Electronic copy available at: http://ssrn.com/abstract=2165584

Expected VIX Option Returns

Zhaogang Songy

Federal Reserve Board

First Draft: March 30, 2012This Draft: October 22, 2012

ABSTRACT

We dissect the ne structure of volatility risks, as an important component of time-varying investment

opportunities, by studying returns on VIX option portfolios. In particular, we provide model-free evidence

regarding the two leading channels in modeling volatility risks: stochastic volatility-of-volatility and volatility

jumps. We nd that zero-delta straddles and delta-hedged portfolios of VIX options, which are neutral to

changes in VIX but sensitive to volatility-of-volatility, underperform zero. In contrast, tail portfolios, which

we construct by out-of-the-money and at-the-money VIX options and are only sensitive to tails risks of

volatility, outperform zero. Therefore, risks in volatility-of-volatility and jump-induced volatility tails are

priced, providing empirical support for both of the two leading channels. We further construct measures

for volatility-of-volatility and volatility tail risks, by the whole set of out-of-the-money VIX options, to

understand how they are priced relative to variance and equity risk premiums. We nd that both measures

predict short-horizon market returns, with the predictability from volatility-of-volatility risks subsumed by

the variance premium but the volatility tail risk standing as a separate channel. Dierent from existing tail

risk measures of merely market returns, our volatility tail index provides important information regarding

how investors gauge the extreme volatility risks.

KEYWORDS: Return predictability, stochastic volatility-of-volatility, variance risk premium, VIX options,

volatility tail risks

I am very grateful to Gurdip Bakshi, Itamar Drechsler, Yang-Ho Park, Andrew Patton, Yintian Wang, Hao Zhou,and also seminar participants in the Sixth Annual Risk Management Conference at National University of Singaporefor constructive comments. I thank Min Wei for sharing her codes, and Jian Du and Nikunj Kapadia for sending metheir data. All remaining errors are solely mine. The views expressed herein are the authors and do not necessarilyreect the opinions of the Board of Governors of the Federal Reserve System.

yMonetary and Financial Market Analysis Section, Board of Governors of the Federal Reserve System, mail stop165, 20th Street and Constitution Avenue, Washington, DC 20551. E-mail: Zhaogang.Song@frb.gov.

Electronic copy available at: http://ssrn.com/abstract=2165584

Expected VIX Option Returns

ABSTRACT

We dissect the ne structure of volatility risks, as an important component of time-varying investment

opportunities, by studying returns on VIX option portfolios. In particular, we provide model-free evidence

regarding the two leading channels in modeling volatility risks: stochastic volatility-of-volatility and volatility

jumps. We nd that zero-delta straddles and delta-hedged portfolios of VIX options, which are neutral to

changes in VIX but sensitive to volatility-of-volatility, underperform zero. In contrast, tail portfolios, which

we construct by out-of-the-money and at-the-money VIX options and are only sensitive to tails risks of

volatility, outperform zero. Therefore, risks in volatility-of-volatility and jump-induced volatility tails are

priced, providing empirical support for both of the two leading channels. We further construct measures

for volatility-of-volatility and volatility tail risks, by the whole set of out-of-the-money VIX options, to

understand how they are priced relative to variance and equity risk premiums. We nd that both measures

predict short-horizon market returns, with the predictability from volatility-of-volatility risks subsumed by

the variance premium but the volatility tail risk standing as a separate channel. Dierent from existing tail

risk measures of merely market returns, our volatility tail index provides important information regarding

how investors gauge the extreme volatility risks.

KEYWORDS: Return predictability, stochastic volatility-of-volatility, variance risk premium, VIX options,

volatility tail risks

1 Introduction

The volatility (variance) risk has been well documented as a priced element of time-varying invest-

ment opportunities in recent years. For example, many studies have constructed variance premium

measures by model-free variance swap rates or delta-hedged gains, and found that volatility risk is

priced negatively for the S&P 500 index and either positively or negatively for individual stocks

(Bakshi and Kapadia, 2003; Carr and Wu, 2009; Driessen et al., 2009; Todorov, 2010; Vedolin,

2009). There is also a large literature of parametric option pricing models with a negatively priced

stochastic volatility factor, including Heston (1993), Bakshi et al. (1997), Bates (2000), Pan (2002);

Eraker (2004), and Broadie et al. (2007) among many others. Most recently, many eorts have

been devoted to building equilibrium models incorporating the volatility risk and exploring its asset

pricing implications, such as Bansal et al. (2012), Bollerslev et al. (2012), Campbell et al.(2012),

Drechsler (2012), and Drechsler and Yaron (2011).

Two leading channels have been employed in modeling volatility risks in both reduced-form

and structural models: stochastic volatility-of-volatility and volatility jumps. The former is used

in Bollerslev et tal. (2009), Eraker (2008), and Mencia and Sentana (2012), while the latter is

considered by Drechsler (2012), Drechsler and Yaron (2011), Eraker (2004), and Eraker et al.

(2003). However, little work has been undertaken to understand the ne structure of volatility

risks and how they are priced without relying on restrictive specications. In particular, are risks

associated with higher moments of volatility priced? If so, do investors obtain compensations for the

stochastic volatility-of-volatility as in Bollerslev et al. (2009), for the volatility tail risks associated

with volatility jumps as in Drechsler and Yaron (2011), or for both?1 How are these two risks priced

relative to variance and equity premiums? And are they spanned by risks regarding expected returns

or standing as separate channels?

Whether stochastic volatility-of-volatility and volatility tail risks are priced will validate which

mechanism is crucial to model volatility risks. And whether they stand as separate channels from

risks regarding expected returns will improve our understanding of the risk structure of equities and

corresponding premiums. In fact, the Chicago Board Options Exchange (CBOE) began to publish

values for a "VIX of VIX" Index on March 14, 2012, obviously to track volatility-of-volatility.2

Moreover, the importance of tail risks of volatility is echoed by the recently (July 29, 2011) intro-

duced VIX Tail Hedge Index using VIX call options also by the CBOE. Measuring volatility tail

risks and corresponding premiums will rene our understanding of extreme volatility risks, dierent

from the existing studies that only focus on tail risks of returns induced by price jumps (Bollerslev

and Todorov, 2011; Du and Kapadia, 2012; Kelly, 2011).

1David and Veronesi (2012) propose a model that endogenously generates time-varying volatility-of-volatility bythe learning process as a result of the the nonlinear updating inherent in Bayes law. In particular, when investorsare more uncertain about fundamentals, they put less weight on their current beliefs and more weight on incomingnews so that revisions to beliefs and hence volatility-of-volatility are high.

2The VIX of VIX is calculated, using out-of-the-money VIX options, in the same way as the method to calculatethe VIX from S&P 500 options. Details can be found on www.cboe.com/micro/VVIX.

1

We provide model-free empirical analysis of these issues by forming appropriate portfolios of

VIX options, including calls, puts, straddles, delta-hedged positions, and tail portfolios. Conducting

empirical studies of volatility risks based on VIX option portfolios enjoys several advantages. First,

as a security associated exclusively with volatility risks, VIX options provide a pure instrument

to gauge volatility premiums. Second, compared to variance swap contracts with linear payos

in the variance, VIX options have nonlinear payos and are sensitive to high-order moments of

volatility. Hence, they are most informative about stochastic volatility-of-volatility and volatility

tail risks. Third, since VIX options were introduced by the CBOE in 2006, they have achieved huge

liquidity, becoming the third-most active contracts at the CBOE as of October 2011.3 Therefore,

VIX option prices provide direct market quotes from investors with respective to their expectations

about volatility shocks. Fourth, forming portfolios of VIX options enables us to study the economic

payo for bearing specic risks in a model-free way. For example, zero-delta straddles are immune to

small volatility movements, but sensitive to variations in volatility-of-volatility. The tail portfolios,

composed of out-of-the-money and at-the-money VIX options, can be made to single out risks

associated with left, right, or both tails. Moreover, the whole set of out-of-the-money VIX options

enables us to construct model-free measures for volatility-of-volatility and volatility tail risks.

Our study of whether there are premiums for the stochastic volatility-of-volatility as in Bollerslev

et al. (2009) makes use of straddles and delta-hedged portfolios of VIX options. We rst construct

zero-delta volatility straddles combinations of long positions in calls and puts with the same strike

and maturity that have osetting covariances with the VIX that are most sensitive to volatility-

of-volatility. Using option delta values from the BlackScholes model, we nd that at-the-money

zero-delta straddles lose 6% per week on average. We further show that this evidence is robust to

perturbations of BlackScholes delta values. Corroborating results from zero-delta straddles, returns

on delta-hedged portfolios a long position in one call or put, delta-hedged with the underlying VIX

until maturity signicantly underperform zero as well. For example, delta-hedged at-the-money

calls and puts with constant maturities of 21, 28, and 35 days lose over 10% per month.

One concern about using straddles and delta-hedged portfolios is that they may reect fears of

investors for large crashes and hence contain premiums for skewness and kurtosis. To conrm that

our results are not subject to such caveats, we construct crash-neutral straddles, that is, a straddle

position with a long position in deeply out-of-the-money calls. This strategy has a net long position

in call options but earns the same return whether the market volatility increases by 20% or 50%.

We nd that returns of crash-neutral, zero-delta straddles still lose over 3% per week. Overall, our

model-free evidence shows that volatility-of-volatility risks are priced with a signicantly negative

premium.

