VIX option‐implied volatility slope and VIX futures returns

• Published date: 01 June 2022
• Author: Jungah Yoon,Xinfeng Ruan,Jin E. Zhang
• Date: 01 June 2022
• DOI: http://doi.org/10.1002/fut.22317

Received: 12 October 2021

|

Accepted: 27 January 2022

DOI: 10.1002/fut.22317

RESEARCH ARTICLE

VIX option‐implied volatility slope and VIX futures returns

Jungah Yoon |Xinfeng Ruan |Jin E. Zhang

Department of Accountancy and Finance,

Otago Business School, University of

Otago, Dunedin, New Zealand

Correspondence

Jungah Yoon, Department of Accountancy

and Finance, Otago Business School,

University of Otago, Dunedin 9054, New

Zealand.

Email: isabella.yoon@postgrad.otago.ac.nz

Funding information

University of Otago

Abstract

This paper documents the dynamics of the term structure of the implied

volatility (IV) smirk of Chicago Board Options Exchange Volatility Index (VIX)

options. Empirical analysis shows that VIX option–IV slope predicts VIX fu-

tures returns over the next day to month, outperforms existing investors'

perception proxies in the stock and option markets. The empirical finding is

rationalized through time‐varying correlation between the VIX and volatility

of VIX (VVIX), VIX jumps, and investors' net positions in VIX futures market.

KEYWORDS

implied volatility slope, VIX futures returns, VIX options

JEL CLASSIFICATION

G12, G13

1|INTRODUCTION

To the best of our knowledge, this is the first paper to document the dynamics of the implied volatility (IV) smirk of the

options of the Chicago Board Options Exchange (CBOE) Volatility Index (VIX), a benchmark gauge of Standard & Poor's

500 Index (SPX) volatility, and evaluate the information embedded in the IV smirk of VIX options. The VIX options are one

of the most actively traded derivatives contracts that allow market participants to trade based on anticipated movement in

stock market volatility. Previous research has documented the IV smirk of options; however, less of which focuses on the

VIX options. This paper documents the IV smirk of VIX options following Zhang and Xiang (2008)andanalyzesin-

formation contents of the IV smirk factors and rationalize the empirical findings through the channels of time‐varying

correlation between the return of the VIX and volatility of VIX (VVIX), VIX jumps, and investors' trading activity.

The VIX options are one of the most actively traded derivatives contracts written on the VIX that allows market

participants to trade based on anticipated movement in stock market volatility. Since the VIX itself is not directly

tradable, investors rely on derivatives to obtain volatility exposure. A key aspect of the IV of VIX options is that it is a

forward‐looking measure of VVIX (fourth moment of the index return), which may provide useful information to both

the stock market investors and volatility traders. Despite the importance of its existence, there remains a paucity of

studies on the IV of the VIX options. Many existing studies, such as Foresi and Wu (2005), Zhang and Xiang (2008),

Xing et al. (2010), Yan (2011), Kelly et al. (2016), Tian and Wu (2020), among others, focus either on the IV smirk on

individual stock options or stock index options. They find the shape of the IV smirk is negatively skewed, on average,

with positive curvature. Johnson (2017) studies the term structure of volatility of SPX options. The author uses

weighted prices of SPX options to obtain VIX term structure and find a significant return predictability from the second

principal component (SLOPE). Huang et al. (2019) focus heavily on the impact of the VVIX level on delta‐hedged gains

J Futures Markets. 2022;42:1002–1038.1002

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wileyonlinelibrary.com/journal/fut

This is an open access article under the terms of the Creative Commons Attribution‐NonCommercial‐NoDerivs License, which permits use and distribution in any

medium, provided the original work is properly cited, the use is non‐commercial and no modifications or adaptations are made.

© 2022 The Authors. The Journal of Futures Markets published by Wiley Periodicals LLC.

of S&P 500 and VIX options. The author uses (rough) slope measure derived from the IV of VIX options as a control

variable in predicting delta‐hedged gains. Gehricke and Zhang (2020) study the VXX options market and provide a

benchmark for developing a VXX option pricing model. Eraker and Yang (2020) explain observed characteristics in

equity return, VIX futures, and options data through model fitting and simulation processes. Surprisingly, the empirical

documentation of the IV smirk of VIX options and evaluating their embedded information have not been closely

examined. Thus, this study provided an important opportunity to advance our understanding of the role of the IV smirk

of VIX options in valuing volatility‐sensitive assets.

