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

• Author: Jean‐Guy Simonato,Simon Lalancette
• Published date: 01 March 2017
• DOI: http://doi.org/10.1111/eufm.12096
• Date: 01 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,

Canada

E-mails: simon.lalancette@hec.ca, jean-guy.simonato@hec.ca

ABSTRACT

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; Demeterfiet 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 classified into five 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, 325–354

doi: 10.1111/eufm.12096

© 2016 John Wiley & Sons, Ltd.

survey). A first group of articles, based on Whaley (2000) and others, investigates the

financial 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 specifications 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 fifth 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 fifth 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 specifications of the conditional volatility attempting

to successfully replicate the attributes of the VIX must have enough flexibility to capture

the time series dynamics of these two elements. Perhaps as a reflection 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,

1

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, finds

that a square root specification for the stochastic variance is incapable of generating

realistic behaviour while the data are better represented by a non-affine 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 fit

of a term structure of VIX and that the inclusion of jumps is less important than allowing

for non-affine 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 specifications to accomplish this task, although their study

exclusively resorts to GARCH specifications with Gaussian innovations. Specifically,

they find that such GARCH processes, when estimated from stock return data only, tend

1

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