A Very Simple Model for Pricing VIX Futures

Updated: Dec 27th, 2015 | Vance Harwood

Serious volatility watchers are always observing a three-ring circus. The left ring holds the general market. Center ring has options on the S&P 500 and the various CBOE VIX® style indexes and to the right are VIX futures, Volatility Exchange Traded Products like VXX, UVXY, TVIX, and XIV plus associated options.

Activities in the three rings usually follow a familiar choreographed pattern. The VIX moves in opposition to the market while VIX futures and their kin trail the VIX unenthusiastically. VIX futures converge to the VIX’s value at expiration but prior to that they following their own path—usually charging a premium to the VIX, but sometimes offering steep discounts. Meanwhile, in the background, the VIX maintains its reversion to mean behavior, a macro cycle the short term moves modulate.

One of my ongoing interests is monitoring the Volatility Circus’ rings two and three—the family ensembles of VIX and VIX Futures.  I note unusual movements and try to determine which one of them is “right” more often—perhaps foreshadowing market moves. Recently I’ve developed a model that helps describe this relationship. It is presented later in this post.

Interpreting the values of VIX futures has been especially challenging. The price relationship of the next to expire VIX future and the VIX tends to be very dynamic in the last few weeks before its expiration.  With only a single data point, the one active future with less a month to expiration, there hasn’t been much data to work with.

Of course, there are mind-bending mathematical models available for VIX Future pricing—but unless you have a Ph.D. in quantitative finance they are probably too complex to be helpful.

Enter the CBOE’s Weekly Futures

By introducing VIX futures with weekly expiration dates the CBOE boosted the number of close-in data points from one to five—a dramatic improvement. One day while looking at Eli Mintz’s vixcental.com chart on these new futures a light bulb lit up in my head


The green dots are the newly introduced futures. Taken together the leftmost part of this curve looked logarithmic to me.

Sure enough, when plotted in Excel the logarithmic trendline match to the first two months of the futures was very good.

ln match to VIX Futures


However around month 4 the trendline starts seriously understating VIX futures prices.

7mo Trend Projection

Apparently there’s an additional mechanism that boosts the futures’ value over time.

The Model

Using VIX Futures data from 2004 on I developed the following equation which does a surprisingly good job of estimating VIX futures’ prices given its simplicity.  The only inputs are the current VIX value, the number of days (X) until the future expires, and the historical median value of the VIX.

VS-VX_FUT version B equation

The VIX closing median value from January 1990 through October 2015 is 18.01.

Example calculation: if the VIX is at 16 and a VIX future has 10 days before expiration this model predicts a price of 16.93.

16+ (1-16/18.01)* Ln (10+1) +3.1623*0.23 = 16+ 0.1116* 2.3979 + 3.1623*.21 = 16.93

A near real-time chart of the VIX Futures values predicted by the equation is posted here.

A Few Notes on the Equation

  • At VIX Future expiration (X = 0), the equation sets the VIX futures price equal to the VIX. The convergence term in the middle is forced to zero because Ln (0+1) equals zero and the carry cost term on the right is forced to zero by X being zero.
  • If the VIX matches its historical median price the convergence term is canceled out by the expression in front of the natural log, and the only difference in prices from the VIX will be the square root of time scaled factor on the right.
  • If the VIX is relatively low (below the historic median) the equation predicts the typical premium prices of the VIX futures relative to the VIX. The market is in this state 75% to 85% of the time.
  • Conversely, if the VIX is high, the equation predicts the VIX futures will be cheap relative to the VIX levels.
  • Since volatility increases with the square root of time, the term on the right side of the equation suggests a time scaled volatility component.  The 0.21 factor was determined empirically by adjusting its value until the average errors for the 3rd, 4th, and 5th-month futures from March 2004 through September 2015 were less than a percent.  The resulting 0.21 factor is quite close to the historical VIX median volatility of 0.18, so it’s possible that it is an implied volatility factor that rests a few percentage points above the historical value.

Why A Model?