To begin the analysis, the paper documents the dynamics of theIVsmirkofVIXoptionsbyregressingtheIVagainstits

moneyness for each date and maturity. The moneyness of options is defined following Carr and Wu (2003)asitallows

simple interpretation of moneyness measure. That is, it gives an approximate standard deviation on how far the log strike is

away from the log implied forward price in the Black–Scholes model. Carr and Wu (2003) suggest that the standardized

moneyness helps prevent issues with misstated junction between the IV smirk parameters and the risk‐neutral distribution.

1

The level, slope and curvature of IV smirk of VIX options are then constructed on a daily basis following a simple, but

parsimonious methodology proposed by Zhang and Xiang (2008) which involves much fewer option quotes than that of

Bakshi et al. (2003). The constant maturity IV smirk factors are developed either by interpolation or extrapolation to the

target maturity to analyze the term structure of VIX options' IV smirk comprehensively.

2

To evaluate whether the IV smirk

factors contain information regarding volatility‐sensitive asset, we run predictive regressions on the VIX futures returns

based on in‐sample and out‐of‐sample tests. The IV slope is a unique factor in the sense that it provides a direction of the

investors' perceptions observed in the volatility market.

3

Therefore, risk source(s) of the IV slope is(are) further examined by

analyzing the IV slope's relation with (1) correlation between changes in the VIX and VVIX, (2) aggregate jump risk in the

VIX, then further decomposing the aggregate jump risk into (3) positive, and (4) negative jump risk, respectively.

Additionally, we consider trading activity variables in understanding the forecasting power of the IV slope.

The present paper is closely related to Johnson (2017), Huang et al. (2019), Cheng (2019), Chuang et al. (2020),

Eraker and Yang (2020), Jacobs and Mai (2020), among others. There are similarities, however, the study differs

from previous studies in a number of important ways. First, unlike Johnson (2017), who studies the term structure

of volatility of equity index options (SPX options), we examine the term structure and dynamic process of VIX

options. Interestingly, we too find that only the slope of IV smirk of VIX options is informative about subsequent

VIX futures returns. In an analysis of variance risk premium (VRP), Todorov (2010) and Li and Zinna (2018)find

jump (slope) in volatility does matter while the diffusion term (level) does not.

4

Our paper also deviates from

Huang et al. (2019), who focus heavily on the impact of VVIX level on delta‐hedged gains of S&P 500 and

VIX options. The author uses (rough) slope measure derived from the IV of VIX options as a control variable while

we construct the IV smirk factors on a daily basis which provides factors that reflect the true market data with

close proximity.

5

Second, we focus on empirical documentation of the IV smirk of VIX options to derive the IV

slope which is then studied in relation to trading activity in the VIX futures market. This differs from that of Cheng

(2019), who examines the impact of traders' position in the VIX futures market directly on the sensitivity of VIX

premium (i.e., change in VIX premium).

6

Third, unlike Eraker and Yang (2020), who show the basic properties

1

There are alternative moneyness measures, such as Black–Scholes's delta ( Nd

Δ

=()

1) and the ratio of the strike to forward price

(

)

K

F. However,

they often lead to unnecessary complications in interpretation (Jia et al., 2021) or suffer from the distorted link between the IV parameters and that of

the risk‐neutral moments, as seen in Aït‐Sahalia and Lo (1998) as discussed in Carr and Wu (2003).

2

Studying the IV smirk factor is important and intuitive because each factor plays a different role in providing investors' perception in the volatility

market. The level of the IV smirk provides the overall perceptions in the volatility market. The IV smirk slope, which measures the IV's sensitivity

concerning changes in its moneyness, provides a direction of the overall perceptions in the volatility market. Finally, the IV smirk's curvature

provides how intense the given perceptions are in the volatility market.

3

For example, the slope of the IV smirk is found to be a useful factor in pricing uncertain market conditions (Foresi & Wu, 2005; Kelly et al., 2016;

Xing et al., 2010; Yan, 2011).

4

Note, however, the level of the IV smirk of VIX options is insignificant in predicting subsequent VIX futures returns in our study whereas it is found

to be an important factor in Huang et al. (2019). The potential driver of such difference may come from the definition of dependent variable. Todorov

(2010) finds that the jumps play a key role in explaining the observed risk premium. He finds that the occurrence of jump in the market level is

observed and is usually linked with a spike in volatility, however, it dies out quickly while the occurrence of jumps leads to a persistent increase

in VRP.

