A New Class of Stochastic Volatility Models with Jumps ... · Mots Clés : Processus à sauts,...

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MontréalDécembre 1999

Série ScientifiqueScientific Series

99s-48

A New Class of StochasticVolatility Models with Jumps:

Theory and Estimation

Mikhail Chernov, A. Ronald Gallant,Eric Ghysels, George Tauchen

CIRANO

Le CIRANO est un organisme sans but lucratif constitué en vertu de la Loi des compagnies du Québec. Lefinancement de son infrastructure et de ses activités de recherche provient des cotisations de ses organisations-membres, d’une subvention d’infrastructure du ministère de la Recherche, de la Science et de la Technologie, demême 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 andresearch activities are funded through fees paid by member organizations, an infrastructure grant from theMinistère de la Recherche, de la Science et de la Technologie, and grants and research mandates obtained by itsresearch teams.

Les organisations-partenaires / The Partner Organizations

•École des Hautes Études Commerciales•École Polytechnique•Université Concordia•Université de Montréal•Université du Québec à Montréal•Université Laval•Université McGill•MEQ•MRST•Alcan Aluminium Ltée•Banque Nationale du Canada•Banque Royale du Canada•Bell Québec•Développement des ressources humaines Canada (DRHC)•Egis•Fédération des caisses populaires Desjardins de Montréal et de l’Ouest-du-Québec•Hydro-Québec•Imasco•Industrie Canada•Microcell Labs inc.•Raymond Chabot Grant Thornton•Téléglobe Canada•Ville de Montréal

© 1999 Mikhail Chernov, A. Ronald Gallant, Eric Ghysels et George Tauchen. Tous droits réservés. All rightsreserved.Reproduction partielle permise avec citation du document source, incluant la notice ©.Short sections may be quoted without explicit permission, provided that full credit, including © notice, is given tothe source.

ISSN 1198-8177

Ce document est publié dans l’intention de rendre accessibles les résultats préliminairesde la recherche effectuée au CIRANO, afin de susciter des échanges et des suggestions.Les idées et les opinions émises sont sous l’unique responsabilité des auteurs, et nereprésentent pas nécessairement les positions du CIRANO ou de ses partenaires.This paper presents preliminary research carried out at CIRANO and aims atencouraging discussion and comment. The observations and viewpoints expressed are thesole responsibility of the authors. They do not necessarily represent positions of CIRANOor its partners.

A New Class of Stochastic Volatility Models with Jumps:Theory and Estimation*

Mikhail Chernov†, A. Ronald Gallant‡, Eric Ghysels§, George Tauchen¶

Résumé / Abstract

Nous présentons une nouvelle classe de processus à sauts avec volatilitéstochastique. Cette nouvelle classe généralise les modèles affinés proposés parDuffie, Pan et Singleton (1998). La généralité se manifeste par une représentationgénérique des sauts par un processus de Lévy. La classe des processus que nousprésentons nous fournit également des prix d’options. Une application empiriquedémontre la présence de sauts dans des séries financières telles le S&P500 et leDow Jones. De plus, les processus n’ont pas une intensité constante. Nousanalysons plusieurs spécifications empiriques.

The purpose of this paper is to propose a new class of jump diffusionswhich feature both stochastic volatility and random intensity jumps. Previousstudies have focussed primarily on pure jump processes with constant intensityand log-normal jumps or constant jump intensity combined with a one factorstochastic volatility model. We introduce several generalizations which can betteraccommodate several empirical features of returns data. In their most generalform we introduce a class of processes which nests jump-diffusions previouslyconsidered in empirical work and includes the affine class of random intensitymodels studied by Bates (1998) and Duffie, Pan and Singleton (1998) but alsoallows for non-affine random intensity jump components. We attain the generalityof our specification through a generic Lévy process characterization of the jumpcomponent. The processes we introduce share the desirable feature with the affineclass that they yield analytically tractable and explicit option pricing formula. Thenon-affine class of processes we study include specifications where the randomintensity jump component depends on the size of the previous jump whichrepresent an alternative to affine random intensity jump processes which featurecorrelation between the stochastic volatility and jump component. We also allowfor and experiment with different empirical specifications of the jump sizedistributions. We use two types of data sets. One involves the S&P500 and theother comprises of 100 years of daily Dow Jones index. The former is a return

