Transcript High Frequency Finance

Semivariance Significance
in the S&P500
Baishi Wu, 4/7/08
Outline
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Motivation
Background Math
Data Information
Summary Statistics
Regressions
Appendix
Introduction
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Want to examine predictive regressions for realized
variance by using realized semi-variance as a regressor
Test significance of realized semi-variance and realized upvariance by correlation with daily open-close returns
Regressions are of the HAR-RV form from Corsi (2003)
Semi-variance from Barndorff-Nielsen, Kinnebrock, and
Shephard (2008)
Equations
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Realized Volatility (RV)
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Bipower Variance (BV)
Equations
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Realized Semivariance (RS)
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Realized upVariance (upRV)
upRV = RV - RS
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Bipower Downard Variance (BPDV)
Equations
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Daily open to close returns (ri)
ri = log(priceclose) – log(priceopen)
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The daily open to close returns are correlated with the
RV, upRV, and RS to determine whether market volatility
is dependent on direction
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This statistic is also squared to determine if the size of
the open to close price shift correlates with the
magnitude of realized volatility
Equations
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Heterogenous Auto-Regressive Realized Volatility
(HAR-RV) from Corsi, 2003:
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Multi-period normalized realized variation is defined as
the average of one-period measures. The model is using
rough daily, weekly, monthly periods.
Equations
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Extensions of HAR-RV
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Created different regressions using lagged RS and lagged
upRV in predicting RV creating HAR-RS and HAR-upRV
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Compared to original HAR-RV model
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Created combined regressions of a combination of both
RS and upRV to predict RV using HAR-RS-upRV
Equations
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Tri-Power Quarticity
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Relative Jump
Equations
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Max Version z-Statistic (Tri-Power)
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The max version Tri-Power z-Statistic is used to measure
jumps in the data in this case
Take one sided significance at .999 level, or z = 3.09
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Data Preparation
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Collected at five minute intervals
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S&P 500 Data Set
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1985 to late 2007 (5751 Observations) – Included large spike
in RV/BV, less sampling days in this data set
1990 to late 2007 (4487 Observations) – Largely influenced by
upward trend of S&P 500 in the 1990s
2000 to late 2007 (1959 Observations) – Possibly examines a
period of the greatest market volatility
Chose different sample lengths in order to test the
consistency in correlations and regressions
Data Preparation
S&P500, 1985-2007
Summary Statistics
1985-2007
1990-2007
2000-2007
Mean
(x 1e-4)
Std
Mean
(x 1e-4)
Std
Mean
(x 1e-4)
Std
1.8151
0.0104
1.1894
0.0090
-0.3733
0.0097
ri2
1.0846
0.0012
0.8167
0.0002
0.9365
0.0002
RV
0.9735
0.0008
0.8130
0.0001
0.9350
0.0001
upRV
0.4956
0.0005
0.4040
0.0001
0.4703
0.0001
RS
0.4778
0.0003
0.4089
0.0001
0.4647
0.0001
BV
0.8860
0.0005
0.7680
0.0001
0.8761
0.0001
BPDV
0.0348
0.0001
0.0249
0.0000
0.0266
0.0000
ri
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Numbers are similar except for daily returns
Correlation
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Semi-variance correlates the highest with squared daily
returns; is this indicative of higher volatility in a down
market?
Realized up-variance is not higher than Realized Variance
S&P500, 1985-2007
Correlation
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This segment has the lowest correlation of semi-variance
with realized up-variance
Semi-variance does not have a higher correlation with
squared daily returns than either RV or upRV
S&P500, 1990-2007
Correlation
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Only segment where daily squared returns are positively
correlated (though slightly) with daily returns
S&P500, 2000-2007
Correlation Summary
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Anticipate positive correlations of realized up-variance
with daily returns, negative correlations of semi-variance
Both semi-variance statistics ought to have a higher
correlation with the daily returns than the realized
variance (found untrue in 1985-2007 dataset)
Expected to see a higher correlation with semi-variance
and daily squared returns in order to indicate higher
volatility in a down market (not the case)
Bipower Downward Variation is a combination of
Bipower Variation and Semivariance; correlates very
negatively with daily returns (why?)
