Hostname: page-component-745bb68f8f-v2bm5 Total loading time: 0 Render date: 2025-01-12T10:40:45.488Z Has data issue: false hasContentIssue false
  • English
  • Français

ESTIMATING VOLATILITY FUNCTIONALS WITH MULTIPLE TRANSACTIONS

Published online by Cambridge University Press:  15 February 2016

Bing-Yi Jing ,
Zhi Liu  and
Xin-Bing Kong
Bing-Yi Jing
Affiliation:
Hong Kong University of Science and Technology, Hong Kong
Zhi Liu*
Affiliation:
University of Macau and UMacau Zhuhai Research Institute
Xin-Bing Kong
Affiliation:
Soochow University, China
*
*Address correspondence to Dr. Zhi Liu at [email protected].
Get access Rights & Permissions [Opens in a new window]

Abstract

The phenomenon of multiple transactions at each recording time is a common occurrence for high-frequency financial data because of the heavy trading of the market and limitation of the recording mechanism. This situation has existed for many years, but has become more common in recent years because of heavier trading. Surprisingly, there have been few studies on this important issue, in spite of some ad hoc approaches to treat multiple transactions. In this paper we investigate how to handle multiple transactions, particularly in the context of estimating the integrated volatility and integrated quarticity, which are of great interest in financial econometrics. Two approaches are proposed for this purpose, and their asymptotic properties are investigated. Their performances are confirmed by simulation studies. The estimators are also applied to some real world problems. The work represents only the first step in this direction, and some future research problems are discussed.

Type
ARTICLES
Information
Econometric Theory , Volume 33 , Issue 2 , April 2017 , pp. 331 - 365
Copyright
Copyright © Cambridge University Press 2016 

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.)

