio dollar value, not by the number of futures contracts. For example, assume the value at market close of a VIX futures portfolio was $2,020,000, and it was composed of 75 M1 contracts valued at 12 and 80 M2 contracts at 14 (VIX futures contracts have a notional value of $1K times the trading value). To shift that portfolio to a 9/21 mix for M1 and 12/21 for M2 you should take the entire value of the portfolio and multiply it by 9/21 to get the new dollar allocation for M1, $865,714 (72.14 contracts) and 12/21 times the entire portfolio value to get the dollar allocation for M2, $1,154,286 (82.45 contracts).
Value weighting gives the index a consistent volatility horizon (e.g., 30 calendar days)—otherwise higher valued futures would be disproportionately weighted.
The next section is for people that want to compute the index themselves. Yes, there are people that do that. If you are interested in the supposed “buy high, sell low” theory of roll loss you should check out the “Contango Losses” topic at the bottom of this post.
The Variables
Lower case “t” stands for the current trading day, “t1” stands for the previous trading day.
The index level for today ( IndexTR_{t }) is equal to yesterday’s index (IndexTR_{t1}) multiplied by a one plus a complex ratio plus the Treasury Bill Return TBR_{t. }The index creators arbitrarily set the starting value of the index to be 100,000 on December 20^{th}, 2005.
_{ }The number of trading days remaining on the M1 contract is designated by “dr” and the total number of trading days on the M1 contract is “dt.”
M1 and M2 are the daily markto market settlement values, not the close values of the VIX futures. The CBOE provides historical data on VIX futures back to 2004 here.
The Equations
When dr is not equal to dt:
When dr = dt (the day the previous M1 expires):
Yes, this equation could be simplified, but then it wouldn’t fit as nicely into the equation below which uses a little logic to combine both cases:
The equation assumes that the entire index value is invested in treasury bills.
Contango Losses
For more information:
]]>A chart showing volatility expectations vs time is called a volatility term structure. The updating chart below uses indexes published by the CBOE to provide up to 6 different points on the current VIX term structure. The green dots show the numbers published by the CBOE.
The black vertical bar shows the level of the older style VIX calculation (VIXMO) and the top of the purple outline around it shows the VIX value.
All VIX style volatility calculations are annualized—they indicate how much the market would be expected to vary in a year if the volatility stayed at that level. So for example if the volatility number is 15 then the model predicts that the market will stay between +15% of the current value in the next year with a 68% probability.
The annualization process assumes that volatility drops off with the square root of time. This is a good assumption, however there is a question of what sort of time should you use. For example the CBOE uses calendar time but I think there is a good case for using the actual amount that the market will be open instead—not counting evenings, weekends and holidays. The triangles shown on the graph show the CBOE index values annualized with market time instead of calendar time. Although the two calculations often agree sometimes there are significant deviations.
Additional Resources:
]]>Disagreements between the two indexes are not due to only one factor, but clearly one improvement was to eliminate the week of the month where the VIX was calculated using extrapolation.
The VIX provides a 30 day estimate of volatility, but the S&P 500 options (SPX) used in the VIX calculation have fixed expiration dates. The CBOE transforms the options’ data into the VIX by using volatility specific interpolation / extrapolation. For example, if you have a volatility number for options that expire in 10 days and another for options that expire in 38 days you can reasonably assume that the VIX level should be between those two numbers.
One requirement with the old VIX calculation was that it couldn’t use options with less than 7 days to expire. When the 7 day restriction was reached the calculation switched to using options that had more than 30 and 58 days until expiration. The chart below shows the calculation right after that switch on January 12^{th}, 2015.
The blue and red bars are the volatility numbers (VINMO & VIFMO) from the S&P options and the green bar is the VIX calculation extrapolated from those two numbers. For details on that process see: Calculating the VIX Index—the Easy part
Normally this extrapolation was reasonable, but if the market is nervous the shorter term volatility values climb. January 12^{th}, 2015 was a case in point—notice how the light blue VXST bar (9 day volatility) is higher than the VIX estimate.
