VIX futures pricing with conditional skewness

• Published date: 01 September 2018
• Date: 01 September 2018
• DOI: http://doi.org/10.1002/fut.21925
• Author: Xinglin Yang,Peng Wang

Received: 5 July 2017

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Revised: 22 March 2018

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Accepted: 1 April 2018

DOI: 10.1002/fut.21925

RESEARCH ARTICLE

VIX futures pricing with conditional skewness

Xinglin Yang

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Peng Wang

Institute of Chinese Financial Studies,

Southwestern University of Finance and

Economics, Chengdu, China

Correspondence

Xinglin Yang, Institute of Chinese

Financial Studies, Southwestern

University of Finance and Economics,

555 Liutai Avenue, 611130 Chengdu,

Sichuan, China.

Email: xinglinyang@126.com

Funding information

National Natural Science Foundation of

China, Grant/Award Number: 71473200

We develop a closed‐form VIX futures valuation formula based on the inverse

Gaussian GARCH process by Christoffersen et al. that combines conditional

skewness, conditional heteroskedasticity, and a leverage effect. The new model

outperforms the benchmark in fitting the S&P 500 returns and the VIX futures

prices. The fat‐tailed innovation underlying the model substantially reduced

pricing errors during the 2008 financial crisis. The in‐and out‐of‐sample pricing

performance indicates that the new model should be a default modeling choice

for pricing the medium‐and long‐term VIX futures.

KEYWORDS

closed‐form formula, conditional skewness, GARCH, pricing kernel, VIX futures

JEL CLASSIFICATION

G13

1

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INTRODUCTION

The VIX, introduced by the Chicago Board Options Exchange (CBOE) in 1993, is an important risk indicator. It is often

referred to as the “fear index”by leading financial publications and business news shows. The purpose of the VIX is to

measure S&P 500 volatility over the next month as implied by stock index option prices. The VIX has become a

particularly popular measure of the U.S. stock market volatility. CBOE introduced exchange‐traded VIX futures and

options as tradable assets in March 2004 and February 2006, respectively. In the decade since their launch, their

combined trading activity has grown markedly, now totaling approximately 800,000 contracts per day.

The fast development of the financial market has increased the importance of speculation and hedging against

unexpected changes in volatility. The crucial roles of VIX derivatives in such hedging have been studied widely in the

academic literature; see, for example, Grünbichler and Longstaff (1996), Dotsis, Psychoyios, and Skiadopoulos (2007), and

Mencía and Sentana (2013). To obtain protection from volatility risk, strategies for managing the risk should be built on

reliable VIX futures and option valuation models that describe the empirical properties of asset returns adequately.

Compared with the approaches of options valuation, the literature still offers relatively few pricing formulas of

futures valuation. However, several meaningful valuation models for VIX futures have been proposed. Zhang and Zhu

(2006) price VIX futures assuming stochastic instantaneous variance in the diffusion process of the S&P 500 returns.

Considering the importance of jump diffusion, Lin (2007) constructs a valuation model of VIX futures with

simultaneous jumps in the price and variance processes. Zhu and Lian (2012), combining stochastic volatility and

random jumps, derive a closed‐form and exact formula to evaluate VIX futures. To avoid pricing distortions during the

financial crisis, Mencía and Sentana (2013) find that a VIX process combining central tendency and stochastic volatility

has a better pricing performance for VIX futures and options.

The GARCH process can simultaneously capture the correlation of volatility with returns and the path dependence

in volatility, and it has a more parsimonious description in many situations (Bollerslev, 1986; Heston & Nandi, 2000).

Compared with continuous models, GARCH‐type models readily compute volatility through historical asset prices and

J Futures Markets. 2018;38:1126–1151.wileyonlinelibrary.com/journal/fut1126

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© 2018 Wiley Periodicals, Inc.

conveniently implement maximum likelihood estimation (MLE). Therefore, GARCH‐type models have been used

extensively for options valuation (Christoffersen, Feunou, Jacobs, & Meddahi, 2014; Christoffersen, Heston, & Jacobs,

2006; Dorion, 2016; Heston & Nandi, 2000). To the best of our knowledge, only Wang, Shen, Jiang, and Huang (2017)

have proposed a closed‐form pricing formula for VIX futures based on the Heston–Nandi (HN) GARCH process.