We then study volatility tail risks by forming tail portfolios. These portfolios are sensitive to

left, right, and both tails, respectively, but all positively correlated with positive volatility shocks.

3Average daily trading volumes of VIX options can be found on the CBOE website atwww.cboe.com/data/pdf/QRGVol2010.pdf.

2

As increases of volatility imply a deterioration of investment opportunities, the tail portfolios are

good hedges against extreme volatility risks and should earn positive returns. Moreover, they are

all made neutral to changes in the underlying with a zero delta and to changes in the volatility-

of-volatility with a zero Vega, isolating the eect of tail risks induced by volatility jumps. We

nd that all three portfolios earn signicantly positive returns held to expiration, averaging over

17% and 10% per month for 25% and 15% increases (decreases) in the right (left) tail of volatility,

respectively. The portfolio sensitive to both left and right tails of volatility receives returns of 90%

and 220% correspondingly. Therefore, our results demonstrate that the volatility tail risk premium

is signicantly positive indeed, lending support to the volatility jump channel in modeling volatility

risks.

Given that risks in volatility-of-volatility and volatility tails are both priced, we next investigate

how they are associated with variance and equity premiums. A stylized consumption-based asset

pricing model with stochastic volatility-of-volatility and volatility jumps shows that these two risks

are components of both equity and variance premiums. Therefore, measures of volatility-of-volatility

and volatility tail risks are expected to predict market returns. Moreover, although the variance

premium contains components from volatility-of-volatility and volatility jumps, it does not exclude

the possibility that these two channels have stand-alone contributions to equity premiums. Finally,

adding jumps into the dynamics of market returns brings a return tail premium to both equity and

variance premiums. Given the negative correlation between returns and volatility, it is essential to

check whether volatility tail risks are spanned by risks regarding expected returns.4

Although the variance premium has been well documented to have strong return predictability

by Bollerslev et al. (2009) and Drechsler and Yaron (2011) motivated by channels of stochastic

volatility-of-volatility and volatility jumps, respectively, no studies have deciphered the ne structure

of variance premiums and their connection to equity premiums raised above. We study these issues

by constructing measures for volatility-of-volatility and volatility tail risks using the whole set of

out-of-the-money VIX options. The former is similar to the variance premium measure in Carr and

Wu (2009), and the latter is analogous to the tail measure in Du and Kapadia (2012) for S&P 500

returns.

Results based on a series of return predictive regressions give answers to the questions raised

above. First, both volatility-of-volatility and volatility tail measures predict short-horizon market

returns as expected. Second, the predictability of the volatility-of-volatility risk is subsumed by the

variance premium but the volatility tail risk stands as a separate channel. Third, the predictability

of the return tail measure is also subsumed by the variance premium, while the volatility tail

index is still signicant in predicting returns controlling the return tail index. Therefore, although

both Bandi and Reno (2011) and Todorov and Tauchen (2011) document the importance of co-

4Todorov and Tauchen (2011), using high-frequency VIX data, suggest that volatility is a pure jump process andjumps in the volatility and price level are strongly dependent with opposite signs, providing an important channel forgenerating leverage eect.

3

jumps of volatility and price level, our results show that large volatility-specic jumps that are

not associated with extreme price changes are also priced. Hence, dierent from existing tail risk

measures of merely market returns, our volatility tail index provides important information for

investors to gauge extreme volatility shocks.

Finally, we also study expected returns of VIX calls and puts to appreciate them in the risk-

return space. Motivated by the U-shape of the pricing kernel as a function of index returns (Ait-

Sahalia and Lo, 2000; Bakshi et al., 2010a; Chabi-Yo et al., 2008), we also investigate the shape of the

volatility pricing kernel given the negative correlation between volatility and returns. Results show

that call (put) options earn returns that are signicantly negative (positive), averaging 6% (4%)

per week out-of-the-money (at all moneyness levels). This is broadly consistent with an increasing

pricing kernel of volatility. However, while both call and put returns decrease with strikes at

medium-to-large strike levels, they are increasing at low strikes, indicating a decreasing region of

the pricing kernel and hence a U-shape. The model of Bakshi et al. (2010b) with investors displaying

heterogeneous beliefs over volatility is able to accommodate such a U-shape and is supported by

our results.

To date, most studies of variance premiums only employ synthetic variance swap contracts with

linear payos on variance (Carr and Wu, 2009; Todorov, 2010). Although several works have used

VIX futures and options to study some pricing models (Sepp, 2008; Mencia and Sentana, 2011),

this paper is the rst to employ VIX options to investigate the ne structure of volatility risks by

exploring the unique nonlinear payos of options on volatility. In fact, our study makes use of a

more comprehensive dataset of VIX options than most previous studies, daily data from February

2006 when VIX options are introduced by the CBOE to April 2011.

Our approach of examining returns on option portfolios is motivated by Coval and Shumway

(2001) and Bakshi and Kapadia (2003), which validate negative volatility premiums using S&P 500

index options. Our analysis, by forming portfolios of VIX options, instead focuses on deciphering

volatility risks, motivated by state-of-the-art equilibrium models. Our study of volatility pricing

kernels is closely related to Bakshi et al. (2010b) that also uses VIX options. However, they only

examine returns on out-of-the-money options, while we document returns on calls and puts for all

the moneyness levels.

Our investigation of volatility tail risks contributes to the recent literature of tail risks. For

example, tail risk measures of returns have been proposed by Bollerslev and Todorov (2011) using

deeply out-of-the-money S&P 500 puts, by Kelly (2011) using observations of extreme returns, and

by Du and Kapadia (2012) using the dierence between the integrated variance and variance of

holding period log returns. Instead, our volatility tail index focuses on the tail risks of volatility, as

a priced element of time-varying investment opportunities. Therefore, combining our volatility tail

index and the existing return tail measures can deliver a more complete picture of extreme event

risks.5

5A contemporaneous literature focuses on equilibrium asset pricing models with tail event risks. For example,

4

The rest of the paper is as follows. Section 2 rst motivates the issues we focus on by a

stylized consumption-based model with volatility risks and then constructs portfolios and indices

for volatility-of-volatility and volatility tail risks. The data is described in Section 3. Section 4

presents returns on the VIX option portfolios, while Section 5 investigates how risks in volatility-

of-volatility and volatility tails are priced relative to equity and variance premiums. Robustness

checks and some further results are in Section 6. Section 7 concludes. An appendix provides some

technical details.

2 A Model of Volatility Risk and Volatility Option Portfolios

This section rst presents a stylized consumption-based asset pricing model with both stochastic

volatility-of-volatility and volatility jumps. We next discuss the advantage of volatility options to

analyze these two channels of volatility risks and then introduce the portfolios we focus on.

2.1 Time-Varying Volatility Risk in a Consumption-Based Framework

The basic setup is a stylized general equilibrium model with both stochastic volatility-of-volatility

and volatility jumps. It is a simplied combination of Bollerslev et al. (2009) and Drechsler and

Yaron (2011) that both build on the discrete-time long-run risk model of Bansal and Yaron (2004).

In particular, the representative agents preferences over consumption are recursive (Epstein and

Zin, 1989). Therefore, the log pricing kernel at time t+1 is

mt+1 = log 1ct+1 + ( 1)rc;t+1; (1)

where = 11 1 , 0 < < 1 is the subjective discount factor, is the risk aversion coe cient,

is the intertemporal elasticity of substitution, ct+1 is the growth rate of log consumption, and

rc;t+1 is the time t to t+1 return on the aggregate wealth claim. We assume > 1 and > 1

throughout the paper, which implies < 0. These restrictions ensure that the representative agent

has a preference for early resolution of uncertainty, which is the key for volatility risks to be priced.

The state vector of the economy is described as the following endowment processes:

ct+1 = c + c;tzc;t+1 + Jc;t+1;

2c;t+1 = + 2c;t +

pqtz;t+1 + J;t+1; and

qt+1 = q + qqt + 'qpqtzq;t+1; (2)

where fzc;tg, fz;tg, and fzq;tg are independent i.i.d. N(0,1) processes, Jc;t+1 is a compound PoissonBarro (2006) examines the eect of disaster risks on equity premiums using international data. Gabaix (2012) andWachter (2012) show that time variations in rare event risks are important to explain many stylized facts of equitypremiums. Our results, however, imply that accounting for time-varying risks of extreme volatility movements inthese asset-pricing models would improve their power in matching the data.

5

process with intensity c;t and i.i.d. jump size ci , J;t+1 is a compound Poisson process with

intensity ;t and i.i.d. jump size i , and both jump processes are independent of each other and

of the Gaussian shocks. Moreover, ci and i have mean and variance as (

cJ ; g

cJ) and (

J ; g

J ),

respectively.

Observe that two shocks, zc;t+1 and Jc;t+1, drive the randomness of the log consumption growth

rate. Furthermore, except a Gaussian shock z;t+1 to the level of volatility directly, the model

has two additional sources for temporal variation of the economic uncertainty proxied by 2c;t,

i.e., the stochastic volatility-of-volatility qt and volatility jump shock J;t+1. They are the two

leading mechanisms to model volatility risks and analyze variance premiums in a consumption-

based framework. The stochastic volatility-of-volatility qt is the focus of Bollerslev et al. (2009)

in justifying the return predictability of variance premiums, while the volatility jump J;t+1 is

documented by Drechsler and Yaron (2011) as indispensable to quantitatively match both moments

of variance premiums and standard characteristics of equity premiums.

