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

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  • David E. Allen

    ()
    (Centre for Applied Financial Studies, University of South Australia, and School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia)

  • Michael McAleer

    (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)

  • Robert Powell

    ()
    (School of Business, Edith Cowan University, Western Australia 6027, Australia)

  • Abhay K. Singh

    ()
    (School of Business, Edith Cowan University, Western Australia 6027, Australia)

Abstract

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.

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Bibliographic Info

Article provided by MDPI, Open Access Journal in its journal Journal of Risk and Financial Management.

Volume (Year): 6 (2013)
Issue (Month): 1 (October)
Pages: 6-30

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Handle: RePEc:gam:jjrfmx:v:6:y:2013:i:1:p:6-30:d:29740

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Keywords: S&P 500; VIX; entropy; non-parametric estimation; quantile regressions;

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  1. Caporin, M. & McAleer, M.J., 2010. "Do We Really Need Both BEKK and DCC? A Tale of Two Multivariate GARCH Models," Econometric Institute Research Papers EI 2010-13, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
  2. Chia-Lin Chang & Juan-Ángel Jiménez-Martín & Michael McAleer & Teodosio Pérez Amaral, 2011. "The Rise and Fall of S&P500 Variance Futures," Working Papers in Economics 11/32, University of Canterbury, Department of Economics and Finance.
  3. Tristen Hayfield & Jeffrey S. Racine, . "Nonparametric Econometrics: The np Package," Journal of Statistical Software, American Statistical Association, vol. 27(i05).
  4. Isao Ishida & Michael McAleer & Kosuke Oya, 2011. "Estimating the Leverage Parameter of Continuous-time Stochastic Volatility Models Using High Frequency S&P 500 and VIX," KIER Working Papers 759, Kyoto University, Institute of Economic Research.
  5. C. W. Granger & E. Maasoumi & J. Racine, 2004. "A Dependence Metric for Possibly Nonlinear Processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 25(5), pages 649-669, 09.
  6. Chatayan Wiphatthanananthakul & Michael McAleer, 2009. "A Simple Expected Volatility (SEV) Index: Application to SET50 Index Options," CIRJE F-Series CIRJE-F-672, CIRJE, Faculty of Economics, University of Tokyo.
  7. Brenner, Menachem & Ou, Ernest Y. & Zhang, Jin E., 2006. "Hedging volatility risk," Journal of Banking & Finance, Elsevier, vol. 30(3), pages 811-821, March.
  8. Engle, Robert F & Granger, Clive W J, 1987. "Co-integration and Error Correction: Representation, Estimation, and Testing," Econometrica, Econometric Society, vol. 55(2), pages 251-76, March.
  9. Koenker, Roger W & Bassett, Gilbert, Jr, 1978. "Regression Quantiles," Econometrica, Econometric Society, vol. 46(1), pages 33-50, January.
  10. Racine, Jeffrey S., 2008. "Nonparametric Econometrics: A Primer," Foundations and Trends(R) in Econometrics, now publishers, vol. 3(1), pages 1-88, March.
  11. Maasoumi, Esfandiar & Racine, Jeff, 2002. "Entropy and predictability of stock market returns," Journal of Econometrics, Elsevier, vol. 107(1-2), pages 291-312, March.
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