VIX Futures Indexes

8/13/2019 VIX Futures Indexes

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Author: M.R. JahanbinDate: 27 May 2009

S&P 500 VIX Futures Indexes

Int roduct ion

We discuss features and applications of the recently launched S&P 500 VIX FuturesIndexes. These tradable benchmarks have returns equal to VIX futures of short andmedium term maturities. They are rolled systematically in fixed increments.

The positions in the different contracts held by SP5VFI is rebalanced daily so as toensure they remain self funded.

These indexes have a negative correlation to most asset classes and so tend to performbetter 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 orexpressing opinions around the richness or cheapness of certain parts of the volatilityterm structure.

The strength of the negative correlation with underlying equity returns, especially duringcrises, 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 calcu lated

VIX is a minute-by minute snapshot of expected stock market volatility over the next 30calendar days. It is calculated in real-time from the options on the S&P 500 index in awide range of strike prices.

The VIX is an amalgam of the information reflected in the prices of all of the optionsused. The contribution of a single option to the VIX value is proportional to the price ofthat option and inversely proportional to the options strike price.

The calculation uses both the near term as well as the next term options to arrive at a 30day implied volatility.

See Appendix for a detailed description.

Background

Initial approach for obtaining volatility exposure was through listed options but theperiodic delta hedging can be challenging.


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The advent of variance swaps allows for a cleaner expression of views variance, which isclose to realized volatility but not exactly there. In addition these instruments are onlytrade OTC and so suffer from counterparty risk.

Although the VIX is not directly tradable, futures and options on it have made it easy totrade around expectations of future implied volatility. While the VIX futures do not needdaily hedging, some active management is necessary for longer holding periods to havethem roll over to a longer dated contracts as expiration nears.

The roll makes it difficult to gauge long term performance of a buy-and-hold strategy justby viewing a history of a given future contract. This is a similar problem to plotting theCL1 contract for Oil and trying to figure out the return for a buy-and-hold investor.

The SP5VFI series addresses these issues by creating tradable benchmarks whose returnsare precisely mimic the payoff from managing a VIX futures position at the given

expirations. The SP5k VIX Short-Term Futures Index (SP5STFI) maintains a rolling longposition in the first and second month contracts while the Medium Term Futures Indexholds the 4

th, 5

th, 6

thand 7

thmonth contracts.

The Short Term Futures Index is better suited for expressing directional views onvolatility expectation, while the Medium Term Futures Index is a better fit as a portfoliohedge (explained in Choice of Maturities section)

Oct05 May06 Nov06 Jun07 Dec07 Jul08 Jan09 Aug090

0.5

1

1.5

2

2.5

3x 10

5

S&P 500 VIX Medium-Term Futures Index TR

S&P 500 VIX Short-Term Futures Index TR


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Advantages:

Offers all advantages enjoyed by the VIX futures investors

No need to worry about rolling to new contracts

Transparent exchange-based 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 notfree and comes at a price which manifests itself via the costs of rolling the contracts inlow 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 monthsATM implied volatility for the past year. Clearly the term structure is generally upwardsloping for long periods of time, punctuated by episodes of inversion.

-0.2 -0.15 -0.1 -0.05 0 0.050

10

20

30

40

50

60

70

80

90Histogram of SPX 6m vs 2m term spread from 16-May-2008 to 19-May-2009

Notes on th e VIX Futures con tracts

Futures contracts on the VIX settle against the level of the spot VIX at the time of thefutures expiration. The settlement date falls on the Wednesday 30 days prior to the thirdFriday of the month following the expiration month. The final settlement price is


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determined from a special opening quotation calculated from the opening prices of thebatch 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 ofSPX 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 theMTM changes of a futures contract need not have anything to do with changes in the spotVIX, although the two get asymptotically closer as expiration approaches.

The simple nature of the contracts makes daily P&L calculations easier a futurecontract 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 1-point increase in implied volatility for that contract.

Rebalancin g (rol l ing)

The SP5VFI series is based on the performance of a strip of VIX futures, which offerconstant vega exposure to changes in volatility at a forward point.

Short Term futures index starts entirely in the first month VIX futures contract on eachmonthly rebalance date. An equal fraction of the position is rolled to the second monthcontract each business day, so that by the subsequent rebalance date (corresponding to theday before the following months VIX expiration) only the second month is held. At thatpoint the second month becomes the front month and the process continues.

The Medium Term follows a similar methodology and starts with holding an equalnumber of contacts in the 4

th, 5

thand 6

thmonths and then rolls the fourth month VIX

futures into the 7th

month contract in equal increments each day. The proportions of the5thand 6thmonth futures are left unchanged.

The daily rebalancing is generally better than monthly. This is because the shape of thevolatility term structure is typically upward-sloping, the ideal roll frequency would beone that minimizes the roll costs during these periods.

The shape of the curve is non-linear and concave when its is upward sloping and convexwhen in backwardation. This implies that for an upward sloping term, rolling a fraction ofthe position more frequently is better than moving the entire position at smaller intervals.


