Electronic Journal of Statistics

Nonparametric estimation of the volatility function in a high-frequency model corrupted by noise

Axel Munk and Johannes Schmidt-Hieber

Full-text: Open access

Abstract

We consider the models Yi,n=0i/nσ(s)dWs+τ(i/n)εi,n, and i,n=σ(i/n)Wi/n+τ(i/n)εi,n, i=1,,n, where (Wt)t[0,1] denotes a standard Brownian motion and εi,n are centered i.i.d. random variables with E (εi,n2)=1 and finite fourth moment. Furthermore, σ and τ are unknown deterministic functions and (Wt)t[0,1] and (ε1,n,,εn,n) are assumed to be independent processes. Based on a spectral decomposition of the covariance structures we derive series estimators for σ2 and τ2 and investigate their rate of convergence of the MISE  in dependence of their smoothness. To this end specific basis functions and their corresponding Sobolev ellipsoids are introduced and we show that our estimators are optimal in minimax sense. Our work is motivated by microstructure noise models. A major finding is that the microstructure noise εi,n introduces an additionally degree of ill-posedness of 1/2; irrespectively of the tail behavior of εi,n. The performance of the estimates is illustrated by a small numerical study.

Article information

Source
Electron. J. Statist., Volume 4 (2010), 781-821.

Dates
First available in Project Euclid: 8 September 2010

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1283952132

Digital Object Identifier
doi:10.1214/10-EJS568

Mathematical Reviews number (MathSciNet)
MR2684388

Zentralblatt MATH identifier
1329.62366

Subjects
Primary: 62M09: Non-Markovian processes: estimation 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]
Secondary: 62G08: Nonparametric regression 62G20: Asymptotic properties

Keywords
Brownian motion variance estimation minimax rate microstructure noise Sobolev embedding

Citation

Munk, Axel; Schmidt-Hieber, Johannes. Nonparametric estimation of the volatility function in a high-frequency model corrupted by noise. Electron. J. Statist. 4 (2010), 781--821. doi:10.1214/10-EJS568. https://projecteuclid.org/euclid.ejs/1283952132


