Hostname: page-component-8448b6f56d-cfpbc Total loading time: 0 Render date: 2024-04-24T07:24:25.866Z Has data issue: false hasContentIssue false
  • English
  • Français

INFERENCE FOR THE JUMP PART OF QUADRATIC VARIATION OF ITÔ SEMIMARTINGALES

Published online by Cambridge University Press:  18 August 2009

Almut E.D. Veraart
Get access Rights & Permissions [Opens in a new window]

Abstract

Recent research has focused on modeling asset prices by Itô semimartingales. In such a modeling framework, the quadratic variation consists of a continuous and a jump component. This paper is about inference on the jump part of the quadratic variation, which can be estimated by the difference of realized variance and realized multipower variation. The main contribution of this paper is twofold. First, it provides a bivariate asymptotic limit theory for realized variance and realized multipower variation in the presence of jumps. Second, this paper presents new, consistent estimators for the jump part of the asymptotic variance of the estimation bias. Eventually, this leads to a feasible asymptotic theory that is applicable in practice. Finally, Monte Carlo studies reveal a good finite sample performance of the proposed feasible limit theory.

Type
Research Article
Information
Econometric Theory , Volume 26 , Issue 2 , April 2010 , pp. 331 - 368
Copyright
Copyright © Cambridge University Press 2009

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.)

Footnotes

This paper is a revised part of my D.Phil. thesis and, therefore, I wish to thank my supervisors Neil Shephard and Matthias Winkel for their guidance and support throughout this project. Furthermore, I am grateful to Ole Barndorff–Nielsen, Mark Podolskij, and two anonymous referees for constructive comments on an earlier draft of this article. Financial support by the Rhodes Trust and by the Center for Research in Econometric Analysis of Time Series (CREATES), funded by the Danish National Research Foundation, is gratefully acknowledged.

