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The Relationship Between the VIX and the Volatility Risk
Premia
Paul F. Kratz
Abstract
The following paper aims to develop and investigate the relationship between holding returns on
the VIX and the variance risk premia.The paper establishes the presence of a statistically signifcant
relationship between the variance risk premia and returns on squared values of the VIX - (VIX
2
), that
is non-linear in nature but can also be recast as a linear combination of the VIX
2
and CBOE SKEW
Index. In addition, the paper also investigates whether the VIX
2
can predict future realizations of the
variance risk premia and if so whether the predictive power of the VIX
2
is superior to realized values
of the variance risk premia. To estimate the variance risk premia, the paper adopts the variance risk
premia characterization used in Schneider (2012), using S&P 500 Index spot options data from 2004
to 2012.
1
Contents
1 Introduction 3
2 Litreature Review 3
3 Data and Methodology 5
3.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.2 VIX
2
- Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.3 Variance Risk Premia - Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.4 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4 Results and Analysis 13
4.1 The Relationship between the VIX and Variance Risk Premia . . . . . . . . . . . . . . . . 13
4.2 Forecasting Properties of the VIX
2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5 Summary & Conclusion 19
2
1 Introduction
The motivation for the aforementioned research project lies in both the recent practical and theoretical
developments within the volatility space. From a practical perspective, though volatility as an asset has
proved its utility within an institutional setting for many years (Derman.E, Kamal.M, Zou.J. 1999), the
emergence of ETF/ETN linked to the VIX in recent years, followed by the reinstatement of the VSTOXX
and VDAX series of indices in 2009, has demonstrated the broader practicality of an often misinterpreted
asset class (Whalley, 2009). The product innovation seen within the volatility sphere has been met with
equally rigorous academic work. Of note within the volatility sphere is the work of Neuberger (1994)
providing the first closed form solution to the variance swap, now the basis for the revised VIX. Further
works based off of Britten-Jones & Neuberger (2000), Jiang& Tian (2005) have only served to further
open the subject area with the work of Carr & Wu (2008) providing one of the first comprehensive reviews
of the variance risk premia, subsequently analysis of the variance risk premia has been further enriched
by bothBollerslev & Todorov (2012) and Schneider (2012), by analyzing higher moment associated with
the variance risk premia. As a result this paper aims to contribute to the study of the volatility/variance
asset class through extending the study of the variance risk premia while analyzing the linkage between
the variance risk premia and related assets in the volatility/variance asset class, providing the potential
for small but interesting contributions to both the interpretation and understanding of the variance risk
premia and related assets.
2 Litreature Review
Volatility as a measure of risk and uncertainty is one of the founding principals in modern finance, used
rigourously in both modern portfolio theory and options pricing. However, the understanding of volatility
and subsequently variance as an asset class is in many ways a novel yet increasingly important concept
within modern finance. Initial attempts aimed at capturing volatility or constant vega exposure
1
came
in the form of delta hedged portfolios
2
or straddle position
3
involving holdings in both options on the
underlying as well as in the underlier itself. However, despite the early popularity of both strategies, both
suffered from a variety of pitfalls. Firstly, the effectiveness of either strategy is limited to the range of
strikes covered by either strategy, once the price of the underlier exits the strategies effective strike range
it becomes ineffective in capturing volatility. Secondly, as cited by Neuberger (1994) the final payout
of a delta hedged portfolios is dependent not only on difference in realized and expected volatility but
also on the difference between the final payout and the value of the claim itself, resulting in an imperfect
exposure to volatility. Lastly, disregarding questions of trading infrastructure(Carr,Lee 2009), exposure
to volatility through such strategies, is dependent on both the initial level at which the option is struck
(e.g. ATM or OTM) and the terminal value of the underlying asset(Neuberger,1994). The inherent
flaws of option based strategies led to the ground breaking work of Neuberger(1994) proposing the use
of the Log-Contract. Unlike options based strategies, the Log-Contract proved to be revolutionary by
providing a constant vega exposure, with its payoff dependent only on the terminal value of the asset
(Neuberger,1994). However, despite the revolutionary nature of the Log-Contract, the framework on
which the vega netural result is based on is highly restrictive, in particular the absence of jumps within
the intial framework, leads to potential errors in estimation of the variance and volatility swap rates.
Subsequent work within the variance/volatility asset class aimed to loosen the framework surrounding the
derivation of the Log-Contract result. The model free derivation of the Log-Contract developed initially
by Carr & Madan (1998) enhanced by Britten-Jones & Neuberger (2000) provided a generally accepted
and consistent method for producing the variance swap rate. The work seen in both papers has been
extended by Jiang & Tian (2005) in a jump diffusion framework, providing a complete characterization
of variance swap rates.
The contributions of (Carr, Madan 1998),(Britten-Jones, Neuberger 2000) and (Jiang, Tian 2005),
have proven to be centerpieces in the seminal work of Carr & Wu (2009) credited with providing the
1
Vega, is defined as the options sensitivity to the volatility of the underlying asset denoted as
∂V
∂σ
2
insert description
3
citation for hull
3
first comprehensive analysis on the variance risk premia
4
,derived from the variance swap payout, for both
indices and single stocks. Using a 30-day horizon from 1996-2003, Carr & Wu (2008) find the variance
risk premia to be negatively related to returns of the underlying assets and more importantly find the
premia itself cannot be fully explained by the classical CAPM model nor by the Fama-French factors
(Fama,French 1993). Furthermore, Carr & Wu (2008) find the variance risk to be time varying through-
out their sample. The implications of the above observations are deep and far reaching in understanding
the variance risk premia and its relation to investor risk aversion; in particular, the lack of explanation
by the three factor model, suggests implicitly the pricing of risk factors beyond those considered in clas-
sical equity return models. The analysis of the variance risk premia by Carr & Wu (2008), has in turn
sparked further research into the possible information contained within the variance risk premia and its
relation to risk averse behaviour of market participants. Of note within this line of research, is the work
Bollerslev & Todorov (2011), who investigate the tail characteristics of the variance risk premia and its
economic significance. In particular Bollerslev & Todorov (2011), assert the statistical behaviour of the
tails contains greater information over the simple variance risk premia,asserting the variance risk premia
provides compensation for risks beyond those commonly associated with investor aversion (e.g. jump
risk)
5
.Bollerslev & Todorov (2011) find jump risk accounts for more than half of the variance risk pre-
mia; furthermore exploiting this finding,the paper goes on to develop using the left and right tails of the
variance risk premia, the Fear Index
6
measuring the skewness of returns, while providing an improved
characterization of jump risk in equity markets and the subsequent cost of insuring against such risk.
