Resumen: This paper features an analysis of the relationship between the S&P500 Index and the VIX using
daily data obtained from both the CBOE website and SIRCA (The Securities Industry Research
Centre of the Asia Pacic). We explore the relationship between the S&P500 daily continuously
compounded return series and a similar series for the VIX in terms of a long sample drawn from the
CBOE running from 1990 to mid 2011
and a set of returns from SIRCA's TRTH datasets running
from March 2005 to-date. We divide this shorter sample, which captures the behaviour of the new
VIX, introduced in 2003, into four roughly equivalent sub-samples which permit the exploration of
the impact of the Global Financial Crisis. We apply to our data sets a series of non-parametric
based tests utilising entropy based metrics. These suggest that the PDFs and CDFs of these two
return distributions change shape in various subsample periods. The entropy and MI statistics
suggest that the degree of uncertainty attached to these distributions changes through time and
using the S&P500 return as the dependent variable, that the amount of information obtained from
the VIX also changes with time and reaches a relative maximum in the most recent period from
2011 to 2012. The entropy based non-parametric tests of the equivalence of the two distributions
and their symmetry all strongly reject their respective nulls. The results suggest that parametric
techniques do not adequately capture the complexities displayed in the behaviour of these series.
This has practical implications for hedging utilising derivatives written on the VIX, which will be
the focus of a subsequent study.
Resumen: This paper examines the asymmetric relationship between price and implied volatility and the associated extreme quantile dependence using a linear and non-linear quantile regression approach. Our goal is to demonstrate that the relationship between the volatility and market return, as quantied by Ordinary Least Square (OLS) regression, is not uniform across the distribution of the volatility-price re-
turn
pairs using quantile regressions. We examine the bivariate relationships of six volatility-return pairs, namely: CBOE VIX and S&P 500, FTSE 100 Volatility and
FTSE 100, NASDAQ 100 Volatility (VXN) and NASDAQ, DAX Volatility (VDAX) and DAX 30, CAC Volatility (VCAC) and CAC 40, and STOXX Volatility (VS-TOXX) and STOXX. The assumption of a normal distribution in the return series
is not appropriate when the distribution is skewed, and hence OLS may not capture a complete picture of the relationship. Quantile regression, on the other hand, can
be set up with various loss functions, both parametric and non-parametric (linear case) and can be evaluated with skewed marginal-based copulas (for the non-linear case), which is helpful in evaluating the non-normal and on-linear nature of the relationship between price and volatility. In the empirical analysis we compare the results from linear quantile regression (LQR) and copula based non-linear quantile regression known as copula quantile regression (CQR). The discussion of the prop-erties of the volatility series and empirical ndings in this paper have signicance for portfolio optimization, hedging strategies, trading strategies and risk management, in general.
