Abstract
A central limit theorem for the realized volatility estimator of the integrated volatility based on a specific random sampling scheme is proved, where prices are sampled with every ‘continued price change’ in bid or ask quotation data. The estimator is shown to be robust to market microstructure noise induced by price discreteness and bid–ask spreads. More general sampling schemes also are treated in case that the price process is a diffusion.
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Aït-Sahalia, Y., Mykland, P.A., Zhang, L.: How often to sample a continuous-time process in the presence of market microstructure noise. Rev. Financ. Stud. 18, 351–416 (2005)
Barndorff-Nielsen, O.E., Shephard, N.: Econometric analysis of realized volatility and its use in estimating stochastic volatility models. J. R. Stat. Soc. Ser. B 64, 253–280 (2002)
Billingsley, P.: Convergence of Probability Measures, 2nd edn. Wiley, New York (1999)
Delattre, S., Jacod, J.: 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 (1997)
Fitzsimmons, P.J., Pitman, J.: Kac’s moment formula and the Feynman–Kac formula for additive functionals of a Markov process. Stoch. Process. Appl. 79, 117–134 (1999)
Fukasawa, M.: Estimation of the integrated volatility from space-discretized data. Unpublished manuscript (2007). http://www-csfi.sigmath.es.osaka-u.ac.jp/faculty/personal/fukasawa/fukasawa_0312.pdf
Genon-Catalot, V., Jacod, J.: On the estimation of the diffusion coefficient for multi-dimensional diffusion processes. Ann. Inst. H. Poincaré Probab. Stat. 29, 119–151 (1993)
Griffin, J.E., Oomen, R.C.A.: Sampling returns for realized variance calculation: tick time or transaction time? Econom. Rev. 27, 230–253 (2008)
Hansen, P.R., Lunde, A.: Realized variance and market microstructure noise. J. Bus. Econom. Stat. 24, 127–218 (2006)
Hasbrouck, J.: The dynamics of discrete bid and ask quotes. J. Finance 54, 2109–2142 (1999)
Jacod, J.: Limit of random measures associated with the increments of a Brownian semimartingale. Preprint number 120, Laboratoire de Probabilités, Université Pierre et Marie Curie, Paris (1994)
Jacod, J.: La variation quadratique du Brownien en présence d’erreurs d’arrondi. Astérisque 236, 155–161 (1996)
Jacod, J., Protter, P.: Asymptotic error distributions for the Euler method for stochastic differential equations. Ann. Probab. 26, 267–307 (1998)
Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin (2003)
Karatzas, I., Shreve, S.: Brownian Motion and Stochastic Calculus, 2nd edn. Springer, Berlin (1991)
Large, J.: Estimating quadratic variation when quoted prices change by a constant increment. Preprint, submitted to J. Econom. (2007). http://www.economics.ox.ac.uk/members/jeremy.large/
Li, Y., Mykland, P.A.: Determining the volatility of a price process in the presence of rounding errors. Preprint (2006). http://galton.uchicago.edu/~mykland/
Mykland, P.A., Zhang, L.: ANOVA for diffusions. Ann. Stat. 34, 1931–1963 (2006)
Niederhoffer, V., Osborne, M.F.M.: Market making and reversal on the stock exchange. J. Am. Stat. Assoc. 61(316), 897–916 (1966)
Oomen, R.C.A.: Properties of realized variance under alternative sampling schemes. J. Bus. Econ. Stat. 24, 219–237 (2006)
Protter, P.: Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin (2004)
Roll, R.: A simple implicit measure of the effective bid–ask spread in an efficient market. J. Finance 39, 1127–1139 (1984)
Zhang, L., Mykland, P.A., Aït-Sahalia, Y.: A tale of two time scales: determining integrated volatility with noisy high-frequency data. J. Am. Stat. Assoc. 100(472), 1394–1411 (2005)
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Fukasawa, M. Central limit theorem for the realized volatility based on tick time sampling. Finance Stoch 14, 209–233 (2010). https://doi.org/10.1007/s00780-008-0087-3
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DOI: https://doi.org/10.1007/s00780-008-0087-3