DOUBLE THE FUN WITH CBOE’s VVIXSM Index
Introduction
CBOE is expanding its suite of volatility benchmarks with a new index called the
VVIX Index, the VVIX for short. The VVIX is a volatility of volatility measure in
that it represents the expected volatility of the 30-day forward price of the CBOE
Volatility Index (the VIX®). It is this expected volatility that drives the price of VIX®
nearby options.
The VVIX is calculated from the price of a portfolio of liquid at- and out-of-themoney VIX® options. The calculation method is the same as for the VIX® and
summarized in Appendix 1. CBOE also calculates a term structure of the VVIX
associated with different VIX option expirations. The VVIX is disseminated in real
time and its term structure is posted on CBOE’s web site at
http://www.cboe.com/micro/VVIX/documents/vixvixtermstructure.xls
The VVIX is designed to guide and inform the growing number of investors in
VIX-based products. The VVIX and its term structure convey:
 The expected volatility that determines VIX® option prices.
 The expected volatility of the VIX® itself to a nearby horizon.
 The mean and standard distribution of settlement values of VIX® futures and
options.
Different points on the VVIX term structure price portfolios of VIX® options (VVIX
portfolio) to different expirations. A position in a VVIX portfolio replicates the
volatility of VIX® forward prices. VVIX portfolio prices have usually been at a
premium relative to future realized volatility. The discount is a volatility risk
premium. For nearby expirations, these prices have also tended to surge at the
same time as VIX®. These features suggest several trading opportunities:
 Buying a VVIX portfolio returns the difference between realized and expected
volatility less the volatility risk premium. Conversely selling a VVIX portfolio
returns the difference between expected and realized volatility plus the
volatility risk.
 To the extent that volatility expectations are unbiased, consistently selling a
VVIX portfolio captures the volatility risk premium.
 Buying a short-dated VVIX portfolio can cushion losses from extreme
increases in VIX® futures prices.
1
1. Historical Behavior of VVIX June 2006 – February 2012
Chart 1a. The VIX® and VVIX, Time Series
160
VIX
VVIX
140
120
100
80
60
40
20
1‐Jun‐06
1‐Sep‐06
1‐Dec‐06
1‐Mar‐07
1‐Jun‐07
1‐Sep‐07
1‐Dec‐07
1‐Mar‐08
1‐Jun‐08
1‐Sep‐08
1‐Dec‐08
1‐Mar‐09
1‐Jun‐09
1‐Sep‐09
1‐Dec‐09
1‐Mar‐10
1‐Jun‐10
1‐Sep‐10
1‐Dec‐10
1‐Mar‐11
1‐Jun‐11
1‐Sep‐11
1‐Dec‐11
0
Chart 1b. The VVIX versus the VIX® Sorted in Ascending Order
160
140
120
VIX
VVIX
100
80
60
40
20
9.89
11.13
12.31
13.97
15.54
16.3
17.06
17.59
18.02
18.6
19.44
20.23
20.97
21.6
22.42
23.08
23.8
24.51
25.26
26.05
27.49
29.68
32.8
37.15
43.37
54.56
0
Chart 1a. is a graph of the time series of the VVIX and VIX between June 2006
and February 2012 ; Chart 1b.shows the variation of the VVIX when the VIX® is
2
sorted in ascending order. Several features are apparent from these charts and
from an analysis of index returns:
 The range of values of the VVIX is at a significantly higher level than
that of the VIX®.
The VVIX ranges between 60 and 145 around an average of 86. The VIX
ranges between 10 and 81 around an average of 24. Also note that the range
of variation of the VVIX tends to widen at higher values of the VIX.
 Except at high values of VIX, there is little correlation between
variations of the VIX® and VVIX.
The VVIX and VIX® both reached local peaks in October 2008, during the
credit crisis of 2008 and in May 2010, the “flash crash” month. In general
however, the relationship between their variations is weaker than the
relationship between the VIX® and the S&P 500.
 Since the flash crash of May 2010, the VVIX has rarely dropped below
80.
