(Volatility),, (High-Frequency Data),,, (Non-parametric) (Realized Variance, RV ) RV,, Andersen, Bollerslev, Diebold & Labys (3) Barndorff-Nielsen & S

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1 (QV), QV, (Realized Variance),,,, RV, Hansen & Lunde (b) RV,, RV Hansen & Lunde RV,, ( ), :,,,, Abstract Theory on quadratic variation which self can be as a good measure of volatility of returns shows that realized variance based on the effective (frictionless) stock prices can be used as an unbiased and consistent estimate of quadratic variation under certain mild conditions. However, the effective price is unobserved in the real market because it is contaminated by the so called microstructure noise and thus realized variance based on observed prices loses its good statistical properties, and even worse with increasing of sampling frequency. The existing disturbing of microstructure noise calls for noise-reduction technique. Based on the unbiased estimate of realized variance under serially correlated noise proposed by Hansen & Lunde (b), we went a further step by deducing a measure of variance of noise. Such measure utilizes the differences between the traditional estimate of RV under noise-i.i.d. assumption and Hansen & Lunde s unbiased estimate of RV under serially correlated noise assumption, all of which are estimated at different frequencies. We also proposed the methods of how to determine those sampling frequencies. Compared to what exist in most current literature that assume noise series are i.i.d., our measure based on a more loose assumption (serially correlated noise) and therefore is more realistic. Keywords: High-Frequency Data, Quadratic Variation, Realized Variance, Microstructure Noise, Signal-to-Noise Ratio

2 (Volatility),, (High-Frequency Data),,, (Non-parametric) (Realized Variance, RV ) RV,, Andersen, Bollerslev, Diebold & Labys (3) Barndorff-Nielsen & Shephard (), RV,, (Microstructure Noise), ;,,, RV,, RV QV;, RV QV,, Bandi & Russell (a) RV Aït-Sahalia, Mykland & Zhang (3), RV, RV Hansen & Lunde (a) i.i.d., RV ; Zhang, Mykland & Aït-Sahalia (3) (subsamples),, RV RV, RV ; Hansen & Lunde (b), RV Hansen & Lunde,,,,, 3.,, Hansen & Lunde (b) RV,.3,,,..,, Andersen, Bollerslev & Diebold ()

3 (Signal-to-Noise Ratio), 3.3, RV. (RV) I ( ) {p(t)} t I, I,, p, : ɛ(t) p(t) p (t), t [ a, b ]. (.) [ a, b ] I, m, a = t < t < t < < t m = b, m (b a)/m 3, b a, ( ) (Intraday Returns) y i,m p(t i ) p(t i ), i =,,, m. (.), yi,m p (t i ) p (t i ), i =,,, m. (.3) (Realized Variance, RV) : RV m i= y i,m, i =,,, m, (.) (semi-martingale), Andersen et al. (3), Barndorff-Nielsen & Shephard (), Andersen, Bollerslev & Diebold () RV (Quadratic Variation, QV), RV (Noise-in-Returns) : e i,m y i,m yi,m, i =,,, m. (.) [.], [.3] [.], e i,m = ɛ(t i ) ɛ(t i ), i =,,, m, (.6) MA(),,,,, 3 m m 3

4 , ɛ(t) i.i.d., p(t) Bandi & Russell (b), RV : [.], m yi,m = i= m i= y i,m + m m e i,m + yi,me i,m. (.7) i= i= (m ), ( m i= y i,m m i= e i,m ); ( m, ), m i= y i,m, RV ; ( ) RV Hansen & Lunde (a), Zhou (996) RV RV AC m m m yi,m + y i,m y i,m + y i,m y i+,m (.8) i= i= i= DJIA 3 RV RV, AC Hansen & Lunde, RV AC Bandi & Russell, Hansen & Lunde, ɛ(t) i.i.d.,, ɛ(t) i.i.d., RV. RV, : : ɛ ( ), ACF π(s) E [ ɛ(t) ɛ(t + s) ]. (.9), π(), i.i.d. [.] RV m i= y i,m RV, RV, y y(t) RV, RV (m) = m i= y i,m, [ a, b ] RV, (m),,, i.i.d., RV (m), RV (m) bias(rv (m) ) E [ RV (m) m i= y i,m ]. (.) i.i.d., Bandi & Russell(a, b) O( m) O(), m (m ), ; m,

