# An Eigenfunction Approach for Volatility Modeling

• 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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## Bibliographic Info

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

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Handle: RePEc:cir:cirwor:2001s-70

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## Related research

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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Find related papers by JEL classification:
• C51 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Construction and Estimation
• C52 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Evaluation, Validation, and Selection
• C50 - Mathematical and Quantitative Methods - - Econometric Modeling - - - General

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