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An E-ARCH model for the term structure of implied volatility of FX options

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  • Yingzi Zhu
  • Marco Avellaneda

Abstract

We construct a statistical model for the term-structure of implied volatilities of currency options based on daily historical data for 13 currency pairs over a 19-month period. We examine the joint evolution of 1 month, 2 month, 3 month, 6 month and 1 year at-the-money (50 δ) options in all the currency pairs. We show that there exist three uncorrelated state variables (principal components) which account for the parallel movement, slope oscillation, and curvature of the term structure and which explain, on average, the movements of the termstructure of volatility to more than 95% in all cases. We test and construct an exponential ARCH, or E-ARCH, model for each state variable. One of the applications of this model is to produce confidence bands for the term structure of volatility.

Suggested Citation

  • Yingzi Zhu & Marco Avellaneda, 1997. "An E-ARCH model for the term structure of implied volatility of FX options," Applied Mathematical Finance, Taylor & Francis Journals, vol. 4(2), pages 81-100.
    • Handle: RePEc:taf:apmtfi:v:4:y:1997:i:2:p:81-100
    DOI: 10.1080/13504869700000001
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    References listed on IDEAS

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    1. Rubinstein, Mark, 1994. "Implied Binomial Trees," Journal of Finance, American Finance Association, vol. 49(3), pages 771-818, July.
    2. Mark Rubinstein., 1994. "Implied Binomial Trees," Research Program in Finance Working Papers RPF-232, University of California at Berkeley.
    3. Marco Avellaneda & Antonio ParAS, 1996. "Managing the volatility risk of portfolios of derivative securities: the Lagrangian uncertain volatility model," Applied Mathematical Finance, Taylor & Francis Journals, vol. 3(1), pages 21-52.
    4. Hull, John C & White, Alan D, 1987. "The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
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    Cited by:

    1. T. F. Coleman & Y. Kim & Y. Li & M. Patron, 2007. "Robustly Hedging Variable Annuities With Guarantees Under Jump and Volatility Risks," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 74(2), pages 347-376, June.
    2. Matthias Fengler & Wolfgang Härdle & Christophe Villa, 2003. "The Dynamics of Implied Volatilities: A Common Principal Components Approach," Review of Derivatives Research, Springer, vol. 6(3), pages 179-202, October.
    3. Philipp Maier & Garima Vasishtha, 2008. "Good Policies or Good Fortune: What Drives the Compression in Emerging Market Spreads?," Staff Working Papers 08-25, Bank of Canada.
    4. Matthias Fengler & Wolfgang Härdle & Enno Mammen, 2005. "A Dynamic Semiparametric Factor Model for Implied Volatility String Dynamics," SFB 649 Discussion Papers SFB649DP2005-020, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    5. Martin Magris & Perttu Barholm & Juho Kanniainen, 2017. "Implied volatility smile dynamics in the presence of jumps," Papers 1711.02925, arXiv.org, revised May 2020.
    6. Jacinto Marabel Romo, 2012. "Volatility Regimes For The Vix Index," Revista de Economia Aplicada, Universidad de Zaragoza, Departamento de Estructura Economica y Economia Publica, vol. 20(2), pages 111-134, Autumn.
    7. Dotsis, George, 2017. "The market price of risk of the variance term structure," Journal of Banking & Finance, Elsevier, vol. 84(C), pages 41-52.
    8. Jin Zhang & Yi Xiang, 2008. "The implied volatility smirk," Quantitative Finance, Taylor & Francis Journals, vol. 8(3), pages 263-284.
    9. Fengler, Matthias R. & Härdle, Wolfgang & Mammen, Enno, 2003. "Implied volatility string dynamics," SFB 373 Discussion Papers 2003,54, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    10. Lovreta, Lidija & Silaghi, Florina, 2020. "The surface of implied firm’s asset volatility," Journal of Banking & Finance, Elsevier, vol. 112(C).

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