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1 Recent Studies on Estimation of Volatility in the Presence of Market Microstructure Noise RV MMN HP: t-kanatani@biwako.shiga-u.ac.jp 1

2 I ARCH Autoregressive Conditional Heteroscedasticity SV Stochastic Volatility Ghysels, Harvey, and Renault [1996] Morimune [2007] Andersen, Bollerslev, Diebold, and Labys [2003] Realized Volatility Realized Variance RV 1) RV RV RV [2007] [2007] RV 2 RV MMN MMN 1) Realized Volatility Realized Variance RV 2

3 Zhou [1996] RV MMN MMN Zhou [1996] Zhou [1996] Bandi and Russell [2008] Zhou [1996] RV MSE MSE RV MSE Zhang, Mykland, and Ait-Sahalia [2005] 2 RV Two Scale Estimator TS Hansen and Lunde [2006] Newey and West [1987] Kernel-Based Estimator TS Hansen and Lunde [2006] 30 MMN Barndorff-Nielsen, Hansen, Lunde, and Shephard [2008] Hansen and Lunde [2006] Kernel-Based Estimator Realized Kernel RK 3

4 Sun [2006] RK RV III RV IV RV V II 2 RV RV 2 MMN p t t 2) p t = p 0 + t 0 σ s dw s, σ t cadlag W V (dp t )=σt 2 [0, 1] Integrated Volatility Integrated Variance IV IV = 1 0 σ2 t dt IV 0=t 0,t 1,..., t n =1 {p ti } n i=0 t i t i 1 =1/n 3) 2) 3) MSE 1 4

5 x i = p ti p ti 1 [t i 1,t i ] p lim n n i=1 x 2 i }{{} RV (n) = 1 0 σ 2 t dt, (1) (1) 2 n i=1 x2 i Realized Volatility IV 1 0 σ2 t dt RV RV (n) = n i=1 x2 i RV n(rv (n) IV ) L MN (0, 2IQ), IQ = 1 0 σ4 t dt 4) MN Barndorff-Nielsen and Shephard [2002] MMN RV 1/n III Zhou [1996] RV p t ɛ t p t = p t + ɛ t, y i = p ti p ti 1 η i = ɛ ti ɛ ti 1 y i = x i + η i 4) IQ Integrated Quarticity (1/3) n i=1 x4 i 5

6 Zhou [1996] ɛ ti p t ɛ tj (j i) ω 2 E(y i y i 1 )= ω 2 Zhou [1996] Dacorogna, Gencay, Muller, Olsen, and Pictet [2001] MMN RV {y i } n i=1 RV RV (n) = n i=1 y2 i RV RV (n) = RV (n) +2 n x i η i + i=1 n ηi 2 (2) 3 (2) 2nω 2 n RV 5) 1 Bandi and Russell [2008] IBM Volatility signature plot Variance signature plot VSP VSP RV RV 6) RV Hansen and Lunde [2006] 30 VSP 1 1 RV VSP 5) Delattre and Jacod [1997] RV MMN 6) Andersen, Bollerslev, Diebold, and Labys [2000] Volatility signature plot Fang [1996] i=1 6

7 1: IBM Volatility signiture plot Bandi and Russell [2008] quote-to-quote RV (2) y i η i (3) RV (n) AC1 Bandi and Russell [2008] RV MMN Bandi and Russell [2008] Zhou [1996] RV MSE MSE n (IQ/E(η 2 ) 2 ) 1/3 n n/n RV MSE 7

8 Bandi and Russell [2008] 1 p lim n n n y q i = E(ηq ), q =2, 3, 4 i=1 IV MMN RV Zhou [1996] RV (n) AC1 = RV (n) + n n 1 y i y i 1 + y i y i+1 (3) i=2 nω 2 RV (n) 2nω 2 MSE 3σ 2 /(2ω 2 ) RV 1 MMN RV 15 Zhang, Mykland, and Ait-Sahalia [2005] K k i=1 y j,k = K y i+k(j 1)+k 1, i=1 j =1, 2,..., m k 8

9 m k =[n/k] [ ] k RV m k RV (K,k) = yj,k 2, {RV (K,k) } K k=1 RV (K,1) RV MMN RV K RV j=1 SRV (K) = 1 K K RV (K,k), (4) k=1 MSE 7) K Zhang, Mykland, and Ait-Sahalia [2005] (4) RV RV TS (K) = SRV (K) m n RV (n), (5) m =(1/K) K k=1 m k 2 Two Scale MMN RV n 1/6 IV MSE m (IQ/(3ω 2 )) 1/3 Zhang [2006] TS Multi-Scale MS n 1/4 Ait-Sahalia, Mykland, and Zhang [2005] Hansen and Lunde [2006] Newey and West [1987] 7) (5) 9

