Ultra High Frequency Volatility Estimation with Market Microstructure Noise. Yacine Aït-Sahalia. Per A. Mykland. Lan Zhang

Transcription

1 Ultra High Frequency Volatility Estimation with Market Microstructure Noise Yacine Aït-Sahalia Princeton University Per A. Mykland The University of Chicago Lan Zhang Carnegie-Mellon University

2 1. Introduction

3 1. Introduction Observed transaction price = unobservable efficient price + some noise component due to the imperfections of the trading process Y τ = X τ + ε τ

4 1. Introduction Observed transaction price = unobservable efficient price + some noise component due to the imperfections of the trading process Y τ = X τ + ε τ ε summarizes a diverse array of market microstructure effects, either informational or not: bid-ask bounces, discreteness of price changes, differences in trade sizes or informational content of price changes, gradual response of prices to a block trade, the strategic component of the order flow, inventory control effects, etc.

5 We study the implications of such a data generating process for the estimation of the volatility of the efficient log-price process dx t = µ t dt + σ t dw t using discretely sampled data on the transaction price process at times 0,,..., n = T.

6 We study the implications of such a data generating process for the estimation of the volatility of the efficient log-price process dx t = µ t dt + σ t dw t using discretely sampled data on the transaction price process at times 0,,..., n = T. Without noise, the realized volatility n i=1 (X ti+1 X ti ) 2 estimates the quadratic variation T 0 σ 2 t dt.

7 We study the implications of such a data generating process for the estimation of the volatility of the efficient log-price process dx t = µ t dt + σ t dw t using discretely sampled data on the transaction price process at times 0,,..., n = T. Without noise, the realized volatility n i=1 (X ti+1 X ti ) 2 estimates the quadratic variation T 0 σ 2 t dt. In theory, sampling as often as possible will produce in the limit a perfect estimate of that quantity.

8 We show that the situation changes radically in the presence of market microstructure noise:

9 We show that the situation changes radically in the presence of market microstructure noise: Computing RV using all the data (say every second) leads to an estimate of the variance of the noise, not the quadratic variation that one seeks to estimate.

10 We show that the situation changes radically in the presence of market microstructure noise: Computing RV using all the data (say every second) leads to an estimate of the variance of the noise, not the quadratic variation that one seeks to estimate. In practice, people sample sparsely at some lower frequency (5 mn to 30 mn usually)

11 We show that the situation changes radically in the presence of market microstructure noise: Computing RV using all the data (say every second) leads to an estimate of the variance of the noise, not the quadratic variation that one seeks to estimate. In practice, people sample sparsely at some lower frequency (5 mn to 30 mn usually) If one insists upon sampling sparsely, what is the right frequency? We show how to determine the optimal sparse frequency.

12 We show that the situation changes radically in the presence of market microstructure noise: Computing RV using all the data (say every second) leads to an estimate of the variance of the noise, not the quadratic variation that one seeks to estimate. In practice, people sample sparsely at some lower frequency (5 mn to 30 mn usually) If one insists upon sampling sparsely, what is the right frequency? We show how to determine the optimal sparse frequency.

13 But even if sampling optimally, one is throwing away a large amount of data.

14 But even if sampling optimally, one is throwing away a large amount of data. For example, if T = 1 NYSE day and transactions occur every δ = 1 second, the original sample size is n = T/δ = 23, 400.

15 But even if sampling optimally, one is throwing away a large amount of data. For example, if T = 1 NYSE day and transactions occur every δ = 1 second, the original sample size is n = T/δ = 23, 400. But sampling sparsely even at the highest frequency used by empirical researchers (once every 5 mn) means throwing away 299 out of every 300 observations: the sample size used is only n sparse = 78.

16 But even if sampling optimally, one is throwing away a large amount of data. For example, if T = 1 NYSE day and transactions occur every δ = 1 second, the original sample size is n = T/δ = 23, 400. But sampling sparsely even at the highest frequency used by empirical researchers (once every 5 mn) means throwing away 299 out of every 300 observations: the sample size used is only n sparse = 78. This violates one of the most basic principles of statistics.