References

REFERENCES

Aït-Sahalia, Y. & Jacod, J. (2009) Testing for jumps in a discretely observed process. Annals of Statistics 37, 184–222.CrossRefGoogle Scholar
Andersen, T.G. & Bollerslev, T. (1998) Answering the skeptics: Yes, standard volatility models do provide accurate forecasts. International Economic Review 39, 885–905.CrossRefGoogle Scholar
Andersen, T.G., Bollerslev, T., & Diebold, F.X. (2003) Some Like It Smooth, and Some Like It Rough: Untangling Continuous and Jump Components in Measuring, Modeling and Forecasting Asset Return Volatility. Duke University.CrossRefGoogle Scholar
Andersen, T.G., Bollerslev, T., & Diebold, F.X. (2007) Roughing it up: Including jump components in the measurement, modeling, and forecasting of return volatility. Review of Economics and Statistics 89(4), 701–720.CrossRefGoogle Scholar
Andersen, T.G., Bollerslev, T., Diebold, F.X., & Ebens, H. (2001) The distribution of realized stock return volatility. Journal of Financial Economics 61, 43–76.CrossRefGoogle Scholar
Andersen, T.G., Bollerslev, T., Diebold, F.X., & Labys, P. (2001) The distribution of exchange rate volatility. Journal of the American Statistical Association 96, 42–55. Correction published in 2003, vol. 98, p. 501.CrossRefGoogle Scholar
Andersen, T., Bollerslev, T., & Meddahi, N. (2005) Correcting the errors: Volatility forecast evaluation using high-frequency data and realized volatilities. Econometrica 73, 279–296.CrossRefGoogle Scholar
Bandi, F.M. & Russell, J.R. (2008) Microstructure noise, realised volatility, and optimal sampling. Review of Economic Studies 75(2), 339–369.CrossRefGoogle Scholar
Barndorff-Nielsen, O.E., Graversen, S.E., Jacod, J., Podolskij, M., & Shephard, N. (2006) A central limit theorem for realised power and bipower variations of continuous semimartingales, In Kabanov, Y. and Lipster, R. (eds.),From Stochastic Analysis to Mathematical Finance, Festschrift for Albert Shiryaev. Springer.Google Scholar
Barndorff-Nielsen, O.E., Graversen, S.E., Jacod, J., & Shephard, N. (2006) Limit theorems for bipower variation in financial econometrics. Econometric Theory 22, 677–719.CrossRefGoogle Scholar
Barndorff-Nielsen, O.E., Hansen, P.R., Lunde, A., & Shephard, N. (2008) Designing realised kernels to measure the ex-post variation of equity prices in the presence of noise. Econometrica 76, 1481–1536.Google Scholar
Barndorff-Nielsen, O.E. & Shephard, N. (2002) Econometric analysis of realised volatility and its use in estimating stochastic volatility models. Journal of the Royal Statistical Society B 64, 253–280.CrossRefGoogle Scholar
Barndorff-Nielsen, O.E. & Shephard, N. (2004) Power and bipower variation with stochastic volatility and jumps (with discussion). Journal of Financial Econometrics 2, 1–48.CrossRefGoogle Scholar
Barndorff-Nielsen, O.E. & Shephard, N. (2006a) Econometrics of testing for jumps in financial economics using bipower variation. Journal of Financial Econometrics 4, 1–30.CrossRefGoogle Scholar
Barndorff-Nielsen, O.E. & Shephard, N. (2006b) Impact of jumps on returns and realised variances: Econometric analysis of time-deformed Lévy processes. Journal of Econometrics 131, 217–252.CrossRefGoogle Scholar
Barndorff-Nielsen, O.E. & Shephard, N. (2007) Measuring the Impact of Jumps in Multivariate Price Processes Using Bipower Covariation. Unpublished paper, Nuffield College, Oxford.Google Scholar
Barndorff-Nielsen, O.E., Shephard, N., & Winkel, M. (2006) Limit theorems for multipower variation in the presence of jumps. Stochastic Processes and Their Applications 116, 796–806.CrossRefGoogle Scholar
Bollerslev, T. & Zhou, H. (2002) Estimating stochastic volatility diffusion using conditional moments of integrated volatility (plus corrections). Journal of Econometrics 109, 33–65.CrossRefGoogle Scholar
Carr, P., Geman, H., Madan, D.B., & Yor, M. (2005) Pricing options on realized variance. Finance and Stochastics 9, 453–475.CrossRefGoogle Scholar
Comte, F. & Renault, E. (1998) Long memory in continuous-time stochastic volatility models. Mathematical Finance 8, 291–257.CrossRefGoogle Scholar
Cont, R. & Mancini, C. (2007) Nonparametric tests for analyzing the fine structure of price fluctuations. Columbia University Financial Engineering Report No. 2007–13.CrossRefGoogle Scholar
Corradi, V. & Distaso, W. (2006) Semiparametric comparision of stochastic volatility models using realized measures. Review of Economic Studies 73(3), 635–667.CrossRefGoogle Scholar
Dette, H., Podolskij, M., & Vetter, M. (2006) Estimation of integrated volatility in continuous time financial models with applications to goodness-of-fit testing. Scandinavian Journal of Statistics 33, 259–278.CrossRefGoogle Scholar
Fleming, J., Kirby, C., & Ostdiek, B. (2003) The economic value of volatility timing using realized volatility. Journal of Financial Economics 67(3), 473–509.CrossRefGoogle Scholar
Giot, P. & Laurent, S. (2004) Modelling daily value-at-risk using realized volatility and arch type models. Journal of Empirical Finance 11(3), 379–398.CrossRefGoogle Scholar
Hansen, P.R. & Lunde, A. (2006) Realized variance and market microstructure noise. Journal of Business and Economic Statistics 24, 127–218.CrossRefGoogle Scholar
Huang, X. & Tauchen, G. (2005) The relative contribution of jumps to total price variance. Journal of Financial Econometrics 3(4), 456–499.CrossRefGoogle Scholar
Jacod, J. (1994) Limit of Random Measures Associated with the Increments of a Brownian Semimartingale. Preprint number 120, Laboratoire de Probabilitiés, Université Pierre et Marie Curie.Google Scholar
Jacod, J. (2006) Asymptotic Properties of Realized Power Variations and Related Functionals of Semimartingales: Multipower Variation. Unpublished paper, Université P. et M. Curie (Paris 6).Google Scholar
Jacod, J. (2007) Statistics and High Frequency Data. Unpublished paper, Université P. et M. Curie (Paris 6).Google Scholar
Jacod, J. (2008) Asymptotic properties of realized power variations and related functionals of semimartingales. Stochastic Processes and their Applications 118, 517–559.CrossRefGoogle Scholar
Jacod, J., Li, Y., Mykland, P.A., Podolskij, M., & Vetter, M. (2007) Microstructure Noise in the Continuous Case: The Pre-Averaging Approach. Unpublished paper, University of Chicago.CrossRefGoogle Scholar
Jacod, J. & Protter, P. (1998) Asymptotic error distribution for the Euler method for stochastic differential equations. The Annals of Probability 26(1), 267–307.CrossRefGoogle Scholar
Jacod, J. & Shiryaev, A.N. (2003) Limit Theorems for Stochastic Processes, 2 ed. Springer.CrossRefGoogle Scholar
Jacod, J. & Todorov, V. (2009) Testing for common arrivals of jumps for discretely observed multidimensional processes. Annals of Statistics 37, 1792–1838.CrossRefGoogle Scholar
Lee, S.S. & Mykland, P.A. (2008) Jumps in financial markets: A new nonparametric test and jump dynamics. Review of Financial Studies 21, 2535–2563.CrossRefGoogle Scholar
Mancini, C. (2001) Does our Favourite Index Jump or Not. Universita di Firenze.Google Scholar
Mancini, C. (2003), Statistics of a Poisson-Gaussian Process. Universita di Firenze.Google Scholar
Mancini, C. (2004) Estimation of the characteristics of the jumps of a general Poisson-diffusion model. Scandinavian Actuarial Journal 1, 42–52.CrossRefGoogle Scholar
Protter, P.E. (2004) Stochastic Integration and Differential Equations, 2nd ed. Springer.Google Scholar
Stentoft, L. (2008) Option Pricing Using Realized Volatility. CREATES Research Paper 2008–13.CrossRefGoogle Scholar
Todorov, V. (2008) Econometric Analysis of Jump–Driven Stochastic Volatility Models. Working paper, Northwestern University.Google Scholar
Todorov, V. (2009) Estimation of continuous–time stochastic volatility models with jumps using high–frequency data. Journal of Econometrics 148, 131–148.CrossRefGoogle Scholar
Todorov, V. & Bollerslev, T. (2007) Jumps and Betas: A New Framework for Disentangling and Estimating Systematic Risks. Working paper, Northwestern University and Duke University.CrossRefGoogle Scholar
Woerner, J.H.C. (2006) Power and multipower variation: Inference for high frequency data, In Shiryaev, A.N., Grossinõ, M.R., Oliviera, P., and Esquivel, M. (eds.), Stochastic Finance, pp. 343–364. Springer.CrossRefGoogle Scholar
Zhang, L., Mykland, P.A., & Aït-Sahalia, Y. (2005) A tale of two time scales: Determining integrated volatility with noisy high–frequency data. Journal of the American Statistical Association 100, 1394–1411.CrossRefGoogle Scholar