5 7,, RV : i.i.d. p,, p 3.,,, : γ E(y i,me i,m ). (.),, 6 Hansen & Lunde (b), RV (m), RV (m) bias(rv (m) ) = mγ + m[π() π( m)]. (.) RV, m bias(rv (m) ), m ( ), bias(rv (m) ), ( ),, ( m ),, :. ρ, s > ρ, π(s) = ;. ρ t < t, s / [t ρ, t + ρ ], E[ɛ(s)(p (t ) p (t ))] = ; 3. i j, E[y i,m y j,m ] =,, ρ ; 3 ( ), RV RV (m) m AC yi,m + i= q m h=, q m, q m /m > ρ, [ bias(rv (m) AC ) = E RV (m) m AC γ,, m m h m y i,m y i+h,m, (.3) m h i= i= y i,m ] =. (.) 6 m m, ;, m,,. 7 ɛ ɛ(t) = R ρ θ(s)db(s), B,,, p p (t) = R t σ(s)dw(s), w, p 3, dp = µ(t)dt + σ(t)dw, m µ(t),

6 [.3] AC (, q m ) Hansen & Lunde (b),, RV,, RV (m) AC m m, m q m ( m, 6),,, m q m, w, ( ) w, m q m q m = (, m = (b a)/m), m, w (b a)/m q m m..3 RV, RV Bandi & Russell (b), ;., Bandi & Russell RV (m) ; RV RV (m) AC ( ) RV RV (m), RV RV (m) [.] [.], [ E RV (m) [ ] m E i= y i,m ] = m [ γ + π() π( m ) ]. (.) [ ], E RV (m) AC ( [.]), [ ] E RV (m) [ ] E RV ( em) AC = m [ γ + π() π( m ) ]. (.6), m m, m RV (m) AC RV, γ ; π() ( [.9] ), ; π( m ), ( ρ, 6 ), m > ρ, π( m ) =, m l, ml = (b a)/m l > ρ, π( ml ) = m l, [.6] [ E RV (m l) ] [ ] E RV ( em) AC = m l [ γ + π() ]. (.7), m, m, π( m ) π(), m h, π() π( mh ) m h, [.6] [ ] [ ] E RV (m h) E RV ( em) AC m h γ. γ = [ E lim m h RV (m h) ] [ ] E RV ( em) AC. (.8) m h 6

7 [.7], : [ ] [ ] E RV (m l) E RV ( em) AC π() = m l lim m h [ E RV (m h) ] [ ] E RV ( em) AC. m h, [a, b], ( ), π() = RV (m l) RV ( em) AC m l RV (m h ) RV ( em) AC, (.9) m h, RV (m l), RV (m h) RV ( em) AC, [.9] : (m l ) ( ml ρ ); (m h ) γ ( [.8]); RV ( ) ( m).... m h m h, (Tick Time Sampling TTS), (Calendar Time Sampling CTS, m ) ( ), TTS CTS,, TTS, m h γ m h,,,, TTS,, ( ),,.. m l m l, m l ml > ρ, m l, ml,,, m l, m l, Hansen & Lunde (a) ρ ρ, ( ) i.i.d p, RV( RV (m) ), MA() 7

8 . Zhou (996) RV RV AC ( [.8]) H : E[ɛ p ] = π(s) = s (, ), m, RV AC ( E[RV AC ] = IV), m, m, d t = RV (m ) AC,t RV (m ) AC,t, (.) t, E[d t ] =, d t ( IV ), d t, t n d t (m,m ) n t= (d t d), (.) n n, d t = n n t= d t, d, n, t (m,m ) N(, ), t (m,m ), t (m,m ) ρ m, m m, m ρ, t (m,m ) ; m, m ρ, t (m,m ), m < ρ m > ρ, t (m,m ), m m ρ, t (m,m ), m l. m, m =,, m > ρ. s, 3 m = m s, 3. t (m,m ),. t (m,m ), m, m m = m s, 3 t (m,m ), m, m < ρ m m m ml > ρ, m l ( )..3 RV m m, IV RV (m) AC, MSE [RV (m) (m) AC ] = E[(RV AC IV) ] RV (m) AC MSE [RV (m) AC ], m w IV,, m, w MSE [RV (m) AC ], m, w, IV,, MSE [RV (m) (m) AC ] m, w RV AC m, w, w ρ.. ml > ρ,, w ml m l 8

9 3, 3 36,,, A,,,,, 3.,,,,,,,,,,,, ( ARMA(p, q) ),, :.,,, e i,m MA() ( )., ( RV,, Bandi & Russell a), RV,, RV,,,, Bandi & Russell (b), [.] RV ( ) : A: 7.6, : 8.93, : 37.3, : 3., : 7., ,, ( ) ɛ(t), ρ 8 Bandi & Russell m m 3 ˆα+m ˆβ h ˆQi = ˆα = Pni= P mj= y j,i nm P ni= P mj= yj,i 3 nm Pni= P mj= y j,i nm «, ˆβ = «, ˆQi = m 3h P m j= y j,i y j,i i j, n ˆQ i, ˆQ i, m Bandi & Russell, ˆα ˆβ m, ˆQi m 9