The next chart shows the new and old VIX calculations together. The black dotted line shows the interpolation used for the new VIX calculation; the blue dotted line shows the old VIX extrapolation. The black vertical bar shows the VIX estimate.
Clearly the two calculations have a major difference of opinion —the new calculation’s result is 7.5% higher (19.60 vs. 18.15).
The CBOE’s new calculation always uses options that expire within a week of the VIX’s 30 days. The red and orange triangles show their values on January 12^{th}.
The old VIX calculation misses the fact that the shorter term volatility has ramped up.
The CBOE provides VIX style calculations on six different sets of options used in their VIX, VIXMO, and VXST calculations. These calculations don’t need time extrapolations / interpolations so that source of potential errors is eliminated. The chart below shows how they mapped out on January 12^{th}, 2015 (click image to enlarge).
The green dots on the top line represent the CBOE VIX style volatilities over time. The vertical black bar shows the old VIX calculation and the purple outline above it shows the new VIX calculation.
In this case the new VIX calculation is more accurate, cleanly mapping into the overall volatility term structure.
Additional Resources:
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There’s nothing magical about this channel. The market will transition from it at some point, and I think it’s important to plan for that, but for the moment the channel is the trend.
This sort of trend channel has characterized the last three bull markets.
The blue line in the chart is the 250 day simple moving average.
I suspect these patterns originate from random market moves—which often look like trend channels— interacting with human / computerized pattern matchers that transform random patterns into a selffulfilling prophecies. When the market starts approaching the top of the channel followers start selling, and they buy when the low channel is breached—reinforcing the pattern.
Anyone using that approach would have done very well the last two years.
As bull markets move into the territory of bubbles and nosebleed valuations the risk of the trend ending increases, but predicting the end is notoriously difficult. You can be years early in calling the top. In Jack Schwinger’s excellent book “Hedge Fund Market Wizards”, he comments, “Predicting the top of a bubble is like trying to predict the weather a year out.” In the book hedge fund manager Colm O’Shea agrees, but adds, “But you can notice when things have changed.”
Colm relates how in 2006 and 2007, “I was thinking the markets were in a completely unsustainable bubble.”, but rather than try to pick the top his firm stayed long. “We were quite happy to be part of the bubble.” However they did limit their risk by doing trades with limited downside (e.g., buying options rather than the securities themselves), and they waited for things to change. When they did notice a big change (LIBOR rates spiking in August 2007) they moved to bearish positions.
In the chart above, the last two bull markets signaled they were really over when the index was below the channel and below the 250 day moving average for more than a couple days. I will certainly be watching those metrics during 2015.
I will also be matching the health of the overall US economy, because fundamentally the market relates to the economy. The next recession and the next bear market will be linked together. Some of the factors I will be watching:
Interest rates
Oil prices
Wage Growth
Stock Buybacks
Geopolitical
In the year or two before the tech crash of 2000 and the financial crisis of 2008 the market felt overheated to me. Before those crashes there was an outrageously overvalued tech sector, and a vastly overheated home building / mortgage industry respectively. So far I don’t see the next bubble forming. Yes, trillion dollar student loan debt is worrisome, but I don’t see it crashing the economy. Yes, oil prices dropping 50%+ will put the pinch on oil business, but again I don’t see the domino effect that goes with the collapse of a bubble. To destroy the momentum of a growing economy requires a collapse on multiple fronts—no single facet has enough impact.
So, for the short term—at least for a week or two—the trend channel is safe.
]]>In many cases data is available from multiple sources. I did not attempt to list all of them.
If you are looking for symbols/tickers for volatility exchange traded products then you should go to this post where I list information on all 23 USA traded volatility style funds. Simulated histories for some of these funds back to 2004 are available here.
Historical data from different sources can differ—often because they use different closing times. Sites like Yahoo and Google Finance use standard NYSE hours, but the CBOE’s hours are different (close is 4:15 ET) and open times vary. For example with the advent of near 24 hour trading on VIX futures the open time for VIX futures for Tuesday through Friday is 4:30PM the previous day and Sunday at 5PM is the opening time for Monday. When used for computing other indexes (e.g., when VIX is used in computing the index used by VQT), the CBOE data should be used.
