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S&P 500 VIX Futures Indexes

|Includes: iPath S&P 500 VIX Short-Term Futures ETN (VXX), VXZ
S&P 500 VIX Futures Indexes

 

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 minute-by minute snapshot of expected stock market volatility over the next 30 calendar days. It is calculated in real-time 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.

Background

Initial 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 buy-and-hold 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 buy-and-hold 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 Short-Term Futures Index (SP5STFI) maintains a rolling long position in the first and second month contracts while the Medium Term Futures Index holds the 4th, 5th, 6th and 7th 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 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 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 1-point 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 4th, 5th and 6th months and then rolls the fourth month VIX futures into the 7th month contract in equal increments each day. The proportions of the 5th and 6th 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 upward-sloping, the ideal roll frequency would be one that minimizes the roll costs during these periods.

 

The shape of the curve is non-linear 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.


Self-funded

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
Correlation

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.

Applications

A 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 rp and a volatility of σp. The expected return after adding the volatility index is:

 

 

Where wp and wvix are the weights of the existing portfolio and the volatility index. In the long run the return from the volatility index rvix, 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 12-month 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 under-hedging.

 

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 out-performance 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

Ki = Strike price of ith out-of-the-money option; a call if Ki > F and a put if Ki < F

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

 

 

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.

K0 = First strike below the Forward index level F

R = Risk-free interest rate to expiration

Q(Ki) = 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. 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 at-the-money option prices. The at-the-money 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 + 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. In this example, K0 = 900 for both expirations.

 

·        Sort all of the options in ascending order by strike price. Select call options that have strike prices greater than K0 and a non-zero 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 K0 and a non-zero 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 K0. Then average the quoted bid-ask prices for each option.

 

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 around K0. In order to avoid double counting, however, the put and call prices at K0 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

Mid-Quote Price

 

Next Term Strike

Option Type

Mid-Quote 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 T1 and T2, 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, ΔKi 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,

ΔK775Put = 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/T1. Likewise, the resulting values for the next term options are summed and multiplied by 2/T2. The table below summarizes the results for each strip of options.


 

Near Term Strike

Option Type

Mid-Quote Price

Contribution by Strike

 

Next Term Strike

Option Type

Mid-Quote 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
σ12and σ22 :

 

 

 


Step 3 – Interpolate σ12and σ22 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)

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)

 


 

 

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: 1-86152-367-X





Disclosure: No Position