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Functional Relationships Between Price and Volatility Jumps and Their Consequences for Discretely Observed Data

Part of: Nonparametric inference Stochastic processes Stochastic analysis Inference from stochastic processes Mathematical finance

Published online by Cambridge University Press:  30 January 2018

Jean Jacod ,
Claudia Klüppelberg  and
Gernot Müller
Jean Jacod*
Affiliation:
Université Pierre et Marie Curie
Claudia Klüppelberg*
Affiliation:
Technische Universität München
Gernot Müller*
Affiliation:
Technische Universität München
*
∗ Postal address: Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie, 4 Place Jussieu, 75 005 Paris, France. Email address: [email protected]
∗∗ Postal address: Centre for Mathematical Sciences, Technische Universität München, 85748 Garching, Germany.
∗∗ Postal address: Centre for Mathematical Sciences, Technische Universität München, 85748 Garching, Germany.
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Abstract

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Many prominent continuous-time stochastic volatility models exhibit certain functional relationships between price jumps and volatility jumps. We show that stochastic volatility models like the Ornstein–Uhlenbeck and other continuous-time CARMA models as well as continuous-time GARCH and EGARCH models all exhibit such functional relations. We investigate the asymptotic behaviour of certain functionals of price and volatility processes for discrete observations of the price process on a grid, which are relevant for estimation and testing problems.

Keywords

Barndorff-Nielsen Shephard modelCARMAcontinuous-time GARCHCOGARCHcommon jumpshigh-frequency dataItô semimartingaleLévy processOrnstein–Uhlenbeck processstochastic volatility

MSC classification

Primary: 60G48: Generalizations of martingales 60H30: Applications of stochastic analysis (to PDE, etc.) 91G70: Statistical methods, econometrics
Secondary: 62G10: Hypothesis testing 62M02: Markov processes
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 4 , December 2012 , pp. 901 - 914
Copyright
© Applied Probability Trust 

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