We now solve for the equity and variance premiums, with the focus on how risks associated

with qt and J;t+1 are priced. The standard approach for solving the model is to use Campbell and

Shillers (1988) log-linearization with rc;t+1 = 0 + 1wt+1 wt +ct+1. The price-dividend ratiowt is conjectured to be a ne in the state vector:

wt = A0 +A2c;t +Aqqt; (3)

where A0 > 0, A < 0, and Aq < 0 are functions of the model parameters we suppress for notational

brevity. With (2) and (3), we have

rc;t+1 = log + 1c + (1A A)2c;t +1Aqq Aq

qt

+c;tzc;t+1 + 1pqtAz;t+1 +Aq'qzq;t+1

+1AJ;t+1 + Jc;t+1: (4)

By (1) and (4), the model-implied equity premium is

r;t Covt (mt+1; rc;t+1)= 2c;t + (1 )21

A2q'

2q +A

2

qt +

1 + 1

gcJc;t + (1 )21A2gJ;t: (5)

The variance risk premium is dened as vpt;t+1 EQt [V art+1 (rc;t+2)] Et [V art+1 (rc;t+2)].Although it is straightforward to obtain Et [V art+1 (rc;t+2)], computing E

Qt [V art+1 (rc;t+2)] involves

specifying risk-neutral dynamics of the state vector. As our model is just for illustration, we avoid

6

this additional complication by following Bollerslev et al. (2009) to use a log-linear approximation:6

EQt [V art+1 (rc;t+2)] logherf;tEt

emt+1+V art+1(rc;t+2)

i 12V art [V art+1 (rc;t+2)]

Et [V art+1 (rc;t+2)] + ( 1)1A +Aq

21

A2 +A

2q'2q

'2qqt

+( 1)1A;tJ +( 1)1A + 1

2( 1)2 21A2

;tg

J

+( 1)c;tcJ +1

2

h2 2 + ( 1)2

ic;tg

cJ ;

which implies that

vpt;t+1 = ( 1)1A +Aq

21

A2 +A

2q'2q

'2qqt

+

( 1)1AJ +

( 1)1A + 1

2( 1)2 21A2

gJ

;t

+

( 1)cJ +

1

2

h2 2 + ( 1)2

igcJ

c;t: (6)

A comparison of (5) and (6) delivers the following testable implications. First, risks of stochastic

volatility-of-volatility and volatility jumps are priced as components of both the equity and volatility

premium. Second, jump risks of log consumption growth rate are also priced in both premiums.

Third, as a combination of premiums for qt, J;t, and Jc;t that enter the equity premium, the variance

premium is expected to predict market returns. Fourth, the return predictability of stochastic

volatility-of-volatility, volatility jump, and consumption jump premiums may or may not be fully

subsumed by the variance premium.

The return predictability of variance premiums have been well documented in Bollerslev et al.

(2009, 2011), Drechsler and Yaron (2011), and Zhou (2010). Moreover, Bollerslev and Todorov

(2011) and Du and Kapadia (2012) propose model-free measures of Jc;t and nd that tail risks

induced by price jumps are important components of r;t and vpt;t+1, while the predictability of

price tail premiums for market returns is only through variance premiums. In contrast, no studies

have proposed model-free measures for qt and J;t+1 and analyzed their connections with equity

and variance premiums, making the picture of variance risks incomplete and vague. We argue that

current studies only based on synthetic variance swaps cannot answer these questions, as the swap

rates mainly reveal the conditional mean of return variation (Carr and Wu, 2009). Instead, we ll in

this gap by utilizing options written on volatility that are sensitive to higher moments of volatility

by design.

6See Drechsler and Yaron (2011) for a full solution by assuming a ne specications for risk-neutral dynamics.

7

2.2 Volatility Option Portfolios

We rst discuss the advantage of volatility options to analyze the role of stochastic volatility-of-

volatility and volatility jumps and then introduce the portfolios we rely on. From now on, let

vt;T 2 (0;1) represent the market return variability from time t to T that can be measured indierent ways. For example, most existing studies of variance premiums measure v2t;T by an estimate

of the quadratic variation of log returns (Bollerslev et al., 2009; Carr and Wu, 2009; Todorov, 2010).7

We denote hv2t;T

the payo function of a volatility-contingent claim. For the model from the last

section, there exists a pricing kernel mt;T that can price hv2t;T

.8 As we mainly focus on volatility

securities, it su ces to work with the projection of the pricing kernel onto the space generated by

vt;T , denoted as m (vt;T ).9 Therefore, the price Zt;T of the volatility-contingent claim hv2t;T

is

Zt;T =1

RfEtm (vt;T )h

v2t;T

; (7)

where Rf is the risk-free rate between time t and T .

Existing studies of variance premiums consider the variance swap contract with a payo to the

long position equal to v2t;T st;T , where st;T is the variance swap rate and vt;T is proxied by thequadratic variation. As a variance swap contract has zero net market value at entry, (1) implies

that 0 = EPthm (vt;T )

v2t;T st;T

iand hence

st;T = Etm (vt;T ) v

2t;T

: (8)

Therefore, st;T can be regarded as the price of a volatility-contingent claim with payo hv2t;T

=

v2t;T . An inspection of (8) shows that variance swaps are generally not informative about higher

moments of volatility due to the linear payo function hv2t;T

= v2t;T .

We turn to volatility options with nonlinear payo functions. Consider a call option with its

payo (vt;T K)+ max (vt;T K; 0) written directly on the same proxy of volatility as that usedin (8).10 An application of (7) shows that the price Ct;T of a volatility call is

Ct;T =1

RfEtm (vt;T ) (vt;T K)+

: (9)

A comparison of (8) with (9) shows that the nonlinear payo function makes the option very

7Alternatively, Sassan et al. (2002) and Brandt and Diebold (2006) proposed range-based volatility measures8The existence of such a pricing kernel can also be justifed by no-arbitrage arguments (Du e, 1992; Harrison and

Kreps, 1979).9The price of a volatility security with payo h

v2t;T

isRf1

Etmt;Th

v2t;T

. By the law of iterated ex-

pectation, the price is equal toRf1

EtEtmt;Th

v2t;T

jv2t;T = Rf1 Et Et mt;T jv2t;T h v2t;T , wherem (vt;T ) = Et

mt;T jv2t;T

. Accordingly, all that is required for volatility securities is m (vt;T ).

10Rigorously, the underlying of volatility options diers from that of variance swap rates by a convexity term.However, both refer to the same stochastic drivers, and ignoring the convexity term will not change our results ofwhether and how risks of stochastic volatility-of-volatility and volatility jumps are priced.

8

sensitive to higher moments of volatility, providing sensible information about stochastic volatility-

of-volatility and volatility jumps. In the next section, we rst document general properties of the

volatility pricing kernel m (vt;T ) and then returns on volatility calls and puts.

2.2.1 Calls and Puts

We study returns on calls and puts to shed light on the risk-return relationship for products of the

volatility market. In particular, we ask whether m (vt;T ) is always monotonically increasing with

volatility or exhibits a U-shape. The former implies a negative variance premium while the latter

predicts a positive sign at low volatility levels. We characterize patterns of expected call and put

returns under both the monotonically increasing and U-shaped pricing kernels of volatility. Let Pt;Tdenote the price of a volatility put. Then the the returns on volatility calls and puts can be dened

as rCt;T (vt;T K)+ =Ct;T 1 and rPt;T (K vt;T )+ =Pt;T 1, respectively. For call returns, wehave

Proposition 1: (i) If m (vt;T ) is increasing in vt;T over (0;1), then the return on a volatilitycall, rCt;T , is negative and decreasing in the strike. (ii) If m (vt;T ) exhibits a U shape, decreasing on

(0;K] and increasing on [K;1), then rCt;T is negative and decreasing at relatively high strikes butcould be positive and increasing at su ciently low strikes.

For put returns, we have

Proposition 2: (i) If m (vt;T ) is increasing in vt;T over (0;1), then the return on a volatilityput, rPt;T , is positive and decreasing in the strike. (ii) If m (vt;T ) exhibits a U shape, decreasing on

(0;K] and increasing on [K;1), then rPt;T is positive and decreasing at relatively high strikes butcould be negative and increasing at su ciently low strikes.

Propositions 1-2 extend the theoretical results on expected returns of market index options in

Coval and Shumway (2001) to the case with volatility options and U-shaped pricing kernels. They

are similar to Theorem 1 of Bakshi et al. (2010b) except that they consider a pricing kernel projected

on both the market return and volatility, while we only study the pricing kernel projected on the

volatility space. This dierence results from the dierent focuses: Bakshi et al. (2010b) try to set

up an equilibrium model of volatility trading that is suited to the market data, while we aim to

decipher the ne structure of variance risks empirically.

Propositions 1-2 imply that volatility call (put) returns are negative (positive) and decreasing

with strikes in general. However, positive returns on deeply in-the-money calls or increasing patterns

with moneyness will support a U-shaped m (vt;T ). Analogously, negative returns of deeply out-of-

the-money puts or increasing patterns with moneyness also support a U shape for m (vt;T ).

2.2.2 Straddles and Delta-Hedged Portfolios

Returns on volatility calls and puts do not allow us to make conclusions about volatility-of-volatility

and volatility jumps, which are mixed with each other and polluted by jumps in the consumption

9

growth rate, as shown in (5) and (6). We purge the compensation for variability of the underlying by

forming portfolios that are most sensitive to changes in volatility-of-volatility. Motivated by Coval

and Shumway (2001) and Bakshi and Kapadia (2003), we propose to employ zero-delta straddles

and delta-hedged portfolios of volatility options.

