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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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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The volatility ETPs (complete list of USA funds) are all based on two or more sets of VIX futures. The CBOE provides historical open/high/low/close/settlement values for these futures starting in March 2004. Since the indicative values (IV) of the volatility ETPs are directly tied to these futures, the futures’ opening values can be used to accurately compute the ETP’s opening values—as long as the VIX futures and ETPs start trading at the same time of day. This was the case until December 10^{th}, 2010 when the CBOE starting shifting the opening times of VIX futures—more on this later.
The ETP intraday high / low values can also be calculated using the appropriate VIX futures intraday values but one additional assumption must be made—that the futures hit their intraday highs and lows at the same time. I didn’t expect that assumption to introduce a huge amount of error with the simulated values, but I wanted to verify that by comparing my simulation results to actual data.
To evaluate the magnitude of these errors I used O/H/L indicative value data from Barclays’ VXX short term volatility fund from June 1^{st}, 2012 through July 16^{th}, 2012. I would have preferred preDecember 2010 data, but I don’t have access to intraday IV data that goes that far back. A chart showing the relative percentage error is shown below.
Considering the uncertainties, worst cases errors in the +3% range seem reasonable. Sixty five percent of the data points had errors less than 1%. Six values had errors less than 0.01%, which suggests to me that my methodology is correct.
The next chart shows the differences between the actual trades (not the IV values) and simulated O/H/L values for VXX, starting January 30, 2009.
This chart illustrates a couple of additional difference terms that emerge when comparing the IV values to real trade data. First of all, there’s no guarantee that a trade will occur coincident with the open or the intraday high / low of the ETP’s IV. For example, the big 25% dip for the highs occurred on 6May2010—the Flash Crash. It’s not surprising that no one traded at the indicative intraday high of 42.13 (open was at 23.34!).
Other differences come from bid/ask spreads and tracking errors. The indicative value is computed from real time VIX futures values and updated every 15 seconds, but volatility fund market makers are not obliged to trade at that value. Unless the fund is heavily traded the spread between the bid and ask price will be at least several cents and if demand is unbalanced on the buy or sell side the offered spread values may be significantly different from the IV value.
This next chart zooms into the +5% portion of the chart.
The 22 trading day moving averages show the impact of the CBOE’s shift in the open time starting in December 2010—the average difference between the simulated IV values and trade data moves from close to zero to somewhere between +0.5% and +1.0%.
I cut off the O/H/L simulation on the 25^{th} of October, 2013 because on the 28^{th} the CBOE changed the Tuesday through Friday opening times to 4:30PM the previous day. This change was in preparation for the eventual move to nearly 24 hour VIX future trading which began June 2014. This change meant that the VIX futures were trading many hours before the volatility ETPs began trading—making VIX futures an unreliable proxy for ETP open/high/low levels. The close time, 4:15PM ET, has remained consistent, so VIX futures can still be used to compute ETP closing values.
I verified with the CBOE that the historic VIX futures data published on its website tracks the shifted opening times and is no longer synchronized with the ETP trading times. In the case of my simulations, there’s really no harm, because their primary value lies in predicting what the ETP’s O/H/L values would have been from March 29^{th} 2004 until the various volatility funds started trading. Actual trade O/H/L values exist for short term volatility ETP types (1X, 2X, 1X) prior to the 10Dec2010 shift in VIX trading hours.
The Spreadsheet
For more information on my ETP O/H/L/C simulation spreadsheet see this readme. The spreadsheet includes the formulas that convert from various indexes (e.g., similar to SPVXSTR, etc.) to the IV values, but it does not include the VIX futures values or the index calculation formulas.
If you purchase the spreadsheet you will be eventually be directed to PayPal where you can pay via your PayPal account or a credit card. When you successfully complete the PayPal portion you will be shown a “Return to Six Figure Investing“ link. Click on this link to reach the page where you can download the spreadsheet. Please email me at vh2solutions@gmail.com if you have problems, questions, or requests.
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Consider this chart:
Will the market bounce off this trend line for the fifth time, or will it go into a correction?
If the market breaks through the trendline it’s likely volatility will really spike. Alternately if the market rallies then volatility will quickly fade, so an asymmetric bet (e.g., call options) is attractive. If volatility spikes you benefit from the rapid runup, but if it’s a false alarm your losses are limited.
The next question is to determine what underlying volatility product is best for this hedge and how large a position is needed to balance the risk in your general market position. Investing in the CBOE’s VIX® would be ideal, but unfortunately there’s no way to directly invest in the VIX, so we’re left with a set of compromised choices—volatility Exchange Traded Products (ETPs) like TVIX, VXX, or VIXM (see volatility tickers for the complete list), or VIX futures. Later in this post I’ll analyze how three specific investments would have performed during an actual correction, but first I’ll examine a key issue—how much will the volatility products move up if the market drops.