The work of Bollerslev & Todorov (2011), though groundbreaking in its characterization of jump risk
and analysis of the variance risk premia, reflects only a sub-segment of the studies aimed at investigating
market implied pricing of jump risk. For instance Pan (2002) uses an affine jump diffusion framework to
investigate the pricing of jump risk in near-the-money short-dated S&P 500 index options. Pan (2002)
finds strong evidence in favour of affine jump diffusion models in explaining the volatility skew for near-
the-money short-dated options, moreover Pan (2002) finds the jump-risk premia, specified in her jump
diffusion model, provides an important link between realized spot and option implied volatility. The
work of both Pan (2002) and Bollerslev & Todorov (2011) is extended by Schneider (2012) providing
an adjoint trading strategy, priced within an affine jump diffusion framework. The strategy presented by
Schneider (2012) exploits the implied skew, akin to the tails of the variance risk premia, through purchas-
ing the VIX variance swap and selling the simple variance swap, deveoped by Martin (2011), providing
pure exposure to the tails of the volatility smile. The results of Schneider (2012) points towards the
presence of a feasible trading strategy capable of exploiting the risk premia priced in the ’Fear-Index’ of
Bollerslev & Todorov (2011) while generating returns in excess of those seen under pure variance swap
strategies. The above studies on the dynamics and decomposition of the variance risk premia, provide
a comprehesive understanding as to what the variance risk premia represents along and its linkage to
the risk premias present in their adjoint asset markets. However, what is absent from existing work is
the linkage between the implied dynamics of the variance risk premia and its subsequent realization in
the market. Though studies on the relationship between implied volatility and its relation to realized
volatility are well known under both model dependent and model free representations (Jiang,Tian 2005);
of particular interest is the relationship between the model free representation of implied variance, cap-
tured through the squared value of the VIX, and the variance risk premia. In theory, though the VIX
as cited by Jiang & Tiang (2005) is a structually flawed measure of model free implied volatility as
noted by Whalley (2009) the VIX does contain information regarding to cost of insuring against risks
typically associated with increases in volatility. In turn, though the VIX may not fully specify market
expectations of risks contained within the variance risk premia, it is likely matched holding returns on the
squared-VIX (VIX
2
) should exhibit a strong relationship with the variance risk premia and thus explain
and possibly forecast the variance risk premia.
Though the relationship between the VIX
2
and variance risk premia is likely to be present, the specific
form of this relationship, whether linear or non-linear, is ambigous. In-spite of this ambiguity, it is possible
to refer to the statistical characterization of the variance/volatility asset class and in turn the variance
4
the variance risk premia is denoted by [RV-SW]x100 = VRP, where SW is the swap rate and RV is realized variance;
note this is the payout for a variance swap contract.
5
explain jump risk - differentiate between peso problem and insurance against jumps
6
Fear Index explanation
4
risk premia, under both the P and Q-measures to infer a the form of the relationship between both
variables. Of particular interest is the affine characterization of option prices in Pan (2002) , implying
related asset classes - namely variance/volatility products and related measures- can be represented
as affine functions under the Q-measure. The importance of affine representations becomes clear in
the context of yield curve modelling, where affine representations of yield curve models have provided
for huge gains in tractability in modelling the yield curve under the Q-measure. More importantly, as
noted by Cheredito et. al (2006), depending on the market price of risk specification, the affine properties
encountered under the Q-measure may also hold under the P-measure, implying the potential for an affine
relationship amongst assets within an affine economy.Schneider (2012) extends the affine framework by
characterizing the variance swap rate within an affine framework
7
. The work of both Pan (2002) and
Schneider (2012) provides a consistent framework for analysis of the variance risk premia,within the Q-
measure which may may be extended equally into the P-measure, depending on the market price of risk
(Cheredito et. al ,2006) . In turn, should the market price of risk provide for an affine representation of
the variance swap rate and thus the variance risk premia under both the P and Q-measures then in theory
an affine relationship under the P-measure should exist across assets within the volatility/variance asset
class and thus linear analysis should prove to be at the very least feasible if not statistically significant
upon implementation.
From the above review of significant literary work and developments in the field of the volatility
asset class and variance risk premia, the paper aims to contribute to this field through investgating the
following. Firstly, given the affine characterization of the variance risk premia and subsequently the VIX
by construction under the Q - measure, the paper aims to exploit this relationship to test whether a
linear relationship exists between the VIX
2
and the variance risk premia, a suffcient but not a necessary
condition for affinity in under the P-measure. Secondly, provided a relationship exists between the VIX
2
and the variance risk premia, the paper aims to develop an economic explanation behind the relationship
between both variables and whether this relationship can be further specified by other measures such
as the implied skewness. Finally, to complement the study the paper will also investigate the predictive
power of the VIX
2
with respect to the variance risk premia relative to lagged value of the variance risk
premia.
3 Data and Methodology
3.1 Data
For the purposes of this paper two data-sets are required, VIX
2
holding returns and the log-variance risk
premia,both estimated using monthly frequencies.
3.2 VIX
2
- Dataset
It is important to note, the paper differentiates between the pure VIX and the V IX
2
data due to the
construction and interpretation of the CBOE VIX Index as defined in Figure.1
8
. From the definition
in Figure.1 the CBOE VIX index measures the 30 day implied volatility on S&P 500 Index, which is
derived from the square root of the 30 day implied variance, interpolated from S&P 500 Index spot option
prices, as demonstrated by Figure.1. As a result, to create a comparable measure of implied variance it
is necessary to square the CBOE VIX generating the implied price of variance from the options set used
to construct the CBOE VIX Index.
To estimate CBOE V IX
2
holding returns, the paper uses squared front month CBOE VIX futures
contract price data, taken from the Bloomberg L.P. database, purchasing in May 2004 the June 2004
VIX contract (Bloomberg Ticker: UXM04 Index) rolling over to the next front month contract upon
expiriation; in those periods where no new front month contract is available, the next nearest contract
is used and the holding period extended to account for the missing front month contract. Additionally,
if the date of reinvestment falls on a vacation or a weekend, the nearest available price is used
9
. The
7
See Appendix #
8
for further details on the construction of the VIX refer to the CBOE VIX white paper, (CBOE,2009)
9
e.g. if the date of reinvesment falls on sunday, the closing price on monday is used to calculate the end of month returns
5
Figure 1: CBOE VIX Definition (CBOE, 2009)
σ
2
1,near−term
=
2
T
1

i
K
i
δ
K
2
i
e

Q(K
i
) −
1
T
_
F
OTM,1
K
0
−1
_
2
σ
2
2,next−term
=
2
T
2

i
K
i
δ
K
2
i
e

Q(K
i
) −
1
T
_
F
OTM,2
K
0
−1
_
2
V IX
T,t
= 100 ∗
¸
_
T
1
σ
2
1,near−term
N
T
1
−N
30
N
T
2
−N
T
1
+T
1
σ
2
1,next−term
N
T
1
−N
30
N
T
2
−N
T
1
_

N
365
N
30
VIX
2
monthly turnover strategy is carried out until January 2012, generating 94 monthly holding period
returns from 2004 to 2012. To provide a more realistic return series the paper uses two return series, one
based on monthly end of day settlement prices and another based on bid and ask values to simulate the
purchase and sale of the contracts for each 30 day period. The returns, themselves are calculated using
log-returns (Figure.2) instead of basic returns due to the favourable statistical properties, with respect
to stationarity of the underlying return series.
