Estimation of High-Frequency Volatility:
An Autoregressive Conditional Duration Approach
Yiu-Kuen Tse
School of Economics, Singapore Management University
Thomas Tao Yang
Department of Economics, Boston College
April 2012
Appendix
A.1 Estimation of the ACD Model
Given an assumed density function f (·) for i , the MLE of the parameters of the ACD equation
can be computed straightforwardly. A particularly simple model is the case when i are assumed
to be standard exponential, giving rise to the QMLE, which is consistent provided the conditional
expected-duration equation is correctly specified (Drost and Werker (2004)), irrespective of the
true duration distribution.
Under the exponential distribution assumption, the conditional intensity is constant. When
this assumption is used to integrate the instantaneous variance, misspecification errors in the conditional intensity may induce errors in the integrated conditional variance. To resolve this issue,
the SP method may be used, for which the standardized-duration distribution is estimated nonparametrically.
A.1.1 Semiparametric Estimation Method
Engle and Gonzalez-Rivera (1991) introduced the use of the SP method for the ARCH model.
They showed that the SP estimator is consistent and more efficient than the QMLE, which assumes
normal errors. Drost and Werker (2004) discussed the conditions under which the QMLE of the
ACD model is consistent. They also derived conditions under which the SP method is adaptive
and efficient. Although the SP method is only efficient under very restrictive conditions, there may
yet be improvements in efficiency over the QMLE.
To compute the SP estimates of the ACD model, we perform the computation as follows. First,
we estimate the ACD model using the QMLE method. We denote the parameter vector of the ACD
model by θ and its QMLE by θ̂. Next, we calculate the estimated conditional expected duration
ψ̂i using θ̂ and thus the standardized duration ˆi as
ˆi =
xi
ψ̂i
.
(A.1)
The unknown density function f (·) of i is then estimated using ˆi by the discrete maximum
penalized likelihood estimation (DMPLE) technique (Tapia and Thompson (1978)), which was also
used by Engle and Gonzalez-Rivera (1991). The estimated density function is denoted by fˆ(·) and
1
the parameters of the ACD equation are then estimated again assuming i are i.i.d. with density
function fˆ(·), resulting in the SP estimates of the ACD model, denoted by θ∗ .
To apply the DMPLE technique we define m − 1 grid knots n1 , n2 , · · · , nm−1 that divide the
support of ˆi into m equal intervals. We then maximize the likelihood of the sample for which the arguments are the heights h1 , h2 , · · · , hm−1 corresponding to the knots n1 , n2 , · · · , nm−1 , respectively.
Specifically, we maximize the function
N
X
i=1
m−1
φX
(hi+1 − 2hi + hi−1 )2
log g(ˆ
i ) −
q
(A.2)
i=1
with respect to h1 , h2 , · · · , hm−1 (with h0 ≡ hm ≡ 0), subject to
q
m−1
X
hi = 1,
hi ≥ 0, i = 1, 2, · · · , m − 1,
(A.3)
i=1
where q is the (equal) subinterval length and

h + hi+1 − hi (y − n ), y ∈ [n , n ),
i
i i+1
i
q
g(y) =

0,
y 6∈ (n0 , nm ),
(A.4)
and φ is the penalty term to ensure the smoothness of hi . In our empirical implementation, we
set φ = 10 for its stability. The grids are constructed to divide the interval (n0 , nm ) into m equal
subintervals, with n0 = min {ˆ
i } − 0.01 and nm = 99.5th percentile of {ˆ
i } + 0.01. We choose
m = 31 for our computation, so that there are 31 subintervals of the duration.1
A.1.2 Monte Carlo Results for the Estimation of the ACD Model
To examine the performance of the estimators of θ in the ACD models, we conduct a MC experiment. We consider both the ACD(1, 1) model and the AACD model, where specific values of
parameters are assigned. For the AACD model, we assign the values based on the illustrative examples reported by Fernandes and Grammig (2006). For the conditional duration distribution, we
consider the Weibull distribution and the Burr distribution. The Weibull distribution has density
function
κ
f () =
µ
κ−1 “ ”κ
− e µ ,
µ
1
> 0,
(A.5)
Apart from the DMPLE method, an alternative is to use the gamma kernel method (Chen (2000)). We find,
however, that the gamma kernel method is generally computationally more costly and is inferior to the DMPLE
method for its relative instability. Thus, we adopt the DMPLE method in the computation of the SP estimates.
2
with parameters µ and κ > 0. We let κ = 2 and µ = 1/Γ(1.5) so that the mean of is unity.2
In the MLE computation we constrain the expected value of to be unity by setting µ̂ equal to
1/Γ(1 + 1/κ̂).
The density function of the Burr distribution is
− γ1 −1
f () = µκκ−1 (1 + µγκ )
,
> 0,
(A.6)
with positive parameters µ, κ and γ. We let κ = 2, γ = 0.2906 and µ = 1, so that the expected
value of is unity. In the MLE computation, we set the free parameters to be κ and γ, and constrain
the value of µ so that the mean of is unity.
As the results for the ACD and AACD models are similar, we report the results of the AACD
model only. We consider estimation sample size of N = 1,000, 10,000 and 50,000. The Monte
Carlo replications are 500, 100 and 50, depending on the values of N . The following estimators are
considered: MLE, QMLE and SP methods.
In Tables A.1 and A.2 we report the means of the Monte Carlo samples as well as the standard
errors and root mean-squared errors. Only results for the parameters of the AACD equation are
summarized. For the Weibull errors we observe that the RMSE of all estimates decreases as N
increases, which suggests that the estimates are consistent. The RMSE of the SP estimate is
uniformly lower than that of the QMLE, showing that the SP estimate is more efficient than the
QMLE. Comparing the RMSE of the MLE and SP estimates, it is found that the SP estimate
has larger RMSE than the MLE for N = 1,000 (except for c). However, for larger values of N ,
there is no uniform order of superiority. The lack of superiority of the MLE, which is theoretically
asymptotically efficient, may be due to the Monte Carlo estimation errors. Nonetheless, the good
performance of the SP estimate versus the MLE is exhibited.
For the Burr errors, the results show that the RMSE of all estimates generally decreases as N
increases (when N increases from 10,000 to 50,000 the only exception is for the MLE value of c;
when N increases from 1,000 to 10,000 the only exception is for the SP estimate of α). This again
suggests the consistency of all estimators. The superiority of the SP estimate over the QMLE is
2
Γ(·) denotes the gamma function.
3
quite clear, although not as strong as for the case of the Weibull errors.
A.2 Monte Carlo Results for the Estimation of Volatility
This section contains some supplemental materials of MC results for the estimation of volatility.
A.2.1 Monte Carlo Results for Deterministic Volatility Models
We consider a deterministic volatility set-up in which the volatility function is fixed with intraday
periodic variation. Following the assumptions underlying the theoretical derivation of the RV
estimates that the logarithmic stock price follows a Brownian semimartingale (BSM), we set up an
artificial data generation process along this line. We assume that the observed price s(u) at time u
(we now use u for time, for reasons to be made clearer later) consists of a BSM component sB (u)
and a jump component sJ (u). Let s̃B (u) = log sB (u), which is generated from the following BSM
Z
s̃B (u) =
u
Z
µ(u) du +
0
u
σ(u) dW (u),
(A.7)
0
where µ(u) is the instantaneous drift, σ 2 (u) is the instantaneous variance and W (u) is a standard
Brownian process. We assume s̃B (u) has no drift so that µ(u) ≡ 0. To specify the time u, we
denote t as the day of trade and τ as the intraday time. Thus, we write σ(u) = σ(t, τ ) and let
σ(t, τ ) = σ1 (t)σ2 (τ ), so that the instantaneous volatility at time u depends on the component σ1 (t)
(representing the average volatility of day t) and the component σ2 (τ ) (capturing the intraday
variations). In our MC study, we consider a period of 150 days. We set σ1 (t) = 20% (annualized
standard deviation of returns) for t = 1, which increases linearly with t over 50 days to reach
30%. It then remains level for 50 days, and after that decreases linearly to 20% over 50 days. The
intraday variation function σ2 (τ ) was estimated empirically as follows. We selected randomly a
large-cap stock from the TAQ Database and used its data from January through March 2005. For
each day we computed the sum of 2-minute squared logarithmic returns over 1 hour and rolled
this forward every 2 minutes, which were used as estimates of the volatility at the center of each
(overlapping) 1-hour interval. We then averaged these estimates over the sample period and did a
cubic spline on the resulting average. Finally, we normalized the average function to have a mean
4
of unity and called this σ2 (τ ).