17 But even if sampling optimally, one is throwing away a large amount of data. For example, if T = 1 NYSE day and transactions occur every δ = 1 second, the original sample size is n = T/δ = 23, 400. But sampling sparsely even at the highest frequency used by empirical researchers (once every 5 mn) means throwing away 299 out of every 300 observations: the sample size used is only n sparse = 78. This violates one of the most basic principles of statistics.

18 We propose a solution which makes use of the full data sample:

19 We propose a solution which makes use of the full data sample: Our estimator (TSRV) is based on subsampling, averaging and bias-correction.

20 We propose a solution which makes use of the full data sample: Our estimator (TSRV) is based on subsampling, averaging and bias-correction. This is the only consistent estimator (we know of) for quadratic variation in the presence of market microstructure noise.

21 We propose a solution which makes use of the full data sample: Our estimator (TSRV) is based on subsampling, averaging and bias-correction. This is the only consistent estimator (we know of) for quadratic variation in the presence of market microstructure noise.

22 2. Outline

23 2. Outline Without market microstructure noise

24 2. Outline Without market microstructure noise What happens when noise is present

25 2. Outline Without market microstructure noise What happens when noise is present The fifth best approach: use all the data to compute RV

26 2. Outline Without market microstructure noise What happens when noise is present The fifth best approach: use all the data to compute RV The fourth best approach: sample sparsely (5, 10 or 15 mn) to compute RV

27 2. Outline Without market microstructure noise What happens when noise is present The fifth best approach: use all the data to compute RV The fourth best approach: sample sparsely (5, 10 or 15 mn) to compute RV The third best approach: sample sparsely at an optimally determined frequency to compute RV

28 2. Outline Without market microstructure noise What happens when noise is present The fifth best approach: use all the data to compute RV The fourth best approach: sample sparsely (5, 10 or 15 mn) to compute RV The third best approach: sample sparsely at an optimally determined frequency to compute RV

29 The second best approach: subsampling and averaging

30 The second best approach: subsampling and averaging The first best approach, TSRV: use the fifth best to bias-correct the second best

31 The second best approach: subsampling and averaging The first best approach, TSRV: use the fifth best to bias-correct the second best Monte Carlos

32 The second best approach: subsampling and averaging The first best approach, TSRV: use the fifth best to bias-correct the second best Monte Carlos Two extensions: time series dependence in the noise, and MSRV

33 The second best approach: subsampling and averaging The first best approach, TSRV: use the fifth best to bias-correct the second best Monte Carlos Two extensions: time series dependence in the noise, and MSRV Data analysis: INTC, MSFT

34 The second best approach: subsampling and averaging The first best approach, TSRV: use the fifth best to bias-correct the second best Monte Carlos Two extensions: time series dependence in the noise, and MSRV Data analysis: INTC, MSFT

35 3. Without Market Microstructure Noise

36 3. Without Market Microstructure Noise Suppose first that σ is constant. Without noise, the log-returns are iid N(0, σ 2 ). The MLE for σ 2 coincides with the realized volatility of the process, ˆσ 2 = 1 T n i=1 (X ti+1 X ti ) 2,

37 3. Without Market Microstructure Noise Suppose first that σ is constant. Without noise, the log-returns are iid N(0, σ 2 ). The MLE for σ 2 coincides with the realized volatility of the process, ˆσ 2 = 1 T n i=1 (X ti+1 X ti ) 2, T 1/2 (ˆσ 2 σ 2) N(0, T 2σ4 )

38 3. Without Market Microstructure Noise Suppose first that σ is constant. Without noise, the log-returns are iid N(0, σ 2 ). The MLE for σ 2 coincides with the realized volatility of the process, ˆσ 2 = 1 T n i=1 (X ti+1 X ti ) 2, T 1/2 (ˆσ 2 σ 2) N(0, T 2σ4 ) Thus selecting as small as possible is optimal for the purpose of estimating σ 2.

39 When volatility is stochastic, dx t = σ t dw t :

40 When volatility is stochastic, dx t = σ t dw t : Realized volatility n i=1 (X ti+1 X ti ) 2 estimates the quadratic variation T 0 σ 2 t dt.