Figure 2: Calculation of Log-Return Series
r(t)
b/a
V IX
2
= log
_
V IX
2
T,T;bid
V IX
2
T,t;ask
_
r(t)
s
V IX
2 = log
_
V IX
2
T,T;settle
V IX
2
T,t;settle
_
The V IX
2
returns as shown in Table.1 exibits significantly negative returns throughout the sample
period, the only exception to this phenomenon occurs during the credit crisis (2007-2010) where returns
on the V IX
2
, during select periods, excceded holding period returns of 80%. However, negative returns
are more the norm rather than the exception, regardless of whether the VIX or V IX
2
is used or whether
end of day settlement prices or end of day bid/ask prices are used, as shown in Table.1. However, as
demonstrated by Table.1, the magnitude and standard deviation of the returns of either series does differ,
this is evident through the kernel density plots in Figure.3.
Table 1: VIX
2
and VIX return statistics
VIX
2
Settlement VIX
2
Bid/Ask VIX Settlement VIX Bid/Ask
Mean -9.560% -10.777% -4.780% -5.389%
St. Dev. 0.3343 0.3510 0.1671 0.1755
Skewness 0.7390 0.9314 0.7390 0.9314
Kurtosis 0.6113 0.7350 0.6113 0.7350
Max 86.39% 89.46% 43.20% 44.73%
Min -79.61% -71.93% -39.80% -35.97%
As demonstrated in Figure.3 the VIX
2
and VIX returns, though exhibiting similar levels of kurtosis
and skewness, the dispersion and magnitude of returns under the VIX
2
is substantially higher than those
seen under the simple VIX returns, implying the potential for substantially higher gains and losses in
any trading strategy linked to the VIX
2
index relative to the VIX index; this effect becomes evident due
to the amplification of the absolute dollar returns
10
, hence, ceteris paribus, the returns on the VIX
2
will
be larger and exhibit equally large levels of dispersion, while retaining identical kurotsis and skewness
10
(x+δx)
2
−x
2
= (2xδx+δx
2
) ≥ δx
2
, thus the magnitude of returns will be larger than the adjoint VIX trading strategy
6
Figure 3: VIX
2
and VIX return kernel density plots
(a) VIX - Settlement Ret. (b) VIX - Bid/Ask Ret. (c) VIX
2
- Settlement Ret.
(d) VIX
2
- Bid/Ask Ret.
statistics seen under either return series in Table.1. Beyond comparing the differences between the VIX
2
and VIX, it is equally important to recognize the differences between the two methods of measuring
returns in the VIX
2
series. In the case of the VIX
2
returns series calculated using end-of-day settlement
prices, the mean returns are greater than the mean returns calculated using the VIX
2
bid/ask price.
The reason for the difference in means is due in large part to the bid/ask bounce present in the VIX
futures market, reflecting transaction costs for dealers in the VIX market, leading to a discounted price
relative to the ask or mid. This line of reasoning is evident upon delving deeper into the measurement
methodology for each price series, in particular, the settlement price reflects the fair value, at closing, of
the VIX futures contract, regardless of whether the final transaction is a sale or purchase, and is thus
not a tradeable price but for the purposes of marking to market is an acceptable means of valuation.
The contrary is true in the case of the last bid/ask quote of the day which are realizeable should the
strategy have been carried out in the open market. The presence of the bid/ask bounce, in turn provides
a plausible explanation for the larger standard deviation, kurtosis and skewness, present in the bid/ask
series.
The difference between the two series is further reflected, albeit to a lesser degree, in the autocor-
relation function plots (Figure.4). Both series, exhibit similar autocorrelation patterns, with the VIX
bid/ask returns series exhibiting lower levels of autocorrelation relative to the returns calculated using
the VIX settlement price. However, despite the difference in absolute autocorrelations, both series gen-
erally exhibit statistically insignificant levels of autocorrelation with the only exception being the first
order autocorrelation of the VIX bid/ask return series, which is marginally significant at the 5% level
(Table.2), using the Ljung-Box Q-statistics.
Table 2: Phillips Perron, Ljung-Box Q & Augmented Dickey Fuller statistics
PP - Z(t) Q - L(1) Q - L(5) Q - L(10) ADF - Z(1) ADF- Z(5)
VIX
2
- Settlement -8.230 3.1545 5.6423 9.4959 -6.163 -4.188
0.000% 7.570% 34.260% 48.580% 0.000% 0.070%
VIX
2
- Bid/Ask -7.911 4.0413 4.725 8.9988 -6.401 -4.209
0.000% 4.440% 45.040% 53.220% 0.000% 0.060%
7
Figure 4: VIX
2
Autocorrelation Function Plots
(a) VIX
2
- Settlement ACF
(b) VIX
2
- Bid/Ask ACF
The inferences based on the Ljung-Box Q-statistic at lags of 1,5 and 10 months is confirmed by the
rejection of the unit root hypothesis, using the Augmented Dickey-Fuller test at equivalent lags of 1 and
5 months. The results of the Augmented Dickey-Fuller test further supported by the Phillips-Perron test,
which avoids potential bias due to the lag specification, rejecting the unit root null.All test statistics are
reported in Table.2 with their adjoint p-values.
3.3 Variance Risk Premia - Dataset
The variance risk premia as developed in Carr & Wu (2008) is defined as the payout on a variance swap
contract. Therefore, to construct the variance risk premia both the variance swap rate, measured at the
beginning of the measurement period, and a measure of realized variance thoughout the measurement
period, are required to determine the swap payout or variance risk risk premia.Carr & Wu (2008) define
two distinct method for calculating the variance risk premia as shown by (Figure.5)
Figure 5: Calculation Methodologies for the Variance Risk Premia
V RP
l
T,t
= log
_
RV
T,t
SW
t,t
_
OR
V RP
s
T,t
= (RV
T,t
−SW
t,t
) ∗ 100
The log-representation used in Carr & Wu (2008), can be interpreted as the excess return or premia
paid by investors for entering into a variance swap contract and fixing their exposure to implied volatility
at the beginning of the measurement period. Conversely, the second definition, also implemented in Carr
& Wu (2008), can be interpreted as the dollar return on a variance swap with a $100 notional. For the
purposes of this paper, only the log-variance risk premia will be used due to both its interpretation as
well as to maintain analytical consistency with the VIX
2
log-returns.
To estimate the variance swap rate component of the variance risk premia, the formula implemented
by Carr & Wu (2008) and Schneider (2012) is used to estimate the swap rate for the relevant holding
8
period, as shown below (Figure.6)
Figure 6: Variance Swap Rate Formula
SW
T,t
=
2
b
T,t
_
_

F
t,T
C(K)
K
2
dK +
_
F
t,T
0
P(K)
K
2
dK
_
Since there does not exist a contiuum of strikes within the S&P 500 index option market, the variance
swap rate formula must be discretized into the weighted sum of the out of the money calls and puts;
whereby each call and put is weighted by the inverse squared of their respective strikes multiplied by
the difference between the current and last strike. Prices for both puts and calls are extrapolated from
OptionMetrics implied volatilities, using the Black-Scholes formula. The necessary zero-rates are interpo-
lated from OptionMetrics zero-rate data with implied dividend yields taken also from the OptionMetrics
database. To avoid microstructure effects, the selection criterion imposed in Carr& Wu (2008) is used
to filter and construct the strip of implied volatilities.
