The Annals of Statistics

Volatility occupation times

Jia Li, Viktor Todorov, and George Tauchen

Full-text: Open access

Abstract

We propose nonparametric estimators of the occupation measure and the occupation density of the diffusion coefficient (stochastic volatility) of a discretely observed Itô semimartingale on a fixed interval when the mesh of the observation grid shrinks to zero asymptotically. In a first step we estimate the volatility locally over blocks of shrinking length, and then in a second step we use these estimates to construct a sample analogue of the volatility occupation time and a kernel-based estimator of its density. We prove the consistency of our estimators and further derive bounds for their rates of convergence. We use these results to estimate nonparametrically the quantiles associated with the volatility occupation measure.

Article information

Source
Ann. Statist., Volume 41, Number 4 (2013), 1865-1891.

Dates
First available in Project Euclid: 5 September 2013

Permanent link to this document
https://projecteuclid.org/euclid.aos/1378386241

Digital Object Identifier
doi:10.1214/13-AOS1135

Mathematical Reviews number (MathSciNet)
MR3127851

Zentralblatt MATH identifier
1277.62196

Subjects
Primary: 62F12: Asymptotic properties of estimators 62M05: Markov processes: estimation
Secondary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60J75: Jump processes

Keywords
Occupation time local approximation stochastic volatility spot variance quantiles nonparametric estimation high-frequency data

Citation

Li, Jia; Todorov, Viktor; Tauchen, George. Volatility occupation times. Ann. Statist. 41 (2013), no. 4, 1865--1891. doi:10.1214/13-AOS1135. https://projecteuclid.org/euclid.aos/1378386241


Export citation

References

  • [1] Bandi, F. M. and Phillips, P. C. B. (2003). Fully nonparametric estimation of scalar diffusion models. Econometrica 71 241–283.
  • [2] Barndorff-Nielsen, O. and Shephard, N. (2004). Power and bipower variation with stochastic volatility and jumps. Journal of Financial Econometrics 2 1–37.
  • [3] Barndorff-Nielsen, O. E. and Shephard, N. (2006). Econometrics of testing for jumps in financial economics using bipower variation. Journal of Financial Econometrics 4 1–30.
  • [4] Dassios, A. (1995). The distribution of the quantile of a Brownian motion with drift and the pricing of related path-dependent options. Ann. Appl. Probab. 5 389–398.
  • [5] de Haan, L. and Ferreira, A. (2006). Extreme Value Theory: An Introduction. Springer, New York.
  • [6] Eisenbaum, N. and Kaspi, H. (2007). On the continuity of local times of Borel right Markov processes. Ann. Probab. 35 915–934.
  • [7] Florens-Zmirou, D. (1993). On estimating the diffusion coefficient from discrete observations. J. Appl. Probab. 30 790–804.
  • [8] Geman, D. and Horowitz, J. (1980). Occupation densities. Ann. Probab. 8 1–67.
  • [9] Hall, P. (1992). The Bootstrap and Edgeworth Expansion. Springer, New York.
  • [10] Jacod, J. and Protter, P. (2012). Discretization of Processes. Stochastic Modelling and Applied Probability 67. Springer, Heidelberg.
  • [11] Jacod, J. and Reiß, M. (2012). A remark on the rates of convergence for integrated volatility estimation in the presence of jumps. Technical report. Available at arXiv:1209.4173v1.
  • [12] Jacod, J. and Rosenbaum, M. (2012). Quarticity and other functionals of volatility: Efficient estimation. Technical report. Available at arXiv:1207.3757.
  • [13] Mancini, C. (2009). Non-parametric threshold estimation for models with stochastic diffusion coefficient and jumps. Scand. J. Stat. 36 270–296.
  • [14] Protter, P. E. (2004). Stochastic Integration and Differential Equations: Stochastic Modelling and Applied Probability, 2nd ed. Applications of Mathematics (New York) 21. Springer, Berlin.
  • [15] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Grundlehren der Mathematischen Wissenschaften 293. Springer, Berlin.
  • [16] Sato, K.-i. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge Univ. Press, Cambridge.
  • [17] van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics 3. Cambridge Univ. Press, Cambridge.
  • [18] van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes: With Applications to Statistics. Springer, New York.
  • [19] Yor, M. (1995). The distribution of Brownian quantiles. J. Appl. Probab. 32 405–416.