This suggests that a new volatility regime came about after the flash crash.
Market participants appear to have become more tentative about the future
value of the VIX®.
The following additional features of VVIX characterize its term structure and its
tendency to mean revert.
 The VVIX tends to revert to its historical mean.
Since the VVIX is a rate variable, it is mean reverting. Table 1 shows the
historical value of the VVIX 30, 60 and 90 days later, starting from different
initial intervals. For example, when the initial value of the VVIX is between 30
and 50, its average value 30 days later is 78. The mean reversion is centered
about the historical mean of VVIX, approximately 85 and the fixed point is the
interval from 80 to 90 volatility points. Starting from this interval, the VVIX
ends in the same interval. Starting from an initial value below its historical
mean, the VVIX tends to end in a higher range in subsequent months.
Conversely, from an initial value above its historical mean, the VVIX tends to
revert to a lower range.
Table 1 Mean Reversion of VVIX
3
VVIX 30 VVIX 60 VVIX 90
VVIX
Days
Days
Days
Range
Later
Later
Later
78
74
71
30 - 50
78
83
82
50 - 60
82
86
87
60 - 70
85
85
86
70 -80
90
86
85
80 - 90
95
90
85
90 - 100
96
89
92
100 - 110
95
92
93
110 - 120
96
94
91
120 - 130
99
93
91
130 - 145
 The term structure of the VVIX is downward sloping
Based on past history, the term structure of the VVIX is downward sloping.
This is normal because a forward price is a forecast, and a short term
forecast tends to be more volatile than a long-term forecast.
Chart 2 Term Structure of the VVIX
Term Structure of VVIX
160
140
2 Months
3 Months
6 Months
1 Month
120
100
80
60
40
20
10/1/2011
6/1/2011
2/1/2011
10/1/2010
6/1/2010
2/1/2010
10/1/2009
6/1/2009
2/1/2009
10/1/2008
6/1/2008
2/1/2008
10/1/2007
6/1/2007
2/1/2007
10/1/2006
6/1/2006
0
2. Applications of the VVIX
 Volatility Premium Capture
The VVIX and its term structure price the volatility of VIX® futures prices. This
is useful information for trading that volatility. VVIX embeds a risk premium
that can be captured by selling the VVIX portfolio on a consistent basis. The
risk premium is illustrated in the bottom panel of Chart 3 where there is a
positive spread between the daily value of the December 2010 VVIX and the
volatility subsequently realized until the December 2010 expiration.
4
The top panel of Chart 3 shows that the risk premium is offset in periods
when the market significantly underestimates future realized volatility, as it did
in September and October 2008, the onset of the credit crisis. An investor
who sensed that VVIX undervalued future volatility prior to that period could
have profited by buying the VVIX portfolio.
Chart 3 VVIX vs. Realized Volatility of VIX® Futures Price
Expected vs Realized Volatility
Dec 08 VIX Futures
140
120
100
80
60
40
20
0
VVIX
12/8/2008
11/24/2008
11/10/2008
10/27/2008
9/29/2008
10/13/2008
9/1/2008
9/15/2008
8/18/2008
8/4/2008
7/21/2008
7/7/2008
6/23/2008
Realized Volatility
Expected vs Realized Volatility
Dec 10 VIX Futures
180
160
140
120
100
80
60
40
20
0
VVIX
12/9/2010
11/25/2010
11/11/2010
10/28/2010
9/30/2010
10/14/2010
9/2/2010
9/16/2010
8/19/2010
8/5/2010
7/8/2010
7/22/2010
6/24/2010
Realized Volatility
 Points on the VVIX Term Structure yield estimates of the fair value of the
VIX® at expiration dates
At date t, VVIX t,T is the point on the VVIX term structure corresponding to
expiration date T. VVIX t,T generates an alternative estimate of the final
settlement value of VIX futures expiring at T. This estimate is an approximate
“fair value” for VIX® futures which is based on VIX® option prices.
