Volatility, the Size Premium, and the Information Quality
of the VIX and VIX Futures: New Evidence
Lorne N. Switzer and Qianyin Shan
Concordia University, John Molson School of Business
2015 Morton Topfer Chair Lecture Series
New York University
Polytechnic School of Engineering
Department of Finance and Risk Engineering
Six Metrotech Center, RH Dibner Auditorium
Tuesday, February 17th 4:30PM.
Abstract
This paper examines the futures market efficiency of the VIX and the relative
merits of the VIX and VIX futures contracts in forecasting future S&P 500
excess returns the future Russell 2000 excess returns, and the future small-cap
premium. We find that the current VIX is significantly negative related to S&P
500 index excess returns and positively related to the Russell 2000 index
excess returns. These results suggest that the VIX predicts asset returns based
on size based portfolios asymmetrically – with higher (lower) values of the VIX
associated with lower (lower) values of large-cap (small-cap) returns in the
future. The VIX and VIX modeled by an ARIMA process are significantly
positive related to future values of the small-cap premium consistent with a
fundamental default risk explanation. In addition, VIX futures show
forecasting prowess for the S&P 500 excess return, the Russell 2000 excess
return and the small-cap premium. The results for the speculative efficiency of
the VIX futures contracts are mixed, however. Overall, the analyses support the
hypothesis of informational advantages of the futures markets relative to the
spot market in the price discovery process not just for sized based asset
returns, but on the size premium as well.
Agenda






Section I
Section II
Section III
Section IV
Section V
Section VI
Motivation
Literature review
Modeling Issues
Data Description
Empirical results
Conclusion
Motivation



Markets are volatile
Volatility affects investor (Switzer et al
2015)
Volatility affects asset prices (Black and
Scholes, 1973, Ross, 1976, ensuing
generations of option pricing models
Is VIX as a leading indicator for returns as well as
the small cap premium? Is the market efficient?
US Market Volatility 1996-2013 (plot from Switzer
et al 2015)
Volatility affects investor behavior
Motivation, cont’d


Perhaps the most widely quoted measure of market
volatility is the CBOE Volatility Index (the VIX) The VIX
is meant to capture the market’s expectation of stock
market volatility over the next 30 calendar days, and has
been disseminated by the CBOE on a real-time basis
since 1993, as a weighted blend of prices for a range of
options on the S&P 500 index. The VIX is quoted in
percentage points and translates, roughly, to the
annualized expected movement in the S&P 500 index
over the upcoming 30-day period.
Many investors consider the VIX Index to be the world's
premier barometer of investor sentiment and market
volatility, and VIX options are very powerful risk
management tools
Is VIX as a leading indicator for returns as well as
the small cap premium? Is the market efficient?
How the VIX is calculated
The VIX formula
Calculating the VIX, cont’d
Trading on the VIX


On March 26, 2004, trading in futures on the VIX began
on CBOE Futures Exchange (CFE). On February 24,
2006, options on the CBOE Volatility Index® (VIX®)
began trading on the Chicago Board Options Exchange.
The VIX options contract is the first product on market
volatility to be listed on an SEC-regulated securities
exchange: the most successful new product in CBOE
history. In just ten years since the launch, combined
trading activity in VIX options and futures has grown to
over 800,000 contracts per day.
Many investors consider the VIX Index to be the world's
premier barometer of investor sentiment and market
volatility, and VIX options are very powerful risk
management tools
The VIX presumably captures fear


The VIX is often referred to as the fear index or
the fear gauge, since high levels of VIX typically
coincide with high degrees of market turmoil
(Whaley, JPM 2000)
“At the first whiff of bad news, stocks tend to
plummet and the CBOE Volatility Index (VIX)
tends to spike.” Steven Sears in Barrons,
January 7, 2015
VIX (the so-called fear guage) as a
forecast of market returns


07/07/14 01:34 PM "Front month VIX testing its
all-time lows," wrote Hedgeye CEO Keith
McCullough in today's Morning Newsletter.
"Closing at 10.32 (it has never held below 10,
sustainably) – contrarian signal: e.g. the adage
"When the VIX is high, it's time to buy.…VIX is
one of the best contrarian indicators in the
business.” (Volnado trades)
But the analysts are not in agreement…..
“Surprise! VIX is too LOW for a Market Crash”
http://finance.yahoo.com/news/surprise-vix-too-low-market-032813345.html
July 25, 2014
Too low for a crash? Oops
VIX and current markets

“U.S. stocks closed at highs on Friday, with the
Dow above 18,000 and S&P 500 setting a new
record as firming oil prices sent the energy
sector higher……The CBOE Volatility Index
(VIX), widely considered the best gauge of
fear in the market, traded near 15..” CNBC
News: http://www.cnbc.com/id/102424287., Feb.
13, 2015
Inefficient pricing of the VIX (weak
form sense)?