The delta of a call (put) option of volatility, Ct;T (Pt;T ), measures the sensitivity of the option

price with respect to the underlying volatility vt;T . We follow Coval and Shumway (2001) and

Bakshi and Kapadia (2003) to compute them as the BlackScholes hedge ratio, hence

Ct;T = N

1

t;TpT t log(vt;T =K) +

1

2t;T

pT t

Pt;T = 1Ct;T ; (10)

whereN [] is the cumulative normal distribution, t;T is the volatility-of-volatility over [t; T ] and thesecond equation follows from putcall parity. Both zero-delta straddles and delta-hedged portfolios

are made neutral to movements of vt;T with a zero delta by combining positions of calls and puts

for the former and positions of calls or puts that are delta-hedged until expiration for the latter.

In particular, the zero-delta straddle comprises buying one call and Ct;T =Pt;T puts with thesame maturity and strike so that the delta of the portfolio equals Ct;T +

Pt;T (Ct;T =Pt;T ) = 0.

The return of this portfolio is hence

rt;T =(vt;T K)+ Ct;T (vt;T K)+ =Pt;T

Ct;T Ct;TPt;T =Pt;T 1

=(vt;T K)+ Ct;T (vt;T K)+ =(1Ct;T )

Ct;T Ct;TPt;T =(1Ct;T ) 1; (11)

where (10) is used in the second equality. Therefore, we only need the delta value of the call option

in computing rt;T .

For delta-hedged portfolios, we consider a call that is hedged discretely N times over [t; T ], where

the hedge position is rebalanced at tn, n = 0; 1; ; N 1 with t0 = t and tN = T . Therefore, thisportfolio is delta neutral and not sensitive to variations of the underlying volatility at all of these

dates until the maturity. The return is hence r;Ct;T = Ct;T =

Ct;T vt;T Ct;T

, where

Ct;T = (vt;T K)+ Ct;T N1Xn=0

Ctn;Tvtn+1;T vtn;T

N1Xn=0

tnCt;T Ctn;T vtn;T

T tN

; (12)

where tn is the risk-free rate and Ctn;T

is the hedge ratio on date tn. Similar expressions for the

delta-hedged gain Pt;T and returns r;Pt;T can be obtained analogously with put option prices and

delta values.

10

2.2.3 Tail Portfolios

Besides the risk in the volatility-of-volatility, returns on volatility options are also sensitive to

even higher (of order larger than 3) moments of volatility, skewness and kurtosis for instance.

These higher moments are closely associated with volatility-jump-induced tail risks because volatility

options provide hedges against large-scale volatility increases and hence market declines because of

the negative correlation between market returns and volatility (French et al., 1987, and Glosten

et al., 1993). We construct portfolios that are mainly subject to these tail risks and investigate

whether they are priced. As usual, we remove the exposure of these tail portfolios to changes in

the underlying volatility with a zero delta and to changes in the volatility-of-volatility with a zero

Vega.

As payo functions of out-of-the-money volatility calls and puts are only determined by the left

and right volatility tails, respectively, we construct three types of portfolios that are sensitive to

only the left tails (L-Tail portfolio), only the right tails (R-Tail portfolio), and both tails (LR-Tail

portfolio). Moreover, we make all three portfolios positively (negatively) correlated to the right

(left) volatility tails and hence their payos increase as the probability of a high (low) realized

volatility is increased (decreased). Hence, we expect returns on them to be positive if volatility tail

risks are indeed priced.

The L-Tail portfolio consists of shorting one out-of-the-money volatility put, buying #OTMP =#ATMP

at-the-money puts, and a volatility position of (P;OTMt;T #OTMP P;ATMt;T =#ATMP ), where #OTMP and#ATMP are Vegas of out-of-the-money and at-the-money volatility puts, respectively.

11 The delta of

this portfolio is equal to

P;OTMt;T + #OTMP P;ATMt;T =#ATMP +P;OTMt;T #OTMP P;ATMt;T =#ATMP = 0: (13)

Moreover its Vega is

#OTMP + (#OTMP =#ATMP )#ATMP = 0: (14)

Hence, L-Tail portfolio is indeed both delta and Vega neutral. To verify its payo is negatively

correlated with the left tail, we rst note that both puts in this portfolio have changes in payos

only when the left tail of the volatility changes. Lowering the left tail, we have a smaller probability

of low volatility levels, which implies smaller payos for the two puts simultaneously. Because

the out-of-the-money put has a larger decrease in payos than the at-the-money put, the shorting

position in the L-Tail portfolio ensures a higher value for the portfolio at expiration, which conrms

that the portfolio payo is decreasing in the left tail. Therefore, this Left-Tail portfolio singles out

the lefttail risk of the volatility.

11Such a strategy is actually a put spread combined with positions in the underlying to make the portfolio bothdelta and Vega neutral. Bali and Murray (2011) employ similar strategies and explain the risk they are exposed to asthe skewness. We treat them as assets sensitive to the tail risk, as they are sensitive to risks associated with not onlyskewness but also kurtosis and even higher moments (of order higher than 2).

11

The R-Tail portfolio consists of buying one out-of-the-money volatility call, shorting #OTMC =#ATMC

at-the-money calls, and a volatility position of (C;OTMt;T +#OTMC C;ATMt;T =#ATMC ). Similar to (13)-(14), we can verify that the R-Tail portfolio is also delta and vega neutral. Moreover, its payo is

increasing in the right tail of volatility, hence isolating the risks associated with the right tail. Finally,

the LR-Tail portfolio consists of buying one out-of-the-money volatility call, shorting #OTMC =#OTMP

out-of-the-money puts, and a volatility position of (C;OTMt;T + #OTMC P;OTMt;T =#OTMP ). It can alsobe veried that the LR-Tail portfolio is delta and vega neutral, and its payo is both negatively

correlated with the left tail and positively correlated with the right tail. Therefore, the LR-Tail

portfolio incorporate risks of both tails of volatility.

The returns on these three portfolios can be calculated by the net payos of the positions held

until maturity divided by the entering costs. They provide us a complete picture of whether and

how investors perceive the tail risks and hence volatility jumps. Furthermore, by changing the

moneyness levels of the out-of-the-money calls and puts in the portfolios, we can examine premiums

of tail risks at dierent scales, for example, a 15% versus 25% volatility increase.

2.3 Measuring Volatility-of-Volatility and Volatility Tail Risks

Portfolios of the volatility options above are most eective to answer whether risks of stochastic

volatility-of-volatility and volatility jumps are priced. However, to gauge the premiums for these

two risks and examine their relationship with equity and variance premiums shown in (5) and (6),

we need direct quantitative measures.

As stochastic volatility-of-volatility is about the temporal variation of return volatility, we dene

a volatility index of volatility (V IXV ) by constructing variance swap contracts written on volatility

using volatility options. In particular, let RV denote the realized annualized variance of the volatilityvt;t+ from time t to T . A variance swap contract on vt;t+ pays the dierence between RV and aswap rate determined at the initiation of the contract. Similar to Carr and Wu (2009), the swap

rate is equal to EQt [RV], i.e., the risk-neutral expectation of the realized variance. Assuming vt;t+follows a diusion process, then EQt [RV] = IV, which is the annualized integrated variance and canbe computed by out-of-the-money volatility options with maturity T ,

IV 2er(Tt)

T

(Z vT;T+0

1

K2P Tt (K)dk +

Z 1vT;T+

1

K2CTt (K)dk er(Tt)

her(Tt) 1 r(T t)

i);

(15)

where r is the risk-free rate from time t to T . Our V IXV is dened as V IXV = IV RV, whichessentially equals the variance risk premium of volatility associated with qt in (2).

An important point emphasized by Du and Kapadia (2012) is that IV diers from EQt [RV]by terms due to the volatility jump J;t+1. Although this dierence is a drawback for measuring

variance risk premiums, it is a benet for us to separate volatility-of-volatility and volatility jumps

and dene appropriate measures for both. Specically, the swap rate IV dened in (15) only includes

12

the volatility-of-volatility qt in computing V IXV and leaves out J;t+1. Therefore, V IXV indeed

mainly captures premiums for the volatility-of-volatility risk.

To quantify the volatility tail risk due to J;t+1, we follow Du and Kapadia (2012) to consider

the following measure for variance premiums of volatility that incorporates volatility jumps:

V 1T

8

We clean the data using several lters. First, we exclude options with zero open interests or

trading volumes. Second, we discard options with bid quotes equal to zero or negative bidask

spreads. To mitigate the eect of price recording errors, observations with bid or ask prices less

than 0.05 are also eliminated. Finally, we only consider options with maturity larger than a week

and less than 60 days for liquidity concerns. Our focus on these short-term options also makes our

results robust to the impact of stochastic interest rates. For each option, we take the mid-quote as

the option price. Our sample ultimately contains a total of 47,256 VIX option prices, with 29,167

calls and the 18,089 puts.

Table 1 provides summary statistics for our sample of VIX options across moneyness, measured

by strike divided by the underlying price.13 As expected, the call (put) price is negatively (posi-

tively) associated with moneyness. Most interestingly, the liquidity pattern of VIX options across

moneyness is substantially dierent from that of S&P 500 options. As is well known, the liquidity of

in-the-money S&P 500 options is seriously worse than near-the-money options. In contrast, Table

1 shows that VIX calls and puts that are 5% to 15% in the money have similar liquidity to their

near-the-money counterparts in terms of both the number of contracts and the average trading

volumes. For example, VIX calls with moneyness in (0.8,0.95] has contracts of 13.81% of the total

call contracts, larger than 9.70% for calls with moneyness in (0.95,1.05].14 The nal observation is

that deep out-of-the-money calls and in-the-money puts are the most liquid VIX options, as they

are closely related to volatility increases and market declines.

