Introduction
We discuss features and applications of the recently launched S&P 500 VIX Futures Indexes. These tradable benchmarks have returns equal to VIX futures of short and medium term maturities. They are rolled systematically in fixed increments.
The positions in the different contracts held by SP5VFI is rebalanced daily so as to ensure they remain self funded.
These indexes have a negative correlation to most asset classes and so tend to perform better in periods when systematic risk concerns dominate in a world with higher correlations across asset classes.
The indexes can also be used for taking directional views on equity volatility or expressing opinions around the richness or cheapness of certain parts of the volatility term structure.
The strength of the negative correlation with underlying equity returns, especially during crises, indicates that volatility should be an important part of the asset allocation process.
We use Matlab to demonstrate some of the results claimed in this report.
How the VIX is calculated
VIX is a minuteby minute snapshot of expected stock market volatility over the next 30 calendar days. It is calculated in realtime from the options on the S&P 500 index in a wide range of strike prices.
The VIX is an amalgam of the information reflected in the prices of all of the options used. The contribution of a single option to the VIX value is proportional to the price of that option and inversely proportional to the option’s strike price.
The calculation uses both the near term as well as the next term options to arrive at a 30 day implied volatility.
See Appendix for a detailed description.
BackgroundInitial approach for obtaining volatility exposure was through listed options but the periodic delta hedging can be challenging.
The advent of variance swaps allows for a cleaner expression of views variance, which is close to realized volatility but not exactly there. In addition these instruments are only trade OTC and so suffer from counterparty risk.
Although the VIX is not directly tradable, futures and options on it have made it easy to trade around expectations of future implied volatility. While the VIX futures do not need daily hedging, some active management is necessary for longer holding periods to have them roll over to a longer dated contracts as expiration nears.
The roll makes it difficult to gauge long term performance of a buyandhold strategy just by viewing a history of a given future contract. This is a similar problem to plotting the CL1 contract for Oil and trying to figure out the return for a buyandhold investor.
The SP5VFI series addresses these issues by creating tradable benchmarks whose returns are precisely mimic the payoff from managing a VIX futures position at the given expirations. The SP5k VIX ShortTerm Futures Index (SP5STFI) maintains a rolling long position in the first and second month contracts while the Medium Term Futures Index holds the 4^{th}, 5^{th}, 6^{th} and 7^{th} month contracts.
The Short Term Futures Index is better suited for expressing directional views on volatility expectation, while the Medium Term Futures Index is a better fit as a portfolio hedge (explained in Choice of Maturities section)
Advantages:
 Offers all advantages enjoyed by the VIX futures investors
 No need to worry about rolling to new contracts
 Transparent exchangebased pricing
 Exposure to pure volatility instrument
 Avoidance for the need for management a hedge
There is no free lunch though! The insurance these indexes provide during crises is not free and comes at a price which manifests itself via the costs of rolling the contracts in low volatility environments when the VIX future term structure is upwards sloping, i.e. when longer term futures contracts are more expensive that shorter term ones (contango)
The histogram shows the distribution of the spread between the SPX 6 and 2 months ATM implied volatility for the past year. Clearly the term structure is generally upward sloping for long periods of time, punctuated by episodes of inversion.
