Variance and skew risk premiums for the volatility market: The VIX evidence

• DOI: http://doi.org/10.1002/fut.21968
• Date: 01 March 2019
• Published date: 01 March 2019
• Author: José Da Fonseca,Yahua Xu

Received: 11 October 2017

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Revised: 9 September 2018

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Accepted: 10 September 2018

DOI: 10.1002/fut.21968

RESEARCH ARTICLE

Variance and skew risk premiums for the volatility market:

The VIX evidence

José Da Fonseca

1,2

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Yahua Xu

3

1

Department of Finance, Business School,

Auckland University of Technology,

Auckland, New Zealand

2

PRISM Sorbonne, Université Paris 1

Panthéon‐Sorbonne, Paris, France

3

China Economics and Management

Academy, Central University of Finance

and Economics, Beijing, China

Correspondence

Yahua Xu, China Economics and

Management Academy, Central

University of Finance and Economics,

No. 39 South College Road, Haidian

District, 100081 Beijing, China.

Email: yahua.xu@cufe.edu.cn

Funding information

University of Technology, Sydney

Abstract

We extract variance and skew risk premiums from volatility derivatives in a model‐

free way and analyze their relationships along with volatility index and equity index

returns. These risk premiums can be synthesized through option trading strategies.

Using a time series of option prices on the VIX, we find that variance swap excess

return can be partially explained by volatility index and equity index excess returns

while these latter variables carry little information for the skew swap excess return.

The results sharply contrast with those obtained for the equity index option market

underlining very specific characteristics of the volatility derivative market.

KEYWORDS

risk premiums, skew swap, VIX option market, variance swap

JEL CLASSIFICATION

G11, G12, G13

1

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INTRODUCTION

The rapid growth of volatility products, among which the VIX index is by far the most well known, has turned volatility

into an asset class (Whaley, 1993; Zhang, Shu, & Brenner, 2010). The availability of VIX options suggests to apply

option‐implied moment estimation methodologies, which were extensively used on equity (index) options, to that

market as they allow the extraction of information on the VIX distribution without specifying any parametric model for

it. They are often qualified as model‐free approaches. These methodologies are favored over more traditional historical

estimation strategies as options embed a more forward looking point of view of asset moments’distribution. Also,

options contain risk‐neutral information and when combined with historical information enables the determination of

risk premiums that are the key variables for risk management.

These methodologies have been extensively applied to equity index options and/or individual stock options, and to

foreign exchange options and are based on the analytical results originally proposed in Carr and Madan (1998) and

Derman, Kani, and Kamal (1997). The literature is so vast that we will restrict ourselves to quote the works of Bakshi,

Kapadia, and Madan (2003) and Neuberger (2012) as entry points in this field. Among many possible applications let us

mention the use of higher risk‐neutral moments for asset pricing models in Bakshi et al. (2003), the estimation of an

investor’s risk aversion from volatility spread in Duan and Zhang (2014) or volatility forecasting using implied moments

in Byun and Kim (2013) and Neumann and Skiadopoulos (2013), and more recently the analysis of variance and skew

swaps for the S&P500 option market in Kozhan, Neuberger, and Schneider (2013). Quite surprisingly, the use of these

results for the VIX option market remains largely unexplored. A noticeable exception is the interesting work of Kaeck

(2014) that analyzes the variance risk premium extracted from VIX options using a model‐free methodology approach

in conjunction with other risk factors.

J Futures Markets. 2019;39:302–321.wileyonlinelibrary.com/journal/fut302

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© 2018 Wiley Periodicals, Inc.

Following Kozhan et al. (2013), we use VIX options to compute variance swap and skew swap excess returns and

analyze their relationships to VIX index excess return and S&P500 index excess return. All the quantities involved are

obtained from options in a model‐free way and correspond to tradable strategies some of which, like the variance swap,

are actively used in the market nowadays. A byproduct of these results is to draw some conclusions on variance and

skew risk premiums for the VIX market that will certainly hold for other less developed volatility markets. The results

obtained also underline certain differences between the equity (index) option market and the volatility (index) option

market that may not be a surprise as equity and volatility dynamics are profoundly different.