As discussed in Section 2, both delta-hedged and tail portfolios require investments in the VIX

that is not tradable directly. As a solution, we use VIX futures as the tool to trade volatility. The

daily VIX futures prices can be obtained directly from the CBOE website, along with the trading

volume and open interest. We apply several exclusion lters to the raw data. For example, we

exclude VIX futures with zero open interests or trading volumes due to liquidity concerns. We also

delete futures contracts with prices less than 0.5 to mitigate the impact of prices discreteness.15

Panel A of Figure 1 plots daily time series of VIX index over our sample period of March 2006 to

April 2011, along with the one-month VIX futures price. It is observed that the futures prices are

close to the spot VIX in general, with discrepancies at regions with peaks. To put the VIX index

of our sample period in historic perspective, Panel B of Figure 1 plots the VIX time series starting

from 1990.13These summary statistics are for our cleaned sample data that only contain VIX options with maturities between

one week and two months. Unreported summary statistics show that the patterns we document here are representativeof VIX options with maturities between one month and half a year that are mostly liquid.14For a fair comparison, we nd that the percentages of VIX calls with moneyness in (0.8, 0.9] that has the same

length as (0.95, 1.05] is equal to 10.53%, which is fairly close to, though not larger than, the percentages of optionswith moneyness in (0.95,1.05].15The underlying value of VIX futures contract is VIX times 10 prior to March 26, 2007. We follow the rescaling

method of the CBOE Futures Exchange to divide the contract prices by 10 and bring the traded futures prices in linewith the underlying VIX index.

14

4 Returns of VIX Option Portfolios

This section documents empirical characteristics of returns on calls and puts, straddles and delta-

hedged portfolios, and tail portfolios. The questions we address are the shape of the volatility

pricing kernel and premiums for risks in volatility-of-volatility and volatility tails.

4.1 Call and Put Returns

Table 2 reports statistics of weekly VIX option returns over the ve-year sample period of March

2006 to April 2011, with results for calls in Panel A and those for puts in Panel B. We follow Coval

and Shumway (2001) to consider weekly returns because of the short sample period of available VIX

options. But dierent from the overlapping returns they employ, our weekly returns are computed

by holding a 30-day option for one week at the weekly frequency, which delivers a non-overlapping

weekly return series. We divide both calls and puts into six groups according to the moneyness,

dened as strike over the underlying VIX value. Note that we consider a large scale of moneyness

level because deeply away-from-the-money VIX options are at least as liquid as near-the-money

contracts, as documented in Table 1.

We record a variety of statistics, including mean, median, minimum, and maximum, for each

of the six moneyness groups. Because of the well-known skewness of option returns and small

sample concern, we report both normal distribution-based t-statistics and p-values, along with

bootstrapped p-values calculated by B=25,000 bootstrap draws.16 Panel A shows that the mean

returns of VIX calls are signicantly negative across moneyness levels, except deeply in-the-money

at 0.750.85. For example, an at-the-money call loses 11.54% per week, both economically and

statistically signicant. In contrast, VIX puts earn positive average returns across all groups of

moneyness. An at-the-money put tends to earn an average return of 6.51%. In general, the signs

of average call and put returns provide no evidence against an increasing volatility pricing kernel.

Moreover, the median, minium and maximum demonstrate that both VIX call and put returns

exhibit a substantial positive skewness, except the deeply in-the-money put returns. As a result,

bootstrapped p-values are mostly larger than the conventional p-values based on normality.

We then study the shape of the volatility pricing kernel from the perspective of option returns

across moneyness levels. Panel A shows that VIX call returns are decreasing with moneyness while

Panel B implies put returns are decreasing with moneyness in general but with reversals at levels of

0.750.85 and 1.051.15. However, we cannot draw conclusions directly about the pattern of option

returns with respect to moneyness without considering statistical uncertainties. To make rigorous

inferences, we use the MR test proposed in Patton and Timmermann (2010) that explores the full

set of implications of monotonic patterns.17

16As the autocorrelation of the weekly returns is fairly small, we perform i.i.d. bootstrap. We also tried theparametric bootstrap based on the BlackScholes model favored by Broadie et al. (2009) and block bootstrap, bothof which can account for time-series correlations. The results are similar to those using the i.i.d. bootstrap.17We also test the inequality relationship of the adjacent pair of moneyness groups. Results show that we cannot

15

In particular, we test the null hypothesis of decreasing in-the-money (out-of-the-money) VIX

call (put) returns with moneyness in the rst three groups. The bootstrapped p-values in Panels

A and B (based on B bootstrap samples) show that the decreasing pattern is rejected for in-the-

money calls at the 7% signicance level but not for out-of-the-money VIX puts. This evidence is

in sharp contrast to the naive observation that deeply in-the-money (out-of-the-money) VIX call

(put) returns exhibit some decreasing (increasing) regions. We also report the MR test for out-

of-the-money (in-the-money) call (put) returns with moneyness in the last three groups and nd

no rejection against the decreasing pattern. Therefore, it is the in-the-money call returns that

cast evidence against a uniformly increasing volatility pricing kernel and support the heterogenous

beliefs model of Bakshi et al. (2010b) that can support a U-shape volatility pricing kernel.18

4.2 Returns on Straddles and Delta-Hedged Portfolios

We direct our attention on the returns of zero-delta straddle to determine whether risks associ-

ated with stochastic volatility-of-volatility are priced, consistent with Bollerslev et al. (2009). As

discussed in Section 2, zero-delta straddle returns are sensitive to volatility-of-volatility but not to

the underlying VIX. Hence, their expected returns are negative if there is a premium for stochastic

volatility-of-volatility.

Table 3 reports statistics of weekly zero-delta straddle returns across ve moneyness groups. As

before, the weekly returns are computed by holding a 30-day zero-delta straddle position for one

week at the weekly frequency. We record the mean, median, minimum and maximum returns, along

with the conventional t-statistic and p-value for the null hypothesis of zero expected returns and

also bootstrapped p-values by B bootstrap draws. The results with Black-Scholes deltas show that

average returns on zero-delta straddles are signicantly negative at the 5% level across dierent

moneyness groups, except that the signicance level for the group with moneyness in (1.15, 1.25] is

10%. At-the-money straddles lose 6.59% per week and deeply in-the-money straddles lose the most.

To check the robustness of our results to beta calculations, we allow for measurement errors that are

10% above and below the Black-Scholes deltas. As we can see, all the returns remain signicantly

negative, suggesting that risks in volatility-of-volatility are indeed priced.

We then consider the second set of portfolios subject to risks in volatility-of-volatility, i.e., the

delta-hedged calls and puts. Results of returns on VIX calls hedged discretely until expiration are

reported in Table 4 while those for VIX puts are reported in Table 5. For each of the ve moneyness

groups, we pick options with maturities that are closest to 30, 44, and 58 days and delta hedge them

reject the null hypothesis that low-moneyness options earn higher returns in each pair for both calls and puts althoughsome negative dierences are observed for puts in a pair.18Our results are very dierent from those of Bakshi et al. (2010b) who nd that out-of-the-money put returns

support the U shape and both call and put returns are not statistically dierent from zero. The dierent resultscould be due to the dierent sample periods (March 2006 to April 2011 versus to September 2010), dierent samplingfrequencies (weekly versus monthly), dierent price quotes (mid-quote versus ask quote), and dierent moneynessschemes (moneyness groups versus options with the moneyness closest to a predetermined level).

16

until expiration.19 This practice delivers three non-overlapping return series at 30-day, 40-day, and

58-day frequencies, respectively.

Looking at returns of delta-hedged calls in Table 4, we nd that they are signicantly negative

across maturities and moneyness, with the only exception of 44-day call with moneyness in (1.15,

1.25]. For example, returns on at-the-money delta-hedged calls lose 11.70%, 10.31%, and 18.65% for

maturities of 30, 44, and 58 days, respectively. For returns of delta-hedged puts in Table 5, we see

that returns of delta-hedged deeply in-the-money puts are not signicantly negative. But both delta-

hedged out-of-the-money and at-the-money puts earn signicantly negative returns. The latter, for

example, lose 11.01%, 18.03%, and 32.94% for maturities of 30, 44, and 58 days, respectively.20

Moreover, we note that delta-hedged deeply in-the-money puts earn puzzling signicantly positive

returns of 12.23%. Similar to Bakshi and Kapadia (2003), we conjecture that the relative illiquidity

for in-the-money puts may upwardly bias the mean returns. This conjecture is veried by observing

that returns on delta-hedged deeply out-of-the-money calls that are equivalent to deeply in-the-

money puts but much more liquid 10,363 contracts versus 4,831 contracts as shown in Table

1 lose 11.37% at the 1% signicance level.