Fair valuet = Et [VIX T ]  Et [ FT ]  Et [ FT2 ]  Ft 2 *
5
T t
VVIX t2,T
365
The derivation of the fair value is based on a result of B. Dupire described in
Appendix 2.
The first term under the square root sign, Et[VIXT 2], is the expected variance
of the S&P 500 from T to T+30. It is estimated from S&P 500 option (SPX)
strips that replicate spot variances to different dates. The second term is a
multiple of VVIXt,T .
Any deviation between the fair value and the price of VIX futures is a potential
signal to buy or sell VIX futures. It also suggests the following quasi-arbitrage:
a spread between the portfolio of SPX options that replicates the forward S&P
500 variance VIXT2 and a position in VIX® futures that approximately
replicates the same forward variance.
The fair value of VIX May 2009 VIX® futures, the VIX® futures price, and the
final settlement value are shown in Chart 4.
Chart 4. Fair Value of May 2009 VIX Futures
May 09 VIX Futures
55
50
Fair Value of VIX May 2009 Futures
VIX Futures Price
Settlement Value of VIX May 09 Futures
45
40
35
30
25
01/20/09
02/20/09
03/20/09
04/20/09
 Points on the VVIX Term Structure proxy for the standard deviation of
the VIX® at expiration dates
The standard deviation expresses the degree of confidence the market has in
forecasts of VIX. For any expiration of VIX® futures, the standard deviation is
obtained by multiplying the value of the corresponding VVIX with the price of
VIX® futures. VVIX is deannualized and expressed in decimals. The
derivation is described in more detail in the Appendix.
Paths that are one standard deviation from the mean are shown in Chart 5 for
December 2008 and December 2009 VIX® futures. Uncertainty about the
expiration value of December 2008 VIX futures started to increase in
6
September 2008 – the onset of the credit crisis - and still prevailed close to
expiration. The picture for December 2009 is closer to the norm. Uncertainty
about the expiration value decreases over the life of the futures contract.
Chart 5. Estimated Standard Deviation of VIX®, Dec 2008 and Dec 2009 VIX®
Dec 2008 VIX Futures
100
90
80
70
60
50
40
30
20
10
0
VIX Futures Price
12/7/09
11/23/09
11/9/09
10/26/09
10/12/09
9/28/09
9/14/09
8/31/09
8/3/09
8/17/09
Futures Price +/- One Std. Deviation
6/22/09
12/8/08
11/24/08
11/10/08
10/27/08
10/13/08
9/29/08
9/15/08
9/1/08
8/18/08
8/4/08
7/21/08
7/7/08
6/23/08
Futures Price +/- One Std. Deviation
7/20/09
VIX Futures Price
7/6/09
100
90
80
70
60
50
40
30
20
10
0
Dec 2009 VIX Futures
Options involve risk and are not suitable for all investors. Prior to buying or selling an option, a person must
receive a copy of Characteristics and Risks of Standardized Options. Copies are available from your
broker,by calling 1-888-OPTIONS, or from The Options Clearing Corporation at www.theocc.com.
Supporting documentation for any claims, comparisons, statistics or other technical data in this document is
available by visiting www.cboe.com or contacting CBOE at www.cboe.com/Contact. The information in this
document is provided solely for informational purposes is not intended and should not be construed to
constitute investment advice or recommendations to purchase or sell securities or futures. Investors should
consult their tax advisor as to how taxes affect the outcome of contemplated options transactions. Past
performance is not indicative of future results. This document contains index performance data based on
back-testing, i.e., calculations of how the index might have performed prior to launch. Backtested
performance information is purely hypothetical and is provided in this document solely for informational
purposes. Backtested performance does not represent actual performance, and should not be interpreted as
an indication of actual performance. The methodologies of the CBOE volatility indexes are owned by
Chicago Board Options Exchange, Incorporated (CBOE) and may be covered by one or more patents or
pending patent applications. CBOE, Chicago Board Options Exchange, CBOE Volatility Index and VIX are
registered trademarks and SPX and VVIX are service marks of CBOE. Standard & Poor's, S&P and S&P
500 are trademarks of Standard & Poor’s Financial Services, LLC and have been licensed for use by CBOE.