It is worth emphasizing the importance of capturing nonnormality and asymmetry in economics and finance, which

has been widely noted (Patton, 2004, 2006; Patton & Timmermann, 2007). However, under the local risk‐neutral

valuation relationship (RNVR) framework proposed by Duan (1995), the existing GARCH‐type models with Gaussian

innovations may not sufficiently allow for all the mass in the tails and the asymmetry in the asset return distribution.

Motivated by the stylized fact of conditional skewness in asset returns, we propose a closed‐form formula for pricing

VIX futures using the discrete‐time IG GARCH model in which the innovation follows an inverse Gaussian distribution

with a nonzero third moment. The IG GARCH process reinforces the pricing accuracy for the VIX futures through

capturing the conditional skewness of the S&P 500 returns. Unlike the approach that obtains risk‐neutral dynamics by

directly assuming the form of the state‐price density, our framework uses only the no‐arbitrage assumption and some

technical conditions on the investment strategies, without attempting to characterize the preferences underlying the

RNVR or fully describe the economic environment. Following the theoretical framework by Christoffersen, Elkamhi,

Feunou, and Jacobs (2010), first we propose a pricing kernel to calculate the equivalent martingale measure (EMM)

coefficient. Then, we derive the risk‐neutral dynamics of the inverse Gaussian process using an EMM coefficient that

fills the gap between the physical and the risk‐neutral measures.

We test the performance of the IG GARCH model and that of the most related benchmark, the HN GARCH model,

using the daily S&P 500 returns and the VIX futures data. First, we estimate the models by fitting the S&P 500 returns

using the maximum likelihood method. The empirical results show that the IG GARCH model outperforms the HN

GARCH model in fitting the returns, in forecasting 1‐day and 1‐week volatility, and in absorbing the heteroskedasticity

of returns. Second, we optimize the likelihood function defined on the VIX futures errors and analyze the out‐of‐sample

performance. We find that, overall, the IG GARCH model that incorporates conditional skewness has a better in‐and

out‐of‐sample VIX futures valuation performance than the HN GARCH model. When we further dissect the pricing

performance of the VIX futures across maturity, bias, and VIX level, the IG GARCH model outperforms the HN

GARCH model for small bias, for high VIX level, and particularly for medium‐and long‐term VIX futures contracts.

Using the futures contracts of 2017 as an out‐of‐sample experiment, the IG GARCH model has futures pricing errors

that are 19% below those of the benchmark model for medium‐and long‐term VIX futures contracts. The empirical

findings in the joint subsamples that lie at the intersection of bias, maturity, and VIX levels indicate that the IG GARCH

model is significantly superior to the HN GARCH model in all categories. Third, the fat‐tailed innovation of the IG

GARCH model offers an improvement of 11.78% on average on the extreme days of the 2008 financial crisis. Fourth, we

jointly fit the S&P 500 returns and the VIX futures. The joint results once again reconfirm that the IG GARCH model

with the nonnormal innovation reduces the pricing errors of the benchmark model with the normal innovation.

The remainder of the paper is organized as follows: Section 2 introduces the inverse Gaussian GARCH process and

the nested HN GARCH process; Section 3 provides the estimation results on daily returns; Section 4 develops the

theoretical framework of risk neutralization and the closed‐form formula for VIX futures valuation; Section 5 analyzes

the in‐and out‐of‐sample pricing performance with the volatility index and the VIX futures data; Section 6 estimates the

models jointly on returns and futures; and Section 7 concludes the paper.

2

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THE MODEL

2.1

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The return process

The daily returns Rt+

1

and the conditional variance

h

t+

1

specified by the dynamic inverse Gaussian GARCH model are:

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è

ç

ç

ç

ö

ø

÷

÷

÷

÷

RS

Srζh ηylog = + +

,

tt

t

tt

+1 +1 +1 +1 (1)

hwbhcy

ah

y

=+ + +

,

tt

t

t

t

+1

2

(2)

YANG AND WANG

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