Although straddles and delta-hedged positions are most sensitive to changes in volatility-of-

volatility, they also hedge risks associated with large market crashes that are accompanied by

extreme increases of volatility. To deal with this concern in concluding nonzero risk premiums

for volatility-of-volatility by returns on straddles and delta-hedged positions, we consider a crash-

neutral zero-delta straddle. Such a portfolio consists of a long straddle position and a short position

of deeply out-of-the-money calls, i.e., buying one at-the-money call, shorting one out-of-the-money

call, and buying (OTMC ATMC )=ATMp at-the-money put. Suppose the out-of-the-money callis p percent larger than the strike. It can be veried that this crash-neutral zero-delta straddle

has a zero delta, and its payo increases as the volatility increases by p percent while remaining

unchanged when the volatility increases further. By changing p, we can control the sensitivity of

the portfolio payo to dierent scales of large market crashes.

Table 6 reports statistics of weekly returns on the 30-day crash-neutral zero-delta straddles with

ve moneyness levels for the out-of-the-money call. To achieve the target moneyness level p, we

select the call option with the highest available strike that is at most p percent higher above the

underlying VIX value. We observe that the reported average moneyness levels of employed out-

of-the-money calls match the target levels well. Most importantly, the returns on crash-neutral

zero-delta straddles are all signicantly negative across p. They range from -2.94% to -3.96% that

are lower than the -6.59% of at-the-money straddles and decrease with the moneyness of the out-

19The results reported here for delta-hedged calls and puts employ the BlackScholes deltas. Similar to the caseof call and put returns, we also experimented with measurement errors in the delta calculations and found similarresults.20Note that returns on delta-hedged calls and puts are much higher than those on zero-delta straddles, although

both are sensitive to risks associated with volatility of volatility. The reason is that delta-hedged positions with 30-day,40-day, and 50-day options held until maturity deliver returns over longer periods than weekly straddle returns.

17

of-the-money call. This pattern implies that (1) the returns of straddles do contain premiums for

market crash risks, and (2) abstracting from crash risks, signicantly negative returns on crash-

neutral zero-delta straddles conrm that risks associated with volatility-of-volatility are indeed

priced. Therefore, our results show that the mechanism through stochastic volatility-of-volatility in

Bollerslev et al. (2009) is empirically valid to endogenize volatility risks of the consumption growth.

4.3 Returns on Tail Portfolios

The anecdotal evidence in the last section that crash-neutral straddles have returns lower than zero-

delta straddles indicates the existence of premiums for volatility tail risks associated with volatility

jumps. We now formally study the tail risk premiums by computing returns of tail portfolios.

We report returns on 30-day L-Tail, R-Tail, and LR-Tail portfolios held until maturity in Panels

A, B, and C of Table 7, respectively. We record the mean returns, positions of options and the

underlying VIX, and average moneyness levels of calls and puts, along with conventional t-statistic

and p-value for the null hypothesis of zero expected returns and also bootstrapped p-values by B

bootstrap draws. For both left and right tails, we consider two levels of tail events, yL = 0:75; 0:85

and yR = 1:15; 1:25. The former corresponds to a 25% and 15% downward jump of volatility levels

while the latter represents a 15% and 25% upward jump. To achieve the target moneyness levels

yL and yR, we select the out-of-the-money calls and puts with the highest available strike that is

not larger than the target levels. The reported average moneyness levels in Table 7 show that they

closely match the objective yL and yR.

We observe that returns on all of the tail portfolios are signicantly positive, mostly at the 5%

signicance levels using bootstrapped p-values. For example, with yL = 0:85 and yR = 1:15, the

L-Tail, R-Tail, and LR-Tail earn 9.90%, 31.19%, and 90.53% per month, respectively. Returns on R-

Tail and LR-Tail portfolios are substantially higher than those of L-Tail portfolios, suggesting that

investors are more sensitive to the right tails of volatility and hence upward volatility jumps that

usually accompany large market declines. Furthermore, for each set of tail portfolios, returns are

increasing substantially with the magnitude of large volatility changes. This pattern implies that the

market prices severe volatility tail risks exceedingly high. Overall, our results empirically conrm

the importance of Drechsler and Yarons (2011) mechanism of volatility jumps in endogenizing

volatility risks of the consumption growth. Of course, it is not clear so far whether the volatility tail

premium is mainly dictated by its correlation with price tails that have been studied in Barro (2006),

Bollerslev and Todorov (2011), Du and Kapadia (2012), Gabaix (2012), and Wachter (2012).21 To

clarify this issue, we need measures for volatility tail risks.

21There are also a few equilibrium option models incorporating rare event risks, including Backus et al. (2011),Benzoni et al. (2011), Du (2010), Liu et al. (2005), and Shaliastovich (2011). However, most of these studies focuson matching empirical characteristics of S&P 500 options such as the implied volatility smirk, and none of theminvestigates variance premiums.

18

5 Volatility-of-Volatility and Volatility Tail Risk Indices

We rst discuss empirical calculations of our proposed volatility-of-volatility and volatility tail risk

indices V IXV and V TIX. Then we investigate how these risks are priced relative to variance and

equity premiums based on return predictive regressions.

5.1 Empirical Methodology and Descriptive Statistics

We focus on a 30-day horizon for the synthetic variance swap rates written on the VIX, i.e., T t =30. At each day, we choose VIX options with exactly 30 days to expire if they are available.

Otherwise, two contracts with the nearest maturities of 30 days with one longer and the other

shorter than 30 days will be chosen. We keep only out-of-the-money options and exclude days with

fewer than two option quotes of dierent moneyness levels for each chosen maturity.

The computation of IV and V relies on a continuum of moneyness levels, as shown by (15)-(16). Following Carr and Wu (2009) and Du and Kapadia (2012), we interpolate implied volatilities

across the range of observed moneyness levels and use the implied volatility of the lowest (highest)

available moneyness for moneyness levels below (above) it. In total, we generate 2,000 implied

volatility points equally spaced over a strike range of zero to three times the current spot price for

each chosen maturity at each date. We then obtain a 30-day implied volatility curve either exactly

or by interpolating the two implied volatility curves of the two chosen maturities. Finally, we use

the generated 30-day implied volatility curve to compute the out-of-the-money VIX option prices

by the BlackScholes formula and then IV and V according to a discretization of (15)-(16). Inaddition, we compute the annualized 30-day realized variance of the VIX RV as

RV =365

T tTtXi=1

lnvt+i;t+i+30vt;t+30

2(17)

using daily VIX levels.22

With the daily indices, we pick end-of-month quantities to make a monthly time series, extending

from March 2006 to April 2011 with 60 months in total. Panels A and B of Figure 2 provide

plots of V IXV and V TIX over our sample period, respectively. The shaded area represents the

NBER-dened recession period December 2007June 2009. It is clear from Panel A that the index

V IXV for volatility-of-volatility remains positive most of the time but becomes deeply negative

occasionally. But Panel B shows that it is the opposite case for the volatility tail index V TIX.

Furthermore, as expected, V TIX has its largest value during the recession period, much larger than

values during non-recession periods. The volatility-of-volatility index, however, has some extreme

values even during non-recession periods although reaching an extreme value during the recession

period as well.

22We also tried ve-minute VIX series in calculating RV and found similar results.

19

For comparison, we plot in Panel C of Figure 2 the return jump and tail index JTIX VSPIVSP , where IVSPSP and VSP are dened similarly to IV and V in (15) and (16), respectively,with the underlying price replaced by S&P 500 index returns. We nd the JTIX also has its

largest value during the recession period, similar to the V TIX. However, one observable large

dierence between the JTIX and V TIX is that the latter has fairly big and volatile series during

the non-recession period relative to its largest value in the recession while values of the former in

the non-recession period are substantially smaller than its peak. Finally, we plot in Panel D two

measures of variance risk premiums, i.e., IVSP RVSP in Bollerslev et al. (2009) and VSP RVSPin Du and Kapadia (2012).23 The former missed price jumps while the latter incorporate them (Du

and Kapadia, 2011). Comparing Panels B and C to D suggests that time variations of variance risk

premiums track those of TIX better than temporal variations in V TIX, providing a preliminary

evidence that V TIX stands as a separate channel, while JTIX may be subsumed relative to

variability risks of returns.

5.2 Return Predictability Regressions

We now study whether measures of stochastic volatility-of-volatility and volatility tails predict

market returns as expected by (5). Then in light of (6), we investigate whether the predictability

of V IXV and V TIX are only through the variance risk premium vpt;T or present additional risks

relative to the channel of return variability.

5.2.1 Setup and Summary of Predictors

Our setup follows Bollerslev et al. (2009) and Du and Kapadia (2012) to estimate a series of

predictive regressions for market returns and control for a large set of traditional predictors. In

particular, monthly P/E and P/D ratios are obtained from Robert Shillers website, while quarterly

CAY data of Lettau and Ludvigson (2001) are downloaded from Lettau and Ludvigsons website.24

The term spread (dened as the dierence between 10-year T-bond and three-month T-bill yields),

the default spread (dened as the dierence between Moodys BAA and AAA corporate bond

spreads), the stochastically detrended risk-free rate (dened as the one-month T-bill rate minus its

backward 12-month moving average), and the three- and one-month T-bill rates are all from the

website of the Federal Reserve Bank of St. Louis.

Panel A of Table 8 reports basic summary statistics for the monthly returns and predictor

variables. The mean excess return is only -0.71% due to our short sample period containing the

recent nancial crisis. The two measures of variance risk premiums both have positive means,

with VSP RVSP higher due to its inclusion of premiums for jump risks in market returns. The23 I am very grateful to Jian Du and Nikunj Kapadia for sharing their data series of VSP RVSP and JTIX, which

are updated until October 2010. Moreover, the data series of IVSP RVSP are downloaded from Hao Zhous website.24We follow Bollerslev et al. (2009) and Du and Kapadia (2012) to use the most recent quarterly CAY data as the

monthly observations.

