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The dynamics of implied volatilities: A common principal components approach

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  • Fengler, Matthias R.
  • Härdle, Wolfgang K.
  • Villa, Christophe

Abstract

It is common practice to identify the number and sources of shocks that move implied volatilities across space and time by applying Principal Components Analysis (PCA) to pooled covariance matrices of changes in implied volatilities. This approach, however, is likely to result in a loss of information, since the surface structure of implied volatilities in the maturities and moneyness dimension is neglected. In this paper we propose to estimate the implied volatility surface at each point in time nonparametrically and to analyze the implied volatility surface slice by slice with a common principal components analysis (CPCA). As opposed to traditional PCA, the basic assumption of CPCA is that the space spanned by the eigenvectors is identical across groups, whereas variances associated with the components are allowed to vary. This allows us to study a p variate random vector of k groups, say the volatility smile at p different grid points of moneyness for k maturities, simultaneously. Our evidence suggests that surface dynamics can indeed be traced back to a common eigenstructure between covariance matrices of the surface slices, which allow for the usual shift, slope, and twist interpretation of shocks to implied volatilities. This insight is a suitable starting point for VaR Monte Carlo Simulations of delta-gamma neutral, vega sensitive option portfolios. --

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Paper provided by Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes in its series SFB 373 Discussion Papers with number 2001,38.

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Date of creation: 2001
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Handle: RePEc:zbw:sfb373:200138

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Keywords: Common Principal Component Analysis; Implied Volatility Surface; Principal Component Analysis; Smile;

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References

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  1. Bernard Dumas & Jeff Fleming & Robert E. Whaley, 1998. "Implied Volatility Functions: Empirical Tests," Journal of Finance, American Finance Association, American Finance Association, vol. 53(6), pages 2059-2106, December.
  2. Rama Cont & Jose da Fonseca, 2002. "Dynamics of implied volatility surfaces," Quantitative Finance, Taylor & Francis Journals, Taylor & Francis Journals, vol. 2(1), pages 45-60.
  3. Farshid Jamshidian & Yu Zhu, 1996. "Scenario Simulation: Theory and methodology (*)," Finance and Stochastics, Springer, Springer, vol. 1(1), pages 43-67.
  4. 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.
  5. Fengler, Matthias R. & Wang, Qihua, 2003. "Fitting the Smile Revisited: A Least Squares Kernel Estimator for the Implied Volatility Surface," SFB 373 Discussion Papers 2003,25, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
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Cited by:
  1. Carol Alexander & Leonardo M. Nogueira, 2004. "Hedging with Stochastic and Local Volatility," ICMA Centre Discussion Papers in Finance, Henley Business School, Reading University icma-dp2004-10, Henley Business School, Reading University, revised Dec 2004.
  2. Ci­zek, P. & Tamine, J. & Härdle, W., 2008. "Smoothed L-estimation of regression function," Computational Statistics & Data Analysis, Elsevier, Elsevier, vol. 52(12), pages 5154-5162, August.
  3. 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.
  4. Fengler, Matthias & Hin, Lin-Yee, 2011. "Semi-nonparametric estimation of the call price surface under strike and time-to-expiry no-arbitrage constraints," Economics Working Paper Series 1136, University of St. Gallen, School of Economics and Political Science, revised May 2013.
  5. Michal Benko & Wolfgang Härdle & Alois Kneip, 2006. "Common Functional Principal Components," SFB 649 Discussion Papers SFB649DP2006-010, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  6. George Skiadopoulos & Dimitris Psychoyios, 2006. "Implied Volatility Process: Evidence from the Volatility Derivatives Markets," Working Papers, Warwick Business School, Finance Group wpn06-17, Warwick Business School, Finance Group.
  7. Szymon Borak & Matthias Fengler & Wolfgang Härdle, 2005. "DSFM fitting of Implied Volatility Surfaces," SFB 649 Discussion Papers SFB649DP2005-022, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  8. Michal Benko & Alois Kneip, 2005. "Common functional component modelling," SFB 649 Discussion Papers SFB649DP2005-016, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  9. Pavel Cizek & Karel Komorad, 2005. "Implied Trinomial Trees," SFB 649 Discussion Papers SFB649DP2005-007, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  10. Ralf Brüggemann & Wolfgang Härdle & Julius Mungo & Carsten Trenkler, 2006. "VAR Modeling for Dynamic Semiparametric Factors of Volatility Strings," SFB 649 Discussion Papers SFB649DP2006-011, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  11. Mónica Fuentes & Sergio Godoy, 2005. "Sovereign Spread in Emerging Markets: A Principal Component Analysis," Working Papers Central Bank of Chile, Central Bank of Chile 333, Central Bank of Chile.
  12. Bernd Engelmann & Matthias Fengler & Morten Nalholm & Peter Schwendner, 2006. "Static versus dynamic hedges: an empirical comparison for barrier options," Review of Derivatives Research, Springer, Springer, vol. 9(3), pages 239-264, November.
  13. Francesco Audrino & Dominik Colangelo, 2009. "Option trading strategies based on semi-parametric implied volatility surface prediction," University of St. Gallen Department of Economics working paper series 2009, Department of Economics, University of St. Gallen 2009-24, Department of Economics, University of St. Gallen.
  14. 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.
  15. 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.
  16. 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, The American Risk and Insurance Association, vol. 74(2), pages 347-376.
  17. Han Shang, 2014. "A survey of functional principal component analysis," AStA Advances in Statistical Analysis, Springer, Springer, vol. 98(2), pages 121-142, April.
  18. Laurini, Márcio P., 2007. "Imposing No-Arbitrage Conditions In Implied Volatility Surfaces Using Constrained Smoothing Splines," Insper Working Papers, Insper Working Paper, Insper Instituto de Ensino e Pesquisa wpe_89, Insper Working Paper, Insper Instituto de Ensino e Pesquisa.
  19. 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.
  20. Fengler, Matthias R. & Wang, Qihua, 2003. "Fitting the Smile Revisited: A Least Squares Kernel Estimator for the Implied Volatility Surface," SFB 373 Discussion Papers 2003,25, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.

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