The Role of the Conditional Skewness and Kurtosis in VIX Index Valuation

Published date01 March 2017
Date01 March 2017
The Role of the Conditional Skewness
and Kurtosis in VIX Index Valuation
Simon Lalancette and Jean-Guy Simonato
Department of Finance, HEC Montr
eal, 3000 C^
ote-Sainte-Catherine, Montr
eal, Qu
ebec, H3T 2A7,
The CBOE VIX index is a widely recognised benchmark measure of expected stock
market volatility. As shown in the literature, probability distributions other than
Gaussian are key features required to describe the dynamics of the S&P 500, the
variable that ultimately determines the VIX index level. As such, it is important to
assess if deviations from the Gaussian distribution have important impacts on the
VIX index level. We examine herein how a model articulated over a time-varying
non-Gaussian distribution with conditional skewness and kurtosis can contribute
to the overall explanation of the VIX dynamics.
Keywords: VIX, GARCH, skewness, kurtosis, risk-neutral valuation
JEL classification: C58, G1
1. Introduction
Stock options is an important category of assets that embodies forecasts of the future
dynamics of the underlying asset. As these forecasts are tinted by the preferences of
market operators, option prices possess a unique forward-looking nature. This is why the
CBOE VIX, a volatility index built from the S&P 500 option pricing data, constitutes a
critical piece of information to market participants. The computation of the VIX index
performed by the CBOE does not rely on any particular option-pricing model. It is based
on the model-free methodology typically used in the variance swap literature (see
Britten-Jones and Neuberger, 2000; Carr and Madan, 1998; Demeteret al., 1999;
amongst others). In this paper, we examine how GARCH models with non-Gaussian
distributions characterised by time-varying skewness and kurtosis can help explain the
level of the VIX index and its related variance premium.
There is an abundance of literature about the VIX index and its uses. This literature can
be roughly classied into ve categories (see Gonzalez-Perez, 2015 for an excellent
The authors thank the two anonymous referees and the Editor for many helpful and
constructive commen ts that greatly improve d earlier versions of th is manuscript.
Correspondence: Simon Lalancette.
European Financial Management, Vol. 23, No. 2, 2017, 325354
doi: 10.1111/eufm.12096
© 2016 John Wiley & Sons, Ltd.
survey). A rst group of articles, based on Whaley (2000) and others, investigates the
nancial leverage effect i.e., the negative correlation between stock returns and
volatility. A second group of studies investigates the ability of the VIX to forecast the
future volatility as in Jiang and Tian (2005), and Bollerslev et al. (2009). A third group of
papers examines the uses of the VIX index as an instrumental variable to study several
empirical issues (for instance, Collin-Dufresne and Goldstein, 2001). A fourth group of
articles investigates the capacity of different dynamic specications of the S&P 500
return and its volatility to explain the VIX level or prices of derivative securities written
on the VIX as in Lin and Chang (2010). Finally, a fth group takes advantage of the VIX
to explore issues about the stock market variance premium as in Bollerslev et al. (2009).
Our study, as we will detail below, belongs to the fourth and fth categories.
As explained in Beakert et al. (2013), the VIX index embodies two components: the
conditional expected variance of the S&P 500 returns under the physical measure, and a
variance premium. Thus, selected specications of the conditional volatility attempting
to successfully replicate the attributes of the VIX must have enough exibility to capture
the time series dynamics of these two elements. Perhaps as a reection in the growth and
importance of the derivative literature, attempts to model the dynamics of the VIX are
typically performed using a continuous time setting. While some studies treat the VIX as
a stand-alone asset,
others like Duan and Yeh (2010), Jones (2003), Kaeck and
Alexander (2012) and Lin and Chang (2010) consider the VIX as the result of a risk-
neutral process obtained from the underlying S&P 500 index returns. Using VIX and
S&P 500 data, these studies attempt to identify the features of the S&P 500 dynamics
capable of generating realistic volatility index levels. Jones (2003), for instance, nds
that a square root specication for the stochastic variance is incapable of generating
realistic behaviour while the data are better represented by a non-afne stochastic
variance model in the CEV class. Duan and Yeh (2010) underline the need to include
jumps in diffusions attempting to explain volatility indexes. More recently, Kaeck and
Alexander (2012) highlight that a stochastic long-term volatility factor improves the t
of a term structure of VIX and that the inclusion of jumps is less important than allowing
for non-afne dynamics.
While most of the literature focuses on continuous time processes, it seems that
relatively few authors have used the GARCH framework to address issues connected to
the VIX dynamics. Nevertheless, as advocated by Christoffersen et al. (2012), the use of
the GARCH framework offers many advantages over continuous-time processes to
explore issues similar to those mentioned herein. For example, the GARCH framework
offers accurate numerical approximations to continuous time models while estimations
are computationally fast. Furthermore, Fleming and Kirby (2003) contend that GARCH
models constitute accurate approximations of continuous time processes with stochastic
volatility while generating similar predictions.
Consistent with this view, Hao and Zhang (2013) examine the capacity of GARCH
models to explain the VIX level and the variance premium. They conclude in the inability
of their different GARCH specications to accomplish this task, although their study
exclusively resorts to GARCH specications with Gaussian innovations. Specically,
they nd that such GARCH processes, when estimated from stock return data only, tend
See for example Becker et al. (2009), Dotsis et al. (2007), Kaeck and Alexander (2013),
Psychoyios and Skiadopoulos (2006), Zhang and Zhu (2006) and Zhu and Zhang (2007).
© 2016 John Wiley & Sons, Ltd.
326 Simon Lalancette and Jean-Guy Simonato

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