References

REFERENCES

Aït-Sahalia, Y., Fan, J., & Xiu, D. (2010) High-frequency covariance estimates with noisy and asynchronous financial data. Journal of the American Statistical Association 105(492), 1504–1517.CrossRefGoogle Scholar
Aït-Sahalia, Y. & Jacod, J. (2009) Estimating the degree of activity of jumps in high frequency financial data. Annals of Statistics 37(5A), 2202–2244.Google Scholar
Aït-Sahalia, Y. & Mykland, P. (2003) The effects of random and discrete sampling when estimating continuous-time diffusions. Econometrica 71(2), 483–549.Google Scholar
Aït-Sahalia, Y., Mykland, P., & Zhang, L. (2011) Ultra high frequency volatility estimation with dependent microstructure noise. Journal of Econometrics 160(1), 160–175.CrossRefGoogle Scholar
Aït-Sahalia, Y. & Mykland, P.A. (2004) Estimators of diffusions with randomly spaced discrete observations: A general theory. Annals of Statistics 32(5), 2186–2222.CrossRefGoogle Scholar
Barndorff-Nielsen, O., Graversen, S., Jacod, J., Podolskij, M., & Shephard, N. (2006) A central limit theorem for realised power and bipower variations of continuous semimartingales. In Kabanov, Yu., Liptser, R., Stoyanov, J. (eds.), From Stochastic Analysis to Mathematical Finance; the Shiryaev Festschrift. Springer.Google Scholar
Barndorff-Nielsen, O.E., Hansen, P.R., Lunde, A., & Shephard, N. (2008a) Designing realized kernels to measure the ex-post variation of equity prices in the presence of noise. Econometrica 76(6), 1481–1536.Google Scholar
Barndorff-Nielsen, O.E., Hansen, P.R., Lunde, A., & Shephard, N. (2008b) Multivariate realised kernels: Consistent positive semi-definite estimators of the covariation of equity prices with noise and non-synchronous trading. Journal of Econometrics 162(2), 149–169.Google Scholar
Barndorff-Nielsen, O.E. & Shephard, N. (2004) Econometric analysis of realized covariation: High frequency based covariance, regression, and correlation in financial economics. Econometrica 72, 885–925.CrossRefGoogle Scholar
Barndorff-Nielsen, O.E. & Shephard, N. (2006) Econometrics of testing for jumps in financial economics using bipower variation. Journal of Financial Econometrics 4(1), 1–30.Google Scholar
Hayashi, T., Jacod, J., & Yoshida, N. (2011) Irregular sampling and central limit theorems for power variations: The continuous case. Annales de l’Institut Henri Poincare Probability and Statistics 47(4), 1197–1218.Google Scholar
Jacod, J. (2012) Statistics and high frequency data. In Kessler, M., Lindner, A., & Sørensen, M. (eds.) Proceedings of the 7th Séminaire Européen de Statistique, La Manga, 2007: Statistical Methods for Stochastic Differential Equations.Google Scholar
Jacod, J., Li, Y., Mykland, P.A., Podolskij, M., & Vetter, M. (2009) Microstructure noise in the continuous case: The pre-averaging approach. Stochastic Processes and Their Applications 119(7), 2249–2276.Google Scholar
Jacod, J., Podolskij, M., & Vetter, M. (2010) Limit theorems for moving averages of discretized processes plus noise. Annals of Statistics 38(3), 1478–1545.CrossRefGoogle Scholar
Jacod, J. & Shiryayev, A.V. (2003) Limit Theorems for Stochastic Processes. Springer.CrossRefGoogle Scholar
Jing, B.Y., Kong, X.B., Liu, Z., & Mykland, P.A. (2012) On the jump activity index for semimartingales. Journal of Econometrics 166(2), 213–223.CrossRefGoogle Scholar
Li, Y., Mykland, P., Renault, E., Zhang, L., & Zheng, X. (2014) Realized volatility when sampling time are possibly endogenous. Econometric Theory 30, 580–605.Google Scholar
Mancini, C. (2009) Nonparametric threshold estimation for models with stochastic diffusion coefficient and jumps. Scandinavian Journal of Statistics 36, 270–296.Google Scholar
Mörters, P. & Peres, Y. (2010) Brownian Motion. Cambridge University Press.Google Scholar
Mykland, P. & Zhang, L. (2009) Inference for continuous semimartingales observed at high frequency: A general approach. Econometrica 77(5), 1403–1445.Google Scholar
Mykland, P.A. & Zhang, L. (2006) ANOVA for diffusion and Itô processes. Annals of Statistics 34(4), 1931–1963.CrossRefGoogle Scholar
Mykland, P.A. & Zhang, L. (2012) The econometrics of high frequency data. In Kessler, M., Lindner, A. and Sorensen, M., (eds.), Statistical Methods for Stochastic Differential Equations, 109–185.Google Scholar
Podolskij, M. & Vetter, M. (2009) Bipower-type estimation in a noisy diffusion setting. Stochastic Processes and Their Applications 119(9), 2803–2831.Google Scholar
Renyi, A. (1963) On stable sequences of events. Sankhya: The Indian Journal of Statistics, Series A 25(3), 293–302.Google Scholar
Todorov, V. & Bollerslev, T. (2010) Jumps and betas: A new framework for disentangling and estimating systematic risks. Journal of Econometrics 157, 220–235.Google Scholar
Todorov, V. & Tauchen, G. (2010) Activity signature functions for high frequency data analysis. Journal of Econometrics 154, 125–138.CrossRefGoogle Scholar
Tsay, R.S. (2005) Analysis of Financial Time Series. Wiley.Google Scholar
Xiu, D. (2010) Quasi-maximum likelihood estimation of volatility with high frequency data. Journal of Econometrics 159, 235–250.CrossRefGoogle Scholar
Zhang, L. (2006) Efficient estimation of stochastic volatility using noisy observations: A multi-scale approach. Bernoulli 12(6), 1019–1043.CrossRefGoogle Scholar
Zhang, L. (2011) Estimating covariation: Epps effect, microstructure noise. Journal of Econometrics 160(1), 33–47.Google Scholar
Zhang, L., Mykland, P., & 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(472), 1394–1411.Google Scholar