To minimize the errors induced through the discretization of the variance swap rate formula, a picewise
cubic hermite polynomial interpolation scheme is used to interpolate implied volatilities from the existing
implied volatility strip to generate a grid of 2000 implied volatilities for both calls and puts; the use of a
piecewise cubic hermite polynomial ensures monotonicity across each knot assuring stictly positive implied
volatilties while satisfying the necessary smoothness conditions. In addition to creating a finer grid of
implied volatilities, the range of strikes is expanded for puts and calls to 50% and 200% , respectively, of
the current spot price as suggested by Demertfi et al. (1999), applying the implied volatility of the furthest
in-sample out of the money put and call to the extended set of strikes. To ensure the option weights
decreases monotonically with moneyness, the extended set of strikes are spaced at .20 index points.The
above methodology, applied in Carr & Wu (2008), provides an accruate measure of the variance swap
rate for a given strip of options and benefits from a approximately constant vega position for a wide
range of possible index values. Since options with maturities of 30 days are generally unavailable in the
S&P 500 Index option market, 30 day swap maturities must be linearly interpolated from neighboring
swap maturies. For the purposes of this paper, the 30 day variance swap rate is interpolated from the
next lowest and the next highest maturity
11
relative to the 30 day swap maturity; as cited by Carr &
Wu (2008), this methodology introduces little to no error in the estimation of the 30 day variance swap
rate. As a proxy for accuracy, the square root of the variance swap rate scaled by a factor of 100, by
construction
12
, should approximately track the value of the front month VIX contract; this relationship
is demonstrated in Figure.7 using the estimated 30 day variance swap rate, exhibiting an R
2
of 82% .
Figure 7: VIX vs. Square-Root of Variance Swap Rate
Throughout the sample period the variance swap rate trades within a relative wide range as shown
by Figure.8. Since the variance swap rate is based on option prices and the weights for each call and put
11
e.g. given the 9,20 and 40 day maturity, the 20 and 40 day maturities are used to interpolate the 30 day swap rate
12
As shown by Figure.1 the first term of the VIX formula approximates the discretized integrals in Figure. 6 - as a result
the VIX gives an approximation,albeit a noisy one, of the variance swap rate
9
are strictly positive, the distribution of the variance swap rate is bounded from below at zero with an
unbounded maximum.
Mean 0.0596
St. Dev. 0.0656
Skewness 2.7293
Kurtosis 7.8931
Max 0.3614
Min 0.0119
Figure 8 & Table 3: Variance Swap Rate - Histogram & Summary Statistics
As shown in Table.3, the broad trading range of the 30 day variance swap rate is borne out by the
high degree of kurtosis and skewness as well as the in-sample maximum, which is 6 times the average
swap rate during the sample period.The sample statistics, are consistent with the stylized facts of implied
volatility, exhibiting mean reversion with generally infrequent but large jumps over time (Cont & Da-
Fonseca, 2002). In sample, large jumps in implied volatility are associated with start and continuation
of the global economic crsis, with the largest jump occuring in 2008 followed by smaller yet large jumps
in implied volatility in 2009 and 2012 as shown in Figure.7.
To construct the realized variance leg of the variance risk premia, the paper adopts the realized-
variance formulation of Neuberger(2012) as shown in Figure.9 The choice of measurement for realized
Figure 9: Variance Swap Rate Formula
RV
T,t
= 2
N

i=1
_
F
T,t+i
F
T,t+i−1
−1 −ln
_
F
T,t+i
F
T,t+i−1
__
variance is determined in relation to the no-arbitrage condition imposed with respect to the fixed leg of
the variance swap. As demonstrated by Schneider (2012), taking the excectation of RV
T,t
under the Q
T
forward measure, the resultant value is equivalent to the risk neutral representation of the variance swap
rate, as shown in Figure.6, satisfying the no-arbitrage and zero upfront cost condition of variance swap
contracts. In turn, since the variance risk premia is the terminal payout on a variance swap contract, both
the fixed and floating legs must be measured consistently. Additionally, as demonstrated by Neuberger
(2011), the above measure of realized variance benefits from the aggregation property and therefore is
independent of sampling frequency. For the purposes of this paper, front month S&P 500 Index futures
prices, consistent with the methodology implemented by Carr & Wu (2008), are used to calculate realized
variance on the S&P 500 Index, with all observations taken from the Bloomberg L.P. database.
Table 4: Annualized Realized Variance - Summary Statistics
Mean 0.0500
St. Dev. 0.1119
Skewness 5.5283
Kurtosis 34.9379
Max 0.8626
Min 0.0035
Throughout the sample period, as shown in Table.4, the mean realized variance is significantly lower
than the mean estimated swap rate in Table.3. Additionally, realized variance exhibits extremely high
10
levels of kurtosis, which combined with high levels of positive skewness, implies the presence of relatively
rare but large jumps within the throughout the sample period, this is evident throughout the crisis period
from 2008 to 2012 as shown in Figure.10 (a,b).
Figure 10: Annualized Realized Variance - Histogram & Plot
(a) Histogram (b) Time-Series Plot (2004-2012)
Given the estimates of both the variance swap rate and realized variance, it is possible to calculate both
the basic risk premia and log variance risk premia using the formulæ provided in Figure.5. Throughout
the sample period the average log-variance risk premia is -63.81% , comparable to the -66.00% noted by
Carr & Wu (2008), thus confirming the presence of a significant and persistant variance risk premia on
the S&P 500 Index; both the simple and log-variance risk premia are plotted in Figure 11 (a,b)
Figure 11: Annualized Variance Risk Premia
(a) Log-Variance Risk Premia (b) Simple Variance Risk Premia
The variance risk premia, as shown in Table.5, exhibits a high degree of variation with maxmimums
and minimums both more than twice the absolute average risk premia during the sample period. Addi-
tionally, as demonstrated by the maxmia, though the risk premia is present on average, there are periods
where the premia turns positive with very large positive premias appearing in and around major crisis
events (e.g. collapse of Lehman Brothers), with the frequency of such occurencs increasing post-2008.
However, despite these violations positive premias are not a persistant phenomenon and are usually
followed by a relatively rapid reversion back to a negative risk premia, as shown in Figure.11b. The
high level of variation and bias towards negative values is consistent with the low levels of kurtosis and
skewness as shown in Figure.12
Tests for autocorrelation show the in-sample variance risk premia exhibits little to no autocorrelation
at any lag, this is borne out in the Ljung-Box Q-statistics at lags of 1,5 and 10 exhibiting insignifinicant
P-values at those lags, as seen in Table.6. Equally, using the Augmented Dickey-Fuller test at lags of
1 and 5 as well as the Phillips-Perron Z-statistics (Table.6), the unit root root null is rejected for the
sample period, establishing stationarity of the variance risk premium.