Copyright © 2012 CBOE. All rights reserved
7
Appendix 1
Calculation of VVIX
VVIX is calculated with the same methodology as VIX. It is derived from the price
of a portfolio of liquid at and out-of-the-money VIX options. That portfolio can be
traded to manage the volatility risk of exposures to the VIX® and to harvest the
risk premium between the expected and realized volatility of VIX® forward prices
VVIX is a 30-day volatility but VIX options expiring in 30 days are usually not
available. VVIX-like values are therefore calculated from VIX® options expiring at
two dates straddling 30 days. VVIX is then interpolated from these numbers.
At each expiration, VVIX is calculated from VIX® options prices using the VIX®
formula:
2
 
T
2

i
K i RT
e Q( K i )
K i2
2

1  F

 1 ,

T  K0

where…
 is
VVIX
T
Time to expiration
F
Forward index level derived from index option prices
K0
First strike below the forward index level, F
Ki
Strike price of ith out-of-the-money option; a call if Ki>K0 and
a put if Ki< K0; both put and call if Ki=K0.
Ki
Interval between strike prices – half the difference between
the strike on either side of Ki:
100
 VVIX =   100
Ki =
K i 1  K i 1
2
(Note: K for the lowest strike is simply the difference
between the lowest strike and the next higher strike.
8
Likewise, K for the highest strike is the difference between
the highest strike and the next lower strike.)
R
Risk-free interest rate to expiration
Q(Ki)
The midpoint of the bid-ask spread for each option with
strike Ki.
Since the step-by-step calculation of VVIX follows that of the VIX, please refer to
the VIX White Paper for additional details
http://www.cboe.com/micro/VIX/VIXwhite.pdf
Note that VIX options settle on Wednesdays that precede SPX expirations by 30
days.
Appendix 2
1. Estimation of Standard Deviation of the VIX® from VVIX
At date t and for expiration date T,
VVIX t2,T 
T
F
365
365
365
* E t [   R2tt dt ] 
* VARt [ln( T )] 
* VARt [ln( FT )]
T t
T t
Ft
T t
t
where Rt is the logarithm of the instantaneous return of the forward price Ft and
FT = VIXT.
A truncated Taylor expansion of the logarithm of FT is
ln( FT )  ln( Ft ) 
FT  Ft
Ft
This implies
VARt [ FT ]
T t
 VARt [ FT ]  VARt [VIX T ]  Ft 2 *
VVIX t2,T
2
365
Ft
T t
VARt [VIX T ]  Ft 2 *
VVIX t2,T
365
VARt [ln( FT )] 
Taking square roots on both sides, the standard deviation of VIXT is linked to
VVIXt,T.
STDVt [VIX T ]  Ft *
9
T t
VVIX t ,T
365
2. Fair Value of VIX Futures
The fair value of VIX® futures prices is
Fair valuet = Et[VIXT]= Et[FT]
This fair value is derived by B. Dupire In “Volatility Derivatives Modeling”
(Bloomberg Presentation, 2005) by expanding the variance of the settlement
price of VIX futures as
VARt[FT] = Et[FT 2] – (Et[FT])2 → Fair valuet =
Et [VIX T2 ]  VARt [VIX T ]
The fair value of VIXT is equal to the square root of the difference between the
date T forward value of VIX2 and the variance of FT (=VIXT). From section 1
above, we can approximate this variance as:
VARt [VIX T ]  Ft 2 *
T t
VVIX t2,T
365
3. Fair Value Strategy
Suppose that the fair value of VIXT is greater than the current price of VIX®
futures.
Et [VART ,T 30 ]  c  Ft
where c is VARt[VIXT] which can be estimated from VVIX.
Then
E t [VART ,T 30 ]  c  Ft 2
This payoff is replicated with a bond and a position in SPX option strips that pays
off the 30-day variance of the S&P 500 from T to T+ 30
10