41 When volatility is stochastic, dx t = σ t dw t : Realized volatility n i=1 (X ti+1 X ti ) 2 estimates the quadratic variation T 0 σ 2 t dt. The sum converges to the integral, with a known distribution: Jacod (1994), Jacod and Protter (1998), etc.

42 When volatility is stochastic, dx t = σ t dw t : Realized volatility n i=1 (X ti+1 X ti ) 2 estimates the quadratic variation T 0 σ 2 t dt. The sum converges to the integral, with a known distribution: Jacod (1994), Jacod and Protter (1998), etc. As in the constant σ case, selecting as small as possible (= n as large as possible) is optimal.

43 4. With Noise

44 4. With Noise When dx t = σ t dw t, but we observe X with noise, the object of interest remains the quadratic variation of X : X, X T = 0 σ2 t dt over a fixed time period [0, T ], or possibly several such time periods. T

45 4. With Noise When dx t = σ t dw t, but we observe X with noise, the object of interest remains the quadratic variation of X : X, X T = 0 σ2 t dt over a fixed time period [0, T ], or possibly several such time periods. T Asymptotics are in 0, with T fixed.

46 4. With Noise When dx t = σ t dw t, but we observe X with noise, the object of interest remains the quadratic variation of X : X, X T = 0 σ2 t dt over a fixed time period [0, T ], or possibly several such time periods. T Asymptotics are in 0, with T fixed. The usual estimator of X, X T is the realized volatility [Y, Y ] T = n i=1 (Y ti+1 Y ti ) 2.

47 4.1. Completely Ignoring the Noise: Fifth Best

48 4.1. Completely Ignoring the Noise: Fifth Best We show that, if one uses all the data (say sampled every second),

49 4.1. Completely Ignoring the Noise: Fifth Best We show that, if one uses all the data (say sampled every second), [Y, Y ] (all) T L X, X T }{{} object of interest + 2nE[ε 2 ] }{{} bias due to noise + 2T T n 0 σ4 t dt ] 1/2 due to noise }{{}} due {{ to discretization} total variance + [ 4nE[ε 4 ] }{{} conditionally on the X process. Z total.

50 4.1. Completely Ignoring the Noise: Fifth Best We show that, if one uses all the data (say sampled every second), [Y, Y ] (all) T L X, X T }{{} object of interest + 2nE[ε 2 ] }{{} bias due to noise + 2T T n 0 σ4 t dt ] 1/2 due to noise }{{}} due {{ to discretization} total variance + [ 4nE[ε 4 ] }{{} conditionally on the X process. Z total.

51 4.2. Sampling Sparsely at an Arbitrary Frequency: Fourth Best

52 4.2. Sampling Sparsely at an Arbitrary Frequency: Fourth Best Of course, sampling as prescribed by [Y, Y ] (all) in practice T is not what people do

53 4.2. Sampling Sparsely at an Arbitrary Frequency: Fourth Best Of course, sampling as prescribed by [Y, Y ] (all) in practice T is not what people do Instead, they use the estimator [Y, Y ] (sparse) T constructed by summing squared log-returns at some lower frequency: 5 mn, or 10, 15, 30 mn, typically.

54 4.2. Sampling Sparsely at an Arbitrary Frequency: Fourth Best Of course, sampling as prescribed by [Y, Y ] (all) in practice T is not what people do Instead, they use the estimator [Y, Y ] (sparse) T constructed by summing squared log-returns at some lower frequency: 5 mn, or 10, 15, 30 mn, typically.

55 Here is the fourth best estimator for different values of, averaged for the 30 DJIA stocks and the last 10 trading days in April 2004: sec 30sec 1mn 2mn 3mn 4mn 5mn As = T/n 0, the graph shows that the estimator diverges as predicted by our result (2nE[ε 2 ]) instead of converging to the object of interest X, X T as predicted by standard asymptotic theory.

