Bernoulli

  • Bernoulli
  • Volume 21, Number 4 (2015), 2393-2418.

Estimation of integrated volatility of volatility with applications to goodness-of-fit testing

Mathias Vetter

Full-text: Open access

Abstract

In this paper, we are concerned with nonparametric inference on the volatility of volatility process in stochastic volatility models. We construct several estimators for its integrated version in a high-frequency setting, all based on increments of spot volatility estimators. Some of those are positive by construction, others are bias corrected in order to attain the optimal rate $n^{-1/4}$. Associated central limit theorems are proven which can be widely used in practice, as they are the key to essentially all tools in model validation for stochastic volatility models. As an illustration we give a brief idea on a goodness-of-fit test in order to check for a certain parametric form of volatility of volatility.

Article information

Source
Bernoulli, Volume 21, Number 4 (2015), 2393-2418.

Dates
Received: July 2012
Revised: March 2014
First available in Project Euclid: 5 August 2015

Permanent link to this document
https://projecteuclid.org/euclid.bj/1438777598

Digital Object Identifier
doi:10.3150/14-BEJ648

Mathematical Reviews number (MathSciNet)
MR3378471

Zentralblatt MATH identifier
1125.62084

Keywords
central limit theorem goodness-of-fit testing high-frequency observations model validation stable convergence stochastic volatility model

Citation

Vetter, Mathias. Estimation of integrated volatility of volatility with applications to goodness-of-fit testing. Bernoulli 21 (2015), no. 4, 2393--2418. doi:10.3150/14-BEJ648. https://projecteuclid.org/euclid.bj/1438777598


Export citation

References

  • [1] Aït-Sahalia, Y. and Kimmel, R. (2007). Maximum likelihood estimation of stochastic volatility models. J. Financial Economics 134 507–551.
  • [2] Alvarez, A., Panloup, F., Pontier, M. and Savy, N. (2012). Estimation of the instantaneous volatility. Stat. Inference Stoch. Process. 15 27–59.
  • [3] Bandi, F. and Renò, R. (2008). Nonparametric stochastic volatility. Technical report.
  • [4] Barndorff-Nielsen, O. and Veraart, A. (2009). Stochastic volatility of volatility in continuous time. Technical report.
  • [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 33–68. Berlin: Springer.
  • [6] Barndorff-Nielsen, O.E., Hansen, P.R., Lunde, A. and Shephard, N. (2011). Multivariate realised kernels: Consistent positive semi-definite estimators of the covariation of equity prices with noise and non-synchronous trading. J. Econometrics 162 149–169.
  • [7] Bollerslev, T. and Zhou, H. (2002). Estimating stochastic volatility diffusion using conditional moments of integrated volatility. J. Econometrics 109 33–65.
  • [8] Chernov, M. and Ghysels, E. (2000). Estimation of stochastic volatility models for the purpose of option pricing. In Computational Finance 1999 (Y. Abu-Mostafa, B. LeBaron, A. Lo and A. Weigend, eds.) 567–581. Cambridge: MIT Press.
  • [9] Comte, F., Genon-Catalot, V. and Rozenholc, Y. (2010). Nonparametric estimation for a stochastic volatility model. Finance Stoch. 14 49–80.
  • [10] Dette, H. and Podolskij, M. (2008). Testing the parametric form of the volatility in continuous time diffusion models – A stochastic process approach. J. Econometrics 143 56–73.
  • [11] Dette, H., Podolskij, M. and Vetter, M. (2006). Estimation of integrated volatility in continuous-time financial models with applications to goodness-of-fit testing. Scand. J. Stat. 33 259–278.
  • [12] Genon-Catalot, V., Jeantheau, T. and Laredo, C. (1999). Parameter estimation for discretely observed stochastic volatility models. Bernoulli 5 855–872.
  • [13] Gloter, A. (2007). Efficient estimation of drift parameters in stochastic volatility models. Finance Stoch. 11 495–519.
  • [14] Heston, S. (1993). A closed-form solution for options with stochastic volatility with applications to bonds and currency options. Rev. Financial Studies 6 327–343.
  • [15] Hoffmann, M. (2002). Rate of convergence for parametric estimation in a stochastic volatility model. Stochastic Process. Appl. 97 147–170.
  • [16] Jacod, J. (1997). On continuous conditional Gaussian martingales and stable convergence in law. In Séminaire de Probabilités XXXI. Lecture Notes in Math. 1655 232–246. Berlin: Springer.
  • [17] Jacod, J. (2008). Asymptotic properties of realized power variations and related functionals of semimartingales. Stochastic Process. Appl. 118 517–559.
  • [18] Jacod, J. and Protter, P. (2012). Discretization of Processes. Stochastic Modelling and Applied Probability 67. Heidelberg: Springer.
  • [19] Jacod, J. and Rosenbaum, M. (2013). Quarticity and other functionals of volatility: Efficient estimation. Ann. Statist. 41 1462–1484.
  • [20] Jacod, J. and Shiryaev, A.N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Berlin: Springer.
  • [21] Jones, C.S. (2003). The dynamics of stochastic volatility: Evidence from underlying and options markets. J. Econometrics 116 181–224. Frontiers of financial econometrics and financial engineering.
  • [22] Mancini, C. (2009). Non-parametric threshold estimation for models with stochastic diffusion coefficient and jumps. Scand. J. Stat. 36 270–296.
  • [23] Podolskij, M. and Vetter, M. (2010). Understanding limit theorems for semimartingales: A short survey. Stat. Neerl. 64 329–351.
  • [24] Renò, R. (2006). Nonparametric estimation of stochastic volatility models. Econom. Lett. 90 390–395.
  • [25] Vetter, M. (2012). Estimation of correlation for continuous semimartingales. Scand. J. Stat. 39 757–771.
  • [26] Vetter, M. (2014). Supplement to “Estimation of integrated volatility of volatility with applications to goodness-of-fit testing.” DOI:10.3150/14-BEJ648SUPP.
  • [27] Vetter, M. and Dette, H. (2012). Model checks for the volatility under microstructure noise. Bernoulli 18 1421–1447.
  • [28] Wang, C.D. and Mykland, P.A. (2014). The estimation of leverage effect with high-frequency data. J. Amer. Statist. Assoc. 109 197–215.

Supplemental materials

  • Additional proofs for claims made in the article. We provide several proofs for either theorems from the main corpus or additional steps discussed in the \hyperref[app]Appendix.