Having determined σ1 (t) and σ2 (τ ), and thus σ(u), we generate observations of s̃B (u) by the
equation
s̃B (u + ∆u) = s̃B (u) + σ(u)ε,
(A.8)
where ε ∼ N (0, 1) and are independent for different values of u. We take ∆u to be one second and
use equation (A.8) to compute a second-by-second series of observations of sB (u) = exp(s̃B (u)),
with the starting price of $65. We further add to the series sB (u) a jump component sJ (u), which
is assumed to follow a Poisson process with a mean of 0.4 per five minutes. When a jump occurs,
it takes value of –$0.05, –$0.03, $0.03 and $0.05 with probabilities of 0.25 each. We also consider a
jump component with a higher jump frequency of 2.72 per five minutes and jump sizes of –$0.02,
–$0.01, $0.01 and $0.02 with equal probabilities. Finally, we consider a price process consisting of a
BSM and a white noise. In sum, we consider four experiments. Experiment 1 has a pure BSM price
process, Experiment 2 consists of a BSM process plus infrequent jumps of large sizes, Experiment
3 uses a BSM with more frequent jumps and smaller jump sizes, and Experiment 4 has a BSM
process with white noise.
We estimate daily volatilities using the AACD model with both the SP and QMLE methods,
for a threshold price range of $0.08, and compare them against the RV methods.3 Figure A.1
presents the daily volatility estimates of one MC sample of Experiments 1 and 2. It can be seen
that the VA estimates track the true daily volatilities quite closely. The RV estimates also follow
the volatility trends, albeit apparently with larger variations. We further simulate 50 samples of
150-day data and estimate the daily volatilities.4 The mean error (ME), standard error (SE) and
root mean-squared error (RMSE) of the volatility estimates are summarized in Table A.3. It can
be seen that the ACD-ICV estimates give lower RMSE than the RV estimates. The results for the
SP and QMLE estimates are quite similar. On the other hand, the results are mixed whether the
conditional mean or threshold price range will give better results.
3
The price event defined in this set of experiments is defined as the cumulative change in price exceeding the dollar
price range, with corresponding modification for the ACD-ICV formula.
4
As the SP method is very computer intensive, the MC sample size is maintained to be small. Increasing the
sample size for some of the models showed that the results are qualitatively similar.
5
We further estimate intraday ICV using VA . Figure A.2 presents the results of the intraday ICV
estimates on days 1, 75 and 150 of a simulated sample based on Experiment 2 (BSM with infrequent
large jumps). We consider ICV over two time intervals: 15 minutes and one hour, beginning from
9:45 and ending at 15:45. The graphs show that VA performs very well, exhibiting a clear intraday
volatility smile and tracking the true instantaneous volatility path quite closely. In contrast, the
errors of VB in tracking the true instantaneous volatility path are much larger. contrast,
A.2.2 More Monte Carlo Results for the Stochastic Volatility Models
We further examine the effect of price-range threshold δ on the estimation of volatility. We perform
a robustness check by varying the target average duration. As Tables 1 through 3 show that the
results are quite stable with respect to the noise structure we examine only the case with white
noise. For NSR = 0.25, we consider average durations of 1, 2, 3 and 4 minutes. Likewise, for NSR
= 1.0, we consider average durations of 4, 5, 6 and 7 minutes. Table A.4 shows that the results
do not change qualitatively across the range of average durations considered. In particular, the
ACD-ICV estimates perform favorably against the RV estimates over a range of values of δ. In
Table A.5 we further report the MC results when NSR = 1.5. It can be seen that VA gives the
lowest RMSE except for the Heston model with no jumps, in which case VB performs the best.
These results confirm that VA is more robust to data with large microstructure noise and jumps.
A.3 Empirical Results for NYSE Stocks
The data used in this paper are extracted and compiled from the New York Stock Exchange
(NYSE) Trade and Quote (TAQ) Database provided through the Wharton Research Data Services.
We downloaded the following data from the Consolidated Trade (CT) file: date, trading time, price
and number of shares traded. Only trades in the NYSE are used. To select the stocks, we rank the
500 component stocks of the S&P500 Index by market capitalization as of September 2008. We
then divide the stocks into three groups and select the largest 10 stocks from each group. We call
these the large stocks, medium stocks and small stocks. The selected companies and their stock
codes are summarized in Table A.6.
6
On each day transaction data from 9:30 to 16:00 are downloaded. Using the transaction data, we
estimate diurnal factors by applying a smoothing cubic spline to the average duration at each time
point. The diurnally adjusted durations are then computed by dividing each transaction duration
by the corresponding diurnal factor, which are used for further analysis. We note that the average
duration increases when the size of the stock decreases. Also, the average duration decreases from
Period 1 to Period 3, indicating that trades in the market are more frequent in later periods.
We estimate the AACD model for the duration data of the NYSE stocks using QMLE. To
target an average price duration of 4 to 5 minutes, we set δ to 0.1% in Periods 1 and 2, and 0.2%
in Period 3. For individual stocks with average duration outside the range of 4 to 5 minutes, we
reset δ accordingly. The issue of the optimal price range is analogous to the problem of determining
the optimal sampling frequency for the VR method as studied by Bandi and Russell (2006a, 2006b
and 2008), which is a future topic of research. The QMLE estimates of the AACD models are
are summarized in Tables A.7 through A.9. As the standard errors of some of the models cannot
be computed, we present the results for the parameter estimates only.5 We use the fitted AACD
model to estimate the daily variance of the stocks, which are then multiplied by a factor of 252
to obtain the annual variance. The square root of this variance, which is the annualized return
standard deviation based on the open-to-close price changes. The complete results of all stocks in
the three period are presented in Figures A.3 through A.11.
It can be seen that the ACD-ICV estimates track the RV estimates very closely. This is especially
true for the stocks in Period 1 (sideways market). During this period there appears to be little
interday variation in volatility. All estimates track each other quite closely, although there are more
variations for the medium and small stocks, especially for the VD estimates. In Period 2 (upward
market), the overall volatility level is higher. We also see more daily fluctuations in the volatility
estimates. In particular, the VD estimates fluctuate quite significantly over this period, especially
for medium and small stocks. Generally, there appears to be a ranking of VD being higher than
VA , which is in turn higher than VB and VK . The latter two estimates appear to be closest to
5
The difficulty in computing the standard errors may be due to the asymmetric function in the AACD equation,
which involve calculating an absolute term. Fernandes and Grammig (2006) also reported difficulty in computing
standard errors for the AACD model.
7
each other. For the medium and small stocks it happens quite a number of times that VD is more
than double the values of the other estimates. Finally, Period 3 (downward market) exhibits an
interesting feature that the volatilities of all stocks trend upwards during this 25-day period in
which the overall market dropped about 9%. Most stocks start with an annualized volatility of
about 20% and trend upwards to 40% or over. Again, the VD estimates appear to be the highest on
average, while VB and VK remain close to each other. Rather interestingly, VA seem to moderate
between VD and the pair of estimates VB and VK .
