Expanding the Explanations for the Return–Volatility Relation

• Author: Robert T. Daigler,A. M. Parhizgari,Bakhtear Talukdar
• Date: 01 July 2017
• DOI: http://doi.org/10.1002/fut.21827
• Published date: 01 July 2017

Expanding the Explanations for the

Return–Volatility Relation

Bakhtear Talukdar, Robert T. Daigler, and A. M. Parhizgari*

We examine the return–volatility relation using three of the CBOE’s recent indices. We

employ more robust estimation techniques that account for the asymmetric relation between

return and volatility. Our findings indicate that contributions of these indices to R

2

are

surprisingly large (7.45–35.54%) when the observations are divided into deciles groups. The

results further indicate that behavioral theories explain the return–volatility relation better

than the fundamental theories. We use daily and high frequency data. The results are

consistent across all data, though the high frequency data seem to provide more support for

the behavioral theories. © 2016 Wiley Periodicals, Inc. Jrl Fut Mark 37:689–716, 2017

1. INTRODUCTION

The Chicago Board Options Exchange (CBOE) has recently released a few new indices that

are intended to strengthen the return–volatility relation by incorporating different aspects of

investors’behaviors as well as the underlying relations of the indices constituents. Notable in

this group of new releases are implied correlation index (ICI), skewness index (SKEW), and

volatility of volatility index (VVIX). The ICI considers the underlying relation of the index

constituent securities, SKEW incorporates the shape of the volatility smiles, and VVIX

measures the expected volatility of the VIX nearby options.

Research on the relation between volatility and return abounds. The existence of a

negative relation between these two measures is already well established (e.g., Badshah,

2013; Dennis, Mayhew, & Stivers, 2006; Fleming, Ostdiek, & Whaley, 1995; Frijns, Tallau,

& Tourani-Rad, 2010; Giot, 2005; Hibbert, Daigler, & Dupoyet, 2008; Low, 2004;

Whaley, 2000). Although the inception of the CBOE’s implied volatility index (VIX), this

domain of research has stayed relatively strong and has expanded into detailed behavioral

aspects of return–volatility relations. One such expansion includes the use of VIX to study

the asymmetric volatility phenomenon (Bekaert & Wu, 2000; Wu, 2001) in relation with

systematic and idiosyncratic risk (Dennis et al., 2006). Another expansion is its employment

Bakhtear Talukdar is Assistant Professor at Department of Finance and Business Law, University of

Wisconsin-Whitewater, Whitewater, Wisconsin. Robert T. Daigler is Knight Ridder Research Professor of

Finance and A. M. Parhizgari is Professor of Finance and International Business at Department of Finance,

The Chapman Graduate School of Business, Florida International University, Miami, Florida. We would like

to express our deep appreciation to the Editor and the anonymous reviewer for their constructive comments

and guidance.

JEL Classification: C12, C51, C58, D82, G12

*Correspondence author, Department of Finance, The Chapman Graduate School of Business, Florida

International University, Miami, FL 33199. Tel: 305-348-3326, Fax: 305-348-4245, e-mail: parhiz@fiu.edu

Received March 2016; Accepted October 2016

The Journal of Futures Markets, Vol. 37, No. 7, 689–716 (2017)

© 2016 Wiley Periodicals, Inc.

Published online 6 December 2016 in Wiley Online Library (wileyonlinelibrary.com).

DOI: 10.1002/fut.21827

in the construction of an investor risk aversion index that can predict future stock market

returns (Bollerslev, Gibson, & Zhou, 2011). The continuing driving interest in VIX arises

from its useful attributes. It provides an estimate of the forward looking volatility based

on options and represents the future volatility of the underlying assets over the life of the

options (Frijns et al., 2010). These features are of high interest to market participants and

academicians who, in turn, will redirect VIX to new venues and to new research in the years

to come.

In line with the above trend, we investigate whether the new CBOE indices noted above

contribute to the current literature in explaining the relation between return and volatility.

The measures of return and volatility that are employed are, respectively, SPX return and

30- and 9-day VIX indices. We examine the contributions of the new indices to the relation

between these measures. In addition, we consider and test for asymmetry through analyzing

the effects of positive and negative SPX returns on the changes in implied volatility. Several

prior theoretical explanations that are forwarded in this regard are reexamined under the

presence of the new indices. We employ daily data and supplement it throughout with 5- and

30-minute intraday data to analyze some of the behavioral perspectives of the investors in

the market.

At the empirical level, we emp loy a heterogeneity-consis tent quantile regression

model (QRM) along with GMM and decile-specified OLS. Although GMM is extensively

employed in various prior appli cations, the use of QRM is relat ively far less. Recently,

Badshah (2013) and Kaurijo ki, Nikkinen, and €

Aij€o (2014) have used QRM in analy zing

return volatility asymmet ries in the equity and curre ncy markets, respectively . The main

advantage of quantile regr ession lies in its design tha t allows for varied degree o f

asymmetry across the quant iles. When asymmetry is pre sent, it is shown that OLS

underestimates the true pa rameters (Badshah, 2013) . Our application of GMM is t o

provide a robust base scenari o for comparison. GMM is far superior to OLS because it i s a

distribution-free estima tion process and accounts fo r heteroskedasticity that a rise from

autocorrelation. We use OLS in ra re cases to identify the individual or group contributions

of the CBOE indices.

1

We contribute to the literature in several fronts. First, we conduct a series of statistical

tests that lead us to consider and recommend implied correlation measures (ICI), SKEW, and

volatility of volatility (VVIX) as new explanatory factors in exploring the return–implied

volatility relation. Second, we show that estimation procedures that are mean-based such as

OLS or GMM are not the best methods when heterogeneity is present in the data.

Heterogeneity that arises from the presence of diverse investor groups in the financial

markets causes the groups to respond differently to similar shocks in the markets. For

instance, it is observed that for positive SPX returns, mean-based models severely

underestimate (overestimate) the coefficients beyond (before) the median. The reverse is

true for negative SPX returns. Third, we show that the contribution of an individual CBOE

index may not appear large when the entire sample is employed. In contrast, when the

observations are grouped into deciles, the contributions of the indices are more pronounced,

particularly at the extreme levels, that is, in the lowest and the highest deciles. Fourth, we

argue that when autocorrelation is present in the data, which is true in the case of VIX,

2

OLS

produces biased estimators since one of its principal assumptions, that is, homoscedasticity,

1

This is measured by R

2

.Ceteris paribus, the R

2

s from OLS and GMM stay the same because the violation of

homoscedasticity assumption in the OLS affects (penalizes) only the standard errors of the estimated coefficients.

Using OLS, Daigler et al. (2012); Hibbert et al. (2008); and Padungsaksawasdi and Daigler (2014) analyze the

asymmetric return–volatility relation by grouping the observations.

2

It is shown that VIX possesses a long memory. See, Bandi and Perron (2006); Corsi (2009); and Fernandes et al.

(2014).

690 Talukdar, Daigler, and Parhizgari