You might reasonably ask why bother with a model when you can just look up the current VIX futures prices on the web. This model is interesting to me because:

  • It helps me understand the underlying mechanisms behind VIX Futures pricing
  • I can determine how current VIX futures prices are behaving compared to their predicted behavior—useful for evaluating situations where event risk is distorting prices or the market is especially panicky
  • I can predict future VIX futures prices for various VIX scenarios

Model Errors

The model is very inaccurate at times, with errors on historic data sometimes exceeding +30%/-15% percent. The chart below shows the model’s error terms for the next to expire futures since 2004.

sqrt-time-VX-FT model pt21

The model tends to overestimate futures prices while in sustained periods of low VIX and underestimate the prices in bear markets. During big volatility spikes (Oct 08, Aug 11, Aug 15) the model predicts VIX futures values that far exceed the actual prices.

It isn’t surprising that the model doesn’t adroitly handle the impact of big jumps or drops in market volatility since it doesn’t incorporate any historical information at all—other than the long-term median VIX value.

The error spike on the far right of the error chart is a whopper, nearly 50%. On August 24th, 2015 the VIX closed up 45% at 40.74, but the front month future (September) only climbed 26% to 25.13. The model predicted a value of 37.16, up 39%.

Despite the chaos prevalent on August 24th the futures market did an impressive job of predicting the eventual (23 days later) expiration value of the September 2015 futures.  Expiration was at 22.38, only 2.75 points away from the August 24th closing value.

A more accurate model would need to incorporate the effects of VIX jumps and slumps. It’s not a trivial problem. In general, the VIX futures seriously lag big jumps in the VIX, but then stay higher than you’d expect after the volatility drops.  Part of this post is an enhancement to the model that  does take VIX’s gyrations into account, but it still leaves a lot to be desired.

Now instead of resembling Fellini’s circus the VIX futures moves in the Volatility Circus feel more rational to me. Their movements are often mysterious and complex—but a simple theme unites.

VIX Futures Prices vs. Predictions from a “Simple” Model

Updated: Dec 9th, 2015 | Vance Harwood

It’s been said that we learn more from failures than success. Hopefully the chart below will be an illustration of that. It displays the near real-time prices of VIX futures vs. the predictions of a “simple” model I’ve created.  My intent with the model is not to achieve high accuracy (it won’t) but rather to distinguish between when VIX futures prices are truly unusual, and when they are displaying typical behavior.


The yellow dots show the percentage error between my estimate and the actual quotes.

I put simple in quotes above because I recognize that most people would not consider the formulas below simple.  But compared to the typical academic treatment of VIX Futures pricing this is a simple model.

SM-VIX futures pricing


The difference between this model and the one I described in the very simple model displayed below is the addition of functionality that addresses VIX jumps and slumps. The VIX itself is quite volatile with daily double-digit percentage changes being relatively common. Typically VIX futures lag these moves by about half (e.g, if VIX moves +10% the VIX futures term structure might shift up 5%).

I modeled this behavior with an exponential moving average of the VIX, using a coefficient of 0.5. This enhancement works well for the longer-dated futures, but shorter-term VIX futures have progressively tighter tracking to the VIX. To address this, I tweaked the jump/slump term, using a natural log function to extinguish the jump/slump lag as the time to expiration approaches zero.

A Very Simple Model

The graphs and descriptions below belong to my very simple model, which does not incorporate the historical patterns of the VIX at all.


The percentage errors are shown in yellow organized by expiration date.

The current VIX futures quotes are from Yahoo Finance (e.g., ^VIXnov).  Unfortunately as far as I know they aren’t providing quotes on the new Weekly VIX Futures yet. The chart below shows the model’s predictions for their values.


that I’ve created

You can check over at vixcentral.com if you want to get the current values for both traditional monthly and the new Weeklys VIX futures (click on the “VIX Term All” tab).

My estimates are produced using this equation:

VS-VX_FUT version B equation

Where VIX is the current VIX index value, VIX Median is the historic VIX Median value (18.01 for March 2004 through September 2015), and X = days until VIX Future expiration.