Notes on the VIX Futures contracts
Futures contracts on the VIX settle against the level of the spot VIX at the time of the futures’ expiration. The settlement date falls on the Wednesday 30 days prior to the third Friday of the month following the expiration month. The final settlement price is determined from a special opening quotation calculated from the opening prices of the batch of SPX options used to compute the VIX on the settlement date.
An key aspect of VIX futures is that they do not reference spot VIX (the expectation of SPX volatility over the next 30 days) but instead are priced off a future expectation (estimate of SPX volatility for the 30 days beginning of the futures expiration). So the MTM changes of a futures contract need not have anything to do with changes in the spot VIX, although the two get asymptotically closer as expiration approaches.
The simple nature of the contracts makes daily P&L calculations easier – a future contract purchased at 20 that currently trades at 22 would be marked at a P&L of $2000 ($2 * 1000 multiplier). Thus each contract has vega notional of 1000, representing a $1000 payoff for each 1point increase in implied volatility for that contract.
Rebalancing (rolling)
The SP5VFI series is based on the performance of a strip of VIX futures, which offer constant vega exposure to changes in volatility at a forward point.
Short Term futures index starts entirely in the first month VIX futures contract on each monthly rebalance date. An equal fraction of the position is rolled to the second month contract each business day, so that by the subsequent rebalance date (corresponding to the day before the following month’s VIX expiration) only the second month is held. At that point the second month becomes the front month and the process continues.
The Medium Term follows a similar methodology and starts with holding an equal number of contacts in the 4^{th}, 5^{th} and 6^{th} months and then rolls the fourth month VIX futures into the 7^{th} month contract in equal increments each day. The proportions of the 5^{th} and 6^{th} month futures are left unchanged.
The daily rebalancing is generally better than monthly. This is because the shape of the volatility term structure is typically upwardsloping, the ideal roll frequency would be one that minimizes the roll costs during these periods.
The shape of the curve is nonlinear and concave when its is upward sloping and convex when in backwardation. This implies that for an upward sloping term, rolling a fraction of the position more frequently is better than moving the entire position at smaller intervals.
Selffunded
The dollar value of the futures deleted from the index is kept equal that invested in the new future. So if the VIX curve is upward sloping, longer maturity futures are more expensive that shorter dated ones which results in the total number of contracts (and hence vega notional) decreasing over time. Converse is true when the curve is downward sloping. From VIX history, The VIX futures term structure is upward sloping during low volatility period and downward sloping in periods of stress when term volatility tends to be more elevated.
An alternative construction of the index could have used a fixed total number of contracts, hence keeping vega notional constant over time. While this may be more intuitive to volatility traders it would result in an index that could potentially reach negative values.
Choice of Maturities
Negative correlation with equity returns is one of the key reasons for considering volatility as a standalone asset class. To this end we would prefer an index that is as inversely related to spot as possible. Hypothetical tradable indexes created with VIX futures of different maturities typically display a decreasing correlation with equity return. The correlations do not differ significantly based on whether we use the S&P 500 or the Russell 2000 as a benchmark, highlighting the broad applicability of these instruments
Return  VIX Short Term Index  VIX23  VIX34  VIX45  VIX56  VIX67  VIX Med. Term Index 
2006 







