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

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Research Methods and Data

**Table 1.**Summary statistics, S&P 500, VIX, Daily S&P 500 returns and daily VIX returns, Jan 1990 June 2011.

S&P 500 | S&P 500 Daily Returns | VIX | VIX Daily Returns | |
---|---|---|---|---|

Min | 295.5 | −0.0947 | 9.31 | −0.3506 |

Median | 1047 | 0.0005 | 18.88 | −0.0031 |

Mean | 939 | 0.00024 | 20.34 | 0.00043 |

Maximum | 1560 | 0.1096 | 80.86 | 0.496 |

Variance | 0.0001356015 | 0.0035794839 |

**Figure 1.**Time series behaviour of the S&P 500 and VIX series from Jan 1990 to June 2011. (

**a**) S&P-500; (

**b**) Vix.

**Figure 2.**Time series behaviour of (

**a**) S&P 500 and (

**b**) VIX logarithmic return series from Jan 1990 to June 2011.

**Table 2.**OLS regression of daily continuously compounded S&P 500 returns on daily continuously compounded VIX returns.

Intercept | Slope | ||
---|---|---|---|

Coefficient | 0.000238 | −0.13527 | |

Standard Error | 0.0001138 | 0.001902 | |

t value | 2.094 * | −71.006 ** | |

Adjusted ${R}^{2}$ | 0.4829 | ||

F Value | 5056 ** |

**Figure 3.**OLS regression of daily continuously compounded S&P 500 returns on daily continuously compounded VIX returns with fitted line.

**Table 3.**Cramer–von Mises tests of the normality of the return series for the S&P 500 and the VIX for the period 1990 to June 2011.

S&P 500 Returns | VIX Returns | |
---|---|---|

Cramer-von Mise statistic | 15.0744 | 5.6701 |

Probability value | 0.00000 | 0.00000 |

#### 2.1. Entropy-Based Measures

#### 2.1.1. Maximum Likelihood Estimation

#### 2.1.2. Miller–Madow Estimator

#### 2.1.3. Bayesian Estimators

#### 2.1.4. The Chao–Shen Estimator

#### 2.1.5. Mutual Information

#### 2.2. Some Preliminary Results

**Table 4.**Common choices for the parameters of the Dirichlet prior in the Bayesian estimators of cell frequencies, and corresponding entropy estimators.

${a}_{k}$ | Cell Frequency Prior | Entropy Estimator |
---|---|---|

0 | no prior | maximum likelihood |

1/2 | Jeffreys prior ([38]) | [39] |

1 | Bayes–Laplace uniform prior | [40] |

1/p | Perks prior ([41]) | [42] |

$\sqrt{n}/p$ | minmax prior ([43]) |

S&P500 returns | VIX returns | |
---|---|---|

Maximum Likelihood Estimate | 2.1434 | 3.8269 |

Miller–Madow Estimator | 2.1462 | 3.8367 |

Jeffrey’s prior | 2.1598 | 3.8846 |

Bayes–Laplace | 2.1747 | 3.9322 |

SG | 2.1444 | 3.8279 |

Minimax | 2.1983 | 3.8780 |

Chao–Shen | 2.1494 | 3.8424 |

Mutual Information $MI$ | ||

$MI$ Dirichlet ( a = 0) | 0.3535395 | |

$MI$ Dirichlet ( a = 1/2) | 0.3465671 | |

$MI$ Empirical ($ML)$ | 0.3535395 |

#### 2.3. Non-Parametric Estimation

#### 2.3.1. Kernel Estimation of a Conditional Quantile

**Figure 5.**Nonparametric conditional PDF and CDF estimates of the joint distribution of S&P500 returns and VIX returns 1990-June 2011. (