* Corresponding Author: Eric Ghysels, Department of Economics and Finance, The Pennsylvania State University,523 Kern Graduate Building, University Park, PA 16802-3305 Tel.: (814) 865-7924 Fax: (814) 863-4775email: eghysels@psu.edu

† Pennsylvania State University‡ University of North Carolina§ Pennsylvania State University and CIRANO¶ Duke University

series often used in the literature and allows us to compare our results withprevious studies. The latter has the advantage to provide a long time series andenhances the possibility of estimating the jump component more precisely. Thenon-affine random intensity jump processes are more parsimonious than theaffine class and appear to fit the data much better.

Mots Clés : Processus à sauts, mesures de Lévy, modèles à volatilité stochastique

Keywords: Efficient method of moments, Poisson processes, jump processes, stochasticvolatility models, filtering

JEL: G13, C14, C52, C53

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Liste des publications au CIRANO *

Cahiers CIRANO / CIRANO Papers (ISSN 1198-8169)

99c-1 Les Expos, l'OSM, les universités, les hôpitaux : Le coût d'un déficit de 400 000 emploisau Québec — Expos, Montréal Symphony Orchestra, Universities, Hospitals: TheCost of a 400,000-Job Shortfall in Québec / Marcel Boyer

96c-1 Peut-on créer des emplois en réglementant le temps de travail ? / Robert Lacroix

95c-2 Anomalies de marché et sélection des titres au Canada / Richard Guay, Jean-FrançoisL'Her et Jean-Marc Suret

95c-1 La réglementation incitative / Marcel Boyer

94c-3 L'importance relative des gouvernements : causes, conséquences et organisationsalternative / Claude Montmarquette

94c-2 Commercial Bankruptcy and Financial Reorganization in Canada / Jocelyn Martel

94c-1 Faire ou faire faire : La perspective de l'économie des organisations / Michel Patry

Série Scientifique / Scientific Series (ISSN 1198-8177)

99s-47 Latent Variable Models for Stochastic Discount Factors / René Garcia et Éric Renault99s-46 Sequential Auctions with Multi-Unit Demand: Theory, Experiments and Simulations /

Jacques Robert et Claude Montmarquette99s-45 American Options: Symmetry Properties / Jérôme Detemple99s-44 What Is Happening in the Youth Labour Market in Canada? / Paul Beaudry, Thomas

Lemieux et Daniel Parent99s-43 The Valuation of Volatility Options / Jérôme Detemple et Carlton Osakwe99s-42 Labour Market Outcomes and Schooling in Canada: Has the Value of a High School

Degree Changed over Time? / Daniel Parent99s-41 Travail pendant les études, performance scolaire et abandon / Marcel Dagenais, Claude

Montmarquette, Daniel Parent et Nathalie Viennot-Briot99s-40 Recursive Intergenerational Utility in Global Climate Risk Modeling / Minh Ha-Duong

et Nicolas Treich99s-39 Transition vers le marché du travail au Canada : Portrait de la situation actuelle et

perspective historique / Daniel Parent99s-38 Program Evaluation Criteria Applied to Pay Equity in Ontario / Morley Gunderson et

Paul Lanoie99s-37 Optimal Justice in a General Equilibrium Model with Non Observable Individual

Productivities / Tarek M. Harchaoui et Pierre Lasserre

* Vous pouvez consulter la liste complète des publications du CIRANO et les publications elles-mêmes sur notre siteInternet à l'adresse suivante :

http://www.cirano.umontreal.ca/publication/documents.html