If you have an account with Fidelity’s Active Trader Pro you can get historical intraday data for many volatility tickers by exporting data from their charts. Schwab’s StreetSmart Edge allows export of watch list information, including option Greeks.
Standard long volatility indexes

Notes
Related Posts
Hedged style volatility fund indexes

Related Posts
VIX style Indexes and Settlement quotes

Related Posts
VIX Style Calculation Indexes (used by CBOE to compute VXST, VIX, VIXMO, VXV, VXMT)

Notes
Related Posts
Support Indexes

VIX Futures

Related posts
Some other interesting indexes currently not used by volatility funds

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Since its inception on October 6th, 2014 the new VIX has often differed significantly from the older calculation, often running 5% or more lower than the legacy number. This is disconcerting and I initially wondered if the reduced volume/open interest of the SPX weekly options used in the new calculation or some other factor was distorting things, but as I look at the data I’m becoming comfortable with the new calculation as a significant improvement in the accuracy of the index.
The dynamically updated chart above uses delayed quotes from Yahoo Finance. For more information on these VIX calculations see Calculating the VIX and Calculating the VIXMO.
The VXST is the CBOE’s 9 day version of the VIX, and VXV is the CBOE’s 93 day version.
There are two somewhat parallel markets associated with general USA market volatility: the S&P 500 (SPX) options market and the VIX Futures market. SPX option prices are used to calculate the CBOE’s family of volatility indexes, with the VIX® being the flagship. VIX futures are priced directly in expected volatility for contracts expiring up to 9 months out. The nearest VIX Future synchronizes with the VIX once a month—on its expiration date.
Additional resources:
In addition to these market driven eccentricities the actual calculation of the VIX has some quirks too. The VIX is calculated using SPX options that have a “use by” date. Every week a series of SPX options expire. This schedule of expirations forces a weekly shift in the VIX calculation to longer dated options. For many years the CBOE’s VIX calculations only used monthly SPX options, but starting October 6^{th}, 2014 it switched to using SPX weekly options when appropriate. See “Why the Switch” section towards the bottom of this post for more information.
The VIX provides a 30 day expectation of volatility, but the volatility estimate from SPX options changes in duration every day. For example, on October 13, 2014 the SPX options expiring on the 7^{th }of November provide a 25 day estimate of volatility, while the November 14^{th} options provide a 32 day estimate. In this case to get a 30 day expectation the VIX calculation uses a weighted average of the volatility estimates from these two sets of November options.
The newly updated S&P 500 VIX calculation is documented in this white paper. It computes a composite volatility of each series of SPX options by combining the prices of a large number of puts and calls. The CBOE updates these intermediate calculations using the ticker VIN for the nearer month of SPX options and VIF for the further away options. The “N” in VIN stands for “Near” and the “F” in VIF stands for “Far”. These indexes are available online under the following tickers:
The final VIX value is determined using the VIN and VIF values in a 30 day weighted average calculation. Graphically this calculation looks like the chart below most of the time:
As shown above the VIX value for October 13^{th} is determined by averaging between the November 7^{th} SPX options (VIN) and the November 14^{th} SPX options (VIF) to give the projected 30 day value. If you look closely you can see that the interpolation algorithm used between VIN and VIF does not give a straight line result; I provide calculation details later in “The Weighted Average Calculation” section.
The chart below shows the special case when the VIX is very close, or identical to the VIF value.
Wednesdays are important days for the VIX calculation:
Although SPX weekly options are available for 5 weeks in the future, the VIX calculation uses the SPX monthly options (expiring the 3rd Friday of the month) instead of the weeklies when they fit into the 24 to 36 day window used by the calculation. The SPX weeklies expire at market close on Friday but the monthly options expire at market open on Friday. By using these monthly options the CBOE keeps the VIX futures / options settlement process identical with the previous month based VIX calculation.
Why the Switch?
The chart below illustrates how the CBOE changed the VIX calculation methodology.
