Hostname: page-component-7dd5485656-tbj44 Total loading time: 0 Render date: 2025-10-30T15:26:51.716Z Has data issue: false hasContentIssue false
  • English
  • Français

Power variation and stochastic volatility: a review and some new results

Published online by Cambridge University Press:  14 July 2016

Ole E. Barndorff-Nielsen ,
Svend Erik Graversen  and
Neil Shephard
Ole E. Barndorff-Nielsen
Affiliation:
Centre for Mathematical Physics and Stochastics (MaPhySto), University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, Denmark. Email address: oebn@imf.au.dk
Svend Erik Graversen
Affiliation:
Department of Mathematical Sciences, University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, Denmark. Email address: matseg@imf.au.dk
Neil Shephard
Affiliation:
Nuffield College, University of Oxford, Oxford OX1 1NF, UK. Email address: neil.shephard@nuffield.oxford.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we review some recent work on limit results on realised power variation, that is, sums of powers of absolute increments of various semimartingales. A special case of this analysis is realised variance and its probability limit, quadratic variation. Such quantities often appear in financial econometrics in the analysis of volatility. The paper also provides some new results and discusses open issues.

Keywords

Bipowermixed Gaussian limitpower variationquadratic variationrealised variancerealised volatilitystochastic volatility

Information

Type
Part 3. Financial mathematics
Information
Journal of Applied Probability , Volume 41 , Issue A: Stochastic Methods and their Applications , 2004 , pp. 133 - 143
Copyright
Copyright © Applied Probability Trust 2004 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Andersen, T. G. and Bollerslev, T. (1997). Intraday periodicity and volatility persistence in financial markets. J. Empirical Finance 4, 115–158.Google Scholar
Andersen, T. G. and Bollerslev, T. (1998a). Answering the sceptics: yes, standard volatility models do provide accurate forecasts. Internat. Econom. Rev. 39, 885–905.Google Scholar
Andersen, T. G. and Bollerslev, T. (1998b). Deutschemark-dollar volatility: intraday activity patterns, macroeconomic announcements, and longer run dependencies. J. Finance 53, 219–265.Google Scholar
Andersen, T. G., Bollerslev, T. and Diebold, F. X. (2004). Parametric and nonparametric measurement of volatility. To appear in Handbook of Financial Econometrics , eds Ait-Sahalia, Y. and Hansen, L. P., North Holland, Amsterdam.Google Scholar
Andersen, T. G., Bollerslev, T., Diebold, F. X. and Ebens, H. (2001). The distribution of realized stock return volatility. J. Financial Econom. 61, 43–76.Google Scholar
Andersen, T. G., Bollerslev, T., Diebold, F. X. and Labys, P. (2001). The distribution of exchange rate volatility. J. Amer. Statist. Assoc. 96, 42–55.Google Scholar
Andersen, T. G., Bollerslev, T., Diebold, F. X. and Labys, P. (2003). Modeling and forecasting realized volatility. Econometrica 71, 579–625.Google Scholar
Andreou, E. and Ghysels, E. (2002). Rolling-sampling volatility estimators: some new theoretical, simulation and empirical results. J. Business Econom. Statist. 20, 363–376.Google Scholar
Barndorff-Nielsen, O. E. and Shephard, N. (2001). Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics (with discussion). J. R. Statist. Soc. B 63, 167–241.Google Scholar
Barndorff-Nielsen, O. E. and Shephard, N. (2002a). Econometric analysis of realised covariation: high frequency covariance, regression and correlation in financial economics. To appear in Econometrica 72.Google Scholar
Barndorff-Nielsen, O. E. and Shephard, N. (2002b). Econometric analysis of realised volatility and its use in estimating stochastic volatility models. J. R. Statist. Soc. B 64, 253–280.Google Scholar
Barndorff-Nielsen, O. E. and Shephard, N. (2003a). Power and bipower variation with stochastic volatility and jumps (with discussion). To appear in J. Financial Econometrics. Google Scholar