56 4.3. Sampling Sparsely at an Optimal Frequency: Third Best

57 4.3. Sampling Sparsely at an Optimal Frequency: Third Best If one insists upon sampling sparsely, what is the right answer? Is it 5 mn, 10 mn, 15 mn?

58 4.3. Sampling Sparsely at an Optimal Frequency: Third Best If one insists upon sampling sparsely, what is the right answer? Is it 5 mn, 10 mn, 15 mn? To determine optimally the sparse sampling frequency, we show that: n sparse = ( T 4 E[ε 2 ] 2 ) T 1/3 0 σ4 t dt.

59 4.3. Sampling Sparsely at an Optimal Frequency: Third Best If one insists upon sampling sparsely, what is the right answer? Is it 5 mn, 10 mn, 15 mn? To determine optimally the sparse sampling frequency, we show that: n sparse = ( T 4 E[ε 2 ] 2 T 0 σ4 t dt ) 1/3. This gives rise to the third best estimator we define as [Y, Y ] (sparse,opt) T.

60 4.4. Correcting for Microstructure Noise: Second Best

61 4.4. Correcting for Microstructure Noise: Second Best We have just argued that one could benefit from using infrequently sampled data.

62 4.4. Correcting for Microstructure Noise: Second Best We have just argued that one could benefit from using infrequently sampled data. Indeed, the fourth and third best estimators do better than the fifth best.

63 4.4. Correcting for Microstructure Noise: Second Best We have just argued that one could benefit from using infrequently sampled data. Indeed, the fourth and third best estimators do better than the fifth best. But this entails discarding a very substantial fraction of the data sample.

64 4.4. Correcting for Microstructure Noise: Second Best We have just argued that one could benefit from using infrequently sampled data. Indeed, the fourth and third best estimators do better than the fifth best. But this entails discarding a very substantial fraction of the data sample. And yet, one of the most basic lessons of statistics is that one should not do this.

65 4.4. Correcting for Microstructure Noise: Second Best We have just argued that one could benefit from using infrequently sampled data. Indeed, the fourth and third best estimators do better than the fifth best. But this entails discarding a very substantial fraction of the data sample. And yet, one of the most basic lessons of statistics is that one should not do this.

66 We present a method to tackle the problem:

67 We present a method to tackle the problem: We partition the original grid of observation times, G = {t 0,..., t n } into subsamples, G (k), k = 1,..., K where n/k as n.

68 We present a method to tackle the problem: We partition the original grid of observation times, G = {t 0,..., t n } into subsamples, G (k), k = 1,..., K where n/k as n. For example, for G (1) start at the first observation and take an observation every 5 minutes; for G (2), start at the second observation and take an observation every 5 minutes, etc.

69 We present a method to tackle the problem: We partition the original grid of observation times, G = {t 0,..., t n } into subsamples, G (k), k = 1,..., K where n/k as n. For example, for G (1) start at the first observation and take an observation every 5 minutes; for G (2), start at the second observation and take an observation every 5 minutes, etc. Then we average the estimators obtained on the subsamples.

70 We present a method to tackle the problem: We partition the original grid of observation times, G = {t 0,..., t n } into subsamples, G (k), k = 1,..., K where n/k as n. For example, for G (1) start at the first observation and take an observation every 5 minutes; for G (2), start at the second observation and take an observation every 5 minutes, etc. Then we average the estimators obtained on the subsamples. To the extent that there is a benefit to subsampling, this benefit can now be retained, while the variation of the estimator can be lessened by the averaging.

71 We present a method to tackle the problem: We partition the original grid of observation times, G = {t 0,..., t n } into subsamples, G (k), k = 1,..., K where n/k as n. For example, for G (1) start at the first observation and take an observation every 5 minutes; for G (2), start at the second observation and take an observation every 5 minutes, etc. Then we average the estimators obtained on the subsamples. To the extent that there is a benefit to subsampling, this benefit can now be retained, while the variation of the estimator can be lessened by the averaging.

