Analytic Evaluation of Volatility Forecasts

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(1)2002s-90. Analytic Evaluation of Volatility Forecasts Torben G. Andersen, Tim Bollerslev, Nour Meddahi. Série Scientifique Scientific Series. Montréal Décembre 2002. © 2002 Torben G. Andersen, Tim Bollerslev, Nour Meddahi. Tous droits réservés. All rights reserved. Reproduction partielle permise avec citation du document source, incluant la notice ©. Short sections may be quoted without explicit permission, if full credit, including © notice, is given to the source..

(2) CIRANO Le CIRANO est un organisme sans but lucratif constitué en vertu de la Loi des compagnies du Québec. Le financement de son infrastructure et de ses activités de recherche provient des cotisations de ses organisationsmembres, d’une subvention d’infrastructure du ministère de la Recherche, de la Science et de la Technologie, de même que des subventions et mandats obtenus par ses équipes de recherche. CIRANO is a private non-profit organization incorporated under the Québec Companies Act. Its infrastructure and research activities are funded through fees paid by member organizations, an infrastructure grant from the Ministère de la Recherche, de la Science et de la Technologie, and grants and research mandates obtained by its research teams. Les organisations-partenaires / The Partner Organizations •École des Hautes Études Commerciales •École Polytechnique de Montréal •Université Concordia •Université de Montréal •Université du Québec à Montréal •Université Laval •Université McGill •Ministère des Finances du Québec •MRST •Alcan inc. •AXA Canada •Banque du Canada •Banque Laurentienne du Canada •Banque Nationale du Canada •Banque Royale du Canada •Bell Canada •Bombardier •Bourse de Montréal •Développement des ressources humaines Canada (DRHC) •Fédération des caisses Desjardins du Québec •Hydro-Québec •Industrie Canada •Pratt & Whitney Canada Inc. •Raymond Chabot Grant Thornton •Ville de Montréal. Les cahiers de la série scientifique (CS) visent à rendre accessibles des résultats de recherche effectuée au CIRANO afin de susciter échanges et commentaires. Ces cahiers sont écrits dans le style des publications scientifiques. Les idées et les opinions émises sont sous l’unique responsabilité des auteurs et ne représentent pas nécessairement les positions du CIRANO ou de ses partenaires. This paper presents research carried out at CIRANO and aims at encouraging discussion and comment. The observations and viewpoints expressed are the sole responsibility of the authors. They do not necessarily represent positions of CIRANO or its partners.. ISSN 1198-8177.

(3) Analytic Evaluation of Volatility Forecasts* Torben G. Andersen†, Tim Bollerslev‡, Nour Meddahi§ Résumé / Abstract Le développement de procédures d’estimation et de prévision utilisant des modèles réalistes, en temps continu et à volatilité stochastique est sévèrement bloqué par la non disponibilité de la densité de transition des rendements. En réponse à ce problème, Andersen, Bollerslev, Diebold et Labys (2002) ont récemment proposé de modéliser et de prévoir la volatilité intégrée (et inobservable), qui est importante pour la valorisation des produits dérivés, à partir de formes réduites simples de la volatilité réalisée, calculée en sommant des carrés des rendements à haute fréquence. En utilisant la classe de modèles à volatilité stochastique basée sur des fonctions propres introduites dans Meddahi (2001), nous présentons les expressions analytiques des fonctions de pertes de ces prévisions simples en fonction du pas de temps des rendements à haute fréquence. En quantifiant ces fonctions de pertes pour les modèles en temps continu usuels comme les modèles de diffusion GARCH, affine à plusieurs facteurs, et lognormal, nous trouvons que les procédures à base de formes réduites se comportent très bien par rapport aux procédures optimales (et infaisables) qui utilisent toute la trajectoire de la volatilité instantanée et inobservable.. The development of estimation and forecasting procedures using empirically realistic continuous-time stochastic volatility models is severely hampered by the lack of closed-form expressions for the transition densities of the observed returns. In response to this, Andersen, Bollerslev, Diebold and Labys (2002) have recently advocated modeling and forecasting the (latent) integrated volatility of primary import from a pricing perspective based on simple reduced form time series models for the observable realized volatilities, constructed from the summation of high-frequency squared returns. Building on the eigenfunction stochastic volatility class of models introduced by Meddahi (2001), we present analytical expressions for the loss in forecast efficiency associated with this easy-to-implement procedure as a function of the sampling frequency of the returns underlying the realized volatility measures. On numerically *. This work was supported by a grant from the National Science Foundation to the NBER (Andersen and Bollerslev), and from IFM2 and MITACS (Meddahi). Discussions with Francis X. Diebold on closely related ideas have been very helpful. We would also like to thank seminar audiences at Université de Montréal, the NSF/NBER Time Series Conference 2002, Philadelphia, and especially Neil Shephard for helpful comments on the first version of the paper. The third author thanks the Bendheim Center for Finance, Princeton University, for its hospitality during his visit and the revision of the paper. †. Department of Finance, Kellogg School of Management, Northwestern University, Evanston, IL 60208, and NBER, USA, phone: 847-467-1285, e-mail: [email protected]. ‡. Department of Economics, Duke University, Durham, NC 27708, and NBER, USA, phone: 919-660-1846, e-mail: [email protected].. §. Département de sciences économiques, CIRANO, CIREQ, Université de Montréal, C.P. 6128, succursale Centreville, Montréal (Québec), H3C 3J7, Canada, phone: 514-343-2399, e-mail: [email protected]..

(4) quantifying this efficiency loss for such popular continuous-time models as GARCH, multifactor affine, and log-normal diffusions, we find that the realized volatility reduced form procedures perform remarkably well in comparison to the optimal (non-feasible) forecasts conditional on the full sample path realization of the latent instantaneous volatility process.. Mots clés: modèles à temps continu; modèles à volatilité stochastique basée sur des fonctions propres; volatilité intégrée, volatilité réalisée; données à haute fréquence; prévision de séries chronologiques; régressions de Mincer-Zarnowitz. Keywords: Continuous-time models; eigenfunction stochastic volatility models; integrated volatility; realized volatility; high-frequency data; time series forecasting; Mincer-Zarnowitz regressions. Codes JEL: C22, C52, G12, G13.

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