11
Mean -63.81%
St. Dev. 72.74%
Skewness 0.5967
Kurtosis 0.9732
Max 162.82%
Min -217.18%
Figure 12 & Table 5: Variance Risk Premia - Kernel Density Plot & Summary Statistics
Table 6: Phillips Perron, Ljung-Box Q & Augmented Dickey Fuller statistics
PP - Z(t) Q - L(1) Q - L(5) Q - L(10) ADF - Z(1) ADF- Z(5)
-7.747 3.461 5.347 9.071 -5.714 -3.438
0.000% 6.280% 37.500% 53.500% 0.000% 0.97%
3.4 Methodology
To exploit the affine characterization of both the VIX
2
and variance risk premia under the Q-measure in
the context of the P-measure, a basic univariate regression scheme to analyze the relationship between
the two variables and assumes the following form (Figure.13).
Figure 13: Univariate Regression - Variance Risk Premia on VIX
2
V RP
T,t
= α +β(r(t)
V IX
2) +
The use of a univariate regression, serves two purposes. Firstly,regressions are affine functions since
they are a linear mapping between R
M
⇒ R
N
. Thus, given a significant relationship between both
variables, then an affine relationship exists under the P-measure, assuming the classical linear regression
assumptions hold. Secondly, the regression denoted in Figure.13 provides a simple yet robust way to
establish the explantory value of the VIX
2
log-returns with respect to the variance risk premia.
For the purposes of this paper, both the VIX
2
bid-ask and settlment returns
13
will be used to analyze
the relationship between log variance risk premia.Furthermore, based on preliminary scatterplot analysis
(Figure.14), the relationship between the log variance risk premia and log VIX
2
returns appears to contain
non-linearities; possibly due to a portion of the premia associated with higher statistical moments, such
as the volatility skew, as presented by Schneider (2012).
Figure 14: VIX vs. Square-Root of Variance Swap Rate
13
see Figure.2 for return calculation methodology
12
As a result the remaining analysis will entail regressions on higher order terms using polynomial
regressions of the form seen in Figure.15.
Figure 15: Polynomial Regression - Variance Risk Premia on VIX
2
V RP
T,t
= α +
N

i=1
β
i
(r(t)
i
V IX
2) +
The aim of the polynomial regression, beyond testing for non-linearity with respect to the VIX
2
log
returns, is to identify whether higher order terms, if significant, coincide with higher cumulants implied
from option volatilities, such as the implied skew, which can be proxied by the CBOE SKEW Index
(CBOE,2010). Finally to complement the analysis, tests on the forecasting power of the VIX
2
will be
carried out in Section 4.2, using lagged values of the VIX
2
and variance risk premia. All regressions
will be estimated using the OLS method, as both variables satistfy the necessary conditions both with
respect to autocorrelation and stationarity.
4 Results and Analysis
4.1 The Relationship between the VIX and Variance Risk Premia
To analyze the realtionship between the VIX and the variance risk premia, both univariate and multi-
variate polynomial regressions are estimated with the results presented in Table.7 & 8
14
.
Table 7: Regression Statistics - VIX
2
Bid/Ask Returns
α r(t)
V IX
2 r(t)
2
V IX
2
r(t)
3
V IX
2
Adjusted R
2
DW-Stat. F- Stat.
Univariate -0.4787 1.5390 53.00% 2.27 60.49
0.00% 0.00% 0.00%
2nd Order* -0.5877 1.4361 0.7624 55.33% 2.20 4.18
0.00% 0.00% 0.70% 1.80%
3rd Order -0.5884 1.4057 0.7412 0.0890 54.84% 2.20 N/A
0.00% 0.00% 4.40% 90.40%
Table 8: Regression Statistics - VIX
2
Settlement Returns
α r(t)
V IX
2 r(t)
2
V IX
2
r(t)
3
V IX
2
Adjusted R
2
DW-Stat. F- Stat.
Univariate* -0.4944 1.5818 52.56% 2.26 62.35
0.00% 0.00% 0.00%
2nd Order* -0.6022 1.5185 0.8642 54.97% 2.23 6.19
0.00% 0.00% 0.30% .278%
3rd Order -0.6053 1.4152 0.8171 0.3111 54.55% 2.21 N/A
0.00% 0.00% 2.10% 68.30%
In the case of the univariate regression, a strong relationship at the 1% level exists between the variance
risk premia and log-returns on the VIX
2
, regardless of whether the Bid/Ask or Settlement methodolgy
is used to compute log-returns. The results of the univariate regression, though substantially larger than
the null of β=0, agrees and incidently extends the findings of Jiang & Tian (2005) through establishing
that VIX
2
returns, a measure of implied variance, contains significant information in explaining the
variance risk premia. However, as demonstrated by the F test statistics, using a null of α=0, β=1,
the returns on the VIX
2
severely underestimates the variance risk premia, implying the VIX
2
is in and
14
Results with * are robust statistics correcting for heteroscedasticity - For the Wald F-statistics the Null Hypothesis :(1)
H
0
: α=0, β=1 (2) H
0
: α = β
1
= 0 β
2
= 0
13
of itself an informationally ineffcient estimate of the prevailing variance risk premia, despite containing
statistically valuable information on the variance risk premia. The results are inconsistent with the
theoretical interpretation of the VIX as a measure of the cost of portfolio insurance (Whalley, 2009),
as the VIX and thus VIX
2
holding returns should imply the expected fair cost of insurance against
time-varying variance and jump-risk on S&P 500 Index; this interpretation though partially valid, as
demonstrated by the univariate model, fails to fully encompass the variance risk premia.
To test for the presence of higher order risks, unaccounted for in VIX
2
(see:Figure.14), second and
third order polynomial regressions are used to test for such risks. For all second order regressions, all
coeffcients are highly signifcant based on the 1% level. Moreover, as demonstrated by the Wald F-statistic,
the second order polynomial regression is preferred over the restricted univariate regression. Despite the
siginificance of the second order polynomial regression, this phenomenon does not hold for third order
polynomial regressions exhibiting both poorer fit and an insignificant coeffcient for cubed VIX
2
returns.
To develop a better understanding of the signifcance of the squared VIX
2
log returns, it is important
to revisit the methodology used by the CBOE used to the construct the VIX Index . As noted by Jiang &
Tian (2005) the process used by the CBOE to construct the VIX index introduces both discretization and
truncation errors leading to inaccurate estimates of actual market implied volatility and in turn market
implied variance through the VIX
2
. Though, discretization errors are present in estimates of both the
variance swap rate and the VIX and thus the VIX
2
, both measures differ in the degree of truncation
error. In particular, because of the liquidity cutoff used by the CBOE (CBOE,2009), the VIX does not
capture options that are outside of the liquidity cutoff, providing only partial information regarding jump
or crash risk, commonly associated with deep out of the money options. The under-estimation of higher
order risks, in particular jump-risk, is clear when comparing Table.1 & Figure.12, whereby the VIX
2
,
though exhibiting a higher degree of positive skewness exhibits a lower level of kurtosis relative to the
variance risk premia as well as a significant tracking error throughout the sample period. In turn,the
VIX and VIX
2
indices provide only a partial estimate of the total variance risk premia by excluding a
significant portion of the variance risk premia attributable to risks theoretically captured by deep out of
the money S& P 500 index options (Figure.16).