References
[1] Bandi, F. M., and J. R. Russell, 2006a, Separating market microstructure noise from volatility,
Journal of Financial Economics, 79, 655-692.
[2] Bandi, F. M., and J. R. Russell, 2006b, Market microstructure noise, integrated variance estimators, and the accuracy of asymptotic approximations, Journal of Econometrics, forthcoming.
[3] Bandi, F. M., and J. R. Russell, 2008, Microstructure noise, realized variance, and optimal
sampling, Review of Economic Studies, 75, 339-369.
[4] Chen, S. X., 2000, Probability density function estimation using gamma kernels, Annals of
Institute of Statistical Mathematics, 52, 471-480.
[5] Drost, F. C., and B. J. M. Werker, 2004, Semiparametric duration models, Journal of Business
and Economic Statistics, 22, 40-50.
[6] Engle, R., and G. Gonzalez-Rivera, 1991, Semiparametric ARCH models, Journal of Business
and Economic Statistics, 9, 345-359.
[7] Fernandes, M., and J. Grammig, 2006, A family of autoregressive conditional duration models,
Journal of Econometrics, 130, 1-23.
[8] Tapia, R. A., and J. R. Thompson, 1978, Nonparametric Probability Density Estimation, Johns
Hopkins University Press.
8
Table A.1:
Monte Carlo results for the AACD model with Weibull errors
Parameter
MLE
QMLE
SPE
Panel A: N = 1,000, MC replications = 500
ω (0.04)
α (0.13)
β (0.80)
λ (0.26)
v (0.50)
b (0.10)
c (0.20)
0.0423
0.1308
0.7972
0.2525
0.4860
0.1006
0.2109
(0.0081)
(0.0260)
(0.0213)
(0.0255)
(0.1032)
(0.0324)
(0.1823)
(0.0084)
(0.0260)
(0.0215)
(0.0266)
(0.1041)
(0.0323)
(0.1825)
0.0492
0.1550
0.7781
0.1070
0.1873
0.0317
0.5020
(0.0170)
(0.0794)
(0.0792)
(0.0663)
(0.0965)
(0.0263)
(0.1960)
(0.0193)
(0.0832)
(0.0821)
(0.1667)
(0.3272)
(0.0732)
(0.3599)
0.0444
0.1563
0.7741
0.2134
0.3741
0.0679
0.2780
(0.0120)
(0.0577)
(0.0587)
(0.0822)
(0.1136)
(0.0323)
(0.1448)
(0.0128)
(0.0633)
(0.0641)
(0.0944)
(0.1695)
(0.0455)
(0.1643)
(0.0052)
(0.0119)
(0.0146)
(0.0514)
(0.0744)
(0.0340)
(0.2464)
(0.0056)
(0.0118)
(0.0165)
(0.0567)
(0.0987)
(0.0432)
(0.2944)
0.0407
0.1351
0.7987
0.2602
0.4959
0.0945
0.2010
(0.0021)
(0.0092)
(0.0096)
(0.0177)
(0.0276)
(0.0087)
(0.0098)
(0.0022)
(0.0105)
(0.0096)
(0.0177)
(0.0278)
(0.0103)
(0.0098)
(0.0034)
(0.0126)
(0.0110)
(0.0270)
(0.0251)
(0.0059)
(0.0599)
(0.0034)
(0.0125)
(0.0111)
(0.0275)
(0.0274)
(0.0102)
(0.0632)
0.0411
0.1360
0.7964
0.2636
0.5088
0.0918
0.2006
(0.0006)
(0.0015)
(0.0013)
(0.0030)
(0.0082)
(0.0015)
(0.0021)
(0.0012)
(0.0062)
(0.0038)
(0.0046)
(0.0120)
(0.0083)
(0.0021)
Panel B: N = 10,000, MC replications = 100
ω (0.04)
α (0.13)
β (0.80)
λ (0.26)
v (0.50)
b (0.10)
c (0.20)
0.0407
0.1288
0.8000
0.2568
0.4956
0.0998
0.2105
(0.0033)
(0.0078)
(0.0083)
(0.0162)
(0.0376)
(0.0112)
(0.0699)
(0.0033)
(0.0078)
(0.0083)
(0.0165)
(0.0377)
(0.0112)
(0.0704)
0.0421
0.1305
0.7922
0.2354
0.4346
0.0732
0.3629
Panel C: N = 50,000, MC replications = 50
ω (0.04)
α (0.13)
β (0.80)
λ (0.26)
v (0.50)
b (0.10)
c (0.20)
0.0403
0.1295
0.8006
0.2584
0.4985
0.1008
0.1948
(0.0015)
(0.0038)
(0.0050)
(0.0091)
(0.0212)
(0.0063)
(0.0518)
(0.0015)
(0.0038)
(0.0050)
(0.0092)
(0.0210)
(0.0063)
(0.0516)
0.0409
0.1297
0.7983
0.2537
0.4883
0.0916
0.2218
λ [|
v
λ
Notes: The AACD equation is ψiλ = ω + αψi−1
i−1 − b| + c(i−1 − b)] + βψi−1 . The figures in
the first column are the true parameter values. The figures in other columns are the means of the
parameter estimates, with standard deviations in the first parentheses and RMSE in the second
parentheses. MLE = maximum likelihood estimate, QMLE = quasi-MLE, SPE = semiparametric
estimate.
Table A.2:
Monte Carlo results for the AACD model with Burr errors
Parameter
MLE
QMLE
SPE
Panel A: N = 1,000, MC replications = 500
ω (0.04)
α (0.13)
β (0.80)
λ (0.26)
v (0.50)
b (0.10)
c (0.20)
0.0432
0.1329
0.7940
0.2582
0.4862
0.1035
0.1969
(0.0124)
(0.0374)
(0.0392)
(0.0945)
(0.1382)
(0.0374)
(0.1697)
(0.0128)
(0.0375)
(0.0396)
(0.0944)
(0.1388)
(0.0375)
(0.1696)
0.0448
0.1192
0.8078
0.2568
0.5850
0.0943
0.1878
(0.0181)
(0.0547)
(0.0574)
(0.1716)
(0.1773)
(0.0400)
(0.2611)
(0.0187)
(0.0557)
(0.0579)
(0.1715)
(0.1965)
(0.0404)
(0.2611)
0.0414
0.1283
0.8009
0.2570
0.5189
0.0979
0.2155
(0.0091)
(0.0222)
(0.0220)
(0.0498)
(0.1029)
(0.0255)
(0.0931)
(0.0092)
(0.0222)
(0.0220)
(0.0498)
(0.1045)
(0.0255)
(0.0943)
(0.0045)
(0.0081)
(0.0100)
(0.0247)
(0.0314)
(0.0274)
(0.0950)
(0.0048)
(0.0084)
(0.0105)
(0.0314)
(0.1803)
(0.0334)
(0.0946)
0.0389
0.1533
0.7923
0.2226
0.5886
0.1005
0.1979
(0.0035)
(0.0104)
(0.0089)
(0.0173)
(0.0323)
(0.0226)
(0.0202)
(0.0036)
(0.0255)
(0.0118)
(0.0411)
(0.0942)
(0.0225)
(0.0202)
(0.0014)
(0.0023)
(0.0043)
(0.0091)
(0.0216)
(0.0055)
(0.0496)
(0.0014)
(0.0023)
(0.0043)
(0.0090)
(0.0214)
(0.0055)
(0.0492)
0.0405
0.1324
0.8005
0.2626
0.5020
0.0980
0.2022
(0.0010)
(0.0019)
(0.0017)
(0.0042)
(0.0158)
(0.0043)
(0.0028)
(0.0011)
(0.0031)
(0.0018)
(0.0049)
(0.0158)
(0.0047)
(0.0035)
Panel B: N = 10,000, MC replications = 100
ω (0.04)
α (0.13)
β (0.80)
λ (0.26)
v (0.50)
b (0.10)
c (0.20)
0.0410
0.1302
0.7981
0.2575
0.4929
0.1013
0.2115
(0.0033)
(0.0058)
(0.0083)
(0.0181)
(0.0424)
(0.0142)
(0.0378)
(0.0034)
(0.0058)
(0.0085)
(0.0182)
(0.0428)
(0.0141)
(0.0394)
0.0418
0.1322
0.7966
0.2403
0.6776
0.0808
0.2037
Panel C: N = 50,000, MC replications = 50
ω (0.04)
α (0.13)
β (0.80)
λ (0.26)
v (0.50)
b (0.10)
c (0.20)
0.0402
0.1302
0.7987
0.2623
0.5001
0.1011
0.2104
(0.0016)
(0.0026)
(0.0030)
(0.0079)
(0.0179)
(0.0055)
(0.0569)
(0.0016)
(0.0025)
(0.0032)
(0.0082)
(0.0177)
(0.0056)
(0.0573)
0.0401
0.1301
0.7995
0.2608
0.5018
0.1007
0.2026
λ [|
v
λ
Notes: The AACD equation is ψiλ = ω + αψi−1
i−1 − b| + c(i−1 − b)] + βψi−1 . The figures in
the first column are the true parameter values. The figures in other columns are the means of the
parameter estimates, with standard deviations in the first parentheses and RMSE in the second
parentheses. MLE = maximum likelihood estimate, QMLE = quasi-MLE, SPE = semiparametric
estimate.