The Choices
The chart below shows how the volatility ETPs have historically reacted during negative S&P 500 (e.g., SPY) market moves. The data uses simulations of ETP prices from 2004 until their inceptions and actual data after that.
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The median value of these ratios stays fairly stable over a wide range of percentage moves. For example the median percentage moves of 1X short term ETPs like VXX will consistently cluster around negative 2.25 times the percentage moves in the S&P. A daily 1% move in SPY typically results in a VXX positive move of around 2.25%.
These ratios aren’t guaranteed—they’re statistics. In fact 20% of the time the volatility products move in the same direction as the S&P 500. Fortunately, when the market is dropping the distribution of ratios tightens up
The chart below shows the historical distribution of VXX percentage moves compared to SPY moves of > 0.1% and > 1%. SPY moves of less than +0.1% are excluded because they can generate high ratios that aren’t meaningful.
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When the S&P makes a 1% or larger negative move the median doesn’t shift much, but the number of results on the positive side drops from 21% of the total down to under 5%.
Since these ratios are relatively stable regardless of the size of the market moves we can view these ratios vs. the various ETPs / indexes.
Remember these are one day relative % ratio numbers. While TVIX & UVXY ratios are close to the VIX’s on this metric, the contango losses in holding these ETPs other than during a market downswing are ruinous. The 1X short term ETPs (e.g., VXX) aren’t much better.
So far I’ve only discussed the CBOE’s indexes and some of the volatility ETPs. There are also VIX futures that have various sensitivities to the moves of the S&P 500. These products differ from the indexes and ETPs in that they have expiration dates like options.
As these futures get closer to expiration their sensitivity increases. Interestingly, a simple natural log relationship (shown on the chart) gives a good match to the data.
There are also VIX weekly futures based on the CBOE’s 9 day VXST index, but I’ll discuss those in a different post.
The Hedge
Circling back to the trend chart at the beginning of this post—what would be a volatility hedge that would protect you if you bet on a 5^{th} upward bounce?
There’re a lot of moving parts here (e.g., security, strike price, expiration date) and a lot of different strategies. I’ll pick one general approach, and work through the details if the hedge had been applied during the 30July2014 through 8Aug2014 period.
My assumptions:
I’ll review the results from three different trades—buying calls on UVXY (2x Short term), August VIX calls (based on next to expire VIX futures or M1 futures), and VXX (1X Short term).


So, in spite of the underlying volatility instruments moving around 2X more than expected, the $1K spent on hedges did not achieve the goal of break even with a 3% decline in the S&P 500—although UVXY was pretty close. During this period the VIX ramped from 13.33 to 15.77—an increase of 18.3% (the expected move was 15%). If the correction had continued volatility would have probably increased rapidly (the intraday option prices spiked > 50% on the 8^{th} –when the VIX climbed to 17.09), so the hedges probably would have worked well protecting the S&P 500 position against further declines.
One of the challenges of trading is wrestling with strategies that work until they don’t. With short term volatility hedges you can bet on the market going up—without paying too much for insurance in case you’re wrong.
]]>To have a good understanding of how the VIX works you need to know how its value is established, what it tracks, what it predicts, and how the CBOE makes money with it.
How is VIX’s value established?
What does VIX track?
How does VIX trade?
What does the VIX predict?
How does the CBOE make money on the VIX?
The VIX frustrates a lot of investors. It’s complicated, you can’t directly trade it, and it’s not useful for predicting future moves of the market. In spite of that, the investment community has adopted it, both as a useful second opinion on the markets, and as the backbone for a growing suite of volatility based products.
But what impresses me is the vision and persistence of the people at the CBOE in advancing the highly theoretical concept of stock market volatility from an academic exercise to an effective commercial product. It was a multidecade project and they were successful.
For more information:
This relationship holds for ATM option prices too. With the Black and Scholes model if an option due to expire in 30 days has a price of $1, then the 60 day option with the same strike price and implied volatility should be priced at sqrt (60/30) = $1 * 1.4142 = $1.4142 (assuming zero interest rates and no dividends).
Underlying the sqrt[t] relationship of time and volatility is the assumption that stock market returns follow a Gaussian distribution (lognormal to be precise). This assumption is flawed (Taleb, Derman, and Mandelbrot lecture us on this), but general practice is to assume that the sqrt[t] relationship is close enough.