Figure 16: VIX
2
vs. Variance Risk Premia - Tracking Error
The consequences of a structural bias in the VIX leads to the initial discrepancy seen under the
univariate model whereby the VIX
2
consistently underestimates the variance risk premia. Despite the
significance of squared returns, the signing and decomposition of the VIX
2
coeffcient in the univariate
model provides an insight into the information generating process between both variables (Figure.17)
15
.
Figure 17: Coeffcient Decomposition of β
V IX
2
β
V IX
2 =
COV
P
(RV, V IX
2
T
) +COV
P
(SW
t
, V IX
2
t
) −[COV
P
(RV, V IX
2
t
) +COV
P
(SW
t
, V IX
2
T
)]
σ
2
V IX
2
T

2
V IX
2
t
−2 ∗ COV
P
(V IX
2
T
, V IX
2
t
)
15
Proof for β decomposition is provided in Appendix.#
14
As demonstrated in Table 7 & 8, the analysis of the univariate regression confirms the VIX
2
coefficient
is not only statistically greater than unity but more importantly strictly positive, exploiting both of these
properties, the decomposition in Figure.17 allows for the following inference. Given the coeffcient is
positively signed the sum of the covariance between realized variance and the VIX
2
value at the end of the
holding period, COV (RV, V IX
2
T
), and the covariance between the swap rate and VIX
2
at the beginning
of the period, COV (SW
t
, V IX
2
t
), is greater than the following COV (RV, V IX
2
t
) +COV (SW
t
, V IX
2
T
).
In turn, the coefficient demonstrates that in theory the variance risk premia should at least be reflected
in the return process of the VIX
2
, as both the VIX
2
at time T and the VIX
2
at time t, covary with
realized variance and the variance swap rate respectively. However, as established by Jiang & Tiang
(2005) the VIX
2
, though structurally similar to the variance swap rate, contains truncation errors, thus
only capturing part of the information generating process between implied variance and the variance risk
premia. Under this premise,using the inferences taken from Jiang & Tian (2005), it is not surprising
the squared-VIX
2
is a significant variable as it captures the marginal information left outside of VIX
2
.
More importantly, in light of potential departures from log-normality, the informational content of higher
order cumulants such as the implied skew and implied kurtosis are likely to be of importance in fully
characterizing the information captured in the variance risk premia (Backus et al., 2011) but also in
capturing premias associated with extreme-value events.
To develop a robust economic understanding of the squared-VIX
2
returns, two theoretical interpre-
tations are presented and tested as follows. Under the first interpretation, since squared returns are
strictly positive the metric, squared returns on the VIX
2
can serve as an approximation for the vari-
ance or implied variance of the VIX
2
Index. To proxy for the implied variance, the CBOE publishes
the VVIX Index measuring the implied volatility of the CBOE VIX Index, the index is designed in the
same fashion as the VIX except the underlying options are based on the VIX options. To estimate
returns on implied variance of variance, the methodology used to calculate VIX
2
returns, discussed in
section 3.2, are used to calculate VVIX
2
returns. Theoretically, the use of implied variance of the VIX
should approximate higher order risks by proxying for probability or expected intensity of large jumps
in volatility, associated with large jumps in value of the underlying asset. Under the second interpre-
tation, the squared-VIX
2
returns can be interpreted as an approximation for higher moments based on
the implied volatility surface. As demonstrated by Bollerslev & Todorov (2011) and Schneider (2012),
higher moments are implicitly priced into the variance risk premia through the presence of the jump risk
premia, as measured by the tails of the variance risk premia. To proxy for the tails of the variance risk
premia, the CBOE publishes the SKEW index, measuring the implied skew of the S&P 500 index using
S&P 500 Index options(CBOE,2010).The SKEW index, provides an appropriate gauge for jump risk
by measuring the risk-adjusted probability of jumps in excess of two-standard deviations of the current
average returns on the S&P 500(CBOE,2010). To test for the presence of the higher moments in the
variance risk premia, log returns on the SKEW Index will be used to gauge the expected cost of insuring
against risks associated with higher moments.
To test the significance of the VVIX
2
and the SKEW Index as substitutes for squared-VIX
2
log
returns, the following regressions in Figure.18 are estimated. Additionally, regressions containing the
Figure 18: Multivariate Regression - SKEW & VVIX
V RP
T,t
= α +β
1
(r(t)
V IX
2) +β
2
(r(t)
V V IX
2) +
V RP
T,t
= α +β
1
(r(t)
V IX
2) +β
2
(r(t)
SKEW
) +
squared-VIX
2
log-returns and both auxiliary variables are used to test for any additional information
encompassed by VVIX, SKEW and squared-VIX
2
returns. The results of these regressions are presented
in Table.9 & 10
16 17
, with their adjoint p-values.
Under both measures of log-returns on the VIX
2
, log-returns on the VVIX
2
fail to provide an economic
substitute for squared log-VIX
2
returns, at all levels of significance. The opposite is true with log-returns
16
Results with * are robust statistics correcting for heteroscedasticity
17
VVIX data is tabulated by the CBOE from 2006 onward - thus all analysis involving the VVIX uses sample sizes
equivalent to the number of monthly VVIX observations.
15
Table 9: SKEW & VVIX Models - VIX
2
Settlement Ret.
α r(t)
V IX
2 r(t)
V V IX
r(t)
SKEW
r(t)
2
V IX
2
R
2
VVIX Model* -0.4703 1.6075 -0.0846 56.45%
0.00% 0.00% 65.60%
SKEW Model* -0.4915 1.6344 2.5225 55.01%
0.00% 0.00% 1.10%
R1 Model* -0.4690 1.6426 -0.1037 1.8687 57.44%
0.00% 0.00% 59.50% 14.70%
R2 Model* -0.6188 1.4870 -0.1701 0.9495 60.73%
0.00% 0.00% 35.70% 0.20%
R3 Model* -0.5852 1.5690 2.0013 0.7452 57.41%
0.00% 0.00% 8.10% 2.70%
Table 10: SKEW & VVIX Models - VIX
2
Bid/Ask Ret.
α r(t)
V IX
2 r(t)
V V IX
r(t)
SKEW
r(t)
2
V IX
2
R
2
VVIX Model* -0.4611 1.5691 -0.0528 57.66%
0.00% 0.00% 77.30%
SKEW Model -0.4768 1.5712 2.3659 54.75%
0.00% 0.00% 3.50%
R1 Model* -0.4606 1.5847 -0.0634 1.2037 58.08%
0.00% 0.00% 73.80% 33.50%
R2 Model* -0.5776 1.4566 -0.1162 0.0731 60.09%
0.00% 0.00% 53.30% 2.70%
R3 Model* -0.5677 1.4784 1.8233 0.6331 56.12%
0.00% 0.00% 10.80% 5.30%
on the SKEW Index, exhibiting significance at the 5% level under both measures of the VIX
2
returns.
