An eigefunction approach for volantility modeling

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(1)CAHIER 29-2001. AN EIGENFUNCTION APPROACH FOR VOLATILITY MODELING. Nour MEDDAHI.

(2) CAHIER 29-2001. AN EIGENFUNCTION APPROACH FOR VOLATILITY MODELING Nour MEDDAHI1. 1. Centre de recherche et développement en économique (C.R.D.E.), CIRANO and Département de sciences économiques (Université de Montréal), and CEPR. October 2001. _______________________ The author would like to thank Torben Andersen, Federico Bandi, Tim Bollerslev, Marine Carrasco, Xiahong Chen, Peter Christoffersen, Frank Diebold, Jean-Marie Dufour, Rob Engle, Jean-Pierre Florens, Ron Gallant, Eric Ghysels, Silvia Gonçalves, Lars Hansen, Éric Jacquier, Benoit Perron, Éric Renault, Enrique Sentana, Neil Shephard, George Tauchen, and Nizar Touzi for helpful discussions when he was writing this paper and/or for comments on an earlier draft. The author is particularly indebted to Bryan Campbell, René Garcia and especially to Christian Gouriéroux. He thanks seminar participants at CIRANO, Triangle (Duke, NCS, and UNC), University of Chicago, MITACS workshop, C.R.D.E. conference on “Modeling, Estimating and Forecasting Volatility”, April 2001, CIRANO conference on “Financial Mathematics and Econometrics”, June 2001, Econometric Society European Meeting, Lausanne, August 2001, and the “International Conference on Modeling and Forecasting Financial Volatility”, Perth, September 2001, for comments. Many thanks also go to Neil Shephard for providing the data and Brahim Boudarbat for excellent research assistance. The author acknowledges financial support from FCAR, IFM2, and MITACS, and is solely responsible for any remaining errors..

(3) RÉSUMÉ 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 (ou générateur infinitésimal) associé à la variable d’état en temps discret (ou continu). Les modèles populaires exponentiels et racine carrée sont des exemples où les fonctions propres sont respectivement les polynômes de Hermite et de Laguerre. L’approche par fonctions propres a au moins six avantages : i) elle est générale puisque toute fonction de carré intégrale 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. Mots clés : volatilité, volatilité stochastique, générateur infinitésimal, espérance conditionnelle, fonctions propres, ARMA, queues épaisses, GMM. 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, hence, 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. Key words : volatility, stochastic volatility, infinitesimal generator, expectation, eigenfunctions, ARMA, fat tails, GMM. conditional.

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