20

V IXV and V TIX have mean values of -0.28 and -0.09, respectively, with the former negatively and

the latter positively skewed. Panel A also reports autocorrelations of the variables up to order 3.

Consistent with the literature, all of the traditional predictors are highly persistent with rst-order

autocorrelations mostly larger than 0.9 and third-order autocorrelations still above 0.7. In contrast,

V IXV and V TIX have much lower autocorrelations. Also note that the autocorrelation of V TIX

is only one-half of that for JTIX. As a precursor for our main results, the correlation matrix in

Panel B shows that both V IXV and V TIX are fairly correlated with the market returns, while

the correlations of traditional predictors are tiny except for the CAY and RREL. Moreover, V TIX

is much less correlated with CAY, log(P/E), and log(P/D) than JTIX.

5.2.2 Results

All regressions are conducted at the monthly frequency, with one-month, three-month, six-month

and 12-month period returns. Hence, observations are overlapping for regressions of the last three

types. To account for the heteroskedasticity and autocorrelation induced by the overlapping ob-

servations, we compute all t-statistics based on the standard errors of Hodrick (1992).25 Table 9

reports results of predictive regressions for one-, three-, six-, and 12- month return horizons.26

Focusing on univariate regressions, Table 9 shows that variance risk premiums are largely sig-

nicant at all horizons we consider, consistent with the literature. And the traditional predictors

have mixed successes, with the CAY signicant at one- and two- month horizons, and the RREL

signicant up to six months. Most importantly, we nd that V IXV has signicant predictability

for short horizons of one and three months with t-statistics equal to 2.04 and 1.91, respectively.

The volatility tail index V TIX is more impressive in predicting returns, being signicant at 5%

levels up to six-month horizons. Therefore, at least for short horizons, the results show that risk

premiums associated with volatility-of-volatility and volatility tails indeed predict returns, as dic-

tated by (5). This pattern provides further support for incorporating both stochastic volatility and

volatility jumps in consumption-based asset pricing models.

We next consider multivariate regressions by adding variance risk premiums. For the regression

with the volatility-of-volatility index, adding variance risk premiums (either IVSP RVSP or VSP RVSP ) makes V IXV insignicant for the three-month horizon. Even for the one-month horizon, thet-statistic is decreased from 2.04 to 1.70. These results suggest that the predictability of V IXV for

market returns is subsumed substantially by the variance risk premiums, implying that volatility-

of-volatility risk does not stand as a separate risk channel from the return variance.

In contrast, the signicance of V TIX to predict market returns survives the adding of variance

risk premiums to the regression although the t-statistics are all decreased. The only exception

is the regression of six-month horizons with IVSP RVSP for which the t-statistic is -1.09. But25Ang and Bekaert (2007) illustrate by a Monte Carlo study that the Hodrick (1992) standard errors are more reliable

than the classical Newey and West (1987) standard errors for predictive regressions with overlapping observations.Wei and Wright (2009) extend the reverse regression approach of Hodrick (1992) to the multivariate case.26For highly persistent predictors, the R2 of regressions has to be explained with caution.

21

even for this case, the V TIX is still signicant with the better measure of variance risk premium

VSP RVSP . These results do not change even when we add traditional predictors, as shown in thelast three columns of Table 9. In fact, after adding log(P/D) and term premium to the regression

with V TIX and IVSP RVSP , the V TIX becomes signicant even at the 10% level for the 12-month horizon.27 Overall, our results suggest that the volatility tail risk presents a dierent channel

from the return variability risk and is closely associated with equity risk premiums.

5.3 Stock Return Tail vs Volatility Tail

The previous section demonstrated the economic signicance of volatility tail risks as a separate

channel. However, one may ask whether volatility tail risks are signicant only because of their

correlation with return tail risks given the evidence that jumps in volatility and price levels are

strongly dependent (Bandi and Reno, 2011 and Todorov and Tauchen, 2011). The preliminary

answer is no by the observable large dierence in time variations of the JTIX and V TIX: the

latter is more volatile during non-recession periods relative to the peak, as discussed in Section

5.1.28 We now address this concern formally by predictive regressions.

Table 10 reports both univariate return predictive regressions with JTIX and multivariate

regressions by adding variance risk premiums and V TIX to the former. The rst two columns

of each cell show that the JTIX is only signicant for the 12-month horizon individually but

achieves better signicance combining with IVSP RVSP , consistent with Du and Kapadia (2012).But it is more reliable to determine whether JTIX is signicant in predicting market returns as

a separate channel using VSP RVSP that incorporates price jump risks. A comparison of thesecond and third columns implies that no strong evidence exists to claim JTIX as a separate

risk from return variability. In particular, adding VSP RVSP makes the signicance of JTIXvanish for the six-month horizon while decreasing the t-statistic from 2.55 to 2.05 for the 12-month

horizon. Therefore, our results conrm the conclusion of Du and Kapadia (2012) that the rst-

order importance of return jump risks are through their impact on market return variability, which

indirectly supports the importance of volatility jumps as a separate channel.

As direct evidence, the last column of each cell in Table 10 reports regression results with both

JTIX and V TIX. We observe that V TIX is signicant at all horizons except the 12-month period,

increasing the adjusted R2 more than 10 times that of the univariate regression with only JTIX.

It also improves the signicance of JTIX for the six-month and 12-month horizons, increasing

the t-statistics from 0.75 and 1.99 to 2.45 and 2.58, respectively. Overall, our results show that

volatility tail risks present dierent channels from both the return variance and equity tail risks.

Consequently, our volatility tail risk index V TIX is useful as an additional index for investors to

27We exclude traditional predictors that are highly correlated with V TIX or variance premium measures to avoidthe multicolinearity problem for regressions reported in the last three columns.28Using S&P 500 futures high-frequency returns from 1982 to 2009, Bandi and Reno (2011) nd that the number

of estimated yearly co-jumps is 89, while that of the idiosyncratic volatility jumps is about 13.

22

gauge risks of extreme events, dierent from all equity tail risk measures proposed in Bollerslev and

Todorov (2011), Du and Kapadia (2012), and Kelly (2011).

6 Robustness Checks and Further Results

We conduct robustness checks for results of returns on VIX option portfolios in this section. We

also present some further results about the V TIX because of its prominent importance documented

above.

6.1 Transaction Costs of VIX Option Portfolio Strategies

It is essential to understand to what extent the transaction costs incurred by buying and selling

VIX options when employing portfolio strategies in Section 3 would aect our results.29 As our data

cleaning procedure has excluded a large set of illiquid options based on open interest and trading

volumes, we focus on the bid-ask spread here. Our results reported in Section 3 are all based on

returns computed using the mid-point of the bid and ask price quotes. We investigate this issue by

re-computing returns on all of the employed portfolios assuming the eective spread is 10%, 25%,

and 50% of the quote spread, respectively. By denition, the eective spread is equal to twice the

dierence between the true execution price and the market quote. For example, suppose the bid

and ask prices of an option are 4 and 5 dollars, respectively. Then a 50% eective spread means

that we buy the option at 4.75 while selling it at 4.25.

Table 11 reports returns of calls, zero-delta straddles, delta-hedged calls, zero-delta crash-neutral

straddles, and tail portfolios computed by an eective spread of 0% (the mid-quote), 10%, 25%,

and 50% respectively. The rst three strategies are all at the money with moneyness levels in (0.95,

1.05], while the out-of-the-money call in zero-delta crash-neutral straddles has moneyness 1.20. For

the tail portfolios, we consider options that are 15% away from the at-the-money strike. Results

of options with other characteristics including both puts and delta-hedged puts are similar and not

reported here to save space.

Observe that all of the option strategies have decreasing magnitudes of returns with respect to

the eective spread. As expected, weekly returns in Panel A are aected more than the held-until-

maturity returns on delta-hedged calls and tail portfolios in Panels B and C because the former

involves more frequent buying and selling. The most severe case is the zero-delta crash-neutral

straddle with returns increasing from -3.6% to -0.82% per week that is insignicantly dierent from

zero with a p-value at 24.63. This is not surprising given that such a portfolio involves three option

contracts. However, zero-delta crash-neutral straddles with the eective spread less than 25% and

all the other strategies are all signicant consistently as results in Section 3. Therefore, our results

29A large literature documents the importance of transaction costs in the options markets; see Figlewski (1989) andOfek et al. (2004) among many others.

23

are robust to reasonable values of eective spreads although the magnitudes of option returns do

decrease after transaction costs are considered.

6.2 Daily V TIX Series

The reported V TIX series in Section 5 are only at the monthly frequency with a maturity of 30

days. Given the documented importance of the V TIX to capture risks of volatility tails and hence

extreme events, it would be important to have the index at daily frequencies and also with dierent

maturities. Figure 3 plots the daily time series of the 30-day and 45-day V TIX series in Panels

A and B, respectively. Certain days have some missing values, especially in the initial period of

20062007 when VIX options were rst introduced. We linearly interpolate them on days with

missing values.

Observe that the daily V TIX series of 30-day maturity are similar to the monthly series in

Panel B of Figure 3 except that the latter misses a spike in April 2010. The V TIX series of 45-day

maturity in Panel B, however, are smoother than the the 30-day V TIX except at peak times,

probably due to the volatility mean reversion. Unreported statistics show that both the 30-day

and 45-day V TIX series are highly persistent, with rst-order autocorrelations of 0.86 and 0.90,

respectively. More interestingly, the correlation between them is only 0.81, showing that the V TIX

may have certain distinct features across maturities. This non-perfect correlation echoes studies

favoring two factors to drive the volatility process, for example, Adrian and Rosenberg (2008) and

Christoersen et al. (2008).