Our paper contributes to the literature by analyzing variance and skew risk premiums for the volatility market and

finds that both variables are negative (on average). For this market, we show that variance swap excess return can be

partially explained by volatility index excess return and equity index excess return but also by skew swap excess return.

However, considering all these explanatory variables together does not fully capture the return of a variance swap

trading strategy, as such the capital asset pricing model (CAPM) does not hold. To explain the skew swap excess return,

the most important variable is the variance swap excess return from which we deduce that higher order moments of the

volatility distribution can hardly be hedged using equity index trading strategies. Overall, our results depict the

volatility index market as being structurally different from the equity index market.

Thepaperisorganizedasfollows.Wepresentthekeyingredients to obtain the variables from option prices in Section 2.

A description of the empirical data used in our analysis is provided in Section 3. Regression tests and analysis are performed

in Section 4 while Section 5 develops robustness checks. Section 6 concludes the paper by providing some open questions.

2

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PRICING FORMULAS

The main purpose of this study is to analyze the variance and skew risk premiums for the volatility market with these

key variables computed using VIX call and put options. To this end, we will denote by CK(

)

tT,and PK()

tT,the European

call and put option prices at time

t

with maturity

T

and strike

K

on the VIX whose value at time

t

is

V

IX

t. It is often

more convenient to use the forward value of the VIX, we write Ft

T

,for the forward value at time

t

with maturity

T

that is

related to the spot value through the standard equality F=VIX

tt

t

,. We will also make use of

rF F=ln −ln

tT TT tT,,,

, the

log return of a position on the forward contract. The availability of these derivative products allows us to compute the

variance and skew risk premiums in a model‐free way as shown in the literature with the important contribution

provided by Kozhan et al. (2013) that we will closely follow (see also Neuberger, 2012).

Extracting distribution information from option prices, like higher moments, has a long history, let us quote without

pretending to be exhaustive the work of Carr and Madan (1998), and has found many applications; see among many

others the works of Bakshi et al. (2003) for individual options; Bakshi and Madan (2006) and Carr and Wu (2009)) for a

variance risk premium analysis of equity index options (options on S&P500, S&P100 and other major indexes as well as

equity); Byun and Kim (2013), Fleming (1998), Konstantinidi and Skiadopoulos (2016), and Neumann and

Skiadopoulos (2013) for forecasting aspects (using S&P500 options); Ammann and Buesser (2013) for variance risk

premium properties for the foreign exchange market; investors’risk aversion analysis as in Duan and Zhang (2014);

Kostakis, Panigirtzoglou, and Skiadopoulos (2011) (using S&P500 options); and variance risk premiums in commodity

markets in Prokopczuk, Symeonidis and Wese‐Simen (2017).

To the best of our knowledge, we are only aware of the work of Huang and Shaliastovich (2014) that exploits VIX

options to extract volatility higher moments (in their case the second moment, that is to say, the volatility of the VIX or

the volatility of volatility) and performs a joint analysis with the second moment extracted from S&P500 options (i.e.,

the square of the VIX) along with high frequency quantities such as realized volatility and bipower variation (these two

latter quantities allow the authors to isolate the role of jumps). As our work follows Kozhan et al. (2013), it differs from

Huang and Shaliastovich (2014) by focusing also on the skewness of the VIX distribution and this aspect is important as

it controls the shape of the VIX option smile and is also related to the “inverse”leverage effect, or positive skew, for the

volatility market.

1

What is more, it is really at the skewness level that the volatility index option market departs from

the equity index option market.

1

Regarding parametric approaches to VIX option pricing or volatility option pricing the literature is substantial, let us mention the works of Detemple andOsakwe (2000), Grünbichler and Longstaff

(1996), Lian and Zhu (2013), Park (2016), and Sepp (2008). For variance, skew and kurtosis swaps within the affine framework see Zhao, Zhang, and Chang (2013). For additional empirical works

using parametric models to analyze the VIX options market, see Branger, Kraftschik, and Völkert, (2016) and Branger, Hulsbusch, and Kraftschik (2017).

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