10 (, ARMA(p, q) ),.. m, m 9, t (ml, 6 ).,,,,,,, ( ρ o ) 3. RV (RV), RV (m) ( [.], y ), RV (m) AC ( [.8]), RV (m) AC ( [.3], w) RV :, RV (m) QV( IV) ;, RV (m), ( ) i.i.d., RV (m) AC, i.i.d., ( ),, RV (m) AC, ( w, RV (m) AC ), (Signature Plot, Andersen, Bollerslev, Diebold & Labys ) RV,., RV, RV RV AC, ( RV 6, RV AC ),,, [.7],, RV m i= e i,m,, ( ), RV, 9,,, m, m m t (3, 6 ) ( ), t ( ml >3, 6 ), ml = 3, 3 ( ),, ml ρ,.. m l, (m), RV, RV AC, RV AC RV

11 . RV RV AC RV 6 ( ), RV AC ( ) ( ) RV AC RV AC, RV, RV AC 3. RV AC, RV AC RV AC,,, w = 3 w = 6 RV AC, w > ρ, RV AC RV,, ρ 3, RV AC, m < 8 m > 8,, ,.. ml, ρ, : A: 8, : 3, :, :, : 3 w = 3 RV AC,,, w = 3 ρ ( m ) ; w = 6 RV AC RV AC, w = 6 RV AC w = 3 ( ). RV 6, RV AC,, RV RV AC, ( RV) ( m > 9 ), RV,,, ( ),, RV AC, RV AC, (, RV AC )., RV AC RV RV AC,, RV RV AC, [.9] ml 3, w = 3 RV AC,

12 3.3, : m l, m h, m m w.3, ( )., m h, : m l, m w m l 3 m w w ml w > ρ, RV AC RV, ml ρ t- m l, m l t ( ml,6 ) ( [.] ) t RV AC ( ) d t ( [.] ) Q(), Q(), Ljung & Box (978) Q(n), n n, d t /,, t d t, t 3 RV ( em) AC m w ( ) RV,, RV ( em) AC, 3,, 3. RV,,,, (3 ) ( ),,,,,,,,,,,,,, Hansen & Lunde (b) RV,, i.i.d.

13 , Hansen & Lunde,,,,,,, Bandi & Russell (b),,, Aït-Sahalia, Y., Mykland, P. A. & Zhang, L. (3), How often to sample a continuous-time process in the presence of market microstructure noise. Working Paper w96, NBER. Andersen, T. G. & Bollerslev, T. (998), Answering the skeptics: Yes, standard volatility models do provide accurate forecasts, International Economic Review 39(), Andersen, T. G., Bollerslev, T. & Diebold, F. X. (), Parametric and nonparametric measurements of volatility, in Y. Aït-Sahalia & L. P. Hansen, eds, Handbook of Financial Econometrics, Amsterdam: North Holland. Forthcoming. Andersen, T. G., Bollerslev, T., Diebold, F. X. & Labys, P. (), Great realizations, Risk 3(3), 8. Andersen, T. G., Bollerslev, T., Diebold, F. X. & Labys, P. (3), Modeling and forecasting realized volatility, Econometrica 7(), Andersen, T. G., Bollerslev, T. & Lange, S. (999), Forecasting financial market volatility: Sample frequency vis-a-vis forecast horizon, Journal of Empirical Finance 6, Andersen, T. G., Bollerslev, T. & Meddahi, N. (), Analytic evaluation of volatility forecasts, International Economic Review. Forthcoming. Andreou, E. & Ghysels, E. (), Rolling sample volatility estimators: Some new theoretical, simulation and empirical results, Journal of Business and Economic Statistics, Bai, X., Russell, J. R. & Tiao, G. C. (), Beyond merton s utopia: Effects of non-normality and dependence on the precision of variance estimates using high-frequency financial data. Manuscript, Graduate School of Business, University of Chicago. Bandi, F. M. & Russell, J. R. (a), Microstructure noise, realized variance, and optimal sampling. Working paper. Bandi, F. M. & Russell, J. R. (b), Separating market microstructure noise from volatility. Working paper. Barndorff-Nielsen, O. E. & Shephard, N. (), Econometric analysis of realized volatility and its use in estimating stochastic volatility models, Journal of the Royal Statistical Society B(6, Part), 3 8. Barndorff-Nielsen, O. E. & Shephard, N. (3), Realized power variation and stochastic volatility models. forthcoming Bernouilli 9. 3