“When the VIX fell as low as 16.6 on July 6 last year, Money Morning
Capital Wave Strategist Shah Gilani warned investors to protect themselves
against potential volatility..The low VIX creates an excellent opportunity for
you to buy put protection at reasonable prices," Gilani said."In the face of
future unknowns, and as long as implied volatility is low, you should take
advantage of cheap puts to add some portfolio protection … just in case.
Exactly two days later the VIX began a relentless climb. Investors who
heeded his advice were glad they did. The VIX jumped to 48 and remained
above 40 throughout October as the S&P 500 tumbled to a yearly low of
around 1100. Meanwhile, Gilani had also advised his Capital Wave
Forecast subscribers to hedge with put options in May and June. On Aug.
8- the day the Dow Jones Industrial Average plunged 635 points –
subscribers to Capital Wave Forecast locked in gains of 456%, 455%,
371%, and 197% on four of those holdings….” Don Miller, Monday
Morning,”The VIX Indicator: What this Contrarian Index is Telling Us Now,”
March 7, 2012 .
VIX and the Size Premium?


Sparse evidence to date
Unanswered question: perhaps the VIX affects
the size premium as a proxy for (or reflection of)
fundamental default risk
small cap premium: 1990-Jul. 2013
Figure 1. Weekly data
.08
.04
.00
-.04
-.08
-.12
90
92
94
96
98
00
02
04
06
08
10
12
06
08
10
12
VIX: 1990 - July 2013
.8
.7
.6
.5
.4
.3
.2
.1
.0
90
92
94
96
98
00
02
04
A few related academic papers

Banerjee et al.(2007) and Kanas(2013) find that VIX
predicts returns on large-cap stock market indices (S&P
500).
Small Cap Premia in the Literature
Risk/Return Profile Studies:
• Fama and French, 1993 – introduce size premium as
a risk factor, Dimson and Marsh (1999), Switzer and
Fan (2007), Switzer and Tahaoglu (IRFA,2015) –
spanning, recursive cointegration
• Performance studies: Switzer (2010, 2012) looks at
alphas for several countries: Small cap anomaly
appears for the post 2001 period for many countries
Governance factors (Switzer and Tang (2009));
Innovation (Switzer (2012))
24
Small cap premium and default risk

Key finding of previous studies: In countries with
common law jurisdictions, the US default risk,
which is uncorrelated with local default risk is
found to be priced in many international markets
where protection of shareholders and creditors
in bankruptcy states is limited. The default risk
factors are distinct from local business cycle
turning points per se (term structure and inflation
risk are related to the business cycle though:
Switzer and Picard (2015) – Markov switching
model))
Link between this study and the
size premium literature

Hypothesize in this paper: just as default risk as
measured by yield spreads can explain the small
cap premium in a consistent way, so should the
VIX
Modeling Approach



Britten-Jones and Neuberger (2000) derive the “modelfree” implied volatility from current option prices.
Jiang and Tian (2005) extend the model-free implied
volatility to asset price processes with jumps.
Carr and Wu (2009) develop a direct and robust method
for quantifying the return variance risk premium on
financial assets - simple principle to develop
estimating equations; no arbitrage dictates that the
variance swap rate equals the risk-neutral expected
value of the realized variance
Modeling, cont’d

excess return (Realized annualized return
variance RV) vs. the fixed variance swap rate
SW) can be measured by a CAPM model:
Modeling, Cont’d

Realized volatility can be efficiently measured
from implied volatility, hence:
2
2
 RV
,T   ' IV ,t
(5)
where  IV2 ,t is the implied volatility at time t prior to time T. Combining equation (4) and
(5), the relation between excess return and implied variances can be written as:
m
r t ,T  ( '  ) /   ( 1) /  IV2 ,t   / 
(6)
which can be rewritten as
m
r t ,T   *   * IV2 ,t   *
(7)
Modeling, cont’d

Link (7) to a GARCH-M model, as per Kanas
(2013)
l
rt  c   ai rt i   ht  ht  t
(8)
i 1
with the conditional variance equation following GARCH (1, 1) by including the squared
implied volatility as an exogenous variable.
GARCH (1,1) : ht     t21   ht 1  gVIX t21
(9)
Rt is the excess total market return, ht is the conditional variance and λ is the risk-return
parameter.
Modeling: approach of this paper