I decided to test this relationship using actual S&P 500 data. Using an Excel based MonteCarlo simulation^{1} I modeled 700 independent stock markets, each starting with their index at 100 and trading continuously for 252 days (the typical number of USA trading days in a year). For each day and for each market I randomly picked an S&P 500 return for a day somewhere between Jan 2, 1950 and May 30, 2014 and multiplied that return plus one times the previous day’s market result. I then made a small correction by subtracting the average daily return for the entire 1950 to 2014 period (0.0286%) to compensate for the upward climb of the market over that time span. Plotting 100 of those markets on a chart looks like this:
Notice the outliers above 160 and below 60.
Volatility is usually defined as being one standard deviation of the data set, which translates into a plus/minus percentage range that includes 68% of the cases. I used two handy Excel functions: large(array,count) and small(array,count) to return the boundary result between the upper 16% and the rest of the results and the lowest 16% for the full 700 markets being simulated. The 16% comes from splitting the remaining 32% outside the boundaries into a symmetrical upper and lower half. Those results are plotted as the black lines below.
The next chart compares those two lines to the theoretical result which takes the annualized standard deviation of the S&P 500 daily returns from 1950 to 2014 and divides it by the square root of time.
Standard Deviation (N) = Annualized Standard Deviation/ sqrt (252/N)
Where N is the N^{th }day of the simulation.
Impressively close.
Since the simulated boundaries vary some from run to run I collected 32 runs and determined the mean
Very, very close.
So, in spite of the S&P 500’s distribution of results not being particularly normally distributed (see chart below), the general assumption that volatility scales with the square root of time is very appropriate.
Notes:
To have a good understanding of how XIV works (full name: VelocityShares Daily Inverse VIX ShortTerm ETN) you need to know how it trades, how its value is established, what it tracks, and how VelocityShares (and the issuer— Credit Suisse) make money running it.
How does XIV trade?
How is XIV’s value established?
What does XIV track?
How do VelocityShares and Credit Suisse make money on XIV?
XIV won’t be on any worst ETF lists like Barclays’ VXX, but its propensity to dramatic drawdowns will keep it out of most people’s portfolios. Not many of us can sit tight with big loses on the hope that this time will not be different.
It’s interesting that an investment structurally a winner albeit with occasional setbacks is normally not as popular as a fund like VXX that is structurally a loser, but holds out the promise of an occasional big win.
Slow and unsteady is trumped by a lotto ticket.
For more information
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It turns out the 21% gain is there, it’s just on such a low basis ($815), that it’s dwarfed by the other curves on the chart (one of the dangers of a chart with linear vertical axis).
My analysis didn’t include dividends nor did I factor in interest that could have been earned while out of the market. I think these two factors would roughly offset each other with the in/out strategies, and including dividends would have boosted the gains of the “always in” strategy.
I also wondered about the choice of 1994 as a starting point. A 20 year time frame is reasonable, and 1994 wasn’t a particularly eventful year, but I repeated the analysis with 63 years of S&P 500 data to see if it made a difference.
Over this timespan the “Sell in May” strategy significantly lagged in the bull markets of the late nineties and 2002 to 2007, and catches up during the depths of the bear markets.
The hold May through September strategy is still a flatliner. It’s hard to see how being in the market during that period helps the buy & hold strategy. A closer look at the yearly distributions yields the answer.
The chart below shows the distribution of the percentage gains/losses by year being invested only May through September.
The average of all the returns is low—a paltry 0.33% yearly gain, but the number of up years out numbers the down years by almost two to one, 40 up years vs 23 down years.
The next chart adds the results (green bars) of being invested except for May through September.
There were only 10 years during this 63 year period where losses during the May through September period weren’t more than offset by the returns from the rest of the year. And in 33 of those years both periods had gains—dramatically compounding the results. It’s this compounding effect that rewards the buy and hold investors.
This final chart shows the performance of the “Sell in May and Go Away” strategy with taxes included, assuming a 28% marginal tax rate.
Since the strategy is never invested for a full year taxes on profits will always be at short term capital gains rates—typically the same as your marginal tax on income. Since you don’t have to pay taxes on gains until the year after you sell the security I assumed that the money earmarked to pay taxes was used for ongoing investment until early April of the next year. Losses were carried forward and used to reduce/ eliminate tax on latter gains.
Including taxes the results of the 63 year “Sell in May” strategy were hammered down 70%—from $100K down to $30K and the 20 year period takes a 30% haircut. So, unless your investments are in a tax protected account the historical performance of this strategy would have been abysmal. Even in nontaxable accounts the long term performance of “Sell in May” would have been inferior.
I think it’s best to stay away from “Sell in May”.
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