5

We find consistent results (as reported in Panel B of Table 7 in Huang et al., 2019) in our predictive regression on subsequent VIX futures return,

though the definition of return in our study is different from delta‐hedged gains used in their study.

6

We run similar regression to eq. (3) in Cheng (2019) and find consistent (negative temporal relationship) result between change in our VIX futures

returns and change in the level of IV smirk of VIX options at daily frequency.

YOON ET AL.

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of the level, slope, and curvature factors in aggregate (average) level, the present paper provides comprehensive

time‐series evolution of the IV smirk.

7

Finally, our study deviates from Chuang et al. (2020), who mainly provide

evidence of intraday impact of net buying pressure (BP) of VIX options on VIX option prices using a 5‐min interval

sample. Our general finding of the relationship between BP and the slope of the IV smirk of VIX options is

consistent with their study, though, our empirical evidence primarily stems from traders' positions in the VIX

futures market.

8

This finding is consistent with that of Jacobs and Mai (2020), who find net option demand is

positively related to option prices in the market for VIX puts and VIX calls.

The remainder of this paper is organized as follows. Section 2describes data for VIX futures returns, IV of VIX

options, and control variables. Section 3presents the methodology for predictive regressions. Sections 4and 5provide

the empirical results and robustness tests, respectively. Section 6concludes.

2|DATA

2.1 |VIX futures returns

The VIX futures are exchange‐traded contracts written on the VIX and are the first‐ever listed futures contract on the CBOE

VIX. It was launched on March 26, 2004, to provide market participants with direct access to investing in the volatility

market. The VIX futures prices are obtained from the CBOE for the sample period: between March 2004 and June 2019.

Figure 1plots the VIX futures' daily average trading volume and open interest over the sample period. The VIX

futures started trading in 2004. In the first year of trading, they had an average daily volume of 461 contracts. It grew

substantially after 2010 as an increasing number of investors sought to hedge against future stock index price fluc-

tuation. The year 2017 had an even more significant increase—23% up from the year prior. There are at least two

potential reasons for the sudden increase we observed in 2016–2017: (1) the referendum referred to as Brexit and (2) the

election of Donald Trump to the US presidency.

9

Table 1reports a summary of the VIX futures market trading activity overall and by maturity category. As the

contracts' maturity increases, the VIX futures prices, mean daily open interest, and mean daily volume decrease. The

contracts with a maturity of less than 90 days are the most liquid and frequently traded across different maturity

categories.

We compute the constant maturity VIX futures returns following Eraker and Wu (2017)as

















































rT w

FT

FT wFT

FT

()= ()

()−1+(1−)()

()−1

,

tt

t

t

tt

t

+1 +1 1

1

+1 2

2(1)

where FT()

tis the day

t

price of a VIX futures contract with maturity date T.T

1and T

2are the two closest times to

maturity to the target time to maturity T.

w

=

t

TT

TT

−

−

t

tt

,2

,1 ,2 and w

(

1−)

tare corresponding weights for the VIX futures price

with

T

t,1

and

T

t,2

, respectively. Note that the weight,

w

t

, is changing with time,

t

, since

T

t,1

and

T

t,2

may differ by

each date.

As shown in Table 2, the VIX futures returns are negative, on average, with a mean of −0.17% (annualized returns

of −41.82%), a standard deviation of 4.07%, and a skewness of 0.91. This is consistent with Carr and Wu (2009) and

Eraker and Wu (2017), who document negative average daily VIX futures returns. Carr and Wu (2009) find that the

VIX futures returns are strongly negative for the S&P 100 and 500 indexes, and the Dow Jones Industrial index based

on January 1996–February 2003 sample period. More recently, Eraker and Wu (2017) document the daily average

arithmetic return of −0.12% and daily average log return of −0.20% using a sample period from January 2006 to May

2013. The average VIX futures returns of some periods, such as 2007–2008 and 2011, are positive indicating a

7

Eraker and Yang (2020) explain observed characteristics in equity return, VIX futures, and options data through model fitting and simulation

processes, which is distinct from the objective of our paper.

8

The sample period of our data ranges from January 2006 to June 2019. Corresponding empirical evidence is based on both weekly and monthly data

frequencies.

9

Historically, the market participants have tended to reduce their futures exposure as risk rises, thus resulting in a reduction in overall trading

activity, consistent with a fall in demand for volatility insurance (Cheng, 2019).

1004

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