Extend model – look at returns and size
premium, and use Asymmetric GARCH:
Russell 2000
t
R
R
sp 500
t
l
2000
 c   ai ( RtRusell
 Rtspi500 )   ht  ht  t
i
(10)
i 1
For the conditional variance equation, we allow the squared VIX as an exogenous
variable, and consider the GJR asymmetric GARCH (1, 1) specification:
ht     t21   ht 1   t21 I t 1  gVIX t21
where It  1 if  t  0 and 0 otherwise.
(11)
Tests for Speculative EfficiencyFama (1984) approach
We implement Fama’s (1984) regression approach to test whether the basis at any
period contains information about future spot prices or contains information about the
risk premium at the expiration of the contract. We estimate two equations:
St 1  St  1  1 ( Ft  St )  1,t 1
(12)
Ft  St 1   2   2 ( Ft  St )   2,t 1
(13)
Tests for Speculative Efficiency
Park and Switzer(1997)
Second, we test market efficiency by examining the prowess of futures prices relative
to random walk predictors using daily data. As per Park and Switzer (1997) we examine
STi  0  1Ft i,T   2 MATt i   ti
(14)
where STi is the prevailing spot price for contract i at time T (when contract i matures);
Ft i,T is the futures price of contract i at time t; MAT is the number of days for contract i to
mature as of time t, and  ti is the error term. If  1 is found to be significantly different
from 0, then the current contract prices are good predictors of future spot prices.
Modeling, cont’d
To eliminate the effect of a momentum factor in the VIX on the estimation result, we
also model VIX as an ARMA (p, q) process:
VIX t  c  ut
ut  1ut 1  2ut 2  ...   p ut  p   t  1 t 1  ...   q t q
Data Description
We obtain data for S&P 500, Russell 2000, and VIX index from
Bloomberg. The sample period is from January 1990 to July 2013.
Dividend yields for the S&P 500 and Russell 2000 are also obtained
from Bloomberg for the same sample period.
To construct the excess returns series of both S&P 500 and Russell
2000, we obtained the data of 3 month Treasury bill of the total sample
period from Federal Reserve Bank Reports.
Data Characteristics

Returns not normally distributed, but they
appear stationary
Data Distributions
Return-SP500
Mean
Standard
deviation
JB test
ADF unit
root test
Daily
0.0004
Return-Russell
2000
0.0005
Small cap
premium
0.00008
VIX
VIX^2
0.20
0.047
Weekly
0.0020
0.0021
0.00016
0.20
0.048
30 day
0.0083
0.0092
0.00083
0.20
0.047
Daily
0.0117
0.0137
0.0068
0.082
0.05
Weekly
0.023
0.029
0.015
0.08
0.05
30 day
0.043
0.056
0.033
0.078
0.043
Daily
18574*
8886*
3868*
16496*
304856*
Weekly
1529*
1050*
485*
3519*
64615*
30 day
32.7*
26.8*
250*
310*
3499*
Daily
-57.89
-77.82
-77.10
-4.77
-5.54
Weekly
-38.21
-35.58
-35.86
-4.86
-6.31
30 day
-15.6
-14.78
-18.06
-4.87
-5.52
Empirical Results: VIX as an
explanatory factor