7 Conclusion

We provide model-free evidence regarding the two leading channels in modeling volatility risks:

stochastic volatility-of-volatility and volatility jumps. We nd that risks of volatility-of-volatility

and jump-induced volatility tails are both priced. We further construct measures for volatility-

of-volatility and volatility tail risks by the whole set of out-of-the-money VIX options. We nd

that both measures predict short-period market returns, with predictability from the volatility-

of-volatility risk subsumed by the variance risk premium but the volatility tail risk standing as a

separate channel. Hence, dierent from existing tail risk measures of merely market returns, our

volatility tail index provides important information about how investors gauge the extreme volatility

risks.

Our results are limited to the short period from May 2006 to April 2011 when VIX options

are available. It would be important to investigate these issues and update the volatility tail index

when more data are available with further developments of the volatility market. Moreover, our

model-free results imply that asset pricing models with both stochastic volatility-of-volatility and

volatility jumps would benet from volatility option prices in deciphering the ne structure of

equity and variance risk premiums. It remains interesting to investigate the empirical performance

24

of models in Bansal et al. (2012), Bollerslev et al. (2009), Campbell et al.(2012), and Drechsler and

Yaron (2011) by adding the dimension of VIX options, especially those about the extreme volatility

risks.

25

Appendix

Proof of Proposition 1: The expected excess return of a volatility call option held untilmaturity is

rCt;T =Et (vt;T K)+

Etm (vt;T ) (vt;T K)+

1=

R1K (v K) [1m (v)] f (v) dvR1K (v K)m (v) f (v) dv;

where equation (7) is used in the rst equality. Taking the rst-order derivative of rCt;T with respect

to K, we have

drCt;TdK

=

R1K (v K) f (v) dv

R1K m (v) f (v) dv

R1K (v K)m (v) f (v) dv [1 F (v)]R1K (v K)m (v) f (v) dv

2=

hR1K (v K) f(v)[1F (v)]dv

i hR1K m (v)

f(v)[1F (v)]dv

ihR1K (v K)m (v) f(v)[1F (v)]dv

ihR1K (v K)m (v) f(v)[1F (v)]dv

i2=

Et [(vt;T K) jvt;T > K]Et [m (vt;T ) jvt;T > K] Et [m (v) (vt;T K) jvt;T > K]hR1K (v K)m (v) f(v)[1F (v)]dv

i2= Covt [(vt;T K) ;m (vt;T ) jvt;T > K]hR1

K (v K)m (v) f(v)[1F (v)]dvi2 :

Therefore, the sign ofdrCt;TdK is determined by the negative of the covariance between vt;T K and

the volatility pricing kernel m (vt;T ) conditional on the option being in the money.

If m (vt;T ) is increasing in vt;T over (0;1), rCt;T will decrease in the strike K and achieves thelargest value when the strike is equal to zero. As the call option with a zero strike has the same

expected net return as that of the underlying volatility that is usually not larger than zero, we

conclude that all call options have negative returns given an increasing volatility pricing kernel. For

the case (ii) of a U-shaped m (vt;T ) with the threshold at K, the conclusion is the same as abovewhen vt;T is larger than K. However, at low strikes when vt;T is smaller than K, we will have theopposite scenario that rCt;T could be positive and has an increasing pattern. This nishes the proof.

Q.E.D.

Proof of Proposition 2: Similar to the derivations for expected call returns, the expectedexcess return of a volatility put is

rPt;T = Covt [(vt;T K) ;m (vt;T ) jvt;T < K]hRK

1 (K v)m (v) f (v) dvi2 :

26

That is, the sign ofdrPt;TdK is determined by the negative of the covariance between vt;T K and the

volatility pricing kernel m (vt;T ) conditional on the put option being in the money.

If m (vt;T ) is increasing in vt;T over (0;1), rCt;T will decrease in the strike K and achieves thesmallest value at an innite strike price. As the put with an innite strike has an expected net

return equal to the risk-free rate, we conclude that all put options have positive returns given an

increasing volatility pricing kernel. For the case (ii) of a U-shaped m (vt;T ) with the threshold at

K, the conclusion does not change vt;T is larger than K. However, at low strikes when vt;T issmaller than K, we will have the opposite scenario that rCt;T could be negative with a decreasingpattern. Q.E.D.

27

References

[1] Adrian, T., and J. Rosenberg (2008): Stock Returns and Volatility: Pricing the Short-Run

and Long-Run Components of Market Risk,Journal of Finance, 63, 29973030.

[2] Ait-Sahalia, Y. and Lo, A. (2000), Nonparametric Risk Management and Implied Risk Aver-

sion," Journal of Econometrics, 94, 9-51.

[3] Ang, A., and G. Bekaert. 2007. Stock return predictability: Is it there?" Review of Financial

Studies, 20(3): 651.

[4] Backus, David, Mikhail Chernov, and Ian Martin, 2011, Disasters Implied by Equity Index

Options, Journal of Finance forthcoming.

[5] Bali, T., and Murray S., 2011, Does Risk-Neutral Skewness Predict the Cross-Section of Equity

Option Portfolio Returns? working paper.

[6] Bakshi, Gurdip, Nikunj Kapadia, 2003, Delta-Hedged Gains and the Negative Market Volatility

Risk Premium, Review of Financial Studies 16 (2), 527-566.

[7] Bakshi, G., Madan, D., and Panayotov, G. (2010a), Returns of claims on the upside and the

viability of U-shaped pricing kernels," Journal of Financial Economics, 97, 130-154.

[8] Bakshi, G., Madan, D., and Panayotov, G. (2010b), An Asset Pricing Theory of Volatility Tail

Behavior, working paper.

[9] Bandi F. M., and R. Renno, (2011), Price and Volatility Co-Jumps, working paper.

[10] Bansal, R. and A. Yaron (2004). Risks for the Long Run: A Potential Resolution of Asset

Pricing Puzzles. Journal of Finance 59, 1481-1509.

[11] Barro, R. J. (2006). Rare Disasters and Asset Markets in the Twentieth Century. Quarterly

Journal of Economics 121, 823-866.

[12] Benzoni, L, P., Collin-Dufresne, and R., Goldstein, 2011, Explaining Asset Pricing Puzzles

Associated With the 1987 Market Crash, Journal of Financial Economics 101, 552-573.

[13] Bollerslev, T., G., Tauchen and H., Zhou, 2009, Expected stock returns and variance risk

premia, Review of Financial Studies 22 (11), 4463-4492.

[14] Bollerslev, T., M., Gibson, and H., Zhou, 2011, Dynamic Estimation of Volatility Risk Premia

and Investor Risk Aversion from Option-Implied and Realized Volatilities, Journal of Econo-

metrics, vol. 160, pages 102-118.

28

[15] Bollerslev, T., and V., Todorov, 2011, Tails, fears and risk premia, Journal of Finance, Vol. 66,

No. 6, pp. 2165-2221.

[16] Brandt, M., and F., Diebold, 2006, A No-Arbitrage Approach to Range-Based Estimation of

Return Covariances and Correlations, Journal of Business 79, 61-73.

[17] Broadie, M., M. Chernov, and M. Johannes (2009). Understanding Index Options Returns.

Review of Financial Studies 22, 4493-4529.

[18] Campbell, J., and R. Shiller. 1988. The dividend-price ratio and expectations of future divi-

dends and discount factors." Review of Financial Studies, 1(3): 195-228.

[19] Carr, P. and Wu, L. (2009), Variance Risk Premiums," Review of Financial Studies, 22, 1311-

1341.

[20] Chabi-Yo, F., Garcia, R., and Renault, E. (2008), State dependence can explain the risk

aversion puzzle," Review of Financial Studies, 21, 973-1011.

[21] Christoersen, P., K. Jacobs, C. Ornthanalai, and Y. Wang, 2008. "Option valuation with long-

run and short-run volatility components," Journal of Financial Economics, 90 (3), 272-297.

[22] Coval, J. D. and T. Shumway, 2001, Expected Option Returns,Journal of Finance 56, 983-

1009.

[23] David, A., and P., Veronesi, 2012, Investor and Central Bank Uncertainty Measures Embedded

in Index Options, working paper.

[24] Drechsler, I., 2012, Uncertainty, Time-Varying Fear, and Asset Prices, Journal of Finance,

forthcoming

[25] Drechsler, I. and A. Yaron (2011). Whats Vol Got to Do with It? Review of Financial Studies

24, 1-45.

[26] Driessen, J., P. Maenhout and G. Vilkov (2009), The Price of Correlation Risk: Evidence from

Equity Options, Journal of Finance, 64(3), p. 1377-1406.

[27] Du, D., 2010, General Equilibrium Pricing of Options With Habit Formation and Event Risks,

Journal of Financial Economics 99.

[28] Du, J., and N., Kapadia, 2012, The Tail in the Volatility Index, working paper.

[29] Du e D., 1992, Dynamic Asset Pricing Theory, Princeton University Press.

[30] Epstein, L., and S. Zin. 1989. Substitution, risk aversion, and the temporal behavior of con-

sumption and asset returns: A theoretical framework." Econometrica, 57(4): 937-969.

29

[31] Eraker, B. (2004), Do Stock Prices and Volatility Jump? R