72 This gives rise to the estimator [Y, Y ] (avg) T = 1 K K k=1 [Y, Y ] (k) constructed by averaging the estimators [Y, Y ] (k) T of average size n. T obtained on K grids

73 This gives rise to the estimator [Y, Y ] (avg) T = 1 K K k=1 [Y, Y ] (k) constructed by averaging the estimators [Y, Y ] (k) T of average size n. T obtained on K grids We show that: [Y, Y ] (avg) T L X, X T }{{} object of interest + 2 ne[ε 2 ] }{{} bias due to noise + 4T T 3 n 0 σ4 t dt ] 1/2 Z total }{{} due } to noise due {{ to discretization} total variance + [ 4 n K E[ε4 ] }{{}

74 This gives rise to the estimator [Y, Y ] (avg) T = 1 K K k=1 [Y, Y ] (k) constructed by averaging the estimators [Y, Y ] (k) T of average size n. T obtained on K grids We show that: [Y, Y ] (avg) T L X, X T }{{} object of interest + 2 ne[ε 2 ] }{{} bias due to noise + 4T T 3 n 0 σ4 t dt ] 1/2 Z total }{{} due } to noise due {{ to discretization} total variance + [ 4 n K E[ε4 ] }{{}

75 4.5. Two Scales Realized Volatility: First Best

76 4.5. Two Scales Realized Volatility: First Best While a better estimator than [Y, Y ] (all) T, [Y, Y ] (avg) T remains biased.

77 4.5. Two Scales Realized Volatility: First Best While a better estimator than [Y, Y ] (all) T, [Y, Y ] (avg) T remains biased. The bias of [Y, Y ] (avg) T is 2 ne[ε 2 ].

78 4.5. Two Scales Realized Volatility: First Best While a better estimator than [Y, Y ] (all) T, [Y, Y ] (avg) T remains biased. The bias of [Y, Y ] (avg) T is 2 ne[ε 2 ]. But recall that E[ε 2 ] can be consistently approximated by the fifth best estimator: Ê[ε 2 ] = 1 [Y, Y ](all) 2n T

79 4.5. Two Scales Realized Volatility: First Best While a better estimator than [Y, Y ] (all) T, [Y, Y ] (avg) T remains biased. The bias of [Y, Y ] (avg) T is 2 ne[ε 2 ]. But recall that E[ε 2 ] can be consistently approximated by the fifth best estimator: Ê[ε 2 ] = 1 [Y, Y ](all) 2n T Hence the bias of [Y, Y ] (avg) can be consistently estimated by n n [Y, Y ](all) T.

80 A bias-adjusted estimator for X, X can thus be constructed as

81 A bias-adjusted estimator for X, X can thus be constructed as X, X T = [Y, Y ] (avg) }{{ T } slow time scale n n [Y, Y ](all) }{{ T } fast time scale

82 A bias-adjusted estimator for X, X can thus be constructed as X, X T = [Y, Y ] (avg) }{{ T } slow time scale n n [Y, Y ](all) }{{ T } fast time scale We call this estimator Two Scales Realized Volatility.

83 We show that if the number of subsamples is optimally selected as K = cn 2/3, then TSRV has the following distribution: X, X T L X, X T }{{} object of interest + 1 n 1/6 [ 8 c 2E[ε2 ] 2 }{{} + c 4T T 3 0 σ4 t dt ] 1/2 Z total }{{}} due to noise due {{ to discretization} total variance

84 We show that if the number of subsamples is optimally selected as K = cn 2/3, then TSRV has the following distribution: X, X T L X, X T }{{} object of interest + 1 n 1/6 [ 8 c 2E[ε2 ] 2 }{{} + c 4T T 3 0 σ4 t dt ] 1/2 Z total }{{}} due to noise due {{ to discretization} total variance Unlike all the previously considered ones, this estimator is now correctly centered

85 We show that if the number of subsamples is optimally selected as K = cn 2/3, then TSRV has the following distribution: X, X T L X, X T }{{} object of interest + 1 n 1/6 [ 8 c 2E[ε2 ] 2 }{{} + c 4T T 3 0 σ4 t dt ] 1/2 Z total }{{}} due to noise due {{ to discretization} total variance Unlike all the previously considered ones, this estimator is now correctly centered To the best of our knowledge, this is the only consistent estimator for X, X T in the presence of market microstructure noise.