10 (3) H HL (H) = γ 0 + H h=1 ( 1 h 1 ) (γ h + γ h ), (6) H γ q = n i=1 y iy i q γ 0 = RV (n) H =1 HL (1) = Kernel-Based Estimator TS RV (n) AC1 Barndorff-Nielsen, Hansen, Lunde, and Shephard [2008] (6) RK (H) = γ 0 + H ( ) h 1 k (γ h + γ h ), (7) H h=1 k(x) [0, 1] Realized Kernel k(x) 1 RK 3 k(0) = 1 k(1) = 0 H = c 0 n 2/3 c 0 RK IV n 1/6 Bartlett Zhang, Mykland, and Ait-Sahalia [2005] TS Hansen and Lunde [2006] Kernel-Based Estimator k (0) = k (1) = 0 H = c 0 n 1/2 n 1/4 Ait-Sahalia, Mykland, and Zhang [2005] Turkey-Hanning 8 Zhang [2006] MS RK Cubic [0, ) γ h + γ h H = (ω/iq)n 1/2 n 1/4 Optimal 10

11 1: k(x) Bartlett(TS) 1 x n 1/ Epanechnikov 1 x 2 n 1/ nd order 1 2x + x 2 n 1/ Cubic(MS) 1 3x 2 +2x 3 n 1/ th order 1 10x 3 +15x 4 6x 5 n 1/ th order 1 15x 4 +24x 5 10x 6 n 1/ th order 1 21x 5 +35x 6 15x 7 n 1/ th order 1 28x 6 +48x 7 21x 8 n 1/ Parzen 1 6x 2 +6x 3 (0 x 1/2) n 1/ (1 x) 3 (1/2 <x 1) Turkey-Hanning1 sin 2 (π(1 x)/2) n 1/ Turkey-Hanning2 sin 2 (π(1 x) 2 /2) n 1/ Turkey-Hanning5 sin 2 (π(1 x) 5 /2) n 1/ Turkey-Hanning16 sin 2 (π(1 x) 16 /2) n 1/ Turkey-Hanning sin 2 (π exp( x)/2) n 1/ Quadratic spectral (3/x 2 )((sin(x)/x) cos x) n 1/ Dirichlet(Fourier) sin(x)/x n 1/ Fejer (sin(x)/x) 2 n 1/ Optimal (1 + x)e x n 1/ RK Ait-Sahalia, Mykland, and Zhang [2005] n 1/4 8 11

12 Ait-Sahalia, Mykland, and Zhang [2005] Turkey-Hanning 8 1 Dirichlet RK Malliavin and Mancino [2002] RK MMN Oya [2006] Nielsen and Frederiksen [2008] Kanatani [2004b] Mancino and Sanfelici [2007] RK 1 RK Barndorff-Nielsen, Hansen, Lunde, and Shephard [2007] Barndorff-Nielsen, Hansen, Lunde, and Shephard [2009] Sun [2006] 8) WRV = n n w ij y i y j, i=1 j=1 RK i j = h Sun [2006] MSE w ij = λ cosh(α(n +1 i j )) cosh(α(n +1 i j)), λ(λ +4) sinh(α(n + 1)) 8) Kanatani [2004a] Weighted Realized Volatility Sun [2006] Quadratic (Form) Estimator 12

13 λ = IV nω 2, ( 1 α = log 2 λ + 1 ) 4λ + λ2 +1, 2 Best Quadratic Unbiased Estimator BQU wij IV IV n 1/2 TS RK BQU n 1/4 V RV RV MMN Zhou [1996] RV Hansen and Lunde [2006] MMN O Hara [1995] 4 Cov(X, Y )=V (X + Y ) V (X Y ) 13

14 . X Y X + Y X Y 9) Hayashi and Yoshida [2005] 2 Cumulative Covariance Estimator CC Griffin and Oomen [2009] Bandi and Russell [2008] RV CC MSE CC Voev and Lunde [2008] CC Kanatani [2008] Kanatani [2009] CC [2007]., 2007, pp [2007]. Realized volatility, 58(4), pp Ait-Sahalia, Y., Mykland, P., and Zhang, L. [2005]. How often to sample a coutinuous -time process in the presence of market microstructure noise, Review of Financial Studies 18, pp Andersen, T. G., Bollerslev, T, Diebold, F. X. and Labys, P. [2000]. Great realizations, Risk 13(3), pp ) Realized Covariance RC Matrix Barndorff-Nielsen and Shephard [2004] RC Bandi and Russell [2005] 14