Furthermore as demonstrated in the R1 model and R3 Model in Table.9 & 10, the SKEW Index fully
specifies the remaining variation, with non significance for the VVIX and squared- VIX
2
log returns in
each of the models, respectively.However, despite the significance of the the SKEW Index as a substitute
for the squared-returns, the constant term in both polynomial and SKEW regressions remains significant
at all levels of significance.
The failure of the VVIX
2
and success of the SKEW Index in specifying the residual variation of the
univariate regression, though surprising, is likely rooted in the specification of the respective indices.
The SKEW Index unlike the VVIX
2
Index measures directly market expectations of extreme returns
using S&P 500 options; conversly the VVIX
2
Index though directly measuring the implied variance
of implied variance, only partially captures information on the shape and tails of the market implied
distribution (e.g. implied kurtosis, implied skewness). The significance of the informational content in
the SKEW relative to the VVIX
2
Index is confirmed through the results in Table.9 & 10, confiming the
statistical significance of the SKEW Index. The presence of the implied skew in the variance risk premia,
as measured by the SKEW Index returns, is consistent with Kohzan et al. (2011); who find not only
the presence of the skew risk premia priced in deep out of the money index options but also a strong
relationship between the variance risk premia and the skew risk premia. In turn, the significance of the
implied skew, demonstrates the variance risk premia is not only composed of the cost of hedging the
risks associated with the VIX
2
but also those risks captured by the SKEW Index (e.g. jump risk). More
importantly the presence of the SKEW can be decomposed using the residual values from the univariate
regression, denoted as
V IX
2 (Figure.19).
From Figure.19, the SKEW coefficient is positive if and only if the covariance between the terminal
values of the skew and the residual values of the univariate model is greater than the covariance between
the value of the skew at the beginning of the holding period and the residuals of the univariate model
18
.
18
Assuming positive covariation between SKEW
t
and
V IX
2 - see Kohzan et al. (2011), for relationship between skew
16
Figure 19: Coeffcient Decomposition of β
SKEW
β
SKEW
=
COV
P
(
V IX
2, SKEW
T
) −COV
P
(
V IX
2, SKEW
t
)
σ
2
SKEW
t

2
SKEW
T
−2 ∗ COV
P
(SKEW
T
, SKEW
t
)
From Table.9 & 10, the SKEW coeffcient under either measurement of the VIX
2
log- returns is signifi-
cantly greater than zero and strictly positive throughout the sample period. The signing of the SKEW
coeffcient, implies the concurrent value of the implied skew explains the concurrent residual value of the
variance risk premia. However,the SKEW Index returns, on its own, cannot fully specify the temporal
dynamics of the variance risk premia, as is the case with the VIX
2
log-returns (Figure.18). Despite the
lack of predictive power, the signing of the coeffcient is consistent with the pricing of jump risk in the
variance risk premia and thus confirms the role of the SKEW Index in specifying the dynamics of the
VIX
2
and variance risk premia.
Despite the significance of both the VIX
2
and SKEW Index log-returns, the constant terms in all mod-
els are not only statisically different from zero but also negative throughout the sample. The significance
of the constant term in the SKEW based regression(Figure.9 & 10) indicates, though both the VIX
2
and
implied skew partially specify the variance risk premia, the combination of both variables under-estimates
the variance risk premia by significant margin. However, despite what is a significant under-estimation of
the variance risk premia, the presence of a significant constant term is consistent with existing theoretical
work, relating implied volatility and spot market volatility dynamics (Jiang & Tiang,2005), (Christensen
& Prabhala,1998). In particular, returning to the covariance decomposition and time series plot presented
in Figure.16 & 17, the significance of the constant coeffcient is not surprising considering the behaviour
of the VIX
2
, which despite exhibiting positive covariation with the variance risk premia, generally un-
derestimates the realized magnitude of the risk premia. More importantly, comparing the results of the
polynomial and SKEW models, the SKEW model decreases the relative underestimation induced by the
VIX
2
log-returns, as demonstrated by a lower constant coeffcient under the SKEW model (Table 9 &
10). The effect of the SKEW on the constant term is of particular interest in light of the findings in
Schneider (2011), as the constant itself may also be related to the skew strategy implement in his paper.
Thus the persistence of the constant term may reflect moreso the presence of the skew risk premia rather
than any characteristic attributable to implied volatility and implied skewness, as suggested in earlier
papers.
4.2 Forecasting Properties of the VIX
2
To exploit the findings developed in Section 4.1, the study of the relationship between the variance
risk premia and the VIX
2
can be extended to test the forecasting power of the VIX
2
with respect
to the variance risk premia. To develop a theoretical understanding as to why the VIX
2
is likely to
subsume at least as much information as the lagged variance risk premia, it is important to return to the
interpretation and decomposition of the the VIX
2
coeffcient in Figure.17. Of particular interest is the
relationship between the terminal value of the VIX
2
and the initial value of the VIX
2
during the holding
period, which covaries strongly with the floating and fixed legs of the variance swap, respectively. In
turn, by lagging the the VIX
2
the coeffcient decomposition should hold (Figure.20).
Figure 20: Coeffcient Decomposition of the lagged V IX
2
β
V IX
2
lag
=
COV
P
(RV, V IX
2
T−1
) +COV
P
(SW
t
, V IX
2
t−1
) −[COV
P
(RV, V IX
2
t−1
) +COV
P
(SW
t
, V IX
2
T−1
)]
σ
2
V IX
2
T−1

2
V IX
2
t−1
−2 ∗ COV
P
(V IX
2
T−1
, V IX
2
t−1
)
From the above decomposition and the results in Section 4.1 upper-bounds bounds for the co-
effcient value can be deduced. Firstly, using the results of Jiang & Tiang (2005), lagged implied
risk premia and variance risk premia
17
volatility subsumes as least as much information regarding future realized volatility as previous real-
izations of realized volatility, thus lagged implied variance should exhibit a positive covariance with
future realized variance. More importantly implied variance measured at time T-1, will subsume at
least as much information as implied variance measured at t-1, thus in theory COV
P
(RV, V IX
2
T−1
) ≥
COV
P
(RV, V IX
2
t−1
). Moreover, considering the remaining covariances, by construction the following
must be true COV
P
(SW
t
, V IX
2
t−1
) ≤ COV
P
(SW
t
, V IX
2
T−1
), as the terminal VIX
2
of the previous hold-
ing period forms directly the swap rate for the beginning of the next period, thus taken together the
signing of the coefficent is ambiguous. However, using the empirical results in Section 4.1 -Table 7 &
8, the value of β
V IX
2
lag
≤ β
V IX
2 as the concurrent values should contain at least as much if not more
information than any of their lagged counterparts. More importantly, despite the ambiguity regarding
the signing of β
V IX
2
lag
, the results in Section 4.1, point towards a positive relationship between the lagged
VIX
2
and the variance risk premia. To test the hypothesized behaviour as well as the relative perfor-
mance of each model the following regressions are estimated (Figure.21) with the results for the VIX
2
returns, calculated using both bid/ask prices and settlement prices referred to as V IX
2
SP and V IX
2
BA
respectively, presented in Table.11.