For more information on this equation see, “A Very Simple Model for Pricing VIX Futures.”

What is the Arbitrage Driven Range for VIX Futures?

Updated: Jan 29th, 2014 | Vance Harwood

The VIX style calculation for the March 22nd SPX options closed today at 17.06.  The data is from the CBOE’s VIX Term Structure website.


The CBOE will use the March SPX options to compute the settlement value for the February VIX Futures when they expire on February 19th.

February VIX futures, on the other hand closed at 16.25, a 5% discount to the March SPX options.  The blue dot on the chart marks the VIX style calculation for the March SPX options.


Normally VIX Futures trade at a 3% to 9% premium to the VIX style IVs of the SPX options that will be used to settle them.


This implies to me that profitable arbitrage of VIX futures using SPX options requires a  difference between the two of around +-5% or more.  Anybody have any insights on this?

For more background see: VIX Futures—Crystal Ball or Insurance Policy?

VIX Futures—Crystal Ball or Insurance Policy?

Updated: Mar 18th, 2015 | Vance Harwood

Many people seem to believe that the CBOE’s VIX Futures market is attempting to predict upcoming CBOE VIX® values.  I think they are mistaken.  Most futures prices have very little to do with predicting the future.

Futures contracts were invented to allow producers/consumer of commodities to limit their business risk by locking in future prices.  In exchange for eliminating price risk they give up the potential for increased profits if markets later moved in their favor.  Traders wasted no time using futures for speculation, not because they somehow foretell the future, but rather because the low margin requirements for futures allows them to heavily leverage their bets.

Prices for futures contracts are constrained by arbitrage.  For example, if an arbitrager sees the premium for Dec 2016 E-mini S&P 500 futures rise above a certain point they will short E-mini contracts and buy the appropriate amount of stocks in the S&P 500 (or just short SPY).  From this point on they don’t care what direction the market moves—they are perfectly hedged—a risk free position.  The transaction is triggered when there is sufficient difference between the current S&P prices (the “spot” price) and the December contract bid price to compensate for their cost of capital to buy the stocks, account for dividends, and deliver the target profit.  Alternatively, if they think the futures price is too low, they reverse the transaction, buying the future and shorting the S&P 500.   Companies will differ in how much premium they require between the spot price and the futures price but the net effect is that E-mini prices trade between these boundaries—they aren’t a divination of the S&P 500’s value in December 2016.

For physical commodities like corn or natural gas the cost of storage is included in the futures pricing and in some cases seasonality.   For example, prices for corn might be depressed during harvest time because some producers want to move the product directly to market so they don’t have to store it.

Arbitrage for VIX futures is a much trickier thing.  You can’t buy volatility on the spot market and store it in your garage for a few months.  The best you can do is buy or sell the appropriate set of S&P 500 (SPX) options—the ones that expire 30 days after the settlement date, and eventually subject yourself to the settlement process.  Settlement is via the tweaky Special Opening Quotation (SOQ) process, which can’t even tell you ahead of time the full set of options that will be used for the settlement and in the last year has differed from the VIX opening price by as much as +4.27% / -3.8% on the morning that VIX Futures settle.

Monthly SPX options are only available for the next 4 or 5 months, so the options associated with further out VIX futures expirations (up to 9 months out) are sometimes not even available.   Clearly market makers in VIX Futures must have other ways to hedge their positions (e.g., VIX futures spreads, VIX options, calendar spreads of SPX options).  In with VIX futures trading virtually 24 hours a day on weekdays it appears they don’t even need the S&P 500, SPX options, or the VIX to operate their market.

A quick look at historical data suggests that VIX futures tend to trade at a 3% to 9% premium to the VIX level of their associated SPX options.  A chart with April 10, 2012 and April 11, 2013 data is shown below:

Since the notional value of the SPX options market is currently much bigger than the VIX futures value it is reasonable to assume that the VIX futures tail does not wag the SPX options dog.   So the question becomes, what sets SPX options prices, and indirectly their implied volatility?  Are the market participants seers or are there other factors at work?