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Apr May Jun Jul Aug Sep Oct Nov37

38

39

40

41

42

43

44Volatlity term structure as of 01 Apr 2009

Futures'Value

Jun Jul Aug Sep Oct Nov Jun Jul Aug Sep Oct29

29.5

30

30.5

31Volatlity term structure as of 20 May 2009

Futures'Value


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Self- funded

The dollar value of the futures deleted from the index is kept equal that invested in thenew future. So if the VIX curve is upward sloping, longer maturity futures are moreexpensive 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 downwardsloping. From VIX history, The VIX futures term structure is upward sloping during lowvolatility period and downward sloping in periods of stress when term volatility tends tobe 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 tovolatility traders it would result in an index that could potentially reach negative values.

Choic e of Matur i t ies

Negative correlation with equity returns is one of the key reasons for consideringvolatility as a standalone asset class. To this end we would prefer an index that is asinversely related to spot as possible. Hypothetical tradable indexes created with VIXfutures 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 theRussell 2000 as a benchmark, highlighting the broad applicability of these instruments

ReturnCorrelation

VIXShortTerm

Index

VIX23 VIX34 VIX45 VIX56 VIX67

VIXMed.Term

Index2006

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 futurescontracts. This idea is extended all the way to the seventh month with VIX67 which is anindex 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 andcan be seen to imply that hedges using these indexes would become more effectiveduring periods of stress.


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Appl icat ions

A tradable index of short or medium term futures on the VIX can be used for expressingviews on the future course of equity implied volatility. This can use other asset classes tojudge the richness or cheapness of volatility or on a standalone basis using its meanreversion 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 Pr and a

volatility of P . The expected return after adding the volatility index is:

VIXVIXnew rwrwr PP +=

Where Pw and VIXw are the weights of the existing portfolio and the volatility index. In

the long run the return from the volatility index VIXr is not expected to be large and is

likely to suffer from a daily roll cost if the term structure remains upward sloping. Thevolatility index should only be traded if there is a belief that its trading cheap or rich. Theeffect we are seeking here is to reduce portfolio risk:

VIXVIX2VIX

2VIX

22new 2 PPPP wwww ++=

Hedging with the volatility index is a double whammy; the negative correlationbetween 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 stressmeans 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 VIXrrP +=

==

VIX

VIX P

Pw

w

The above result is achieved simply by finding the minimum of the new portfoliovolatility.

We backtest the performance of hedges using the S&P 500 VIX Futures Indexes incombination with several underlying assets. In each case we use the trailing beta over a12-month period on the first business day of October, to resize the hedge at annualintervals. The need for rebalancing the hedge arises due to the way the vega notionalreferenced by the volatility index changes, e.g. in a quit period the notional of theunderlying asset would likely increase as it rallies but the normal upward sloping termstructure would reduce the vega exposure of the volatility index, leading to under-hedging.


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There are two indices to chose from; lower roll costs make longer term VIX futuresincrementally more attractive, while decreasing correlation with equity returns has theopposite effect.

Hedging the S&P 500

In this section we consider for an equity portfolio that has a return equivalent to the S&P500. This type of asset class has the highest absolute correlation to the volatility indexes.

May06 Nov06 Jun07 Dec07 Jul08 Jan09 Aug09-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2S&P vs. P/F hedged with VIX indices, rebalanced annually

S&P

S&P + VIX Future Index Med Term Total Ret

S&P + VIX Future Index Short Term Total Ret

Above figure shows the daily performance of the assumed portfolio hedged with each ofthe VIX future indexes. The hedge is assumed to be sized based on the historical beta ofSPX returns relative to returns of the volatility index. The portfolio is rebalanced the firstbusiness 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 $25for the short term index.

For a 20 business day rebalancing, the portfolio does better with the average notionalsincreasing to $57 and $34 for the Medium and Short Term VIX index for a $100 of SPX.


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The performance statistics for the annual rebalancing is as follows:

S&P 500S&P 500 +

VIX MediumTerm Index

S&P 500 + VIXShort 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 thedistribution graph and he statistics in the table. Notice how the volatility and themaximum drawdown of the combined portfolio is almost halved. The reduction in thedrawdown highlights the smaller exposure to very negative returns (thinner left tail)

-0.1 -0.05 0 0.05 0.1 0.150

10

20

30

40

50

60

70

80

90

S&P 500

S&P 500 + Med Term VIX

S&P 500 + Short Term VIX


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Protect ion for Corp orate Credi t

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 asMertons 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 indexlevel. 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 patternsimilar to that for the equity portfolio hedge a performance drag when the spreads weretightening through late 2006 and early 2007, with significant out-performance in late2008 and 2009.

May06 Nov06 Jun07 Dec07 Jul08 Jan09 Aug09-3

-2.5

-2

-1.5

-1

-0.5

0

0.5CDX IG vs. P/F hedged with VIX indices, rebalanced annually

CDX IG

CDX IG + VIX Future Index Med Term

CDX IG + VIX Future Index Short Term

Its should be noted that the maximum drawdown has fallen to about half using either ofthe VIX futures index as a hedge.