Table A.3: Monte Carlo results for deterministic volatility models
Estimation
Method
VA∗S
VAS
VA∗
VA
VB
VD
VK
VR
ME
Experiment 1
SE
RMSE
−0.270
−0.402
−0.027
−0.970
0.283
−0.679
0.127
−0.679
0.889
0.890
0.944
0.911
1.346
1.524
2.538
1.879
1.015
1.265
1.003
1.100
1.390
1.698
2.565
1.602
ME
Experiment 2
SE
RMSE
−0.026
−0.282
0.309
−0.329
0.283
−0.679
0.062
−0.679
0.780
0.784
0.931
0.898
1.346
1.524
1.301
2.537
0.882
1.053
1.058
1.012
1.390
1.698
1.315
2.449
ME
Experiment 3
SE
RMSE
−0.128
−0.332
0.254
−0.446
0.305
−0.664
0.088
−0.664
0.760
0.765
0.984
0.949
1.347
1.532
1.297
2.572
0.884
1.105
1.075
1.052
1.399
1.703
1.313
2.485
ME
Experiment 4
SE
RMSE
0.301
−0.122
0.308
−0.327
0.383
−1.181
0.139
−1.181
0.906
0.915
0.978
0.944
1.344
0.958
1.317
2.750
1.040
1.025
1.087
1.040
1.413
1.567
1.341
2.435
Notes: ME = mean error, SE = standard error (standard deviation of MC samples), RMSE = root mean-squared error.
The results are based on 50 MC replications of 150-day daily volatility estimates. VA∗S and VAS are computed using equation
(12) (SP method). For the former, δ is the average price range conditional on a price event being observed (i.e., price range
being exceeded) and for the latter δ is the threshold price range. VA∗ and VA are obtained by QMLE. For the former, δ
is the average price range conditional on a price event being observed (i.e., price range being exceeded) and for the latter
δ is the threshold price range. In the first experiment there is no Poisson jump and the price process follows a BSM. The
second experiment includes a Poisson jump component with 0.4 jump per 5 minutes and jump size of −0.05, −0.03, 0.03, 0.05
with equal probabilities. The third experiment has a Poisson jump component with jump frequency of 2.72 per 5 minutes
but smaller jump size of −0.02, −0.01, 0.01 and 0.02 with equal probabilities. The last experiment introduces white noise
component to the BSM with annualized standard deviation of 0.0583, which is about 23% of the annualized standard deviation
of the BSM.
Table A.4:
Estimation
Method
Robustness check for Monte Carlo results of stochastic volatility models
ME
Heston
SE
RMSE
Volatility Model
Heston with Jumps
ME
SE
RMSE
ME
LV
SE
RMSE
LV with Jumps
ME
SE
RMSE
Panel A: NSR = 0.25
Average price duration = 1 min
VA∗
VA
0.025
−1.053
0.889
0.954
0.891
1.402
−0.014
−1.412
1.199
1.253
1.201
1.931
0.037
−1.126
0.976
0.948
0.978
1.460
0.042
−1.121
0.973
0.944
0.975
1.455
0.936
1.296
−0.145
−1.175
1.147
1.173
1.166
1.705
−0.113
−0.930
0.790
0.773
0.796
1.196
−0.109
−0.925
0.787
0.770
0.792
1.192
1.155
1.424
−0.249
−1.104
1.430
1.443
1.464
1.858
−0.212
−0.880
1.055
1.037
1.072
1.347
−0.200
−0.868
1.059
1.041
1.075
1.344
1.222
1.471
−0.441
−1.154
1.481
1.482
1.567
1.921
−0.348
−0.925
0.942
0.926
0.997
1.296
−0.333
−0.912
0.949
0.933
1.000
1.292
1.363
1.199
0.398
−0.419
1.478
1.479
1.516
1.555
0.595
−0.092
0.940
0.922
1.123
0.925
0.607
−0.080
0.950
0.931
1.136
0.933
1.294
1.223
0.148
−0.578
1.474
1.469
1.476
1.602
0.391
−0.228
0.888
0.872
0.978
0.896
0.398
−0.221
0.888
0.870
0.979
0.894
1.338
1.358
−0.049
−0.712
1.555
1.545
1.559
1.728
0.170
−0.387
0.931
0.915
0.950
0.986
0.150
−0.400
0.943
0.926
0.958
1.002
1.433
1.486
−0.174
−0.801
1.659
1.647
1.675
1.860
−0.002
−0.518
1.050
1.033
1.050
1.147
−0.001
−0.516
1.056
1.039
1.056
1.152
Average price duration = 2 min
VA∗
VA
−0.089
−0.889
0.934
0.967
Average price duration = 3 min
VA∗
VA
−0.222
−0.866
1.140
1.154
Average price duration = 4 min
VA∗
VA
−0.341
−0.900
1.182
1.188
Panel B: NSR = 1.0
Average price duration = 4 min
VA∗
VA
0.622
−0.038
1.195
1.199
Average price duration = 5 min
VA∗
VA
0.411
−0.178
1.216
1.214
Average price duration = 6 min
VA∗
VA
0.164
−0.362
1.323
1.318
Average price duration = 7 min
VA∗
VA
0.039
−0.458
1.431
1.425
Notes: ME = mean error, SE = standard error (standard deviation of MC samples), RMSE = root mean-squared error. The results
are based on 1,000 MC replications of 60-day daily volatility estimates. All figures are in percentage. VA∗ is computed using equation
(11) with δ being the average price range conditional on a price event being observed, which is defined as the cumulative change in the
logarithmic price exceeding the threshold. VA is computed from equation (11) with δ being the threshold price range. The Heston model
with jumps is the Heston diffusion model with jumps in the volatility, while the LV model with jumps is the LV model with jumps in the
price.
