## Bernoulli

• Bernoulli
• Volume 15, Number 3 (2009), 687-720.

### Integrated volatility and round-off error

Mathieu Rosenbaum

#### Abstract

We consider a microstructure model for a financial asset, allowing for price discreteness and for a diffusive behavior at large sampling scale. This model, introduced by Delattre and Jacod, consists in the observation at the high frequency $n$, with round-off error $α_n$, of a diffusion on a finite interval. We give from this sample estimators for different forms of the integrated volatility of the asset. Our method is based on variational properties of the process associated with wavelet techniques. We prove that the accuracy of our estimation procedures is $α_n∨n^{−1/2}$. Using compensated estimators, limit theorems are obtained.

#### Article information

Source
Bernoulli, Volume 15, Number 3 (2009), 687-720.

Dates
First available in Project Euclid: 28 August 2009

https://projecteuclid.org/euclid.bj/1251463277

Digital Object Identifier
doi:10.3150/08-BEJ170

Mathematical Reviews number (MathSciNet)
MR2555195

Zentralblatt MATH identifier
1200.62132

#### Citation

Rosenbaum, Mathieu. Integrated volatility and round-off error. Bernoulli 15 (2009), no. 3, 687--720. doi:10.3150/08-BEJ170. https://projecteuclid.org/euclid.bj/1251463277

#### References

• [1] Aït-Sahalia, Y., Mykland, P.A. and Zhang, L. (2005). How often to sample a continuous time process in the presence of market microstructure noise. Rev. Financial Stud. 18 351–416.
• [2] Aldous, D.J. and Eagleson, G.K. (1978). On mixing and stability of limit theorems. Ann. Probab. 6 325–331.
• [3] Andersen, T., Bollerslev, T. and Meddahi, N. (2006). Realized volatility forecasting and market microstructure noise. Working paper.
• [4] Bandi, F.M. and Russel, J.R. (2008). Microstructure noise, realized variance and optimal sampling. Rev. Econom. Stud. 75 339–369.
• [5] Bandi, F.M., Russel, J.R. and Yang, C. (2006). Realized volatility and option pricing. Working paper.
• [6] Barndorff-Nielsen, O., Hansen, P., Lunde, A. and Shephard, N. (2008). Designing realised kernels to measure the ex-post variation of equity prices in the presence of noise. Econometrica 76 1481–1536.
• [7] Barndorff-Nielsen, O.E. and Shephard, N. (2002). Econometric analysis of realized volatility and its use in estimating stochastic volatility models. J. Roy. Statist. Soc. Ser. B 64 253–280.
• [8] Delattre, S. (1997). Estimation du coefficient de diffusion d’un processus de diffusion avec erreurs d’arrondi. Ph.D. thesis, University Paris 6.
• [9] Delattre, S. and Jacod, J. (1997). A central limit theorem for normalized functions of the increments of a diffusion process, in the presence of round-off errors. Bernoulli 3 1–28.
• [10] Gayraud, G. and Tribouley, K. (1999). Wavelet methods to estimate an integrated functional: Adaptivity and asymptotic law. Statist. Probab. Lett. 44 109–122.
• [11] Gloter, A. and Jacod, J. (1997). Diffusions with measurement errors, I. Local asymptotic normality, II. Optimal estimators. ESAIM PS 5 225–260.
• [12] Gonçalves, S. and Meddahi, N. (2005). Bootstrapping realized volatility. Econometrica. To appear.
• [13] Hansen, P.R. and Lunde, A. (2006). Realized variance and market microstructure noise. J. Bus. Econom. Statist. 24 127–161.
• [14] Jacod, J. (1997). On continuous conditional Gaussian martingales and stable convergence in law. Séminaire de Probabilités (Strasbourg) 31 232–246.
• [15] Jacod, J., Li, Y., Mykland, P.A., Podolskij, M. and Vetter, M. (2007). Microstructure noise in the continuous case: The pre-averaging approach. Working paper.
• [16] Jacod, J. and Protter, P. (1998). Asymptotic error distributions for the Euler method for stochastic differential equations. Ann. Probab. 26 267–307.
• [17] Jacod, J. and Shiryaev, A.N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. New York: Springer.
• [18] Kosulajeff, P. (1937). Sur la répartition de la partie fractionaire d’une variable aléatoire. Mat. Sb. (N.S.) 2 1017–1019.
• [19] Large, J. (2006). Estimating quadratic variation when quoted prices change by a constant increment. Working paper.
• [20] Li, Y. and Mykland, P. (2006). Determining the volatility of a price process in the presence of rounding errors. Technical Report 573, Univ. Chicago.
• [21] Li, Y. and Mykland, P. (2007). Are volatility estimators robust with respect to modeling assumptions? Bernoulli 13 601–622.
• [22] Meddahi, N. (2002). A theoretical comparison between integrated and realized volatility. J. Appl. Econometrics 17 475–508.
• [23] Gatheral, J. and Oomen, R. (2007). Zero-intelligence realized variance estimation. Working paper.
• [24] Rényi, A. (1963). On stable sequences of events. Sankhyā Ser. A 25 293–302.
• [25] Robert, C.Y. and Rosenbaum, M. (2009). A new approach for the dynamics of ultra high frequency data: The model with uncertainty zone. Working paper.
• [26] Robert, C.Y. and Rosenbaum, M. (2009). Volatility and covariation estimation when microstructure noise and trading times are endogenous. Working paper.
• [27] Rosenbaum, M. (2008). Estimation of the volatility persistence in a discretly observed diffusion model. Stochastic Process. Appl. 118 1434–1462.
• [28] Rosenbaum, M. (2007). Étude de quelques problèmes d’estimation statistique en finance. Ph.D. thesis.
• [29] Tukey, J.W. (1939). On the distribution of the fractional part of a statistical variable. Mat. Sb. (N.S.) 4 561–562.
• [30] Zhang, L. (2006). Efficient estimation of stochastic volatility using noisy observations: A multi-scale approach. Bernoulli 12 1019–1043.
• [31] Zhang, L., Mykland, P.A. and Aït-Sahalia, Y. (2005). A tale of two time scales: Determining integrated volatility with noisy high-frequency data. J. Amer. Statist. Assoc. 100 1394–1411.