with S&P 500  79%  77%  72%  69%  64%  59%  65% 
with Russell 2000  76%  75%  73%  70%  65%  60%  66% 
2007 







with S&P 500  84%  83%  82%  78%  75%  72%  76% 
with Russell 2000  78%  78%  77%  73%  69%  66%  70% 
2008 







with S&P 500  84%  83%  83%  84%  78%  75%  81% 
with Russell 2000  80%  79%  79%  80%  75%  73%  78% 
VIX23 denotes a daily rolling long position in the second and third month VIX futures contracts. This idea is extended all the way to the seventh month with VIX67 which is an index comprised of a rolling position in the sixth and seventh month contracts.
The steady rise in correlations each year since 2006 is also an encouraging feature and can be seen to imply that hedges using these indexes would become more effective during periods of stress.
ApplicationsA tradable index of short or medium term futures on the VIX can be used for expressing views on the future course of equity implied volatility. This can use other asset classes to judge the richness or cheapness of volatility or on a standalone basis using its mean reversion property. A common application is also in combination with other assets where volatility plays a role as a hedge against systematic risk.
We go though a few steps to show the optimal weight for the index when its being used as a hedge. Assume P to be an existing portfolio with expected return of r_{p} and a volatility of σ_{p}. The expected return after adding the volatility index is:
Where w_{p} and w_{vix} are the weights of the existing portfolio and the volatility index. In the long run the return from the volatility index r_{vix}, is not expected to be large and is likely to suffer from a daily roll cost if the term structure remains upward sloping. The volatility index should only be traded if there is a belief that its trading cheap or rich. The effect we are seeking here is to reduce portfolio risk:
Hedging with the volatility index is a double whammy; the negative correlation between returns of the volatility index and most asset classes will result in lowering the risk. But also the asymmetric response of equity volatility in periods of market stress means there are additional benefits in terms of lowering the maximum drawdown.
The optimal hedge ratio that minimizes standard deviation for the combined portfolio is given by the regression beta where
The above result is achieved simply by finding the minimum of the new portfolio volatility.
We backtest the performance of hedges using the S&P 500 VIX Futures Indexes in combination with several underlying assets. In each case we use the trailing beta over a 12month period on the first business day of October, to resize the hedge at annual intervals. The need for rebalancing the hedge arises due to the way the vega notional referenced by the volatility index changes, e.g. in a quit period the notional of the underlying asset would likely increase as it rallies but the “normal” upward sloping term structure would reduce the vega exposure of the volatility index, leading to underhedging.
There are two indices to chose from; lower roll costs make longer term VIX futures incrementally more attractive, while decreasing correlation with equity returns has the opposite effect.
Hedging the S&P 500
In this section we consider for an equity portfolio that has a return equivalent to the S&P 500. This type of asset class has the highest absolute correlation to the volatility indexes.
Above figure shows the daily performance of the assumed portfolio hedged with each of the VIX future indexes. The hedge is assumed to be sized based on the historical beta of SPX returns relative to returns of the volatility index. The portfolio is rebalanced the first business day of October, every year. On average and for the time range in the graph, for a $100 SPX notional, the optimal notional of the Medium Term VIX Index is $42 and $25 for the short term index.
For a 20 business day rebalancing, the portfolio does better with the average notionals increasing to $57 and $34 for the Medium and Short Term VIX index for a $100 of SPX.
The performance statistics for the annual rebalancing is as follows:
 S&P 500  S&P 500 + VIX Medium Term Index  S&P 500 + VIX Short Term Index 
starting 2 Oct 06 



Return  15.5%  13.8%  13.0% 
Volatility  13.1%  7.1%  6.5% 
Max Drawdown  5.4%  2.8%  3.0% 
starting 1 Oct 07 



Return  23.8%  6.6%  13.2% 
Volatility  25.1%  12.3%  13.9% 
Max Drawdown  13.1%  7.2%  8.2% 
starting 1 Oct 08 



Return  29.2%  6.3%  10.2% 
Volatility  51.2%  21.4%  22.9% 
Max Drawdown  20.6%  8.3%  9.9% 
The effect on the risk characteristics of the combined portfolio can be seen from the distribution graph and he statistics in the table. Notice how the volatility and the maximum drawdown of the combined portfolio is almost halved. The reduction in the drawdown highlights the smaller exposure to very negative returns (thinner left tail)
Protection for Corporate Credit
At the level of individual corporate issuers, a relationship exists between bond spreads, stock price returns and equity implied volatility. This is reflected in models such as Merton’s that assumes a diffusion process for the enterprise value of the firm.
The credit spreads vs. equity volatility relationship is stronger when viewed at the index level. Over the last five years, the level of on the run CDX IG spreads has been 60% correlated with the VIX.
The time series of the cumulative returns after adding the volatility index revels a pattern similar to that for the equity portfolio hedge – a performance drag when the spreads were tightening through late 2006 and early 2007, with significant outperformance in late 2008 and 2009.
Its should be noted that the maximum drawdown has fallen to about half using either of the VIX futures index as a hedge.
 CDX IG  CDX IG + VIX Medium Term Index  CDX IG + VIX Short Term Index 
starting 2 Oct 06 



Return  52.0%  27.5%  37.8% 
Volatility  66.3%  34.1%  39.6% 
Max Drawdown  32.8%  19.0%  21.3% 
starting 1 Oct 07 