Table A.5: Monte Carlo results for stochastic volatility models with NSR = 1.50
Estimation
Method
ME
Heston
SE
RMSE
Volatility Model
Heston with Jumps
ME
SE
RMSE
ME
LV
SE
RMSE
LV with Jumps
ME
SE
RMSE
Panel A: Transaction price with white noise
VA∗
VA
VB
VD
VK
VR
1.143
0.538
0.114
0.429
0.490
−0.155
1.254
1.255
1.202
1.484
1.333
2.237
1.697
1.366
1.208
1.545
1.420
2.242
0.763
−0.034
−0.030
−0.185
0.305
−0.298
1.489
1.491
1.660
1.941
1.822
3.059
1.673
1.491
1.660
1.950
1.848
3.073
1.189
0.515
0.065
0.349
0.431
−0.183
0.913
0.895
1.291
1.552
1.430
2.391
1.499
1.032
1.293
1.591
1.493
2.398
1.176
0.506
0.068
0.346
0.422
−0.199
0.914
0.895
1.291
1.552
1.422
2.395
1.490
1.028
1.293
1.590
1.484
2.403
1.710
1.501
1.666
1.944
1.845
3.089
1.251
0.565
0.059
0.368
0.425
−0.192
0.917
0.898
1.295
1.560
1.430
2.410
1.551
1.061
1.296
1.603
1.492
2.418
1.241
0.558
0.061
0.370
0.414
−0.192
0.914
0.895
1.299
1.557
1.428
2.404
1.542
1.055
1.301
1.600
1.487
2.412
1.140
0.472
0.064
0.301
0.411
−0.178
0.901
0.882
1.300
1.562
1.429
2.406
1.453
1.000
1.302
1.591
1.487
2.413
1.120
0.455
0.064
0.305
0.409
−0.180
0.913
0.894
1.294
1.552
1.423
2.393
1.445
1.003
1.295
1.582
1.481
2.400
1.514
1.042
1.302
1.587
1.487
2.402
1.180
0.505
0.057
0.337
0.407
−0.203
0.918
0.898
1.299
1.569
1.431
2.413
1.495
1.030
1.300
1.605
1.487
2.421
Panel B: Transaction price with autocorrelated noise
VA∗
VA
VB
VD
VK
VR
1.209
0.594
0.123
0.474
0.490
−0.155
1.247
1.248
1.197
1.478
1.325
2.227
1.737
1.382
1.203
1.552
1.413
2.232
0.822
0.014
−0.042
−0.134
0.291
−0.308
1.499
1.501
1.665
1.939
1.822
3.074
Panel C: Transaction price with noise correlated with efficient price
VA∗
VA
VB
VD
VK
VR
1.093
0.490
0.102
0.390
0.466
−0.165
1.254
1.255
1.205
1.481
1.338
2.229
1.663
1.347
1.209
1.531
1.417
2.235
0.712
−0.079
−0.041
−0.224
0.291
−0.302
1.491
1.494
1.662
1.933
1.814
3.055
1.652
1.496
1.662
1.945
1.837
3.069
Panel D: Transaction price with autocorrelated noise correlated with efficient price
VA∗
VA
VB
VD
VK
VR
1.146
0.537
0.103
0.430
0.473
−0.153
1.245
1.245
1.201
1.478
1.325
2.230
1.692
1.356
1.205
1.540
1.407
2.235
0.763
−0.037
−0.039
−0.176
0.300
−0.290
1.495
1.498
1.665
1.947
1.811
3.066
1.678
1.498
1.665
1.954
1.835
3.080
1.201
0.521
0.058
0.331
0.415
−0.193
0.922
0.903
1.300
1.552
1.428
2.394
Notes: ME = mean error, SE = standard error (standard deviation of MC samples), RMSE = root mean-squared error.
The results are based on 1,000 MC replications of 60-day daily volatility estimates. All figures are in percentage. VA∗ is
computed using equation (11) with δ being the average price range conditional on a price event being observed, which is
defined as the cumulative change in the logarithmic price exceeding the threshold. VA is computed from equation (11)
with δ being the threshold price range. The Heston model with jumps is the Heston diffusion model with jumps in the
volatility, while the LV model with jumps is the LV model with jumps in the price.
Table A.6: Stocks and codes
Large Stocks
Stock
Code
Medium Stocks
Stock
Code
Exxon Mobil
General Electric
Procter & Gamble
John & Johnson
AT & T
Chevron
JPMorgan Chase
Wal Mart
IBM
Pfizer
XOM
GE
PG
JNJ
T
CVX
JPM
WMT
IBM
PFE
TJX
Tyco International
Viacom (B)
Allergan
Chesapeake Energy
Aon
Loews
Progress Energy
Williams
Baker Hughes
TJX
TYC
VIAB
AGN
CHK
AOC
CG
PGN
WMB
BHI
Small Stocks
Stock
Code
Brown-Forman (B)
Constellation Energy Group
Computer Sciences
Jacobs Engineering Group
American Int’l. Group
Waters
U.S. Steel
Moody’s
McCormick
Comerica
BFB
CEG
CSC
JEC
AIG
WAT
X
MCO
MKC
CMA
Table A.7: Estimated parameters of the AACD model in Period 1
Stock
ω
AACD model parameters
β
λ
v
α
b
c
Panel A: Large stocks in Period 1
XOM
GE
PG
JNJ
T
CVX
JPM
WMT
IBM
PFE
0.2272
0.0230
0.1033
0.0007
0.0154
0.3155
0.0881
0.0773
0.1002
0.1945
0.1819
0.0174
0.0078
0.0064
0.0000
0.0899
0.0283
0.0104
0.0136
0.0297
0.7996
0.9636
0.8950
0.9938
0.9846
0.9081
0.9203
0.9189
0.8959
0.8654
0.3479
0.0164
0.0095
0.0116
0.0000
0.5916
0.0791
0.0142
0.0147
0.0994
0.3029
0.1306
0.2614
0.3503
0.4897
0.8085
0.4836
0.2877
0.2201
0.1280
0.0014
0.0057
0.0334
0.0601
0.0086
0.1435
0.0894
0.0048
0.0231
0.0655
0.0079
−0.2981
0.1462
0.1025
−0.0694
0.0845
0.1024
0.1348
−0.0764
0.2025
0.0136
0.0221
0.0117
0.0348
0.0008
0.0384
0.5910
0.0288
0.0214
0.3279
0.2551
0.2861
0.4078
0.3329
0.0037
0.3004
0.6431
0.2210
0.2977
0.2789
0.0056
0.0229
0.2000
0.0445
0.0080
0.0071
0.0485
0.0032
0.0065
0.1112
0.0199
0.1111
0.0635
0.1374
0.1445
0.0183
0.9475
0.2135
−0.0356
0.0296
4.8849
0.0190
0.0616
0.0182
0.2600
7.1895
0.0799
0.0158
0.0381
0.0055
1.8360
0.1447
0.2747
0.3544
0.2575
1.9322
0.2751
0.2713
0.4220
0.3051
0.0033
0.0923
0.0018
0.0034
0.0539
0.0023
0.0175
0.0220
0.0261
0.1085
1.1692
0.1043
−0.0180
−0.0299
0.0900
1.1342
−0.0327
0.1522
