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Higher-order volatility: time series

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  • Carey, Alexander

Abstract

This paper presents time-series of higher-order volatilities for the S&P 500 and EURUSD. We use a 3-volatility model which accounts for non-normal skewness and kurtosis. The volatilities control the level, slope and curvature of the Black-Scholes implied volatility smile; accordingly we term them "base", "skew" and "smile" volatility. We define instantaneous skewness and kurtosis as simple ratios of the volatilities, and show that when these metrics are held constant, the model is relative sticky-delta. For the S&P 500 in 2008, skew and smile volatility are highly correlated with base volatility. Instantaneous skewness and kurtosis are remarkably stable, including over the market dislocation of the last four months of the year. Daily changes in all three volatilities are correlated with daily returns. For EURUSD in 2006, base and smile volatility are closely correlated, but in contrast to the equity case, skew volatility is independent and changes sign. This change in sign appears to provide advance warning of the two major market moves of the year. However, daily changes in the volatilities are uncorrelated with daily returns.

Suggested Citation

  • Carey, Alexander, 2010. "Higher-order volatility: time series," MPRA Paper 21087, University Library of Munich, Germany.
  • Handle: RePEc:pra:mprapa:21087
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    File URL: https://mpra.ub.uni-muenchen.de/21087/1/MPRA_paper_21087.pdf
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    References listed on IDEAS

    as
    1. Dorje C. Brody & Lane P. Hughston & Andrea Macrina, 2007. "Information-Based Asset Pricing," Papers 0704.1976, arXiv.org.
    2. Toby Daglish & John Hull & Wulin Suo, 2007. "Volatility surfaces: theory, rules of thumb, and empirical evidence," Quantitative Finance, Taylor & Francis Journals, vol. 7(5), pages 507-524.
    3. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters,in: Theory Of Valuation, chapter 8, pages 229-288 World Scientific Publishing Co. Pte. Ltd..
    4. Simon H. Babbs & Michael J. P. Selby, 1998. "Pricing by Arbitrage Under Arbitrary Information," Mathematical Finance, Wiley Blackwell, vol. 8(2), pages 163-168.
    5. Carr, Peter & Wu, Liuren, 2007. "Stochastic skew in currency options," Journal of Financial Economics, Elsevier, vol. 86(1), pages 213-247, October.
    6. Carey, Alexander, 2005. "Higher-order volatility," MPRA Paper 4993, University Library of Munich, Germany.
    7. Carey, Alexander, 2006. "Higher-order volatility: dynamics and sensitivities," MPRA Paper 5009, University Library of Munich, Germany.
    8. Reiswich, Dimitri & Wystup, Uwe, 2009. "FX volatility smile construction," CPQF Working Paper Series 20, Frankfurt School of Finance and Management, Centre for Practical Quantitative Finance (CPQF).
    9. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    10. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
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    More about this item

    Keywords

    higher-order volatility; time series; S&P 500; EURUSD;

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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