Estimation of volatility functionals in the simultaneous presence of microstructure noise and jumps
Mark Podolskij and Mathias Vetter
Source: Bernoulli Volume 15, Number 3
(2009), 634-658.
Abstract
We propose a new concept of modulated bipower variation for diffusion models with microstructure noise. We show that this method provides simple estimates for such important quantities as integrated volatility or integrated quarticity. Under mild conditions the consistency of modulated bipower variation is proven. Under further assumptions we prove stable convergence of our estimates with the optimal rate n−1/4. Moreover, we construct estimates which are robust to finite activity jumps.
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Keywords: bipower variation; central limit theorem; finite activity jumps; high-frequency data; integrated volatility; microstructure noise; semimartingale theory; subsampling
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Permanent link to this document: http://projecteuclid.org/euclid.bj/1251463275
Digital Object Identifier: doi:10.3150/08-BEJ167
Mathematical Reviews number (MathSciNet): MR2555193
Zentralblatt MATH identifier: 05815949
References
[1] Ait-Sahalia, Y. and Jacod, J. (2009). Testing for jumps in a discretely observed process. Ann. Statist. 37 184–222.
Mathematical Reviews (MathSciNet): MR2488349
Zentralblatt MATH: 1155.62057
Digital Object Identifier: doi:10.1214/07-AOS568
Project Euclid: euclid.aos/1232115932
[2] Aldous, D.J. and Eagleson, G.K. (1978). On mixing and stability of limit theorems. Ann. Probab. 6 325–331.
Mathematical Reviews (MathSciNet): MR517416
Digital Object Identifier: doi:10.1214/aop/1176995577
[3] 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.
Mathematical Reviews (MathSciNet): MR1952727
Zentralblatt MATH: 1015.62107
Digital Object Identifier: doi:10.1198/016214501750332965
[4] Bandi, F.M. and Russell, J.R. (2008). Microstructure noise, realised variance, and optimal sampling. Rev. Econom. Stud. 75 339–369.
Mathematical Reviews (MathSciNet): MR2398721
Zentralblatt MATH: 1138.91394
Digital Object Identifier: doi:10.1111/j.1467-937X.2008.00474.x
[5] Barndorff-Nielsen, O.E., Graversen, S.E., Jacod, J., Podolskij, M. and Shephard, N. (2006). A central limit theorem for realised power and bipower variations of continuous semimartingales. In From Stochastic Calculus to Mathematical Finance: The Shiryaev Festschrift (Y. Kabanov, R. Lipster and J. Stoyanov, eds.) 33–68. Springer: Berlin.
Mathematical Reviews (MathSciNet): MR2233534
Digital Object Identifier: doi:10.1007/978-3-540-30788-4_3
[6] Barndorff-Nielsen, O.E., Hansen, P.R., Lunde, A. and Shephard, N. (2009). Designing realised kernels to measure the ex-post variation of equity prices in the presence of noise. Econometrica 76 1481–1536.
Mathematical Reviews (MathSciNet): MR2468558
Digital Object Identifier: doi:10.3982/ECTA6495
[7] Barndorff-Nielsen, O.E. and Shephard, N. (2002). Econometric analysis of realised volatility and its use in estimating stochastic volatility models. J. Roy. Statist. Soc. Ser. B 64 253–280.
Mathematical Reviews (MathSciNet): MR1904704
Zentralblatt MATH: 1059.62107
Digital Object Identifier: doi:10.1111/1467-9868.00336
[8] Barndorff-Nielsen, O.E. and Shephard, N. (2004). Power and bipower variation with stochastic volatility and jumps. J. Financial Econometrics 2 1–48.
[9] Barndorff-Nielsen, O.E. and Shephard, N. (2006). Econometrics of testing for jumps in financial economics using bipower variation. J. Financial Econometrics 4 1–30.
[10] Barndorff-Nielsen, O.E., Shephard, N. and Winkel, M. (2006). Limit theorems for multipower variation in the presence of jumps. Stochastic Process. Appl. 116 796–806.