0.0934
0.0626
Panel B: Medium stocks in Period 1
TJX
TYC
VIAB
AGN
CHK
AOC
CG
PGN
WMB
BHI
0.0002
0.0681
0.0604
0.1443
0.0647
0.0024
2.3295
0.1006
0.0842
0.2126
0.0125
0.0154
0.0048
0.0212
0.0373
0.0275
0.0984
0.0246
0.0156
0.1721
0.9887
0.9257
0.9399
0.8623
0.8983
0.9738
0.7900
0.8923
0.9112
0.8135
Panel C: Small stocks in Period 1
BFB
CEG
CSC
JEC
AIG
WAT
X
MCO
MKC
CMA
0.1404
0.0754
0.0089
0.0001
0.0718
0.4369
0.0807
0.0836
0.0799
0.1088
0.2180
0.0155
0.0387
0.0087
0.0108
0.3304
0.0550
0.0119
0.0235
0.0035
0.3558
0.9181
0.9590
0.9923
0.9739
0.1888
0.8962
0.9129
0.9156
0.8917
Table A.8: Estimated parameters of the AACD model in Period 2
Stock
ω
AACD model parameters
β
λ
v
α
b
c
Panel A: Large stocks in Period 2
XOM
GE
PG
JNJ
T
CVX
JPM
WMT
IBM
PFE
0.0887
0.0454
0.0792
0.1177
0.0011
0.0723
0.0964
0.1135
0.1161
0.1048
0.0145
0.0246
0.0073
0.0269
0.0175
0.0077
0.0109
0.0297
0.0091
0.0084
0.9109
0.9395
0.9178
0.8711
0.9851
0.9255
0.8989
0.8786
0.8805
0.8920
0.0252
0.0280
0.0073
0.0197
0.0218
0.0113
0.0097
0.0282
0.0072
0.0065
0.4129
0.3037
0.3277
0.3059
0.4391
0.3062
0.2575
0.3455
0.2800
0.2911
0.0134
0.0249
0.0620
0.0028
0.0039
0.0961
0.0440
0.0586
0.0518
0.0863
0.1194
0.1438
0.1709
0.1149
−0.0303
0.1306
0.0980
−0.0796
0.0715
0.0714
0.0113
0.0118
0.0063
0.0082
0.0175
0.0121
0.0272
0.0101
0.0336
0.0157
0.3942
0.3062
0.2564
0.2016
0.3902
0.3578
0.8064
0.1591
0.2654
0.1614
0.0716
0.0001
0.0445
0.0019
0.0131
0.0183
0.0552
0.0099
0.0002
0.0211
0.1158
0.0255
0.1471
0.0367
−0.0190
0.1834
0.2635
0.1877
−0.2291
−0.5932
0.0094
0.2058
0.0062
0.1124
0.0081
0.0080
0.0149
0.0376
0.0092
0.0309
0.0991
0.4015
0.2749
0.3667
0.3008
0.2872
0.2581
0.7320
0.3880
0.3979
0.0202
0.0341
0.0400
0.0199
0.0665
0.0566
0.0532
0.0031
0.0192
0.0205
0.0855
−0.8336
0.1634
−0.3139
0.1205
0.1320
0.0808
−0.1115
0.0951
−0.0976
Panel B: Medium stocks in Period 2
TJX
TYC
VIAB
AGN
CHK
AOC
CG
PGN
WMB
BHI
0.0841
0.0042
0.1222
0.1003
0.1671
0.0661
0.0036
0.1240
0.0062
0.0537
0.0076
0.0136
0.0075
0.0111
0.0168
0.0125
0.0054
0.0186
0.0421
0.0277
0.9147
0.9845
0.8755
0.8942
0.8344
0.9273
0.9915
0.8658
0.9611
0.9287
Panel C: Small stocks in Period 2
BFB
CEG
CSC
JEC
AIG
WAT
X
MCO
MKC
CMA
0.2692
0.0758
0.0788
0.1011
0.0762
0.0935
0.0837
0.0015
0.0763
0.1971
0.0237
0.0898
0.0072
0.0781
0.0087
0.0076
0.0140
0.0175
0.0070
0.0307
0.7239
0.9390
0.9176
0.8852
0.9196
0.9041
0.9100
0.9850
0.9214
0.8098
Table A.9: Estimated parameters of the AACD model in Period 3
Stock
ω
AACD model parameters
β
λ
v
α
b
c
Panel A: Large stocks in Period 3
XOM
GE
PG
JNJ
T
CVX
JPM
WMT
IBM
PFE
0.0081
0.1074
0.0693
0.1371
0.0531
0.0317
0.0001
0.1566
0.0190
0.1153
0.0081
0.0142
0.0135
0.0163
0.0095
0.0909
0.0114
0.0520
0.0118
0.0191
0.9855
0.8928
0.9289
0.8558
0.9453
0.9458
0.9901
0.8611
0.9744
0.8782
0.0251
0.0224
0.0260
0.0091
0.0239
0.0990
0.0221
0.0963
0.0283
0.0171
0.7160
0.3790
0.5272
0.1737
0.5643
0.4485
0.4965
0.4733
0.5718
0.2314
0.0405
0.0474
0.0266
0.0119
0.0829
0.0440
0.0000
0.0466
0.1101
0.0029
0.0980
0.1560
0.0992
0.0758
0.1236
−0.8278
0.0058
0.3512
0.0639
0.1634
0.0000
0.0159
0.0212
0.0120
0.0227
0.0549
0.0300
0.0006
0.0096
0.0794
0.4770
0.2798
0.5011
0.5037
0.3309
0.5332
0.4645
0.0077
0.2241
0.5407
0.0063
0.0176
0.0645
0.0306
0.0008
0.0312
0.0069
0.0008
0.0051
0.1428
−0.0432
0.0973
0.0965
0.1264
−0.3277
0.0666
0.0739
−0.0444
0.1472
0.0511
0.0607
0.2339
0.0185
0.0288
0.6670
0.0079
0.0721
0.0134
0.0263
0.8237
0.4735
0.9440
0.2638
0.3969
0.6108
0.2036
0.5379
0.2290
0.4997
0.7003
0.0260
0.1879
0.0652
0.0008
0.1449
0.0261
0.0968
0.0000
0.0097
0.0436
0.1172
−0.0615
0.1062
−0.2703
0.9938
0.0790
0.0349
0.0228
−0.2772
−0.2341
Panel B: Medium stocks in Period 3
TJX
TYC
VIAB
AGN
CHK
AOC
CG
PGN
WMB
BHI
0.0110
0.0029
0.0953
0.1737
0.0134
0.0708
0.1310
0.1174
0.0895
0.0012
0.0000
0.0142
0.0138
0.0086
0.0219
0.0220
0.0166
0.0195
0.0114
0.0291
0.9890
0.9855
0.9035
0.8298
0.9715
0.9279
0.8761
0.8636
0.9045
0.9759
Panel C: Small stocks in Period 3
BFB
CEG
CSC
JEC
AIG
WAT
X
MCO
MKC
CMA
0.1238
0.1943
0.0852
0.0024
0.7601
0.1485
0.0008
0.0641
0.0184
3.3221
0.0305
0.0395
0.0181
0.0168
0.2344
0.0136
0.0328
0.0159
0.0180
0.2674
0.8871
0.9135
0.9070
0.9854
0.7415
0.8459
0.9725
0.9258
0.9712
0.7685
Experiment 1 (pure BSM)
0.4
VA
VB
Annualized Standard Deviation
0.35
VD
VK
True Volatility
0.3
0.25
0.2
0
50
100
150
Day t
Experiment 2 (BSM with infrequent large jumps)
0.36
VA
Annualized Standard Deviation
0.34
VB
0.32
VD
0.3
VK
True Volatility
0.28
0.26
0.24
0.22
0.2
0.18
0.16
0
50
100
Day t
Figure A.1: Estimates of Deterministic Volatility Model
150
Annualized Standard Deviation
0.35
15-min V A
0.3
0.25
0.2
1-hour V A
15-min V B
1-hour V B
Instantaneous Volatility
0.15
0.1
9:45
10:15 10:45 11:15 11:45 12:15 12:45 13:15 13:45 14:15 14:45 15:15 15:45
Intraday time τ on day t = 1