Barndorff-Nielsen, O. E. and Shephard, N. (2003b). Power variation and time change. Unpublished manuscript, Nuffield College, Oxford.Google Scholar
Barndorff-Nielsen, O. E. and Shephard, N. (2003C). Realised power variation and stochastic volatility. Bernoulli 9, 243–265. (Correction: available at http://www.levyprocess.org.)CrossRefGoogle Scholar
Barndorff-Nielsen, O. E. and Shephard, N. (2004). A Continuous-time Approach to Financial Volatility. Cambridge University Press (to appear).Google Scholar
Becker, E. (1998). Theorems limite pour des processus discretisés. Doctoral Thesis, Université Paris 6.Google Scholar
Christensen, B. J. and Prabhala, N. R. (1998). The relation between implied and realized volatility. J. Financial Econom. 37, 125–150.Google Scholar
Dacorogna, M. M., Müller, U. A., Olsen, R. B. and Pictet, ?. V. (1998). Modelling short term volatility with GARCH and HARCH. In Nonlinear Modelling of High Frequency Financial Time Series , eds Dunis, C. and Zhou, B., John Wiley, Chichester, pp. 161–176 Google Scholar
Davis, R. A. and Mikosch, T. (1998). The limit theory for the sample ACF of stationary process with heavy tails with applications to ARCH. Ann. Statist. 26, 2049–2080.Google Scholar
Ghysels, E., Harvey, A. C. and Renault, E. (1996). Stochastic volatility. In Statistical Methods in Finance , eds Rao, C. R. and Maddala, G. S., North-Holland, Amsterdam, pp. 119–191.Google Scholar
Graversen, S. E. (2003). Proofs of two theorems on power variation and stochastic volatility. Unpublished manuscript, Department of Mathematical Sciences, Aarhus University.Google Scholar
Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Applications. Academic Press, San Diego.Google Scholar
Heston, S. L. (1993). A closed-form solution for options with stochastic volatility, with applications to bond and currency options. Rev. Financial Studies 6, 327–343.Google Scholar
Hull, J. and White, A. (1987). The pricing of options on assets with stochastic volatilities. J. Finance 42, 281–300.Google Scholar
Maheswaran, S. and Sims, C. A. (1993). Empirical implications of arbitrage-free asset markets. In Models, Methods and Applications of Econometrics , ed. Phillips, P. C. B. Blackwell, Oxford, pp. 301–316.Google Scholar
Mikosch, T. and Stâricâ, C. (2000). Limit theory for the sample autocorrelations and extremes of a GARCH(1, 1) process. Ann. Statist. 28, 1427–1451.Google Scholar
Nicolato, E. and Venardos, E. (2003). Option pricing in stochastic volatility models of the Ornstein-Uhlenbeck type. Math. Finance 13, 445–466.Google Scholar
Poterba, J. and Summers, L. (1986). The persistence of volatility and stock market fluctuations. Amer. Econom. Rev. 76, 1124–1141.Google Scholar
Renault, E. (1997). Econometric models of option pricing errors. In Advances in Economics and Econometrics: Theory and Applications , eds. Kreps, D. M. and Wallis, K. F., Cambridge University Press, pp. 223–278.Google Scholar
Schwert, G. W. (1989). Why does stock market volatility change over time? J. Finance 44, 1115–1153.Google Scholar
Shephard, N. (1996). Statistical aspects of ARCH and stochastic volatility. In Time Series Models in Econometrics, Finance and Other Fields , eds. Cox, D. R., Hinkley, D. V. and Barndorff-Nielsen, O. E., Chapman and Hall, London, pp. 1–67.Google Scholar
Shiryaev, A. N. (1999). Essentials of Stochastic Finance: Facts, Models and Theory. World Scientific, Singapore.Google Scholar
Taylor, S. J. and Xu, X. (1997). The incremental volatility information in one million foreign exchange quotations. J. Empirical Finance 4, 317–340.Google Scholar
Woerner, J. (2002). Variational sums and power variation: a unifying approach to model selection and estimation in semimartingale models. Statist. Decisions 21, 47–68.Google Scholar
Woerner, J. (2003). Estimation of integrated volatility in stochastic volatility models. Unpublished manuscript, OCIAM, Mathematical Institute, University of Oxford.Google Scholar