**a**) PDF; (

**b**) CDF.

#### 2.3.2. Testing the Equality of Univariate Densities

## 3. Main Results

#### 3.1. Entropy Metrics

S&P500 2005–2006 | S&P500 2007–2008 | S&P500 2009–2010 | S&P500 2011–2012 | |
---|---|---|---|---|

Maximum Likelihood Estimate | 1.6518 | 2.4165 | 2.2611 | 2.1859 |

Miller–Madow Estimator | 1.6596 | 2.4454 | 2.2790 | 2.2211 |

Jeffrey’s prior | 1.6610 | 2.5364 | 2.2956 | 2.2516 |

Bayes–Laplace | 1.6698 | 2.6299 | 2.3262 | 2.3051 |

SG | 1.6542 | 2.4243 | 2.2649 | 2.1935 |

Minimax | 1.6967 | 2.545 | 2.331 | 2.2920 |

Chao–Shen | 1.6527 | 2.4781 | 2.2809 | 2.2282 |

Mutual Information $MI$ S&P500 and VIX | ||||

$MI$ Dirichlet ( a = 0) | 0.09489 | 0.1941 | 0.1927 | 0.2055 |

$MI$ Dirichlet ( a = 1/2) | 0.08485 | 0.1693 | 0.1702 | 0.1746 |

$MI$ Empirical ($ML)$ | 0.09489 | 0.1941 | 0.1927 | 0.2055 |

VIX 2005–2006 | VIX 2007–2008 | VIX 2009–2010 | VIX 2011–2012 | |
---|---|---|---|---|

Maximum Likelihood Estimate | 3.6441 | 4.0114 | 3.7744 | 3.8723 |

Miller–Madow Estimator | 3.7045 | 4.0911 | 3.8421 | 3.9669 |

Jeffrey’s prior | 3.9220 | 5.3321 | 4.0743 | 4.2843 |

Bayes–Laplace | 4.0811 | 5.7334 | 4.2444 | 4.4751 |

SG | 3.6541 | 4.0278 | 3.7835 | 3.8861 |

Minimax | 3.7787 | 4.2524 | 3.9054 | 4.0306 |

Chao–Shen | 3.7086 | 4.1006 | 3.8539 | 3.9573 |

**Figure 6.**Conditional Density Plots PDF and CDF for S&P500 and VIX returns 2005–2006 for our sample intervals. (

**a**) PDF 2005–2006; (

**b**) CDF 2005–2006.

#### 3.2. Non-Parametric Conditional PDF and CDF Estimation

**Figure 7.**Conditional Density Plots PDF and CDF for S&P500 and VIX returns 2005-2006 for our sample intervals. (

**a**) PDF 2007–2008; (

**b**) CDF 2007–2008.

**Figure 8.**Conditional Density Plots PDF and CDF for S&P500 and VIX returns 2009-2010 for our sample interval. (

**a**) PDF 2009–2010; (

**b**) CDF 2009–2010.

**Figure 9.**Conditional Density Plots PDF and CDF for S&P500 and VIX returns 2011-2012 for our sample interval. (

**a**) PDF 2011–2012; (

**b**) CDF 2011–2012.

#### 3.3. Quantile Regression Analysis of Hedge Ratios

**Figure 10.**Quantile regression slope coefficients of S&P 500 daily returns for sample sub-periods regressed in daily continuously compounded VIX returns, sub-samples 2005–2006 and 2007–2008. (

**a**) Quantile regression slope coefficients by decile 2005–2006 with error bands; (

**b**) Quantile regression slope coefficients by decile 2007–2008 with error bands.

**Figure 11.**Quantile regression slope coefficients of S&P500 daily returns for sample sub-periods regressed in daily continuously compounded VIX returns, sub-samples 2009–2010 and 2011–2012. (

**a**) Quantile regression slope coefficients by decile 2009–2010 with error bands; (

**b**) Quantile regression slope coefficients by decile 2011–2012 with error bands.

#### 3.4. Non-Parametric Tests of Density Equalities

**Table 8.**Entropy density equality tests for sub-samples 2005–2006, 2007–2008, 2009–2010, and 2011–2012 for daily continuously compounded S&P 500 returns and VIX returns.

2005–2006 | 2007–2008 | 2009–2010 | 2011–2012 | |
---|---|---|---|---|

Consistent Univariate Entropy Density Equality Test | ||||

Test Statistic ‘Srho’ | 0.4660 | 0.3017 | 0.3219 | 0.3817 |

Probability | 2.22e-16 *** | 2.22e-16 *** | 2.22e-16 *** | 2.22e-16 *** |

**Table 9.**Entropy density asymmetry tests sub-samples 2005–2006, 2007–2008, 2009–2010, and 2011–2012 for daily continuously compounded S&P 500 returns and VIX returns.

