The Annals of Statistics

Testing whether jumps have finite or infinite activity

Yacine Aït-Sahalia and Jean Jacod

Full-text: Open access

Abstract

We propose statistical tests to discriminate between the finite and infinite activity of jumps in a semimartingale discretely observed at high frequency. The two statistics allow for a symmetric treatment of the problem: we can either take the null hypothesis to be finite activity, or infinite activity. When implemented on high-frequency stock returns, both tests point toward the presence of infinite-activity jumps in the data.

Article information

Source
Ann. Statist., Volume 39, Number 3 (2011), 1689-1719.

Dates
First available in Project Euclid: 25 July 2011

Permanent link to this document
https://projecteuclid.org/euclid.aos/1311600280

Digital Object Identifier
doi:10.1214/11-AOS873

Mathematical Reviews number (MathSciNet)
MR2850217

Zentralblatt MATH identifier
1234.62117

Subjects
Primary: 62F12: Asymptotic properties of estimators 62M05: Markov processes: estimation
Secondary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60J60: Diffusion processes [See also 58J65]

Keywords
Semimartingale Brownian motion jumps finite activity infinite activity discrete sampling high frequency

Citation

Aït-Sahalia, Yacine; Jacod, Jean. Testing whether jumps have finite or infinite activity. Ann. Statist. 39 (2011), no. 3, 1689--1719. doi:10.1214/11-AOS873. https://projecteuclid.org/euclid.aos/1311600280


Export citation

References

  • Aït-Sahalia, Y. (2002). Telling from discrete data whether the underlying continuous-time model is a diffusion. J. Finance 57 2075–2112.
  • Aït-Sahalia, Y. and Jacod, J. (2009a). Estimating the degree of activity of jumps in high frequency financial data. Ann. Statist. 37 2202–2244.
  • Aït-Sahalia, Y. and Jacod, J. (2009b). Testing for jumps in a discretely observed process. Ann. Statist. 37 184–222.
  • Aït-Sahalia, Y. and Jacod, J. (2011). Supplement to “Testing whether jumps have finite or infinite activity.” DOI:10.1214/11-AOS873SUPP.
  • Andersen, T. G., Bollerslev, T. and Diebold, F. X. (2007). Roughing it up: Including jump components in the measurement, modeling, and forecasting of return volatility. Rev. Econom. Statist. 89 701–720.
  • Ball, C. A. and Torous, W. N. (1983). A simplified jump process for common stock returns. Journal of Financial and Quantitative Analysis 18 53–65.
  • Barndorff-Nielsen, O. E. and Shephard, N. (2004). Power and bipower variation with stochastic volatility and jumps (with discussion). Journal of Financial Econometrics 2 1–48.
  • Bates, D. S. (1991). The crash of ’87: Was it expected? The evidence from options markets. J. Finance 46 1009–1044.
  • Belomestny, D. (2010). Spectral estimation of the fractional order of a Lévy process. Ann. Statist. 38 317–351.
  • Carr, P. and Wu, L. (2003a). The finite moment log stable process and option pricing. J. Finance 58 753–777.
  • Carr, P. and Wu, L. (2003b). What type of process underlies options? A simple robust test. J. Finance 58 2581–2610.
  • Carr, P., Geman, H., Madan, D. B. and Yor, M. (2002). The fine structure of asset returns: An empirical investigation. Journal of Business 75 305–332.
  • Cont, R. and Mancini, C. (2009). Nonparametric tests for probing the nature of asset price processes. Technical report, Univ. Firenze.
  • Eberlein, E. and Keller, U. (1995). Hyperbolic distributions in finance. Bernoulli 1 281–299.
  • Huang, X. and Tauchen, G. T. (2005). The relative contribution of jumps to total price variance. Journal of Financial Econometrics 4 456–499.
  • Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Springer, Berlin.
  • Jacod, J., Li, Y., Mykland, P. A., Podolskij, M. and Vetter, M. (2009). Microstructure noise in the continuous case: The pre-averaging approach. Stochastic Process. Appl. 119 2249–2276.
  • Jiang, G. J. and Oomen, R. C. A. (2008). Testing for jumps when asset prices are observed with noise—a “swap variance” approach. J. Econometrics 144 352–370.
  • Lee, S. S. and Hannig, J. (2010). Detecting jumps from Lévy jump diffusion processes. Journal of Financial Economics 96 271–290.
  • Lee, S. and Mykland, P. A. (2008). Jumps in financial markets: A new nonparametric test and jump dynamics. Review of Financial Studies 21 2535–2563.
  • Madan, D. B., Carr, P. P. and Chang, E. E. (1998). The Variance Gamma process and option pricing. European Finance Review 2 79–105.
  • Madan, D. B. and Seneta, E. (1990). The Variance Gamma (V.G.) model for share market returns. Journal of Business 63 511–524.
  • Mancini, C. (2001). Disentangling the jumps of the diffusion in a geometric jumping Brownian motion. Giornale dell’Istituto Italiano degli Attuari LXIV 19–47.
  • Merton, R. C. (1976). Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics 3 125–144.
  • Todorov, V. and Tauchen, G. (2010). Activity signature functions for high-frequency data analysis. J. Econometrics 154 125–138.

Supplemental materials

  • Supplementary material: Supplement to “Testing whether jumps have finite or infinite activity”. This supplementary article contains a few additional technical details about the assumptions made in this paper, and the proofs of all results.