Return  141.4%  33.9%  64.3% 
Volatility  76.0%  24.3%  32.8% 
Max Drawdown  41.0%  14.4%  19.9% 
starting 1 Oct 08 



Return  1.4%  38.5%  54.5% 
Volatility  67.1%  23.7%  31.0% 
Max Drawdown  28.4%  10.6%  13.4% 
Appendix  VIX Calculation Details
The formula for calculating the VIX as reported by the CBOE is (see [3] for a derivation of this formula)
where:
VIX = 100 * σ
T = time to expiration
F = Forward index level derived from index option prices
K_{i }= Strike price of i^{th} outofthemoney option; a call if K_{i }> F and a put if K_{i }< F
ΔK_{i }= Interval between strike prices = half the distance between the strike on either side of K_{i }:
Note: K for the lowest strike is simply the difference between the lowest strike and the next higher strike. Likewise, ΔK for the highest strike is the difference between the highest strike and the next lower strike.
K_{0 }= First strike below the Forward index level F
R = Riskfree interest rate to expiration
Q(K_{i}) = The midpoint of the bidask spread for each (outofthemoney) option with strike K_{i }
Calculating the VIX from the above formula is best done using an example. The following explains the process in 3 steps.
Step 1  Select the options to be used in the new VIX calculation
For each contract month:
· Determine the forward index level, F, based on atthemoney option prices. The atthemoney strike is the strike price at which the difference between the call and put prices is smallest. As shown in the following table, the difference between the call and put prices is smallest at the 900 strike in both the near and next term.
Near Term Options  Next Term Options  
Strike Price  Call  Put  Difference  Strike Price  Call  Put  Difference 
775  125.48  0.11  125.37  775  128.78  2.72  126.06 
800  100.79  0.41  100.38  800  105.85  4.76  101.09 
825  76.7  1.3  75.39  825  84.14  8.01  76.13 
850  54.01  3.6  50.41  850  64.13  12.97  51.16 
875  34.05  8.64  25.42  875  46.38  20.18  26.2 
900  18.41  17.98  0.43  900  31.4  30.17  1.23 
925  8.07  32.63  24.56  925  19.57  43.31  23.73 
950  2.68  52.23  49.55  950  11  59.7  48.7 
975  0.62  75.16  74.53  975  5.43  79.1  73.67 
1000  0.09  99.61  99.52  1000  2.28  100.91  98.63 
1025  0.01  124.52  124.51  1025  0.78  124.38  123.6 
The formula used to calculate the forward index level is:
F = Strike Price + e^{RT} * (Call Price – Put Price)
Using the 900 call and put in each contract month, the forward index prices, F1 and F2, for the near and next term options, respectively, are:
F1 = 900 + e^{(0.01162 }^{× }^{0.041095890 )} * (18.41 – 17.98) = 900.43
F2 = 900 + e^{(0.01162 }^{× }^{0.117808219)} * (31.40 – 30.17) = 901.23
· Next, determine K_{0}  the strike price immediately below the forward index level, F. In this example, K_{0} = 900 for both expirations.
· Sort all of the options in ascending order by strike price. Select call options that have strike prices greater than K_{0} and a nonzero bid price. After encountering two consecutive calls with a bid price of zero, do not select any other calls. Next, select put options that have strike prices less than K_{0} and a nonzero bid price. After encountering two consecutive puts with a bid price of zero, do not select any other puts. Select both the put and call with strike price K_{0}. Then average the quoted bidask prices for each option.
Notice that two options are selected at K_{0}, while a single option, either a put or a call, is used for every other strike price. This is done to center the strip of options around K_{0}. In order to avoid double counting, however, the put and call prices at K_{0} are averaged to arrive at a single value. The price used for the 900 strike in the near term is, therefore, (18.41 + 17.98)/2 = 18.19; and the price used in the next term is (31.40 + 30.17)/2 = 30.78.