First of all, by any measure SPX option implied volatility is terrible at predicting future volatility in any way other than general reversion to mean.  If current volatilities are very low they predict an increase in volatility, if really high they predict it will drop.  Short term they predict that tomorrow will be like today—brilliant…

Insurance is a better model for predicting SPX option prices.  Investors use option strategies for price protection (e.g., fully hedge any market moves below a certain price), and as assets that reliably go up during serious market corrections or panics.  Even if your puts aren’t in-the-money, they still go up a lot when volatility spikes.

An insurance company doesn’t try to predict when you will have losses on your house or car.  It looks at the statistics and then charges a constant monthly rate they believe in aggregate will cover the claims they receive and over time deliver a reasonable profit.   I think the VIX Futures market looks a lot like an insurance provider.

Two final notes:

  • VIX Futures do have one seasonality pattern —the Christmas effect.  Typically December VIX futures are slightly cheaper than the surrounding contracts because volatility around that time tends to be low—a lot of people are on vacation, and the market is closed for several holidays
  • The VIX Futures term structure chart during quiet markets has gotten more linear in the last couple of years.  This shift has dramatically increased the contango in the mid-term futures—much to the chagrin of the XVIX, VXZ, and XVZ ETNs. On the other (inverse) hand ZIV is a happy camper.   Eli, from VIX Central speculates the curve straightened because there used to be a low risk trade of shorting a VIX future month right before it started sliding down the steeper portions of the curve, while being long the month after it as a hedge.  This trade might have just gotten crowded or perhaps the increased volume in the later month futures driven by the volatility ETPs made the market more competitive.


A 3D View of the S&P 500: Price, Time, and Markets

Updated: Nov 6th, 2015 | Vance Harwood

There are lots of valid ways to look at the market.  Obviously price is important, but a number alone (e.g., 1487.85—Feb 25th‘s S&P 500 close) doesn’t mean much.  Price related metrics like percentage changes and support/resistance levels add value, but adding time as a factor enables some really interesting measures like moving averages and momentum trackers.    My favorite time-related metric is term structure—how the prices of various options and futures vary depending on their expiration dates.

February 25th‘s market action provided a very interesting set of views.

On the price dimension, the percentage moves of the S&P 500 (SPX) and the CBOE’s VIX® index were unusual.  At 3:41 EST the SPX was at 1497.7 down 1.21% from the previous close.  The VIX on the other hand, was up 26.7%, which was -22X the SPX move.  The average move is -4.77%, so this was a tad unusual.

On the time dimension the term structure of SPX options was moving rapidly into backwardation—the implied volatility of the near term options climbing dramatically compared to longer dated options.

At open:

At 3:26 EST:

Options in general and SPX options, in particular, are thought to be traded by more sophisticated investors and institutions.  This sort of term structure change indicates they were buying option based protection like mad—even though the market had only fallen 1.21% at the time.

The third dimension that I like to watch is how different markets react to the same set of circumstances.   At 2:44 EST the VIX futures market was still unimpressed with the SPX price action of the day—with the term structure looking almost ruler straight.

The March front month VIX future was only up 5.4% even though the VIX was up 16.7% at that point—and almost equal to the 3rd-month future price.   The term structure looked a little different at market close.

It shows a dramatic shift into backwardation.

Although VIX futures are also linked to SPX options, the linkage is pretty weak.  The front month futures only reliably align with SPX options once a month—on the future’s expiration date.  Far fewer investors trade VIX futures and they are likely more sophisticated than SPX option traders.   Clearly the VIX futures market at 2:44 EST was much less impressed with the general market pullback than the SPX options market.

By market close the VIX futures market was serious about the market move, but judging by the position of the VIX close above the 6th (!) month VIX future value the VIX futures market is still not that worried.

The chart below shows the situation at close on February 26th:




The VIX dropped 11% and although lumpy the VIX Futures term structure straightened out.  For at least 26th the VIX Futures crowd called it right.