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Journal of Financial Econometrics Vol. 2, No. 1, pp. 1-37
© 2004 Oxford University Press; all rights reserved.

Power and Bipower Variation with Stochastic Volatility and Jumps

Ole E. Barndorff-Nielsen
     University of Aarhus

Neil Shephard
     University of Oxford, Nuffield College

Address correspondence to Ole Barndorff-Nielsen, The Centre for Mathematical Physics and Stochastics, University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, Denmark, or e-mail: oebn{at}imf.au.dk.

This article shows that realized power variation and its extension, realized bipower variation, which we introduce here, are somewhat robust to rare jumps. We demonstrate that in special cases, realized bipower variation estimates integrated variance in stochastic volatility models, thus providing a model-free and consistent alternative to realized variance. Its robustness property means that if we have a stochastic volatility plus infrequent jumps process, then the difference between realized variance and realized bipower variation estimates the quadratic variation of the jump component. This seems to be the first method that can separate quadratic variation into its continuous and jump components. Various extensions are given, together with proofs of special cases of these results. Detailed mathematical results are reported in Barndorff-Nielsen and Shephard (2003a).

KEYWORDS: bipower variation, integrated variance, jump process, power variation, quadratic variation, realized variance, realized volatility, semimartingale, volatility

Received October 25, 2002; revised August 1, 2003; accepted October 31, 2003


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