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

rτ

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

rτ

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

+σ

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

+σ

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

+σ

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

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