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The Shape and Term Structure of the Index Option Smirk: Why Multifactor Stochastic Volatility Models Work So Well

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  • Peter Christoffersen

    ()
    (Desautels Faculty of Management, McGill University, Montreal, Quebec H3A 1G5, Canada; Copenhagen Business School, 2000 Frederiksberg, Denmark; and CREATES, University of Aarhus, 8000 Aarhus, Denmark)

  • Steven Heston

    ()
    (Robert H. Smith School of Business, University of Maryland, College Park, Maryland 20742)

  • Kris Jacobs

    ()
    (Desautels Faculty of Management, McGill University, Montreal, Quebec H3A 1G5, Canada; and C. T. Bauer College of Business, University of Houston, Houston, Texas 77204)

Abstract

State-of-the-art stochastic volatility models generate a "volatility smirk" that explains why out-of-the-money index puts have high prices relative to the Black-Scholes benchmark. These models also adequately explain how the volatility smirk moves up and down in response to changes in risk. However, the data indicate that the slope and the level of the smirk fluctuate largely independently. Although single-factor stochastic volatility models can capture the slope of the smirk, they cannot explain such largely independent fluctuations in its level and slope over time. We propose to model these movements using a two-factor stochastic volatility model. Because the factors have distinct correlations with market returns, and because the weights of the factors vary over time, the model generates stochastic correlation between volatility and stock returns. Besides providing more flexible modeling of the time variation in the smirk, the model also provides more flexible modeling of the volatility term structure. Our empirical results indicate that the model improves on the benchmark Heston stochastic volatility model by 24% in-sample and 23% out-of-sample. The better fit results from improvements in the modeling of the term structure dimension as well as the moneyness dimension.

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File URL: http://dx.doi.org/10.1287/mnsc.1090.1065
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Bibliographic Info

Article provided by INFORMS in its journal Management Science.

Volume (Year): 55 (2009)
Issue (Month): 12 (December)
Pages: 1914-1932

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Handle: RePEc:inm:ormnsc:v:55:y:2009:i:12:p:1914-1932

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Keywords: stochastic correlation; stochastic volatility; equity index options; multifactor model; persistence; affine; out-of-sample;

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References

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Citations

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Cited by:
  1. Tiberiu Socaciu & Bogdan Patrut, 2010. "Algorithm for Financial Derivatives Evaluation in Generalized Double-Heston Model," BRAND. Broad Research in Accounting, Negotiation, and Distribution, EduSoft Publishing, vol. 1(1), pages 5-10, September.
  2. Audrino, Francesco & Fengler, Matthias, 2013. "Are classical option pricing models consistent with observed option second-order moments? Evidence from high-frequency data," Economics Working Paper Series 1311, University of St. Gallen, School of Economics and Political Science.
  3. Alessandro Gnoatto & Martino Grasselli, 2011. "The explicit Laplace transform for the Wishart process," Papers 1107.2748, arXiv.org, revised Aug 2013.
  4. Andreou, Panayiotis C. & Charalambous, Chris & Martzoukos, Spiros H., 2010. "Generalized parameter functions for option pricing," Journal of Banking & Finance, Elsevier, vol. 34(3), pages 633-646, March.
  5. Torben G. Andersen & Nicola Fusari & Viktor Todorov, 2012. "Parametric Inference and Dynamic State Recovery from Option Panels," NBER Working Papers 18046, National Bureau of Economic Research, Inc.
  6. De Col, Alvise & Gnoatto, Alessandro & Grasselli, Martino, 2013. "Smiles all around: FX joint calibration in a multi-Heston model," Journal of Banking & Finance, Elsevier, vol. 37(10), pages 3799-3818.
  7. Peter Christoffersen & Kris Jacobs & Chayawat Ornthanalai, 2012. "GARCH Option Valuation: Theory and Evidence," CREATES Research Papers 2012-50, School of Economics and Management, University of Aarhus.
  8. Peter Christoffersen & Bruno Feunou & Kris Jacobs & Nour Meddahi, 2012. "The Economic Value of Realized Volatility: Using High-Frequency Returns for Option Valuation," Working Papers 12-34, Bank of Canada.
  9. Peter Christoffersen & Kris Jacobs & Bo Young Chang, 2011. "Forecasting with Option Implied Information," CREATES Research Papers 2011-46, School of Economics and Management, University of Aarhus.
  10. Bretó, Carles, 2014. "On idiosyncratic stochasticity of financial leverage effects," Statistics & Probability Letters, Elsevier, vol. 91(C), pages 20-26.
  11. Kaeck, Andreas & Alexander, Carol, 2012. "Volatility dynamics for the S&P 500: Further evidence from non-affine, multi-factor jump diffusions," Journal of Banking & Finance, Elsevier, vol. 36(11), pages 3110-3121.
  12. Alejandro Bernales & Massimo Guidolin, 2012. "Can We Forecast the Implied Volatility Surface Dynamics of Equity Options? Predictability and Economic Value Tests," Working Papers 456, IGIER (Innocenzo Gasparini Institute for Economic Research), Bocconi University.
  13. Panayiotis Andreou & Chris Charalambous & Spiros Martzoukos, 2014. "Assessing the performance of symmetric and asymmetric implied volatility functions," Review of Quantitative Finance and Accounting, Springer, vol. 42(3), pages 373-397, April.
  14. Yang-Ho Park, 2013. "Volatility of volatility and tail risk premiums," Finance and Economics Discussion Series 2013-54, Board of Governors of the Federal Reserve System (U.S.).
  15. Li, Gang & Zhang, Chu, 2013. "Diagnosing affine models of options pricing: Evidence from VIX," Journal of Financial Economics, Elsevier, vol. 107(1), pages 199-219.
  16. Alvise De Col & Alessandro Gnoatto & Martino Grasselli, 2012. "Smiles all around: FX joint calibration in a multi-Heston model," Papers 1201.1782, arXiv.org, revised Jun 2013.
  17. Chulmin Kang & Wanmo Kang, 2013. "Exact Simulation of Wishart Multidimensional Stochastic Volatility Model," Papers 1309.0557, arXiv.org.
  18. Wallmeier, Martin, 2012. "Smile in Motion: An Intraday Analysis of Asymmetric Implied Volatility," FSES Working Papers 427, Faculty of Economics and Social Sciences, University of Freiburg/Fribourg Switzerland.
  19. Alfredo Ibáñez, 2008. "The cross-section of average delta-hedge option returns under stochastic volatility," Review of Derivatives Research, Springer, vol. 11(3), pages 205-244, October.
  20. Lorenz Schneider, 2014. "A Stochastic Volatility Model for Crude Oil Futures Curves and the Pricing of Calendar Spread Options," Papers 1401.7913, arXiv.org.
  21. Sang Byung Seo & Jessica A. Wachter, 2013. "Option Prices in a Model with Stochastic Disaster Risk," NBER Working Papers 19611, National Bureau of Economic Research, Inc.
  22. Cho-Hoi Hui & Tsz-Kin Chung, 2010. "The Risk of Sudden Depreciation of the Euro in the Sovereign Debt Crisis of 2009-2010," Working Papers 252010, Hong Kong Institute for Monetary Research.

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