Annualized Standard Deviation
0.5
0.45
0.4
0.35
0.3
0.25
0.2
9:45
10:15 10:45 11:15 11:45 12:15 12:45 13:15 13:45 14:15 14:45 15:15 15:45
Intraday time τ on day t = 75
Annualized Standard Deviation
0.35
0.3
0.25
0.2
0.15
0.1
9:45
10:15 10:45 11:15 11:45 12:15 12:45 13:15 13:45 14:15 14:45 15:15 15:45
Intraday time τ on day t = 150
Figure A.2: Intraday V A Estimates, Experiment 2 (BSM with infrequent large jumps)
XOM Stock
Annualized Std Dev
0.4
VA
VB
0.3
VD
VK
0.2
0.1
5
10
15
20
25
30
35
40
45
50
55
GE Stock
Annualized Std Dev
0.25
VA
0.2
VB
VD
0.15
VK
0.1
0.05
5
10
15
20
25
30
35
40
45
50
55
PG Stock
Annualized Std Dev
0.25
VA
0.2
VB
VD
0.15
VK
0.1
0.05
5
10
15
20
25
30
35
40
45
50
55
JNJ Stock
Annualized Std Dev
0.4
VA
0.3
VB
VD
VK
0.2
0.1
5
10
15
20
25
30
35
40
45
50
55
T Stock
Annualized Std Dev
0.4
VA
0.3
VB
VD
VK
0.2
0.1
5
10
15
20
25
30
35
40
45
50
55
Figure A.3: Volatility estimates of large stocks in Period 1, 06/01/11 - 06/03/31, 56 days
CVX Stock
Annualized Std Dev
0.4
VA
VB
0.3
VD
VK
0.2
0.1
5
10
15
20
25
30
35
40
45
50
55
Annualized Std Dev
JPM Stock
VA
VB
0.2
VD
VK
0.1
5
10
15
20
25
30
35
40
45
50
55
WMT Stock
Annualized Std Dev
0.25
VA
0.2
VB
VD
0.15
VK
0.1
0.05
5
10
15
20
25
30
35
40
45
50
55
IBM Stock
Annualized Std Dev
0.4
VA
0.3
VB
VD
VK
0.2
0.1
5
10
15
20
25
30
35
40
45
50
55
PFE Stock
Annualized Std Dev
0.4
VA
0.3
VB
VD
VK
0.2
0.1
5
10
15
20
25
30
35
40
45
50
55
Figure A.3 (Continue): Volatility estimates of large stocks in Period 1, 06/01/11 - 06/03/31, 56 days
TJX Stock
Annualized Std Dev
0.5
VA
0.4
VB
VD
0.3
VK
0.2
0.1
5
10
15
20
25
30
35
40
45
50
55
Annualized Std Dev
TYC Stock
VA
VB
0.4
VD
VK
0.2
5
10
15
20
25
30
35
40
45
50
55
Annualized Std Dev
VIAB Stock
VA
0.3
VB
VD
0.2
VK
0.1
5
10
15
20
25
30
35
40
45
50
55
AGN Stock
Annualized Std Dev
0.4
VA
0.3
VB
VD
VK
0.2
0.1
5
10
15
20
25
30
35
40
45
50
55
Annualized Std Dev
CHK Stock
0.5
VA
VB
0.4
VD
VK
0.3
0.2
5
10
15
20
25
30
35
40
45
50
55
Figure A.4: Volatility estimates of medium stocks in Period 1, 06/01/11 - 06/03/31, 56 days
Annualized Std Dev
AOC Stock
VA
VB
0.4
VD
VK
0.2
5
10
15
20
25
30
35
40
45
50
55
Annualized Std Dev
CG Stock
VA
VB
0.4
VD
VK
0.2
5
10
15
20
25
30
35
40
45
50
55
Annualized Std Dev
PGN Stock
VA
0.3
VB
VD
0.2
VK
0.1
5
10
15
20
25
30
35
40
45
50
55
WMB Stock
Annualized Std Dev
0.6
VA
VB
0.4
VD
VK
0.2
5
10
15
20
25
30
35
40
45
50
55
BHI Stock
Annualized Std Dev
0.6
VA
0.5
VB
VD
0.4
VK
0.3
0.2
5
10
15
20
25
30
35
40
45
50
55
Figure A.4 (Continue): Volatility estimates of medium stocks in Period 1, 06/01/11 - 06/03/31, 56 days
BFB Stock
Annualized Std Dev
0.5
VA
0.4
VB
VD
0.3
VK
0.2
0.1
5
10
15
20
25
30
35
40
45
50
55
Annualized Std Dev
CEG Stock
VA
VB
0.2
VD
VK
0.1
5
10
15
20
25
30
35
40
45
50
55
CSC Stock
Annualized Std Dev
0.6
VA
VB
0.4
VD
VK
0.2
5
10
15
20
25
30
35
40
45
50
55
JEC Stock
Annualized Std Dev
0.6
VA
VB
0.4
VD
VK
0.2
5
10
15
20
25
30
35
40
45
50
55
Annualized Std Dev
AIG Stock
VA
0.3
VB
VD
0.2
VK
0.1
5
10
15
20
25
30
35
40
45
50
55
Figure A.5: Volatility estimates of small stocks in Period 1, 06/01/11 - 06/03/31, 56 days
Annualized Std Dev
WAT Stock
VA
VB
0.4
VD
VK
0.2
5
10
15
20
25
30
35
40
45
50
55
Annualized Std Dev
X Stock
0.8
VA
VB
0.6
VD
VK
0.4
0.2
5
10
15
20
25
30
35
40
45
50
55
MCO Stock
Annualized Std Dev
0.5
VA
0.4
VB
VD
0.3
VK
0.2
0.1
5
10
15
20
25
30
35
40
45
50
55
Annualized Std Dev
MKC Stock
0.4
VA
VB
0.3
VD
VK
0.2
0.1
5
10
15
20
25
30
35
40
45
50
55
CMA Stock
Annualized Std Dev
0.4
VA
0.3
VB
VD
VK
0.2
0.1
5
10
15
20
25
30
35
40
45
50
55
Figure A.5 (Continue): Volatility estimates of small stocks in Period 1, 06/01/11 - 06/03/31, 56 days
XOM Stock
Annualized Std Dev
0.4
VA
VB
0.3
VD
VK
0.2
0.1
5
10
15
20
25
30
35
40
45
50
55
Annualized Std Dev
GE Stock
VA
VB
0.2
VD
VK
0.1
5
10
15
20
25
30
35
40
45
50
55
PG Stock
Annualized Std Dev
0.3
VA
VB
0.2
VD
VK
0.1
5
10
15
20
25
30
35
40
45
50
55
JNJ Stock
Annualized Std Dev
0.4
VA
0.3
VB
VD
VK
0.2
0.1
5
10
15
20
25
30
35
40
45
50
55
T Stock
Annualized Std Dev
0.4
VA
0.3
VB
VD
VK
0.2
0.1
5
10
15
20
25
30
35
40
45
50
55
Figure A.6: Volatility estimates of large stocks in Period 2, 07/03/13 - 07/06/04, 58 days
CVX Stock
Annualized Std Dev
0.4
VA
VB
0.3
VD
VK
0.2
0.1
5
10
15
20
25
30
35
40
45
50
55
Annualized Std Dev
JPM Stock
VA
0.3
VB
VD
0.2
VK
0.1
5
10
15
20
25
30
35
40
45
50
55
WMT Stock
Annualized Std Dev
0.4
VA
0.3
VB
VD
VK
0.2
0.1
5
10
15
20
25
30
35
40
45
50
55
IBM Stock
Annualized Std Dev
0.4
VA
0.3
VB
VD
VK
0.2
0.1
5
10
15
20
25
30
35
40
45
50
55
PFE Stock
Annualized Std Dev
0.4
VA
0.3
VB
VD
VK
0.2
0.1
5
10
15
20
25
30
35
40
45
50
55
Figure A.6 (Continue): Volatility estimates of large stocks in Period 2, 07/03/13 - 07/06/04, 58 days
TJX Stock
Annualized Std Dev
0.5
VA
0.4
VB
VD
0.3
VK
0.2