14 Bollerslev, T. (986), Generalized autoregressive conditional heteroscedasticity, Journal of Econometrics 3, Engle, R. F. (98), Autoregressive conditional heteroscedasticity with estimates of the variance of u.k. inflation, Econometrica, Galassi, M., Davies, J., Theiler, J., Gough, B., Jungman, G., Booth, M. & Rossi, F. (), GNU Scientific Library Reference Manual. Greene, W. H. (), Econometric Analysis, th edn, Prentice Hall, Upper Saddle River, New Jersey. Hansen, P. R. & Lunde, A. (a), Realized variance and market microstructure noise. Brown University Working Paper. Hansen, P. R. & Lunde, A. (b), An unbiased measure of realized variance. Working paper. Intel Inc (), Intel Math Kernel Library Reference Manual. Ljung, G. & Box, G. E. P. (978), On a measure of lack of fit in time series models, Biometrika 66, Lo, A. W. & MacKinlay, A. C. (99), An econometric analysis of nonsynchronous-trading, Journal of Econometrics, 8. Oomen, R. A. C. (), Modelling realized variance when returns are serially correlated. Manuscript, Warwick Business School, The University of Warwick. Roll, R. (98), A simple implicit measure of the effective bid-ask spread in an efficient market, Journal of Finance 39(), Taylor, S. J. (99), Modeling stochastic volatility: A review and comparative study, Mathematical Finance, 83. Tsay, R. S. (), Analysis of Financial Time Series, John Wiley & Sons, Inc, 6 Third Avenue, New York. Zhang, L., Mykland, P. A. & Aït-Sahalia, Y. (3), A tale of two time scales: Determining integrated volatility with noisy high frequency data. Working Paper w, NBER. Zhou, B. (996), High-frequency data and volatility in foreign-exchange rates, Journal of Business & Economic Statistics (),.

15 : Bin /7/8 9 3// /6/ 37 3// // 7 3/3/ //6 9 3// // 3// //9 6 3/3/ // 9 3// //9 83 3// //9 66 3/8/ /7/7 9 3// //9 3/3/ //9 3 3// //6 73 3/8/ //8 7 3/9/ /6/ 73 3// /7/3 89 3// /9/7 6 3/8/ /8/ 79 3/3/ //9 8 3/9/ /6/ 7 3/3/ /3/ // /6/ 98 3/8/ // 6 3/9/ /6/ 3// /7/7 79 3// //9 9 3// /7/8 98 3// /6/9 9 3// /8/ 9 3// /7/ 37 3// //3 89 3// //3 3/7/ /7/ 7 3/3/ /6/ 9 3// //7 69 3// // 7 3/3/ :. ; Bin 3. bin : 3,,, bin ;, bin, bin,

16 : : t (x, 6 ), x, tick-by-tick, 6

17 3:, RV, ml t- Q() Q() em w RVAC em % π()% % : m l, t- m l, ( [.]),., m l Q(), Q() d t = RV (m l) AC RV (min) AC Ljung-Box,. em, w RV ( em) AC RV ( em) AC RV π(), RV ( em) AC 7

18 . Highest(tick by tick). 3 seconds. 3 seconds Autocorrelation.. (a) A.. Highest(tick by tick). 3 seconds. 3 seconds Autocorrelation.. (b).. Highest(tick by tick). 3 seconds. 3 seconds Autocorrelation.. (c).. Highest(tick by tick). 3 seconds. 3 seconds Autocorrelation.. Lag (d). Lag. Highest(tick by tick). 3 seconds. 3 seconds Autocorrelation. Lag. Lag (e). Lag : 8

19 x x 3. Realized Variance 8 6 Realized Variance Sampling Frequency in Seconds Sampling Frequency in Seconds (a) A (b) Realized Variance x 8 6 Realized Variance x Sampling Frequency in Seconds Sampling Frequency in Seconds (c) (d) Realized Variance Sampling Frequency in Seconds (e) RV (m) RV (m) AC RV (m) AC, w = 3 x RV (m) AC, w = 6 : (Signature plot) 9

20 x 7 x 7 8 Sample Variance 3 Sample Variance Sampling Frequency in Seconds Autocorrelation Window Sampling Frequency in Seconds Autocorrelation Window 8 (a) A: m = 7, w = 8 (b) : m =, w = 3 x 7 x 6.. Sample Variance 8 6 Sample Variance Sampling Frequency in Seconds Autocorrelation Window Sampling Frequency in Seconds Autocorrelation Window 8 (c) : m =, w = (d) : m =, w = 78.8 Sample Variance Sampling Frequency in Seconds Autocorrelation Window 8 (e) : m = 7, w = 3 3: RV AC w > ρ, RV AC ( m ) (w)

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