This particular snapshot shows the old VIX calculation (ticker: VIXMO) doing an extrapolation using SPX monthly options expiring November 22^{nd} and December 20^{th} (11 and 39 days away from the 30 day target)—a hefty distance. If you would like more details about the old VIX calculation see “Computing the VIXMO—the easy part“. The new VIX calculation on the other hand always does an interpolation over a much shorter period of time—never using options with expirations more than +7 days from the 30 day target. This CBOE article gives a good overview of the advantages of the new approach.
If you look closely at the chart, you can see that in this case the VIX calculation using the two methods arrives at slightly different answers (black line). The new method gives a result of 21.16, 1.5% higher than the old method’s 20.85. While I’m confident that the new calculation will be better in the long run because of the tighter VIN / VIF brackets I do have some concerns about the current volumes and low open interest in the SPX weekly options that are 4 to 5 weeks out. I have seen the VIX / VIXMO differ by up to 5%—so for the time being I’m keeping both indexes on my watch lists.
The Weighted Average Calculation
If you want to compute the VIX yourself using the VIN and VIF values you can’t just do a linear interpolation / extrapolation because volatility does not vary linearly with time. Instead you have to convert the volatility into variance, which does scale linearly with time, do the averaging, and then convert back to volatility. The equation below accomplishes this process.
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Wish List
When it comes to option decay most people, including the gurus, believe that option values decay when the markets are closed—a position I believe conflicts with the 252 day approach to annualizing volatility.
The experimental discovery that led to the current theory of option decay occurred in 1825 when the botanist Robert Brown looked through his microscope at pollen grains suspended in water and noticed they were moving in an irregular pattern. He couldn’t explain the motion but later physicists including Albert Einstein showed it was the result of water molecules randomly colliding with the pollen. This effect was named “Brownian Motion” in honor of Mr. Brown.
If you effectively stop time in Mr. Brown’s experiment (e.g., freeze the sample), the pollen will stop moving. Or if you close a casino for a day (probably a better model for the market) the net worth of the associated gamblers stops dropping.
Defenders of the calendar time approach point out there are many activities / events with broadband impact that can move the value of the underliers while the market is closed. Things like extended trading hours, activity in foreign markets, corporate announcements, geopolitical events, and natural disasters.
However it occurs to me that most noteworthy events that happen outside of market hours tend to be bad news. For example, I’m not expecting to see headlines any time soon stating, “ISIS disbands, ‘We realized it was all a terrible misunderstanding’”, or “Harmless landslide reveals huge cache of gold”. This tendency towards negative moves is reflected in the average annual growth rate of off market hours for the last 20 years, 0.37% vs +9.59% for market hours. And bad news tends to make option prices go up…
If option time is still running when the markets are closed I would expect the market’s opening value to be different from the closing value. Below is a quick look at the last 20 years of data:
S&P 500 Returns 1Jan1994 through 22Aug2014 (5197 market days)

I was surprised how often the market opened at nochange from the previous close (3046 times) and how seldom it has gapped overnight more than +1% (3 times).
So what?
So far my armwaving arguments give the edge to market time over calendar time, but really, so what?
Practically there are two things where this makes a difference: the dynamics of option decay and the accuracy of implied volatility calculations on soon to expire options.
Option Decay
Novice options traders are usually disappointed if they try to profit from Theta decay over the weekend. If the underlying doesn’t move, options prices typically open on Monday unchanged from the Friday close. Commentators explain this phenomena noting that market makers, not wanting to be stuck with Theta losses over the weekend, discount prices, overriding their models before the weekend to move their inventory—just like a fruit vendor would.
I think the market makers are right for the wrong reason. Their computer models are (or at least were) based on calendar day assumptions—which assume option decay during the weekend. By overriding their models they are pricing according to what really happens—no decay when the market is closed.
Annualizing factors
For longer term expectations of volatility it doesn’t matter much which approach you use. For options expiring a month from now the differences in implied volatility are only a few percent between the 365 vs 252 day models. However for shorter expirations the differences can be dramatic.
The chart below compares per minute values between the two annualizing approaches and shows the percentage difference. The calendar based approach is the black line and the green line is the market time. Notice how the difference peaks at Monday open and drops to near agreement at Friday close.