Following is a table that contains the options used to calculate the VIX in this example:
Near Term Strike  Option Type  MidQuote Price 
 Next Term Strike  Option Type  MidQuote Price 
775  Put  0.11 
 775  Put  2.72 
800  Put  0.41 
 800  Put  4.76 
825  Put  1.3 
 825  Put  8.01 
850  Put  3.6 
 850  Put  12.97 
875  Put  8.64 
 875  Put  20.18 
900  Put/Call Average  18.19 
 900  Put/Call Average  30.78 
925  Call  8.07 
 925  Call  19.57 
950  Call  2.68 
 950  Call  11 
975  Call  0.62 
 975  Call  5.43 
1000  Call  0.09 
 1000  Call  2.28 
1025  Call  0.01 
 1025  Call  0.78 
Step 2 – Calculate volatility for both near term and next term options
Applying VIX formula for calculating the VIX to the near term and next term options with time to expiration of T_{1} and T_{2}, respectively, yields:
The VIX is an amalgam of the information reflected in the prices of all of the options used. The contribution of a single option to the VIX value is proportional to the price of that option and inversely proportional to the option’s strike price. For example, the contribution of the near term 775 Put is given by
Generally, ΔK_{i} is half the distance between the strike on either side of Ki, but at the upper and lower edges of any given strip of options, Ki is simply the difference between Ki and the adjacent strike price. In this case, 775 is the lowest strike in the strip of near term options and 800 happens to be the adjacent strike. Therefore,
ΔK_{775Put} = 25 (i.e., 800 – 775), and
A similar calculation is performed for each option. The resulting values for the near term options are then summed and multiplied by 2/T_{1}. Likewise, the resulting values for the next term options are summed and multiplied by 2/T_{2}. The table below summarizes the results for each strip of options.
Near Term Strike  Option Type  MidQuote Price  Contribution by Strike 
 Next Term Strike  Option Type  MidQuote Price  Contribution by Strike 
775  Put  0.11  0.000005 
 775  Put  2.72  0.000113 
800  Put  0.41  0.000016 
 800  Put  4.76  0.000186 
825  Put  1.3  0.000048 
 825  Put  8.01  0.000295 
850  Put  3.6  0.000125 
 850  Put  12.97  0.000449 
875  Put  8.64  0.000282 
 875  Put  20.18  0.00066 
900  Put/Call Average  18.19  0.000562 
 900  Put/Call Average  30.78  0.000951 
925  Call  8.07  0.000236 
 925  Call  19.57  0.000573 
950  Call  2.68  0.000074 
 950  Call  11  0.000305 
975  Call  0.62  0.000016 
 975  Call  5.43  0.000143 
1000  Call  0.09  0.000002 
 1000  Call  2.28  0.000057 
1025  Call  0.01  0 
 1025  Call  0.78  0.000019 
 0.066478 



 0.063683 
Next, calculate for the near term (T1) and next term (T2):
Now calculate σ_{1}^{2}and σ_{2}^{2} :
Step 3 – Interpolate σ_{1}^{2}and σ_{2}^{2} to arrive at a single value with a constant maturity of 30 days to expiration. Then take the square root of that value and multiply by 100 to get VIX.
Where:
NT1 = number of minutes to expiration of the near term options (21,600)
NT2 = number of minutes to expiration of the next term options (61,920)
N_{30 = }number of minutes in 30 days (30 x 1,440 = 43,200)
N_{365} = number of minutes in a 365day year (365 x 1,440 = 525,600)
References
[1] M. Deshpande et. Al, “Towards an Investable Volatility Index”, Special Report, Barclay Capital Equity Research, 3 Feb 2009
[2] “CBOE Volatility Index” The New VIX White Paper”
[3] Paul Staneski, “Understanding the new VIX formula”, CSFB Quantitative Trading and Derivative strategy, 12 Sep 2005.
[4] T. Watshma, K. Parramore, “Quantitative Methods in Finance”, Thomson Business Press, 1998, ISBN: 186152367X
Disclosure: No Position