The squared current VIX in the conditional
variance equation is a significant explanatory
factor for the small cap size premium. The
coefficient λ is positive and significant for 30 day
return, and the coefficient g of VIX in the
conditional variance equation is significant for all
the returns of different horizons.
Estimation Results: Small Cap Premium
Table 2: Estimation results for the small cap size premium with squared current VIX in
the conditional variance equation for the sample period from 1990 to July 2013. (***, **,
* denote significant at 1%, 5%, and 10% level, respectively. t statistics in parentheses)
daily return
Weekly return
30 day return
Conditional mean equation parameters
0.0000(0.53)
c
-0.02(-1.32)
a1
0.005(0.36)
a2
-0.003(-0.18)
a3
0.01(0.60)
a4
-0.001(-0.10)
a5
-0.02(-1.55)
a6
-0.02(-1.35)
a7
0.003(0.27)
a8
0.0009(1.20)
-0.06*(-1.91)
0.03(0.86)
0.05*(1.80)
0.01(0.35)
-0.04(-1.36)
-0.05(-1.51)
-0.003(-0.105)
-0.003(-0.09)
-0.01**(-2.05)
-0.06(-1.19)
0.07(1.13)
-0.06(-1.14)
-0.16***(-2.69)
-0.04(-0.63)
-0.16***(-2.71)
0.002(0.03)
0.04(0.49)
λ
-3.36(-0.90)
0.38**(1.97)
Conditional variance equation parameters
0.006***(4.07)
ω(*10,000)
0.063***(7.99)
α
0.88***(75.30)
β
0.016(1.63)
γ
0.138**(2.47)
0.114***(3.22)
0.692***(8.95)
-0.005(-0.125)
0.029(0.23)
-0.087***(-167.3)
0.995***(56.91)
0.147***(5.45)
g (*10,000)
6.17***(3.24)
2.86*(1.85)
1.17(0.35)
0.35***(6.47)
Estimation Results: Small Cap Premium- cont’d
Table 3: Estimation results for the small cap size premium with squared VIX modeled by
ARMA (5, 3) process in the conditional variance equation for the total sample period
from 1990 to July 2013. (***, **, * denote significant at 1%, 5%, and 10% level,
respectively. t statistics in parentheses)
daily return
Weekly return
30 day return
Conditional mean equation parameters
0.0002(1.33)
c
-0.02(-1.53)
a1
0.004(0.32)
a2
-0.007(-0.50)
a3
0.008(0.58)
a4
-0.0003(-0.02)
a5
-0.02(-1.63)
a6
-0.02(-1.23)
a7
0.004(0.30)
a8
0.0006(0.77)
-0.06*(-1.88)
0.04(1.18)
0.04(1.45)
0.01(0.38)
-0.06*(-1.91)
-0.05*(-1.67)
0.02(0.67)
0.006(0.19)
-0.003(-1.49)
-0.16***(-3.57)
-0.03(-0.52)
-0.12**(-2.26)
-0.10*(-1.92)
-0.05(-0.80)
-0.15***(-2.81)
-0.01(-0.10)
0.07(1.00)
λ
-1.22(-0.32)
0.14**(2.55)
Conditional variance equation parameters
0.05***(23.33)
ω(*10,000)
0.056***(9.77)
α
0.92***(149.4)
β
0.025***(3.21)
γ
1.14***(7.22)
0.095***(4.66)
0.858***(37.62)
0.015(0.67)
-10.14***(-350.7)
-0.051***(-4.67)
1.007***(4766.8)
0.090***(4.00)
g (*10,000)
-1.11***(-20.5)
-26.18***(-6.32)
251.97***(348.9)
Log likelihood
Durbin-Watson stat
20988
2.00
3386
1.94
571
1.86
-2.87(-0.87)
VIX as a predictor
VIX Basis is not an unbiased predictor of Futures Spot
Change (eqn 12), time varying risk premium (eqn 13)
Table 12. Results of Fama (1984) Model
Estimated period:
April 2004-July 2013
Regression (12)
St 1  St  1  1 ( Ft  St )  1,t 1
Regression (13)
Ft  St 1   2   2 ( Ft  St )   2,t 1
1
0.148349
[0.574634]
1
F-Stat
-0.264397
[0.181089]
2.131705
2
2
F-Stat
-0.148349
[0.574634]
1.264397*
[0.181089]
48.75081*
Note: Robust standard errors are reported inside parentheses. The * denotes significance at a 1% level.
Test of Market Efficiency of VIX –time varying risk
premium supported(eqn. 13)
Table 13. Wald Test Results of the Fama (1984) Model
Estimated period:
April 2004-July 2013
Regression (12)
St 1  St  1  1 ( Ft  St )  1,t 1
Regression (13)
Ft  St 1   2   2 ( Ft  St )   2,t 1
1  0, 1  1
1  1
1  0
27.18831*
[0.0000]
48.75081*
[0.0000]
0.066648
[0.7968]
 2  0, 2  1
2  1
2  0
1.104387
[0.3352]
2.131705
[0.1473]
0.066648
[0.7968]
Note: F values reported. p-values reported in parentheses.
VIX Futures as Predictor of Future
Spot VIX
Table 14. VIX Futures Contracts as Predictors of Futures Spot VIX: Daily Data
Independent Variable
Coefficient
t-Statistics
OLS estimates of STi  0  1Ft i,T  2 MATt i   ti
Estimation period: April 2004-December 2012
Ft i,T
0.999243**
[0.013601]
73.47
MAT
-0.019037*
[0.011416]
-1.67
0
0.115214
[0.377704]
0.31
F-statistic
Prob(F-statistic)
2714.689
0.0000
