Bernoulli

  • Bernoulli
  • Volume 15, Number 3 (2009), 634-658.

Estimation of volatility functionals in the simultaneous presence of microstructure noise and jumps

Mark Podolskij and Mathias Vetter

Full-text: Open access

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.

Article information

Source
Bernoulli, Volume 15, Number 3 (2009), 634-658.

Dates
First available in Project Euclid: 28 August 2009

Permanent link to this document
https://projecteuclid.org/euclid.bj/1251463275

Digital Object Identifier
doi:10.3150/08-BEJ167

Mathematical Reviews number (MathSciNet)
MR2555193

Zentralblatt MATH identifier
1200.62131

Keywords
bipower variation central limit theorem finite activity jumps high-frequency data integrated volatility microstructure noise semimartingale theory subsampling

Citation

Podolskij, Mark; Vetter, Mathias. Estimation of volatility functionals in the simultaneous presence of microstructure noise and jumps. Bernoulli 15 (2009), no. 3, 634--658. doi:10.3150/08-BEJ167. https://projecteuclid.org/euclid.bj/1251463275


Export citation

References

  • [1] Ait-Sahalia, Y. and Jacod, J. (2009). Testing for jumps in a discretely observed process. Ann. Statist. 37 184–222.
  • [2] Aldous, D.J. and Eagleson, G.K. (1978). On mixing and stability of limit theorems. Ann. Probab. 6 325–331.
  • [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.
  • [4] Bandi, F.M. and Russell, J.R. (2008). Microstructure noise, realised variance, and optimal sampling. Rev. Econom. Stud. 75 339–369.
  • [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.
  • [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.
  • [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.
  • [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.
  • [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.
  • [13] Gloter, A. and Jacod, J. (2001). Diffusions with measurement errors. I – local asymptotic normality. ESAIM Probab. Statist. 5 225–242.
  • [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.
  • [16] Hayashi, T. and Yoshida, N. (2005). On covariance estimation of non-synchronously observed diffusion processes. Bernoulli 11 359–379.
  • [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.
  • [19] Jacod, J. and Protter P. (1998). Asymptotic error distributions for the Euler method for stochastic differential equations. Ann. Probab. 26 267–307.
  • [20] Jacod, J. and Shiryaev, A.N. (2003). Limit Theorems for Stochastic Processes. Springer: Berlin.
  • [21] Protter, P. (2005). Stochastic integration and differential equations. Springer: Berlin.
  • [22] Renyi, A. (1963). On stable sequences of events. Sankhya Ser. A 25 293–302.
  • [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.
  • [24] Zhang, L. (2006). Efficient estimation of stochastic volatility using noisy observations: A multiscale approach. Bernoulli 12 1019–1043.
  • [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.