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CDXIG

CDX IG + VIXMedium Term

Index

CDX IG + VIXShort 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%

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3

0

5

10

15

20

25

30

35

CDX IG

CDX IG + Med Term VIX

CDX IG + Short Term VIX


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Append ix -- VIX Calculat ion Detai ls

The formula for calculating the VIX as reported by the CBOE is (see [3] for a derivationof this formula)

=

2

02

2 11

)(2

K

F

TKQe

K

K

T i

RT

i

i

where:100VIX =

T = time to expirationF = Forward index level derived from index option prices

Ki= Strike price of i

th

out-of-the-money option; a call if Ki> F and a put if K

i< F

Ki= Interval between strike prices = half the distance between the strike on either side

of Ki :

2

11 + =

ii

i

KKK

Note: K for the lowest strike is simply the difference between the lowest strike and thenext higher strike. Likewise, K for the highest strike is the difference between thehighest strike and the next lower strike.K

0= First strike below the Forward index level F

R = Risk-free interest rate to expirationQ(K

i

) = The midpoint of the bid-ask spread for each (out-of-the-money) option with

strike Ki

Calculating the VIX from the above formula is best done using an example. Thefollowing explains the process in 3 steps.


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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 at-the-money option prices. The at-

the-money strike is the strike price at which the difference between the call and putprices is smallest. As shown in the following table, the difference between the calland put prices is smallest at the 900 strike in both the near and next term.

Near Term Options Next Term Options

StrikePrice

Call Put DifferenceStrikePrice

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 51.16875 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 + eRT

*(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 K0- the strike price immediately below the forward index level, F. Inthis example, K0= 900 for both expirations.

Sort all of the options in ascending order by strike price. Select call options that havestrike prices greater than K0and a non-zerobid price. After encountering twoconsecutive calls with a bid price of zero, do not select any other calls. Next, selectput options that have strike prices less than K0and a non-zerobid price. Afterencountering two consecutive puts with a bid price of zero, do not select any otherputs. Select both the put and call with strike price K0. Then average the quoted bid-ask prices for each option.


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Notice that two options are selected at K0, 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 aroundK0. In order to avoid double counting, however, the put and call prices at K0areaveraged to arrive at a single value. The price used for the 900 strike in the near termis, 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 thisexample:

NearTermStrike

OptionType

Mid-QuotePrice

NextTermStrike

OptionType

Mid-QuotePrice

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

900Put/Call

Average18.19 900

Put/CallAverage

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


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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 optionswith time to expiration of T1and T2, respectively, yields:

=

2

0

1

12

1

2

1 11)(2 1

K

F

TKQe

K

K

T i

RT

i

i

=

2

0

2

22

2

2

2 11

)(2

2

K

F

TKQe

K

K

T i

RT

i

i

The VIX is an amalgam of the information reflected in the prices of all of the optionsused. The contribution of a single option to the VIX value is proportional to the price ofthat option and inversely proportional to the options strike price. For example, thecontribution of the near term 775 Put is given by

Put)775(12

put775

Put775 QeK

K RT

Generally, iK 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 betweenKi 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, Put775K = 25 (i.e.,

800 775), and

0.000005)11.0(775

25Put)775( 00.041095890.0116222put775

Put775 1 ==

eQeKK RT

A similar calculation is performed for each option. The resulting values for the near termoptions are then summed and multiplied by 2/T1. Likewise, the resulting values for thenext term options are summed and multiplied by 2/T2. The table below summarizes theresults for each strip of options.


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NearTermStrike

OptionType

Mid-QuotePrice

Contributionby Strike

NextTermStrike

OptionType

Mid-QuotePrice

Contributionby 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

900Put/Call

Average18.19 0.000562 900

Put/CallAverage

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

2

0

11

K

F

Tfor the near term (T1) and next term (T2):

0.0000161900

23.90190.11780821

111

0.0000061900

43.900

00.04109589

11

1

22

0

2

2

22

0

1

1

=

=

=

=

KF

T

K

F

T

Now calculate 21 and2

2 :

0.0636670.000016-0.06368311

)(2

0.0664720.000006-0.06647811

)(2

2

0

2

22

2

22

2

0

1

12

1

2

1

2

1

==

=

==

=

K

F

TKQe

K

K

T

K

F

TKQe

K

K

T

i

RT

i

i

i

RT

i

i

)(

22 i

RT

i i

i

KQeK

K

T


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Step 3 Interpolate 21 and2

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.

30

365302

32

302

11

12

1

12

2

N

N

NN

NNT

NN

NNT

TT

T

TT

T

+

=

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)N30 = number of minutes in 30 days (30 x 1,440 = 43,200)N365= number of minutes in a 365-day year (365 x 1,440 = 525,600)

43,200

525,600

21,600-61,920

21,600-43,2000.063667

525,600

61,920

21,600-61,920

43,200-61,9200.066472

525,600

21,600

+

=

25.36100VIX

0.253610

==

=

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 Tradingand Derivative strategy, 12 Sep 2005.

[4] T. Watshma, K. Parramore, Quantitative Methods in Finance, Thomson BusinessPress, 1998, ISBN: 1-86152-367-X

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