Figure 21: Forecast Regressions - Variance Risk Premia & VIX
2
V RP
T,t
= α +β
V IX
2
lag
(r(t −1)
V IX
2) +
V RP
T,t
= α +β
V RP
lag
(r(t −1)
V RP
) +
Table 11: Forecasting Model - Coeffcient Estimates
α V IX
2
SP V IX
2
BA V RP
T−1
DW-Stat Root-MSE
R1 Model* -0.5843 0.5347 2.0073 0.71101
0.00% 3.70%
R2 Model* -0.5793 0.5185 2.0012 0.71075
0.00% 3.50%
R3 Model* -0.5111 0.1945 1.9935 0.71963
0.00% 9.30%
As demonstrated in Table.11
19
, the coeffcient estimates for the lagged VIX
2
are consistent with the
hypothesized behaviour derived through the covariance decomposition in emphFigure.20, regardless of
the return calculation methodology. Moreover, the VIX
2
appears to outperform the lagged variance risk
premia both in terms of statistical significance and mean square error, with the lagged variance risk
premia only marginally significant at the 10% level compared to the lagged VIX
2
which is significant
at the 5% level. The results in Table.11 contains some similarities with respect to the works of Jiang
& Tiang (2005) and Christensen & Prabhala (1998); in particular the in-sample performance of the
lagged VIX
2
returns relative to the lagged variance risk premia, implies the implied variance subsumes
more information regarding the variance risk premia than than past values of the variance risk premia.
However, as compared to existing studies analyzing the relationship between spot volatility and implied
volatility, realized values of the variance risk premia provide little to no explanatory power as to future
values of the variance risk premia which is in stark contrast to the behaviour of realized volatility. From
the above investigation it is possible to infer, though VIX
2
returns are not a perfect predictor of the
variance risk premia as judged by the in-sample MSE and significant constant term, returns on the VIX
2
do contain valuable information regarding the formation of the variance risk premia and more importantly
plays a role in the formation of future values of the variance risk premia.
19
Results with * are robust statistics correcting for heteroscedasticity
18
5 Summary & Conclusion
In conclusion, the findings of this paper demonstrates the following regarding the relationship between
the variance risk premia and the VIX. Firstly, the relationship between the VIX and the variance risk
premia, though present, is characterized by a non-linear second order polynomial relationship. Secondly,
the second order polynomial relationship between the VIX and variance risk premia can be re-cast as a
linear combination of both the CBOE VIX and SKEW indicies. Finally, with respect to the forecasting
ability of VIX, the VIX by itself is a subsumes more information regarding future realizations of the
variance risk premia rather than past values of the variance risk premia. The above results present a
set of interesting conclusions, first and foremost the VIX does not subsume all risks inherent within the
variance risk premia. Moreover, provided the significance of the SKEW Index this implies the VIX fails
to capture at least some element of jump risk, priced explicitly in the SKEW Index. Finally, despite
the short comings of the VIX, it is clear the VIX subsumes valuable information regarding the variance
risk premia and does contain some predictive power over the variance risk premia. In turn, for future
studies, two possible areas of interest can be pursued with greater academic rigour. Firstly, given the
significance of the SKEW Index further studies should focus on the relationship between the skew and
the variance risk premia and whether different measures of implied skewness provide greater or lesser
explanatory power with respect to the variance risk premia. Secondly, given the significance of the VIX
as a predictor of the variance risk premia, future studies should focus on whether this forecast can be
improved through the addition of variables other than the VIX.
19
References
[1] Backus.D , Chernov.M, Martin.I. (2011). Disasters implied by equity index options. Journal of Fi-
nance. 66 (0), pg. 1969-2012.
[2] Bollerslev.T , Todorov. V. (2011). Tails, Fears and Risk Premia. Journal of Finance. LXVI (6),
p.2165-2211.
[3] Britten-Jones. M Neuberger.A. (2000). Option Prices, Implied Price Processes and Stochastic Volatil-
ity. Journal of Finance. 4 (2), p.839-866.
[4] Carr.P Lee. R. (2009). Volatility Derivatives. Annual Review of Finance. 1 (0), p.1-21.
[5] Carr.P , Madan.D (1998). Towards a Theory of Volatility Trading, in R.Jarrow(ed.), Risk Book on
Volatility. New York, pp.417-427.
[6] Carr. P Wu. L. (2008). Variance Risk Premiums. Review of Financial Studies. 22 (0), p. 1311-1341.
[7] CBOE (2010). The CBOE SKEW Index – SKEW
R
.
[8] CBOE (2009). The CBOE Volatility Index – VIX
R
.
[9] Cheredito.P, Filipovic.D, Kimmel. R. (2007). Market Price of Risk Specifications for Affine Models:
Theory and Evidence. Journal of Financial Economics. 83 (0), pg.123-170.
[10] Christensen. B.J. , Prabhala. N.R.. (1998). The Relation between implied and realized volatility.
Journal of Financial Economics. 50 (0), pg. 125-150.
[11] Cont. R , DaFonseca.J. (2002). Dynamics of Implied Volatility Surfaces. Quantitative Finance -
Institute of Physics Publishing. 2 (0), pg.45-60.
[12] Demeterfi. K , Derman.E, Kamal.M, Zou.J. (1999). More Than You Ever Wanted To Know About
Volatility Swaps. Goldman Sachs: Quantitative Strategies Research Notes. 0 (0), p.1-52.
[13] Fama, Eugene F.; French, Kenneth R. (1992). ”The Cross-Section of Expected Stock Returns”.
Journal of Finance 47 (2): 427–465
[14] Hull. J.C. (2009). Ch. 10, Trading Strategies Involving Options. In: Clinton, D. Options Futures
and Other Derivatives. 7th ed. New Jersey: Pearson - Prentice Hall. p. 230-231.
[15] Jiang.G Tian.Y. (2005). The Model Free Implied Volatility and Its Informational Content. The
Review of Financial Studies. 18 (4), p.1305-1342.
[16] Jiang.G , Tiang.Y. (June. 2005). Gauging the ”Investor Fear Gauge”: Implementation problems in
the CBOE’s new volatility index and a simple solution. University of Arizona - Working Paper.
[17] Kohzan.R , Neuberger.A, Schneider.P. (August. 2011). The Skew Risk Premium in Index Option
Prices. Working Paper - Warwick Business School.
[18] Martin.I (April,2012). Simple variance swaps. Working Paper, Stanford University
[19] Neuberger. A (1994). The Log Contract: A new instrument to hedge to hedge volatility, Journal of
Portfolio Management, Winter, p. 74-80
[20] Neuberger.A. (April.2011). Realized Skewness. Working Paper - Warwick Business School.
[21] Pan.J. (2002). The Jump-Risk Premia Implicit in Options: Evidence from an Integrated Time-Series
Study. Journal of Financial Economics. 63 (0), pg.3-50.
[22] Schneider.P. (2012). Fear Trading.Working Paper - Warwick Business School.
[23] Whalley.R. E. (2009). Understanding the VIX. Journal of Portfolio Management. 35 (3), p.98-105
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