Calculating the New VIX—The Easy Part

Tuesday, October 21st, 2014 | Vance Harwood

The movements of the CBOE’s VIX® are often confusing.  It usually moves the opposite direction of the S&P 500 but not always.  On Fridays the VIX tends to sag and on Mondays it often climbs because S&P 500 (SPX) option traders are adjusting prices to mitigate value distortions caused by the weekend.

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 6th, 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 7th of November provide a 25 day estimate of volatility, while the November 14th 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:

  • Yahoo Finance as ^VIN, ^VIF
  • Schwab $VIN, $VIF; historical data available
  • Google Finance INDEXCBOE:VIN, INDEXCBOE:VIN; historical data available
  • Fidelity:  .VIN, .VIF;  limited historical data

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:

VIX-calc-typ

 

As shown above the VIX value for October 13th is determined by averaging between the November 7th SPX options (VIN) and the November 14th 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.

VIX-calc-Wed

 

Wednesdays are important days for the VIX calculation:

  • The VIX calculation is dominated by the VIF values.
  • The SPX options used switch such that the old VIF becomes VIN and the options with 36 days to expiration become VIF.
  • Once a month on a Wednesday VIX futures and options expire (expiration calendar).  Soon after market open a special opening quotation of VIX called SOQ is generated.  Its ticker is VRO and it’s used as the settlement value for the futures and options.  Unlike the VIX’s normal calculation, the SOQ uses actual trade values of the underlying SPX options not the mid-price between the bid and ask.  Only one series of options, the ones with exactly 30 days to expiration are used.

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.

VIX-new-VS-old



This particular snapshot  shows the old VIX calculation (ticker: VIXMO) doing an extrapolation using SPX monthly options expiring November 22nd and December 20th (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 timenever 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.

VIX-VIN-VIF-eq

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The Volatility Landscape—October 2014

Friday, October 10th, 2014 | Vance Harwood

VIX + VIX Future Term Structure May 2011- March 2012

News

  • CBOE
    • On October 6th, the CBOE introduced an updated methodology for computing the VIX index.  Previously the index was computed using SPX options with 3rd Friday of the month expirations, but now SPX weekly options are used when they bracket the 30 day expectation maturity of the VIX.  For example the VIX for 9-October-2014 computes an implied volatility value for 8-November—since options don’t exist that expire on that date an interpolation is used between the 7-Nov-2014 and the 14-Nov-2014 SPX weekly options.  Calculations via the old methodology are reported under the VIXMO ticker.    While technically I think this is a sound move, the currently much lower volumes on SPX weekly options that far out may cause the new VIX to significantly diverge from the old VIX  (so far I’ve seen variations as high as +5%).   The CBOE’s new methodology does not impact the expiration process for VIX futures and options; they will continue to use the 3rd Friday options for that calculation.
    • The CBOE’s expansion to near 24 hour trading for VIX futures has gone well and they are planning to expand the trading hours of VIX and SPX options starting October 21st to the same Sunday Afternoon 6pm ET to Friday Afternoon at 4:15 ET span.  For more information see this note from the CBOE.
    • VXST futures have been trading since February 13th, 2014.  So far their reception has been lukewarm with volumes running around 50 per day, but their prices do seem to track the VIX index pretty well, significantly better than VIX futures.   For more see VXST FuturesNot a Bad Proxy for the VIX.
  • PHDG gains momentum
    • This PowerShares fund uses the same VEQTOR methodology as VQT, but it’s an ETF rather than an ETN.  Its assets under management have climbed to $431 million, gaining momentum in its quest to top Barclay’s $640 million VQT (source ETFdb.com).
    • PHDG distributes a dividend (currently around 1.6% per year).  Of the 22 USA volatility based funds only PHDG and VelocityShares’ SPXH and TRSK distribute dividends.
    • Recently options became available (30-July-2014) on PHDG.  I’ve wanted options on the VEQTOR based funds (VQT and PHDG) for a long time because they raise the possibility of a Covered Call Strategy That’s Long Volatility.   Unfortunately the market maker isn’t showing much enthusiasm, with huge spreads and no bid prices on ATM options.

