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An Eigenfunction Approach for Volatility Modeling

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Author Info
Nour Meddahi ()
Abstract

In this paper, we introduce a new approach for volatility modeling in discrete and continuous time. We follow the stochastic volatility literature by assuming that the variance is a function of a state variable. However, instead of assuming that the loading function is ad hoc (e.g., exponential or affine), we assume that it is a linear combination of the eigenfunctions of the conditional expectation (resp infinitesimal generator) operator associated to the state variable in discrete (resp continuous) time. Special examples are the popular log-normal and square-root models where the eigenfunctions are the Hermite and Laguerre polynomials, respectively. The eigenfunction approach has at least six advantages : i) it is general since any square integrable function may be written as a linear combination of the eigenfunctions; ii) the orthogonality of the eigenfunctions leads to the traditional interpretations of the linear principal components analysis; iii) the implied dynamics of the variance and squared return processes are ARMA and therefore simple for forecasting and inference purposes; iv) more importantly, this generates fat tails for the variance and returns processes; v) in contrast to popular models, the variance of the variance is a flexible function of the variance; vi) these models are closed under temporal aggregation.

Dans cet article, nous proposons une nouvelle approche pour la modélisation de la volatilité en temps discret et continu. Nous adoptons la même approche que la littérature de la volatilité stochastique en supposant que la volatilité est une fonction d'une variable d'état. Néanmoins, au lieu de supposer que la fonction de lien est donnée de manière ad hoc (par exemple, exponentielle ou affine), nous supposons que c'est une combinaison linéaire des fonctions propres de l'opérateur espérance conditionnelle (générateur infinitésimal, respectivement) associé à la variable d'état en temps discret (continu, respectivement). Les modèles populaires exponentiels et racine carrée sont des exemples où les fonctions propres sont respectivement les polynomes de Hermite et de Laguerre. L'approche par fonctions propres a au moins six avantages : i) elle est générale puisque toute fonction de carré intégrable peut être écrite comme combinaison linéaire des fonctions propres; ii) l'orthogonalité des fonctions propres permet d'utiliser les interprétations usuelles de l'analyse en composantes principales linéaires; iii) les dynamiques induites de la variance et du carré de l'innovation sont des ARMA et donc sont simples pour la prévision et l'inférence statistique; iv) plus important, cette approche génère des queues épaisses pour les processus de volatilité et de rendements; v) à l'opposé des modèles usuels, la variance de la variance est une fonction flexible de la variance; vi) ces modèles sont robustes vis-à-vis de l'agrégation temporelle.

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Paper provided by CIRANO in its series CIRANO Working Papers with number 2001s-70.

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Date of creation: 01 Oct 2001
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Handle: RePEc:cir:cirwor:2001s-70

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Keywords: volatility stochastic volatility infinitesimal generator conditional expectation eigenfunctions ARMA fat tails GMM volatilité volatilité stochastique générateur infinitésimal espérance conditionnelle fonctions propres ARMA queues épaisses GMM

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Full references

Cited by:
(explanations, Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.)

  1. Torben G. Andersen & Tim Bollerslev & Nour Meddahi, 2002. "Analytic Evaluation of Volatility Forecasts," CIRANO Working Papers 2002s-90, CIRANO. [Downloadable!]
    Other versions:
  2. Torben G. Andersen & Tim Bollerslev & Nour Meddahi, 2002. "Correcting the Errors: A Note on Volatility Forecast Evaluation Based on High-Frequency Data and Realized Volatilities," CIRANO Working Papers 2002s-91, CIRANO. [Downloadable!]
    Other versions:
  3. Nour Meddahi, 2002. "ARMA Representation of Integrated and Realized Variances," CIRANO Working Papers 2002s-93, CIRANO. [Downloadable!]
  4. Werker, B. & Meddahi, N. & Renault, E., 2003. "Garch and irregularly spaced data," Discussion Paper 27, Tilburg University, Center for Economic Research. [Downloadable!]
    Other versions:
  5. Peter F. Christoffersen & Francis X. Diebold, 2004. "Financial Asset Returns, Direction-of-Change Forecasting, and Volatility Dynamics," CFS Working Paper Series 2004/08, Center for Financial Studies. [Downloadable!]
    Other versions:
  6. Nour Meddahi, 2002. "A theoretical comparison between integrated and realized volatility," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 17(5), pages 479-508. [Downloadable!]
  7. Valentina Corradi & Norman Swanson & Geetesh Bhardwaj, 2006. "A Simulation Based Specification Test for Diffusion Processes," Departmental Working Papers 200614, Rutgers University, Department of Economics. [Downloadable!]
  8. Yacine Ait-Sahalia & Robert Kimmel, 2004. "Maximum Likelihood Estimation of Stochastic Volatility Models," NBER Working Papers 10579, National Bureau of Economic Research, Inc. [Downloadable!] (restricted)
  9. Nour Meddahi, 2001. "A Theoretical Comparison Between Integrated andRealized Volatilities / A Theoretical Comparison Between Integrated and Realized Volatilities," CIRANO Working Papers 2001s-71, CIRANO. [Downloadable!]
  10. Valentina Corradi & Norman Swanson & Walter Distaso, 2006. "Predictive Density Estimators for Daily Volatility Based on the Use of Realized Measures," Departmental Working Papers 200620, Rutgers University, Department of Economics. [Downloadable!]
  11. Valentina Corradi & Norman R. Swanson, 2003. "Bootstrap Specification Tests for Diffusion Processes," Departmental Working Papers 200321, Rutgers University, Department of Economics. [Downloadable!]
    Other versions:
  12. Mark J. Jensen & John M. Maheu, 2008. "Bayesian semiparametric stochastic volatility modeling," Working Paper 2008-15, Federal Reserve Bank of Atlanta. [Downloadable!]
    Other versions:
  13. Ole E. Barndorff-Nielsen & Neil Shephard, 2003. "Impact of jumps on returns and realised variances: econometric analysis of time-deformed Levy processes," Economics Papers 2003-W12, Economics Group, Nuffield College, University of Oxford. [Downloadable!]
    Other versions:
  14. Mikhail Chernov & A. Ronald Gallant & Eric Ghysels & George Tauchen, 2002. "Alternative Models for Stock Price Dynamics," CIRANO Working Papers 2002s-58, CIRANO. [Downloadable!]
    Other versions:
  15. Xiaohong Chen & Lars Peter Hansen & Jos´e A. Scheinkman, 2005. "Principal Components and the Long Run," Levine's Bibliography 122247000000000997, UCLA Department of Economics. [Downloadable!]
  16. Peter Christoffersen & Francis X. Diebold, 2002. "Financial Asset Returns, Market Timing, and Volatility Dynamics," CIRANO Working Papers 2002s-02, CIRANO. [Downloadable!]
  17. Nour Meddahi, 2002. "ARMA Representation of Two-Factor Models," CIRANO Working Papers 2002s-92, CIRANO. [Downloadable!]
  18. Neil Shephard, 2005. "Stochastic Volatility," Economics Papers 2005-W17, Economics Group, Nuffield College, University of Oxford. [Downloadable!]
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