86 5. Monte Carlo Simulations RV TSRV Fifth Best Fourth Best Third Best Second Best First Best [Y, Y ] (all) T [Y, Y ] (sparse) T [Y, Y ] (sparse,opt) T [Y, Y ] (avg) T X, X (adj) T Small Sample Bias Asymptotic Bias Small Sample Variance Asymptotic Variance Small Sample RMSE Asymptotic RMSE Small Sample Relative Bias Small Sample Relative Variance Small Sample Relative RMSE

87 6. Data Analysis Here is a comparison of RV to TSRV for INTC, last 10 trading days in April 2004: TSRV and RV for INTC on April 19, s 1mn 2mn 3mn 4mn 5mn 6mn 7mn 8mn 9mn 10mn

88 Zooming around the 5 minutes sampling frequency: RV TSRV mn 5mn 6mn

89 7. Dependent Market Microstructure Noise

90 7. Dependent Market Microstructure Noise So far, we have assumed that the noise ε was iid. In that case, log-returns are MA(1): Y τi Y τi 1 = τi τ i 1 σ t dw t + ε τi ε τi 1

91 For example, here is the autocorrelogram for AIG transactions, last 10 trading days in April 2004: 1 AIG 0.8 Sample autocorrelation transaction lag -0.4

92 But here is the autocorrelogram for INTC transactions, same last 10 trading days in April 2004: 1 INTC Sample autocorrelation transaction lag -0.5

93 A simple model to capture this higher order dependence is ε ti = U ti + V ti where U is iid, V is AR(1) and U V.

94 A simple model to capture this higher order dependence is ε ti = U ti + V ti where U is iid, V is AR(1) and U V. Fitted autocorrelogram for INTC: 1 INTC 0.75 Sample autocorrelation Fitted autocorrelation transaction lag -0.5

95 The TSRV Estimator with (J, K) Time Scales X, X T = [Y, Y ] (K) }{{ T } slow time scale n K n J [Y, Y ] (J) }{{ T } fast time scale

96 The TSRV Estimator with (J, K) Time Scales X, X T = [Y, Y ] (K) }{{ T } slow time scale n K n J [Y, Y ] (J) }{{ T } fast time scale We show that if we select J/K 0 when n, then this estimator is robust to (essentially) arbitrary time series dependence in microstructure noise.

97 The TSRV Estimator with (J, K) Time Scales X, X T = [Y, Y ] (K) }{{ T } slow time scale n K n J [Y, Y ] (J) }{{ T } fast time scale We show that if we select J/K 0 when n, then this estimator is robust to (essentially) arbitrary time series dependence in microstructure noise. Specifically, we let the noise process ε ti be stationary and strong mixing with exponential decay. We also suppose that E [ ε 4+κ] < for some κ > 0.

98 Robustness to the selection of the slow (K) and fast (J) time scales, INTC again: Robustness of TSRV with Time Dependent Noise for INTC 5mn mn 7mn 1mn30s K 8mn 9mn 30s 1mn J 10mn 5s

99 8. Further Refinement: Multiple Scales Realized Volatility

100 8. Further Refinement: Multiple Scales Realized Volatility We have seen that TSRV provides:

101 8. Further Refinement: Multiple Scales Realized Volatility We have seen that TSRV provides: the first consistent and asymptotic (mixed) normal estimator of the quadratic variation X, X T ;

102 8. Further Refinement: Multiple Scales Realized Volatility We have seen that TSRV provides: the first consistent and asymptotic (mixed) normal estimator of the quadratic variation X, X T ; that it can be made robust to arbitrary serial dependence in market microstructure noise;

103 8. Further Refinement: Multiple Scales Realized Volatility We have seen that TSRV provides: the first consistent and asymptotic (mixed) normal estimator of the quadratic variation X, X T ; that it can be made robust to arbitrary serial dependence in market microstructure noise; and that it has the rate of convergence n 1/6.