 

Predictions 

 

White Papers

  • Volatility: A New Return Driver?
    • A good non-mathematical overview of volatility, volatility products including futures and a couple example trading strategies using volatility Exchange Traded Products
  • The VIX-VIX Futures Puzzle?”
    • A technical paper testing the forecast accuracy of VIX futures that includes a comprehensive technical overview of the VIX, VIX Futures, and volatility term structures.  It skips the calculus but provides a clear description and comprehensive formulas.
  • Variance and Convexity: A Practitioner’s Approach
    •  My favorite paper from the CBOE’s 2013 Risk Management Conference.  Sparse and very technical it addresses some of the differences between variance and volatility with regards to VIX futures.  Most of the other papers from the conference are posted here with links associated to the agenda items.
  • VIX White Paper
    • Complete details on the VIX calculation, recently updated (8-Oct-2014) to reflect the new methodology that utilizes SPX weekly options

 

Tools

Wish List



The Myth of Option Weekend Decay

Sunday, October 5th, 2014 | Vance Harwood

While doing simulations on volatility and the square root of time I started thinking about how options experience time—is it calendar time, market time, or something in-between?  The CBOE’s VIX® calculations use calendar time, a 365 day year, but most option gurus recommend using a 252 day year for volatility calculations—the typical number of trading days per year in the USA markets.

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 1-Jan-1994 through 22-Aug-2014 (5197 market days)

Market Time: Open to Close (occurrences) Market Time: Close to Open (occurrences)
No change 0.1%  (5) 58% (3046)
Change less than 0.05% 5.2%  (270) 81% (4249)
Changes >= 1% 27% (1396) 0.04%  (3)



I was surprised how often the market opened at no-change from the previous close (3046 times) and how seldom it has gapped overnight more than +-1% (3 times).

So what?

So far my arm-waving 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.

CalvsMrkt-ann



This “weekend” effect is sometimes visible in the CBOE’s VIX index, and is pretty dramatic with their new shorter term VXSTSM 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 simulation1 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 returns2 over the last 64 years. The result:

Sim-Ann-Factors



The square of the annualizing factor comes is only 0.87% from the theoretical median value3 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:

  1. For each day of the simulation I used the standard deviation of the previous 252 days natural log of daily returns for the short term volatility number.  For the yearly return I used the simulated market value one year hence divided by the current day’s market value.  Volatility drag is an important second order effect that needs to be included in the calculations.
  2. I offset the actual results by the average annualized growth rate to compensate for the non-zero mean of actual returns over the last 64 years
  3. My simulation results have a median value of 252.2 (0.08% error) if I use a volatility drag coefficient of 0.6 instead of the standard 0.5.  I believe my model slightly under corrects for volatility drag.



VXST Futures—Not a Bad Proxy for the VIX index

Wednesday, September 17th, 2014 | Vance Harwood

The CBOE”s VXST futures have been trading for over six months now—enough time get a feel for how they behave. The CBOE provides historical data —on a per future basis (VSW), which requires some work to get it into a consolidated format.   I organized the data by weeks, with the next to expire future prices labeled week 1, the next to expire week 2, and so on.  Typically there are four sets of VXST futures active at any point in time, with a set expiring every week on Wednesday morning.

VXST-data

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 VXSTsm 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:

VXSTvsVXST-futures

 

Visually the two look like they’re tracking reasonably well, but from a percentage basis it’s not all that great.

VXST-VXSTfutures-percent

 

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.

VIXvsVXST-futures

 

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 non-tradable 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.

VSXXvsVXX

 

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.

VSXXvsUVXY

 

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 in-between 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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Simulating Volatility ETP Open and Intraday High / Low Values

Tuesday, September 2nd, 2014 | Vance Harwood

Previously I’ve done simulations, based on VIX futures, of volatility Exchange Traded Products (ETPs) back to 2004.  In these simulations I only generated the closing values, but I’ve had requests for open / high / low (O/H/L) values.   Now I’ve extended my backtests to generate ETP opening and intraday highs and lows for many of the short and medium term volatility funds—specifically VXX, VIXY, TVIX, UVXY, XIV, SVXY, VXZ, and ZIV,  in addition to the closing values.

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 10th, 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 1st, 2012 through July 16th, 2012.   I would have preferred pre-December 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.

VXX-OHL-errors

 

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.

IVvsOHLactual

 

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 6-May-2010—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.

5perZoomOHL



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 25th of October, 2013 because on the 28th 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 29th 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 10-Dec-2010 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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