2005–2006 | 2007–2008 | 2009–2010 | 2011–2012 | |
---|---|---|---|---|

Consistent entropy asymmetry test | ||||

S&P 500 Test Statistic ‘Srho’ | 0.00419 | 0.034547 | 0.01239 | 0.01904 |

Probability | 0.52525 | 0.0 *** | 0.19191 | 0.09090 |

VIX Test Statistic ‘Srho’ | 0.00923 | 0.01278 | 0.02485 | 0.03018 |

Probability | 0.1818 | 0.15151 | 0.0 *** | 0.0 *** |

## 4. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

- K. Demeterfi, E. Derman, M. Kamal, and J. Zou. “More than you ever wanted to know about volatility swaps.” Quantative Strategies Research Notes. New York, NY, USA: Goldman Sachs, 1999. available online at http://www.emanuelderman.com/writing/entry/more-than-you-ever-wanted-to-know-about-volatility-swaps-the-journal-of-der (accessed on 18 October 2013).
- C.-L. Chang, J.-A. Jiminez-Martin, M. McAleer, and T. Perez-Amaral. “The rise and fall of S&P500 variance futures.” North Am. J. Econ. Financ. 25 (2013): 151–167. [Google Scholar]
- A. Sepp. “VIX Option Pricing in a Jump-Diffusion Model.” Risk Magazine, April 2008, 84–89. [Google Scholar]
- M. Caporin, and M. McAleer. “Do we really need both BEKK and DCC? A tale of two multivariate GARCH models.” J. Econ. Surv. 26 (2012): 736–751. [Google Scholar] [CrossRef]
- M. Brenner, E.Y. Ou, and J.E. Zhang. “Hedging volatility risk.” J. Bank. Financ. 30 (2006): 811–821. [Google Scholar] [CrossRef]
- B. Huskaj. A value-at-risk analysis of VIX futures: Long memory, heavy tails, and asymmetry, 2009. Working Paper; Rochester, NY, USA: Social Science Research Network, 2009. [Google Scholar] [CrossRef]
- M. McAleer, and C. Wiphatthanananthakul. “A simple expected volatility (SEV) index: Application to SET50 index options.” Math. Comput. Simul. 80 (2010): 2079–2090. [Google Scholar] [CrossRef]
- I. Ishida, M. McAleer, and K. Oya. “Estimating the leverage parameter of continuous-time stochastic volatility models using high frequency S&P500 and VIX.” Manag. Financ. 37 (2011): 1048–1067. [Google Scholar]
- C. Alexander, and D. Korovilas. The Hazards of Volatility Diversification, the University of Reading. ICMA Centre Discussion Papers in Finance DP2011-04; Reading, UK: ICMA, 2011. [Google Scholar]
- B. Lui, and S. Dash. “Volatility ETFs and ETNs.” J. Trading 7 (2012): 43–48. [Google Scholar]
- Available online at: http://www.cboe.com/micro/VIX/vixintro.aspx (accessed on 1 October 2013).
- L. Edrington. “The hedging performance of the new futures markets.” J. Financ. 34 (1979): 157–170. [Google Scholar] [CrossRef]
- R.F. Engle, and C.W.J. Granger. “Co-integration and error correction: representation, estimation and testing.” Econometrica 55 (1987): 251–276. [Google Scholar] [CrossRef]
- C. Alexander. “Optimal hedging using cointegration.” Philos. Trans. R. Soc. Lond. A 357 (1999): 2039–2058. [Google Scholar] [CrossRef]
- R.E. Whaley. “Understanding the VIX.” J. Portf. Manag. 35 (2009): 98–105. [Google Scholar] [CrossRef]
- H.C. Thode Jr. Testing for Normality. New York, NY, USA: Marcel Dekker, 2002. [Google Scholar]
- C.E. Shannon. “A mathematical theory of communication.” Bell Syst. Tech. J. 27 (1948): 623–656. [Google Scholar] [CrossRef]
- National Science Foundation. Report of the National Science Foundation Workshop on Information Theory and Computer Science Interface, Workshop, Chicago, IL, USA, 17–18 October 2003.
- A.N. Kolmogorov. “Three approaches to the quantitative definition of information.” Probl. Inf. Transm. 1 (1965): 1–7. [Google Scholar] [CrossRef]
- G.J. Chaitin. “On the simplicity and speed of programs for computing infinite sets of natural numbers.” J. ACM 16 (1969): 407–422. [Google Scholar] [CrossRef]
- R. Solomonoff. A Preliminary Report on a General Theory of Inductive Inference. Report V-131; Cambridge, MA, USA: Zator Co., 1960. [Google Scholar]
- E.T. Jaynes. Probability Theory: The Logic of Science. Cambridge, UK: Cambridge University Press, 2003, ISBN 0-521- 59271-2. [Google Scholar]
- A. Golan. “Information and entropy econometrics—Editor’s view.” J. Econom. 107 (2002): 1–15. [Google Scholar] [CrossRef]
- C. Granger, E. Maasoumi, and J. Racine. “A dependence metric for possibly nonlinear time series.” J. Time Ser. Anal. 25 (2004): 649–669. [Google Scholar] [CrossRef]
- S. Pincus. “Approximate entropy as an irregularity measure for financial data.” Econom. Rev. 27 (2008): 329–362. [Google Scholar] [CrossRef]