0.1
5
10
15
20
25
30
35
40
45
50
55
Annualized Std Dev
TYC Stock
VA
VB
0.4
VD
VK
0.2
5
10
15
20
25
30
35
40
45
50
55
VIAB Stock
Annualized Std Dev
0.4
VA
0.3
VB
VD
VK
0.2
0.1
5
10
15
20
25
30
35
40
45
50
55
AGN Stock
Annualized Std Dev
0.5
VA
0.4
VB
VD
0.3
VK
0.2
0.1
5
10
15
20
25
30
35
40
45
50
55
Annualized Std Dev
CHK Stock
0.4
VA
VB
0.3
VD
VK
0.2
0.1
5
10
15
20
25
30
35
40
45
50
55
Figure A.7: Volatility estimates of medium stocks in Period 2, 07/03/13 - 07/06/04, 58 days
Annualized Std Dev
AOC Stock
VA
VB
0.4
VD
VK
0.2
5
10
15
20
25
30
35
40
45
50
55
Annualized Std Dev
CG Stock
VA
VB
0.4
VD
VK
0.2
5
10
15
20
25
30
35
40
45
50
55
PGN Stock
Annualized Std Dev
0.4
VA
0.3
VB
VD
VK
0.2
0.1
5
10
15
20
25
30
35
40
45
50
55
WMB Stock
Annualized Std Dev
0.6
VA
VB
0.4
VD
VK
0.2
5
10
15
20
25
30
35
40
45
50
55
BHI Stock
Annualized Std Dev
0.5
VA
0.4
VB
VD
0.3
VK
0.2
0.1
5
10
15
20
25
30
35
40
45
50
55
Figure A.7 (Continue): Volatility estimates of medium stocks in Period 2, 07/03/13 - 07/06/04, 58 days
BFB Stock
Annualized Std Dev
0.5
VA
0.4
VB
VD
0.3
VK
0.2
0.1
5
10
15
20
25
30
35
40
45
50
55
CEG Stock
Annualized Std Dev
0.4
VA
0.3
VB
VD
VK
0.2
0.1
5
10
15
20
25
30
35
40
45
50
55
CSC Stock
Annualized Std Dev
0.6
VA
VB
0.4
VD
VK
0.2
5
10
15
20
25
30
35
40
45
50
55
JEC Stock
Annualized Std Dev
0.6
VA
VB
0.4
VD
VK
0.2
5
10
15
20
25
30
35
40
45
50
55
Annualized Std Dev
AIG Stock
VA
0.3
VB
VD
0.2
VK
0.1
5
10
15
20
25
30
35
40
45
50
55
Figure A.8: Volatility estimates of small stocks in Period 2, 07/03/13 - 07/06/04, 58 days
Annualized Std Dev
WAT Stock
0.4
VA
VB
0.3
VD
VK
0.2
0.1
5
10
15
20
25
30
35
40
45
50
55
Annualized Std Dev
X Stock
0.8
VA
VB
0.6
VD
VK
0.4
0.2
5
10
15
20
25
30
35
40
45
50
55
Annualized Std Dev
MCO Stock
0.5
VA
VB
0.4
VD
0.3
VK
0.2
0.1
5
10
15
20
25
30
35
40
45
50
55
Annualized Std Dev
MKC Stock
0.4
VA
VB
0.3
VD
VK
0.2
0.1
5
10
15
20
25
30
35
40
45
50
55
CMA Stock
Annualized Std Dev
0.4
VA
0.3
VB
VD
VK
0.2
0.1
5
10
15
20
25
30
35
40
45
50
55
Figure A.8 (Continue): Volatility estimates of small stocks in Period 2, 07/03/13 - 07/06/04, 58 days
Annualized Std Dev
XOM Stock
VA
0.4
VB
VD
VK
0.2
5
10
15
20
25
15
20
25
15
20
25
15
20
25
15
20
25
GE Stock
Annualized Std Dev
0.5
0.4
0.3
0.2
VA
VB
VD
VK
0.1
5
10
PG Stock
Annualized Std Dev
0.5
0.4
0.3
0.2
VA
VB
VD
VK
0.1
5
10
JNJ Stock
Annualized Std Dev
0.4
VA
0.3
VB
VD
0.2
VK
0.1
5
10
T Stock
Annualized Std Dev
0.6
VA
0.4
VB
VD
VK
0.2
5
10
Figure A.9: Volatility estimates of large stocks in Period 3, 07/07/13 - 07/08/16, 25 days
Annualized Std Dev
CVX Stock
0.6
VA
VB
0.4
VD
VK
0.2
5
10
15
20
25
15
20
25
15
20
25
15
20
25
15
20
25
Annualized Std Dev
JPM Stock
0.6
VA
VB
0.4
VD
VK
0.2
5
10
WMT Stock
Annualized Std Dev
0.5
0.4
0.3
0.2
VA
VB
VD
VK
0.1
5
10
IBM Stock
Annualized Std Dev
0.4
VA
0.3
VB
VD
0.2
VK
0.1
5
10
PFE Stock
Annualized Std Dev
0.5
0.4
0.3
0.2
VA
VB
VD
VK
0.1
5
10
Figure A.9 (Continue): Volatility estimates of large stocks in Period 3, 07/07/13 - 07/08/16, 25 days
Annualized Std Dev
TJX Stock
0.6
VA
VB
0.4
VD
VK
0.2
5
10
15
20
25
15
20
25
15
20
25
15
20
25
15
20
25
Annualized Std Dev
TYC Stock
0.6
VA
VB
0.4
VD
VK
0.2
5
10
Annualized Std Dev
VIAB Stock
0.6
VA
VB
0.4
VD
VK
0.2
5
10
AGN Stock
Annualized Std Dev
0.6
VA
0.4
VB
VD
VK
0.2
5
10
CHK Stock
Annualized Std Dev
0.6
0.5
0.4
0.3
VA
VB
VD
VK
0.2
5
10
Figure A.10: Volatility estimates of medium stocks in Period 3, 07/07/13 - 07/08/16, 25 days
Annualized Std Dev
AOC Stock
0.6
VA
VB
0.4
VD
VK
0.2
5
10
15
20
25
15
20
25
15
20
25
15
20
25
15
20
25
Annualized Std Dev
CG Stock
VA
0.4
VB
VD
VK
0.2
5
10
Annualized Std Dev
PGN Stock
0.6
VA
VB
0.4
VD
VK
0.2
5
10
Annualized Std Dev
WMB Stock
0.6
VA
VB
0.4
VD
VK
0.2
5
10
Annualized Std Dev
BHI Stock
0.6
0.5
0.4
VA
VB
VD
VK
0.3
0.2
5
10
Figure A.10 (Continue): Volatility estimates of medium stocks in Period 3, 07/07/13 - 07/08/16, 25 days
Annualized Std Dev
BFB Stock
0.6
VA
VB
0.4
VD
VK
0.2
5
10
15
20
25
15
20
25
15
20
25
15
20
25
15
20
25
Annualized Std Dev
CEG Stock
0.6
0.5
0.4
VA
VB
VD
VK
0.3
0.2
5
10
CSC Stock
Annualized Std Dev
0.8
VA
0.6
VB
VD
0.4
VK
0.2
5
10
Annualized Std Dev
JEC Stock
0.8
0.6
0.4
VA
VB
VD
VK
0.2
5
10
AIG Stock
Annualized Std Dev
0.8
VA
0.6
0.4
VB
VD
VK
0.2
5
10
Figure A.11: Volatility estimates of small stocks in Period 3, 07/07/13 - 07/08/16, 25 days
Annualized Std Dev
WAT Stock
0.6
VA
VB
0.4
VD
VK
0.2
5
10
15
20
25
15
20
25
15
20
25
15
20
25
15
20
25
Annualized Std Dev
X Stock
0.8
0.6
0.4
VA
VB
VD
VK
0.2
5
10
Annualized Std Dev
MCO Stock
0.8
0.6
0.4
VA
VB
VD
VK
0.2
5
10
MKC Stock
Annualized Std Dev
0.6
VA
0.4
VB
VD
VK
0.2
5
10
CMA Stock
Annualized Std Dev
0.8
VA
0.6
0.4
VB
VD
VK
0.2
5
10
Figure A.11 (Continue): Volatility estimates of small stocks in Period 3, 2007/07/13 - 2007/08/16, 25 days