104 8. Further Refinement: Multiple Scales Realized Volatility We have seen that TSRV provides: the first consistent and asymptotic (mixed) normal estimator of the quadratic variation X, X T ; that it can be made robust to arbitrary serial dependence in market microstructure noise; and that it has the rate of convergence n 1/6.

105 At the cost of higher complexity, it is possible to generalize TSRV to multiple time scales, by averaging not on two time scales but on multiple time scales (Zhang 2004).

106 At the cost of higher complexity, it is possible to generalize TSRV to multiple time scales, by averaging not on two time scales but on multiple time scales (Zhang 2004). The resulting estimator, MSRV has the form of X, X (msrv) T = M a i [Y, Y ] (K i) T } i=1 {{} weighted sum of M slow time scales + 1 n [Y, Y ](all) }{{ T } fast time scale

107 At the cost of higher complexity, it is possible to generalize TSRV to multiple time scales, by averaging not on two time scales but on multiple time scales (Zhang 2004). The resulting estimator, MSRV has the form of X, X (msrv) T = M a i [Y, Y ] (K i) T } i=1 {{} weighted sum of M slow time scales + 1 n [Y, Y ](all) }{{ T } fast time scale TSRV corresponds to the special case where M = 1, i.e., where one uses a single slow time scale in conjunction with the fast time scale to bias-correct it.

108 con- For suitably selected weights a i and M = O(n 1/2 (msrv) ), X, X T verges to the X, X T at rate n 1/4.

109 con- For suitably selected weights a i and M = O(n 1/2 (msrv) ), X, X T verges to the X, X T at rate n 1/4. Optimal weights are given in closed-form.

110 con- For suitably selected weights a i and M = O(n 1/2 (msrv) ), X, X T verges to the X, X T at rate n 1/4. Optimal weights are given in closed-form. We also provide an analysis of this estimator under dependence of the noise.

111 9. Conclusions

112 9. Conclusions Two Scales Realized Volatility

113 9. Conclusions Two Scales Realized Volatility In the limit where all the data is used, realized volatility converges to the variance of the noise, not the quadratic variation of the log-returns

114 9. Conclusions Two Scales Realized Volatility In the limit where all the data is used, realized volatility converges to the variance of the noise, not the quadratic variation of the log-returns The practical response so far has been to use sparse sampling: once every 5 or 10 minutes.

115 9. Conclusions Two Scales Realized Volatility In the limit where all the data is used, realized volatility converges to the variance of the noise, not the quadratic variation of the log-returns The practical response so far has been to use sparse sampling: once every 5 or 10 minutes. But it is possible instead to correct for the noise by subsampling, averaging and bias-correcting and obtain a well behaved estimator that makes use of all the data: TSRV

116 9. Conclusions Two Scales Realized Volatility In the limit where all the data is used, realized volatility converges to the variance of the noise, not the quadratic variation of the log-returns The practical response so far has been to use sparse sampling: once every 5 or 10 minutes. But it is possible instead to correct for the noise by subsampling, averaging and bias-correcting and obtain a well behaved estimator that makes use of all the data: TSRV

117 The difference matters:

118 The difference matters: In Monte Carlo simulations, the RMSE of TSRV is orders of magnitude smaller than that of RV

119 The difference matters: In Monte Carlo simulations, the RMSE of TSRV is orders of magnitude smaller than that of RV In empirical examples, the difference is also meaningful: for INTC, TSRV = vs. RV in the range (0.0029, ).

120 The difference matters: In Monte Carlo simulations, the RMSE of TSRV is orders of magnitude smaller than that of RV In empirical examples, the difference is also meaningful: for INTC, TSRV = vs. RV in the range (0.0029, ).

121 And one final important message:

122 And one final important message: Any time one has an impulse to discard data, one can usually do better: using likelihood corrections in the parametric volatility case or subsampling and averaging in the stochastic volatility case.

123 And one final important message: Any time one has an impulse to discard data, one can usually do better: using likelihood corrections in the parametric volatility case or subsampling and averaging in the stochastic volatility case. No matter what the model is, no matter what quantity is being estimated.