The Annals of Statistics

Efficient estimation of integrated volatility in presence of infinite variation jumps

Jean Jacod and Viktor Todorov

Full-text: Open access

Abstract

We propose new nonparametric estimators of the integrated volatility of an Itô semimartingale observed at discrete times on a fixed time interval with mesh of the observation grid shrinking to zero. The proposed estimators achieve the optimal rate and variance of estimating integrated volatility even in the presence of infinite variation jumps when the latter are stochastic integrals with respect to locally “stable” Lévy processes, that is, processes whose Lévy measure around zero behaves like that of a stable process. On a first step, we estimate locally volatility from the empirical characteristic function of the increments of the process over blocks of shrinking length and then we sum these estimates to form initial estimators of the integrated volatility. The estimators contain bias when jumps of infinite variation are present, and on a second step we estimate and remove this bias by using integrated volatility estimators formed from the empirical characteristic function of the high-frequency increments for different values of its argument. The second step debiased estimators achieve efficiency and we derive a feasible central limit theorem for them.

Article information

Source
Ann. Statist., Volume 42, Number 3 (2014), 1029-1069.

Dates
First available in Project Euclid: 20 May 2014

Permanent link to this document
https://projecteuclid.org/euclid.aos/1400592651

Digital Object Identifier
doi:10.1214/14-AOS1213

Mathematical Reviews number (MathSciNet)
MR3210995

Zentralblatt MATH identifier
1305.62146

Subjects
Primary: 60F05: Central limit and other weak theorems 60F17: Functional limit theorems; invariance principles
Secondary: 60G51: Processes with independent increments; Lévy processes 60G07: General theory of processes

Keywords
Quadratic variation Itô semimartingale integrated volatility central limit theorem

Citation

Jacod, Jean; Todorov, Viktor. Efficient estimation of integrated volatility in presence of infinite variation jumps. Ann. Statist. 42 (2014), no. 3, 1029--1069. doi:10.1214/14-AOS1213. https://projecteuclid.org/euclid.aos/1400592651


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References

  • [1] Aït-Sahalia, Y. and Jacod, J. (2009). Estimating the degree of activity of jumps in high frequency data. Ann. Statist. 37 2202–2244.
  • [2] Barndorff-Nielsen, O. E. and Shephard, N. (2004). Power and bipower variation with stochastic volatility and jumps. J. Financial Econom. 2 1–48.
  • [3] 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.
  • [4] Belomestny, D. (2011). Spectral estimation of the Lévy density in partially observed affine models. Stochastic Process. Appl. 121 1217–1244.
  • [5] Belomestny, D. and Panov, V. (2013). Abelian theorems for stochastic volatility models with application to the estimation of jump activity. Stochastic Process. Appl. 123 15–44.
  • [6] Bollerslev, T. and Todorov, V. (2011). Tail, fears, and risk premia. J. Finance 66 2165–2211.
  • [7] Chen, S. X., Delaigle, A. and Hall, P. (2010). Nonparametric estimation for a class of Lévy processes. J. Econometrics 157 257–271.
  • [8] Jacod, J. and Protter, P. (2012). Discretization of Processes. Stochastic Modelling and Applied Probability 67. Springer, Heidelberg.
  • [9] Jacod, J. and Reiss, M. (2014). A remark on the rates of convergence for integrated volatility estimation in the presence of jumps. Ann. Statist. To appear.
  • [10] Kappus, J. and Reiß, M. (2010). Estimation of the characteristics of a Lévy process observed at arbitrary frequency. Stat. Neerl. 64 314–328.
  • [11] Mancini, C. (2001). Disentangling the jumps of the diffusion in a geometric jumping Brownian motion. Giornale dell’Istituto Italiano degli Attuari LXIV 19–47.
  • [12] Mancini, C. (2011). The speed of convergence of the threshold estimator of integrated variance. Stochastic Process. Appl. 121 845–855.
  • [13] Neumann, M. H. and Reiß, M. (2009). Nonparametric estimation for Lévy processes from low-frequency observations. Bernoulli 15 223–248.
  • [14] Reiß, M. (2013). Testing the characteristics of a Lévy process. Stochastic Process. Appl. 123 2808–2828.
  • [15] Todorov, V. and Tauchen, G. (2011). Limit theorems for power variations of pure-jump processes with application to activity estimation. Ann. Appl. Probab. 21 546–588.
  • [16] Todorov, V. and Tauchen, G. (2012). The realized Laplace transform of volatility. Econometrica 80 1105–1127.
  • [17] Todorov, V. and Tauchen, G. (2012). Realized Laplace transforms for pure-jump semimartingales. Ann. Statist. 40 1233–1262.
  • [18] Vetter, M. (2010). Limit theorems for bipower variation of semimartingales. Stochastic Process. Appl. 120 22–38.