This “weekend” effect is sometimes visible in the CBOE’s VIX index, and is pretty dramatic with their new shorter term VXST^{SM} index—not surprising since this new index is based on S&P 500 (SPX) option prices with at most 9 days until expiration.
There are good reasons to use a calendar day approach to annualization. It isn’t sensitive to holidays, unexpected market stoppages, or differences in trading calendars between countries. I expect that’s why it became a de facto standard in the volatility world. But the rise of shorter term volatility products like weekly options has shifted the volatility landscape enough that I think we need to at least know what is technically correct.
An analytic approach to a solution
Normally we take a shorter term (e.g., daily) volatility and multiply it by the appropriate annualizing factor to get the annualized volatility. Since the annualizing factor is the thing in question I decided to take the historical annual volatility for the last 64 years of the S&P 500 and divide it by the daily volatility to solve for the actual historical annualizing factor.
First I validated this approach with a Monte Carlo simulation^{1} that computed the theoretical annualizing factor for a simulated 64 year market period—and then repeated that exercise 10000 times to get the statistics of the calculation. I then applied the same calculation to the S&P 500’s returns^{2} over the last 64 years. The result:
The square of the annualizing factor comes is only 0.87% from the theoretical median value^{3} of 252 and the actual S&P 500 result of 243.5 is only 2.5% from the median value. The S&P result of 243.5 is almost 3 sigma away from the competing answer of 365.
The S&P 500 data is consistent with a 252 day based annualizing model—which doesn’t support option decay while the market is closed. The data also indicates that when you see suspiciously high short term volatility numbers at the beginning of the week you should chalk it up to flawed algorithms, not anything real in the market.
Notes:
The “#NA”s occur on week 4 futures because the CBOE currently waits a day after expiration day before initiating trading on futures that are 4 weeks out.
The expiration value of a VXST future is tied to a special quote of the VXST^{sm} index (SVRO), which is linked to actual bid/ask values of SPX options near the market opening on Wednesday mornings. This process is important because among other things it enables VXST futures market makers to hedge their positions with SPX options
I charted how the futures were tracking the underlying VXST index:
Visually the two look like they’re tracking reasonably well, but from a percentage basis it’s not all that great.
There are frequent differences greater than +10%, and the 20 day moving average error is around 5%.
I also looked at the VXST futures values compared to the VIX® index.
These traces are considerably closer to each other, with only 3 occasions having greater than +10% error and a 20 day average error of around 1%. This relationship isn’t too surprising because volatility futures tend to trade at a premium to their indexes, and the longer the time horizon (e.g., 9 days vs 30 days) the higher the futures tend to be priced.
Bottom line, the next to expire VXST futures look like a decent proxy to the nontradable VIX index. Unfortunately this is only useful if your timeframe is pretty short (e.g., a week) — otherwise the carry costs of the futures are probably prohibitive.
VXST Exchange Traded Products
Currently there are no Exchange Traded Funds (ETF) or Exchanged Traded Notes (ETN) using VXST futures, but that situation could change quickly. The chart below shows the simulated performance of a very short term volatility fund that uses the same rolling futures strategy that VXX uses—except it uses VXST futures instead of VIX futures.
The simulated very short term fund behaves as you would expect—more volatile than VXX and larger contango losses during the quiet periods.
I then compared the very short term fund to UVXY, a 2X leveraged short term volatility fund.
Surprisingly similar. If this behavior continues (likely) there won’t be an advantage for an Exchange Traded Product based on VXST futures versus the existing 2X leveraged UVXY and TVIX funds. Bummer.
Of course, there is nothing set in concrete that the exact same futures rolling strategy that the existing short term funds use must be used in a very short term fund. For example, mixes of the first four weeks’ futures could be used, but I suspect that would just end up with performance inbetween VXX and UVXY—not something particularly useful.
VXST futures have not been a great success so far, with volumes for the nearest two week contracts combined averaging around 50 contracts per day, and open interest of twice that, but if they continue to show a good short term correspondence to the VIX then I can imagine their popularity will grow.
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