15 Andersen, T. G., Bollerslev, T, Diebold, F. X. and Labys, P. [2003]. Modeling and forecasting realized volatility, Econometrica 71(2), pp Bandi, F. M. and Russell, J. R. [2005]. Realized covariation, realized beta and microstructure noise, Working Paper, University of Chicago. Bandi, F. M. and Russell, J. R. [2008]. Microstructure noise, realized variance, and optimal sampling, The Review of Economic Studies 75, pp Barndorff-Nielsen, O., Hansen, P. R., Lunde, A., and Shephard, N. [2007]. Subsampling realized kernels, Journal of Econometrics, forthcoming. Barndorff-Nielsen, O., Hansen, P. R., Lunde, A., and Shephard, N. [2008]. Designing realized kernels to measure the ex post variation of equity prices in the presence of noise, Econometrica 76(6), pp Barndorff-Nielsen, O., Hansen, P. R., Lunde, A., and Shephard, N. [2009]. Realized kernels in Practice: Trades and Quotes, Econometrics Journal, forthcoming. Barndorff-Nielsen, O. and Shephard, N. [2002]. Econometric analysis of realized volatility and its use in estimating stochastic volatility models, Journal of the Royal Statistical Society, Series B 64, pp Barndorff-Nielsen, O. and Shephard, N. [2004]. Econometric analysis of realized covariation: high frequency covariance, regression and correlation in financial economics, Econometrica 72, pp Dacorogna, M. M., Gencay, R., Muller, U., Olsen, R. B., and Pictet, O. V. [2001]. An Introduction to High-Frequency Finance, Academic Press, London. Delattre, S. and Jacod, J. [1997]. A central limit theorem for nomalized functions of the increments of a diffusion process in the presence of round off errors, Bernoulli 3, pp

16 Fang, Y. [1996]. Volatility modeling and estimation of high-frequency data with Gaussian noise, Ph. d. thesis, MIT, Sloan School of Management. Ghysels, E., Harvey, A. C., and Renault, E. [1996]. Stochastic volatility, in Handbook of Econometrics, Vol. 14, Statistical Methods in Finance, ed. by G. S. Maddala and C. R. Rao. Amsterdam: North-Holland, Griffin, J. E. and Oomen, R. C. A. [2009]. Covariance measurement in the presence of non-synchronous trading and market microstructure noise, Working Paper, University of Warwick. Hansen, P. R. and Lunde, A. [2006]. Realized variance and market microstructure noise (with discussions), Journal of Business & Economic Statistics 24, pp Hayashi, T. and Yoshida, N. [2005]. On covariance estimation of nonsynchronously observed diffusion processes, Bernoulli 11, pp Kanatani, T. [2004a]. High frequency data and realized volatility, Ph. d. thesis, Graduate School of Economics, Kyoto University. Kanatani, T. [2004b]. Integrated volatility measuring from unvenly sampled observations, Economics Bulletin 3(36), pp Kanatani, T. [2008]. Finite sample analysis of weighted realized covariance with noisy asynchronous observations, Working Paper, Kyoto University. Kanatani, T. [2009]. Subsampling cumulative covariance estimator, Working Paper, Shiga University. Malliavin, P. and Mancino, M. E. [2002]. Fourier series method for measurement of multivariate volatilities, Finance and Stochastics 6, pp Mancino, M. E. and Sanfelici, S. [2007]. Robustness of Fourier estimator of in the presence of microstructure noise, Computational Statistics and Data Analysis 52, pp

17 Morimune, K. [2007]. Volatility models, The Japanese Economic Review 58(1), pp Nielsen, M. O. and Frederiksen, P. [2008]. Finite sample accuracy and choice of sampling frequency in integrated volatility estimation, Journal of Empirical Finance 15, pp Newey, W. and West, K. [1987]. A simple positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix, Econometrica 55, pp O Hara, M. [1995]. Market microstructure theory, Oxford: Blackwell Publishers. Oya, K. [2005]. Measurement of volatility of diffusion processes with noisy high frequency data, Proceeding of MODSIM05, pp Sun, Y. [2006]. Best quadratic unbiased estimators of integrated variance in the presence of market microstructure noise, Working Paper, University of California, San Diego. Voev, V. and Lunde, A. [2007]. Integrated covariance estimation using highfrequency data in the presence of the noise, Journal of Financial Econometrics 5, pp Zhang, L. [2006]. Efficient estimation of stochastic volatility using noisy observations: a multi-scale approach, Bernoulli 12, pp Zhang, L., Mykland, P., and Ait-Sahalia, Y. [2005]. A tale of two time scales: determining integrated volatility with noisy high-frequency data, Journal of the American Statistical Association 100, pp Zhou, B. [1996]. High-frequency data and volatility in foreign exchange rates, Journal of Business & Economic Statistics 14(1), pp

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