- A.K. Bera, and S.Y. Park. “Optimal portfolio diversification using the maximum entropy principle.” Econom. Rev. 27 (2008): 484–512. [Google Scholar] [CrossRef]
- C.A. Sims. Rational Inattention: A Research Agenda. Discussion Paper Series 1: Economic Studies No 34/2005. Frankfurt-am-Main, Germany: Deutsche Bundesbank, 2005. [Google Scholar]
- E. Maasoumi. “A compendium to information theory in economics and econometrics.” Econom. Rev. 12 (1993): 137–181. [Google Scholar] [CrossRef]
- E. Soofi. “Information Theoretic Regression Methods.” In Advances in Econometrics - Applying Maximum Entropy to Econometric Problems. Edited by T. Fomby and R. Carter Hill. London, UK: Jai Press Inc., 1997, Volume 12. [Google Scholar]
- N. Ebrahimi, E. Maasoumi, and E. Soofi. “Ordering univariate distributions by entropy and variance.” J. Econom. 90 (1999): 317–336. [Google Scholar] [CrossRef]
- E. Maasoumi, and J.S. Racine. “Entropy and predictability of stock market returns.” J. Econom. 107 (2002): 291–312. [Google Scholar] [CrossRef]
- J. Hausser, and K. Strimmer. “Entropy inference and the James-Stein estimator, with application to nonlinear gene association networks.” J. Mach. Learn. Res. 10 (2009): 1469–1484. [Google Scholar]
- G.A. Miller. “Note on the Bias of Information Estimates.” In Information Theory in Psychology II-B. Edited by H. Quastler. Glencoe, IL, USA: Free Press, 1955, pp. 95–100. [Google Scholar]
- A. Chao, and T.J. Shen. “Nonparametric estimation of Shannon’s index of diversity when there are unseen species.” Environ. Ecol. Stat. 10 (2003): 429–443. [Google Scholar] [CrossRef]
- D.G. Horvitz, and D.J. Thompson. “A generalization of sampling without replacement from a finite universe.” J. Am. Stat. Assoc. 47 (1953): 663–685. [Google Scholar] [CrossRef]
- I.J. Good. “The population frequencies of species and the estimation of population parameters.” Biometrika 40 (1953): 237–264. [Google Scholar] [CrossRef]
- A. Orlitsky, N.P. Santhanam, and J. Zhang. “Always good Turing: asymptotically optimal probability estimation.” Science 302 (2003): 427–431. [Google Scholar] [CrossRef] [PubMed]
- H. Jeffreys. “An invariant form for the prior probability in estimation problems.” Proc. Royal. Soc. Lond. A 186 (1946): 453–461. [Google Scholar] [CrossRef]
- R.E. Krichevsky, and V.K. Trofimov. “The performance of universal encoding.” IEEE Trans. Inf. Theory 27 (1981): 199–207. [Google Scholar] [CrossRef]
- D. Holste, I. Groï¿½e, and H. Herzel. “Bayes’ estimators of generalized entropies.” J. Phys. A 31 (1998): 2551–2566. [Google Scholar] [CrossRef]
- W. Perks. “Some observations on inverse probability including a new indifference rule.” J. Inst. Actuar. 73 (1947): 285–334. [Google Scholar]
- T. Schurmann, and P. Grassberger. “Entropy estimation of symbol sequences.” Chaos 6 (1996): 414–427. [Google Scholar] [CrossRef] [PubMed]
- S. Trybula. “Some problems of simultaneous minimax estimatio.” Ann. Math. Stat. 29 (1958): 245–253. [Google Scholar] [CrossRef]
- T. Hayfield, and J.S. Racine. “Nonparametric econometrics: The np package.” J. Stat. Softw. 27 (2008): 1–32. [Google Scholar] [CrossRef]
- J.S. Racine. Nonparametric Econometrics: A Primer. Hanover, MA, USA: Now Publishers Inc., 2008, Volume 3, pp. 1–88. [Google Scholar]
- R.W. Koenker, and G. Bassett Jr. “Regression quantiles.” Econometrica 1 (1978): 33–50. [Google Scholar] [CrossRef]
- E. Maasoumi, and J.S. Racine. “Robust Entropy-Based Test of Asymmetry for Discrete and Continuous Processes.” Econom. Rev. 28 (2009): 246–261. [Google Scholar] [CrossRef]

© 2013 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license ( http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

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.
https://doi.org/10.3390/jrfm6010006

**AMA Style**

Allen DE, McAleer M, Powell R, Singh AK. A Non-Parametric and Entropy Based Analysis of the Relationship between the VIX and S&P 500. *Journal of Risk and Financial Management*. 2013; 6(1):6-30.
https://doi.org/10.3390/jrfm6010006

**Chicago/Turabian Style**

Allen, David E., Michael McAleer, Robert Powell, and Abhay K. Singh. 2013. "A Non-Parametric and Entropy Based Analysis of the Relationship between the VIX and S&P 500" *Journal of Risk and Financial Management* 6, no. 1: 6-30.
https://doi.org/10.3390/jrfm6010006