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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.

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Bibliographic Info

Paper provided by University Library of Munich, Germany in its series MPRA Paper with number 21087.

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Date of creation: 12 Jan 2010
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Handle: RePEc:pra:mprapa:21087

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

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References

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  1. Peter Carr & Liuren Wu, 2004. "Stochastic Skew in Currency Options," Finance 0409014, EconWPA.
  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. Merton, Robert C., 1975. "Option pricing when underlying stock returns are discontinuous," Working papers 787-75., Massachusetts Institute of Technology (MIT), Sloan School of Management.
  4. 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 637-54, May-June.
  5. Carey, Alexander, 2005. "Higher-order volatility," MPRA Paper 4993, University Library of Munich, Germany.
  6. Simon H. Babbs & Michael J. P. Selby, 1998. "Pricing by Arbitrage Under Arbitrary Information," Mathematical Finance, Wiley Blackwell, vol. 8(2), pages 163-168.
  7. 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).
  8. Carey, Alexander, 2006. "Higher-order volatility: dynamics and sensitivities," MPRA Paper 5009, University Library of Munich, Germany.
  9. Dorje C. Brody & Lane P. Hughston & Andrea Macrina, 2007. "Information-Based Asset Pricing," Papers 0704.1976, arXiv.org.
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