Mathematical Reviews (MathSciNet): MR2218336
Zentralblatt MATH: 1096.60022
Digital Object Identifier: doi:10.1016/j.spa.2006.01.007
[11] Christensen, K. and Podolskij, M. (2006). Range-based estimation of quadratic variation. Working paper.
[12] Fan, J. and Wang, Y. (2007). Multi-scale jump and volatility analysis for high-frequency financial data. J. Amer. Statist. Assoc. 102 1349–1362.
Mathematical Reviews (MathSciNet): MR2372538
Zentralblatt MATH: 05564455
Digital Object Identifier: doi:10.1198/016214507000001067
[13] Gloter, A. and Jacod, J. (2001). Diffusions with measurement errors. I – local asymptotic normality. ESAIM Probab. Statist. 5 225–242.
Mathematical Reviews (MathSciNet): MR1875672
Digital Object Identifier: doi:10.1051/ps:2001110
[14] Gloter, A. and Jacod, J. (2001). Diffusions with measurement errors. II – measurement errors. ESAIM Probab. Statist. 5 243–260.
[15] Hansen, P.R. and Lunde, A. (2006). Realised variance and market microstructure noise. J. Bus. Econom. Statist. 24 127–161.
Mathematical Reviews (MathSciNet): MR2234447
[16] Hayashi, T. and Yoshida, N. (2005). On covariance estimation of non-synchronously observed diffusion processes. Bernoulli 11 359–379.
Mathematical Reviews (MathSciNet): MR2132731
Digital Object Identifier: doi:10.3150/bj/1116340299
Project Euclid: euclid.bj/1116340299
[17] Jacod, J. (1994). Limit of random measures associated with the increments of a Brownian semimartingale. Preprint number 120, Laboratoire de Probabilitiés, Univ. P. et M. Curie.
[18] Jacod, J., Li, Y., Mykland, P.A., Podolskij, M. and Vetter, M. (2009). Microstructure noise in the continuous case: The pre-averaging approach. Stoch. Process. Appl. To appear.
Mathematical Reviews (MathSciNet): MR2531091
Zentralblatt MATH: 1166.62078
Digital Object Identifier: doi:10.1016/j.spa.2008.11.004
[19] Jacod, J. and Protter P. (1998). Asymptotic error distributions for the Euler method for stochastic differential equations. Ann. Probab. 26 267–307.
Mathematical Reviews (MathSciNet): MR1617049
Zentralblatt MATH: 0937.60060
Digital Object Identifier: doi:10.1214/aop/1022855419
Project Euclid: euclid.aop/1022855419
[20] Jacod, J. and Shiryaev, A.N. (2003). Limit Theorems for Stochastic Processes. Springer: Berlin.
Mathematical Reviews (MathSciNet): MR1943877
[21] Protter, P. (2005). Stochastic integration and differential equations. Springer: Berlin.
Mathematical Reviews (MathSciNet): MR2273672
[22] Renyi, A. (1963). On stable sequences of events. Sankhya Ser. A 25 293–302.
Mathematical Reviews (MathSciNet): MR170385
[23] Woerner, J. H. C. (2006). Power and multipower variation: Inference for high frequency data. In Proceedings of the International Conference on Stochastic Finance 2004 (A.N. Shiryaev, M. do Rosario Grossinho, P. Oliviera and M. Esquivel, eds.) 343–364. Springer: Berlin.
Mathematical Reviews (MathSciNet): MR2230770
Zentralblatt MATH: 1142.62095
Digital Object Identifier: doi:10.1007/0-387-28359-5_12
[24] Zhang, L. (2006). Efficient estimation of stochastic volatility using noisy observations: A multiscale approach. Bernoulli 12 1019–1043.
Mathematical Reviews (MathSciNet): MR2274854
Digital Object Identifier: doi:10.3150/bj/1165269149
Project Euclid: euclid.bj/1165269149
[25] Zhang, L., Mykland, P.A. and Ait-Sahalia, Y. (2005). A tale of two time scales: Determining integrated volatility with noisy high-frequency data. J. Amer. Statist. Assoc. 472 1394–1411.
Mathematical Reviews (MathSciNet): MR2236450
Zentralblatt MATH: 1117.62461
Digital Object Identifier: doi:10.1198/016214505000000169
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