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A Non-Parametric and Entropy Based Analysis of the Relationship between the VIX and S&P 500

Centre for Applied Financial Studies, University of South Australia, and School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
Department of Quantitative Finance, National Tsing Hua University, Taiwan
Econometric Institute, Erasmus School of Economics, Erasmus University Rotterdam, 3000 DR Rotterdam, The Netherlands
Tinbergen Institute, 1082 MS Amsterdam, The Netherlands
School of Business, Edith Cowan University, Western Australia 6027, Australia
Author to whom correspondence should be addressed.
J. Risk Financial Manag. 2013, 6(1), 6-30;
Received: 3 October 2013 / Revised: 7 October 2013 / Accepted: 8 October 2013 / Published: 21 October 2013
(This article belongs to the Collection Feature Papers of JRFM)
This paper features an analysis of the relationship between the S&P 500 Index and the VIX using daily data obtained from the CBOE website and SIRCA (The Securities Industry Research Centre of the Asia Pacific). We explore the relationship between the S&P 500 daily return series and a similar series for the VIX in terms of a long sample drawn from the CBOE from 1990 to mid 2011 and a set of returns from SIRCA’s TRTH datasets from March 2005 to-date. This shorter sample, which captures the behavior of the new VIX, introduced in 2003, is divided into four sub-samples which permit the exploration of the impact of the Global Financial Crisis. We apply a series of non-parametric based tests utilizing 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&P 500 return as the dependent variable, that the amount of information obtained from the VIX 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 behavior of these series. This has practical implications for hedging utilizing derivatives written on the VIX. View Full-Text
Keywords: S&P 500; VIX; entropy; non-parametric estimation; quantile regressions S&P 500; VIX; entropy; non-parametric estimation; quantile regressions
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Allen, D.E.; McAleer, M.; Powell, R.; Singh, A.K. A Non-Parametric and Entropy Based Analysis of the Relationship between the VIX and S&P 500. J. Risk Financial Manag. 2013, 6, 6-30.

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