Note: ** denotes significance at a 1% level. * denotes significance at 10%. Robust standard errors are
i
reported in parentheses. ST is the prevailing spot price for contract i that matures at time T;
Ft i,T is the
futures price of contract i at time; MAT is the number of days for contract i to mature as of time t, and
is the error term.
 ti
Lagged futures VIX vs. of lagged current VIX and the
small-cap premium
We replace the lagged VIX by lagged VIX futures price in
equation (11). The estimation sample period is from April
2004 to July 2013. Table 15 provides the result of the
estimation of future small-cap size premium with the
squared current VIX futures price in the conditional
variance equation. Including squared current VIX futures in
the conditional variance equation does improve the
prediction of future small cap size premium. The coefficient
of λ is significantly positive at 1% level for 30 day returns,
and significantly negative at 1% level for weekly returns.
Estimation Results: VIX futures and
the small cap premium
Table 15: Estimation result for the future small cap premium with squared current VIX
futures price in the conditional variance equation for the sample period from April 2004
to July 2013. (***, **, * denote significant at 1%, 5%, and 10% level, respectively. t
statistics in parentheses)
daily return
Weekly return
30 day return
Conditional mean equation parameters
-0.0000(-0.01)
c
-0.03(-1.44)
a1
-0.02(-1.06)
a2
0.01(0.30)
a3
-0.03(-1.51)
a4
-0.03(-1.52)
a5
-0.06***(-2.73)
a6
-0.01(-0.69)
a7
0.01(0.66)
a8
0.007***(3.96)
-0.15***(-3.52)
-0.04(-0.76)
-0.08*(-1.80)
-0.05(-1.12)
-0.01(-0.26)
-0.09**(-1.97)
-0.05(-0.94)
0.02(0.47)
-0.115***(-13.38)
-0.8***(-158.5)
-0.44***(-5.78)
-0.01(-0.09)
0.36***(3.64)
0.04***(2047.8)
0.15*(1.72)
-0.21**(-2.15)
0.13(1.24)
λ
-35.16***(-3.20)
5.68***(90.2)
Conditional variance equation parameters
0.009***(3.02)
ω(*10,000)
0.026**(2.09)
α
0.873***(44.16)
β
0.099***(4.72)
γ
1.59***(5.18)
-0.074**(-2.49)
-0.788***(-10.77)
0.05(1.07)
2.11***(20.9)
0.159***(263.8)
0.611***(200812)
0.059***(8.28)
g (*10,000)
0.242***(2.99)
30.02***(3.71)
-135.5***(-18466)
Log likelihood
Durbin-Watson stat
8565
2.11
1411
1.95
-3759
0.41
3.77(0.72)
VIX Futures as a predictor of future
spot VIX
Table 14. VIX Futures Contracts as Predictors of Futures Spot VIX: Daily Data
Independent Variable
Coefficient
t-Statistics
OLS estimates of STi  0  1Ft i,T  2 MATt i   ti
Estimation period: April 2004-December 2012
Ft i,T
0.999243**
[0.013601]
73.47
MAT
-0.019037*
[0.011416]
-1.67
0
0.115214
[0.377704]
0.31
F-statistic
Prob(F-statistic)
2714.689
0.0000
Note: ** denotes significance at a 1% level. * denotes significance at 10%. Robust standard errors are
i
reported in parentheses. ST is the prevailing spot price for contract i that matures at time T;
Ft i,T is the
futures price of contract i at time; MAT is the number of days for contract i to mature as of time t, and
is the error term.
 ti
Conclusions




VIX predicts asset returns on size based portfolios asymmetrically for daily
returns. VIX is significantly negative related to S&P 500 excess returns and
positively related to Russell 2000 excess returns.
VIX futures show forecasting prowess for small-cap premium and asset
returns on size based portfolios. VIX futures are negatively related to these
series. It supports the hypothesis of informational advantages of the futures
markets relative to the spot market in the price discovery process.
The speculative efficiency results for the VIX futures are mixed.
Further work to be explored – 1. directly link fundamental default risk and
other risk factors to the VIX – e.g. Market liquidity issues (Switzer and
Picard (2015) ; 2. Look at effects of changes in estimation of VIX in 2014–
more precise estimates of future 30 day volatility with the introduction of
SPX Weeklys allows the VIX Index to be calculated with S&P 500 Index
option seriesthat most precisely match the 30-day target timeframe for
expected volatility that the VIX Index is intended to represent.
Short term vs. long term investment
horizon
Thank you!