Export citation

References

  • [1] Alvarez, A., Panloup, F., Pontier, M., and Savy, N. Estimation of the instantaneous volatility. 2010. arxiv:0812.3538, Math arXiv, Preprint.
  • [2] Bandi, F. and Russell, J. Microstructure noise, realized variance, and optimal sampling., Rev. Econom. Stud., 75:339–369, 2008.
  • [3] Barndorff-Nielsen, O., Hansen, P., Lunde, A., and Stephard, N. Designing realised kernels to measure the ex-post variation of equity prices in the presence of noise., Econometrica, 76(6) :1481–1536, 2008.
  • [4] Barucci, E., Malliavin, P., and Mancino, M.E. Harmonic analysis methods for nonparametric estimation of volatility: theory and applications. In, Stochastic processes and applications to mathematical finance. Proceedings of the 5th Ritsumeikan international symposium, Kyoto, Japan, March 3-6, 2005, pages 1–34, 2006.
  • [5] Bissantz, N., Hohage, T., Munk, A., and Ruymgaart, F. Convergence rates of general regularization methods for statistical inverse problems and applications., SIAM J. Numerical Analysis, 45 :2610–2636, 2007.
  • [6] Britanak, V., Yip, P.C., and Yao, K.R., Discrete Cosine and Sine Transforms: General Properties, Fast Algorithms and Integer Approximations. Academic Press, 2006.
  • [7] Butucea, C. and Tsybakov, A.B. Sharp optimality for density deconvolution with dominating bias, I., Theory Probab. Appl., 52(1):111–128, 2007.
  • [8] Butucea, C. and Tsybakov, A.B. Sharp optimality for density deconvolution with dominating bias, II., Theory Probab. Appl., 52(2):336–349, 2007.
  • [9] Cai, T., Munk, A., and Schmidt-Hieber, J. Sharp minimax estimation of the variance of Brownian motion corrupted with Gaussian noise., Statist. Sinica, 2009. Forthcoming.
  • [10] Delaigle, A. and Gijbels, I. Practical bandwidth selection in deconvolution kernel density estimation., Comput. Stat. Data Anal., 45(2):249–267, 2004.
  • [11] Dey, A.K., Mair, B.A., and Ruymgaart, F.H. Cross-validation for parameter selection in inverse estimation problems., Scand. J. Statist., 23:609–620, 1996.
  • [12] Fan, J. On the optimal rates of convergence for nonparametric deconvolution problem., Ann. Statist., 19(3) :1257–1272, 1991.
  • [13] Fan, J., Jiang, J., Zhang, C., and Zhou, Z. Time-dependent diffusion models for term structure dynamics., Statist. Sinica, 13:965–992, 2003.
  • [14] Gloter, A. and Jacod, J. Diffusions with measurement errors. I. Local asymptotic normality., ESAIM Probab. Stat., 5:225–242, 2001.
  • [15] Gloter, A. and Jacod, J.. Diffusions with measurement errors. II. Optimal estimators., ESAIM Probab. Stat., 5:243–260, 2001.
  • [16] Jacod, J., Li, Y., Mykland, P.A., Podolskij, M., and Vetter, M. Microstructure noise in the continuous case: The pre-averaging approach., Stochastic Process. Appl., 119(7) :2249–2276, 2009.
  • [17] Jacod, J., Podolskij, M., and Vetter, M. Limit theorems for moving averages of discretized processes plus noise., Ann. Statist., 38(3) :1478–1545, 2010.
  • [18] Kinnebrock, S. Asymptotics for semimartingales and related processes with applications in econometrics. 2008. PhD, thesis.
  • [19] Madahavan, A. Market microstructure: A survey., Journal of Financial Markets, 3:205–258, 2000.
  • [20] Munk, A. and Schmidt-Hieber, J. Lower bounds for volatility estimation in microstructure noise models., Borrowing Strength: Theory Powering Applications - A Festschrift for Lawrence D. Brown, IMS Collection, 6:43–55, 2010.
  • [21] Neubauer, A. The convergence of a new heuristic parameter selection criterion for general regularization methods., Inverse Problems, 24 :055005, 2008.
  • [22] Pereverzev, S. and Schock, E. On the adaptive selection of the parameter in regularization of ill-posed problems., SIAM J. Numer. Anal., 43(5) :2060–2076, 2005.
  • [23] Podolskij, M. and Vetter, M. Estimation of volatility functionals in the simultaneous presence of microstructure noise and jumps., Bernoulli, 15(3):634–658, 2009.
  • [24] Reiß, M. Asymptotic equivalence and sufficiency for volatility estimation under microstructure noise. 2010. arxiv:1001.3006, Math arXiv, Preprint.
  • [25] Rosenbaum, M. Integrated volatility and round-off error., Bernoulli, 15(3):687–720, 2009.
  • [26] Steele, M., Stochastic Calculus and Financial Applications. Springer, New York, 2001.
  • [27] Steele, M. Minimum norm quadratic estimation of spatial variograms., J. Amer. Statist. Assoc., 82(399):765–772, 1987.
  • [28] Taylor, M., Partial Differential Equations III: Nonlinear Equations. Springer, New York, 1996.
  • [29] Tsybakov, A.B., Introduction to Nonparametric Estimation (Springer Series in Statistics XII). Springer-Verlag, New York, 2009.
  • [30] Zhang, L. Efficient estimation of stochastic volatility using noisy observations: A multi-scale approach., Bernoulli, 12 :1019–1043, 2006.
  • [31] Zhang, L., Mykland, P., and Ait-Sahalia, Y. A tale of two time scales: Determining integrated volatility with noisy high-frequency data., J. Amer. Statist. Assoc., 472 :1394–1411, 2005.