HAR-RV
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R2 = 0.1088
Monthly regressor not significant, very low
correlation
S&P500, 1985-2007
HAR-RV
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R2 = 0.3648
Daily lag not significant
S&P500, 1990-2007
HAR-RV
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R2 = 0.4972
Daily, monthly not significant
S&P500, 2000-2007
HAR-RS
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R2 = 0.2110
Weekly lag very insignificant, monthly lag also
insignificant
S&P500, 1985-2007
HAR-RS
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R2 = 0.3158
Daily lag not significant
S&P500, 1990-2007
HAR-RS
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R2 = 0.4221
Daily, monthly not significant
S&P500, 2000-2007
HAR-upRV
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R2 = 0.0616
Very low R2 value, monthly regressor very
insignificant, daily insignificant
S&P500, 1985-2007
HAR-upRV
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R2 = 0.2600
Daily insignificant
S&P500, 1990-2007
HAR-upRV
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R2 = 0.3985
Daily, weekly (slightly) insignificant
S&P500, 2000-2007
Normal Regressions Summary
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Low R2 coefficient in 1985-2007 S&P 500 dataset seems
largely caused by the realized up-variance. This is also the
only dataset that has the R2 value of the RV greater than
the average of its parts
Observe similar levels of correlation, similar significant
variables despite specific statistic (RV, RS, or upRV)
Generally there do not seem to be any noticeable trends
that are specific to any individual test statistic; the
significances of the regressors seem to be a function of
the data set and not the test statistic
RV Regressed with RS
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R2 = 0.2191
Only monthy lags not significant
S&P500, 1985-2007
RV Regressed with RS
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R2 = 0.3950
Daily lags are not as significant
S&P500, 1990-2007
RV Regressed with RS
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R2 = 0.5134
Monthly lags not significant
S&P500, 2000-2007
RV Regressed with upRV
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R2 = 0.0565
Very low correlation, monthly lags not significant
S&P500, 1985-2007
RV Regressed with upRV
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R2 = 0.3034
Daily lags not significant
S&P500, 1990-2007
RV Regressed with upRV
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R2 = 0.4398
Daily and weekly (to a lesser extent) are not
significant
S&P500, 2000-2007
RV Regressed with RS and upRV
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R2 = 0.5910
Both monthly lags in general not significant
S&P500, 1985-2007
RV Regressed with RS and upRV
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R2 = 0.3957
Semi-variance statistics much more significant than
realized up-variance statistics
S&P500, 1990-2007
RV Regressed with RS and upRV
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R2 = 0.5194
Semi-variance statistics much more significant than
realized up-variance statistics
S&P500, 2000-2007
Combined Regressors Summary
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Highest R2 values were found for the HAR-RS-upRV
regression combination of using both the semi-variances
and the realized-upvariances (could this be the zeros?)
In general, semi-variance is a better predictor of RV than
realized up-variance and even RV itself; does this indicate
that the down market predicts overall volatility best? (or
am I over interpreting the value of R2?)
For the combined regression, the semi-variance
coefficients were found to be much more significant
Summary Statistics
R2 values
1985-2007
1990-2007
2000-2007
HAR-RV
0.1088
0.3648
0.4972
HAR-RS
0.2110
0.3158
0.4221
HAR-upRV
0.0616
0.2600
0.3985
HAR-RV/RS
0.2190
0.3950
0.5134
HAR-RV/upRV
0.0565
0.3034
0.4398
HAR-RV/RS/upRV
0.5910
0.3957
0.5194
Summary Statistics
Test Statistics
1985-2007
1990-2007
2000-2007
L1
L5
L22
L1
L5
L22
L1
L5
L22
HAR-RV
5.78
4.91
1.35
2.67
4.32
5.67
3.14
3.32
2.75
HAR-RS
7.52
0.40
2.26
2.92
4.36
5.55
2.82
3.41
2.57
HAR-upRV
3.03
5.02
0.55
1.39
5.77
5.89
2.45
3.15
3.49
HAR-RV/RS
6.55
4.00
0.94
2.85
4.27
5.13
3.25
3.33
2.30
HAR-RV/upRV
5.16
5.58
1.47
1.63
6.05
6.15
2.20
2.83
3.72
HAR-RV/RS/upRV
7.00
3.09
-0.48
2.47
3.27
3.05
3.78
2.02
3.28
HAR-RV/RS/upRV
-5.57
-2.52
0.39
-0.10
-0.46
-0.75
1.50
-0.80
-1.81
Appendix
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Graphs for 1990-2007 S&P 500 Data Set
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Realized Variance and Bipower Variation
Z-Scores with 0.001 Significance
Semivariance, Realized upVariance
Bipower Variation and Bipower Downward Variation
Autocorrelation Plots for 1990-2007
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Realized Variance
Semivariance
Realized upVariance
Realized and Bipower Variance
S&P500, 1990-2007
Statistic
Value
mean(RV)
8.1299e-05
std(RV)
1.2352e-04
mean(BV)
7.6804e-05
std(BV)
1.1303e-04
Z-Scores
S&P500, 1990-2007
Statistic
Value
days
4509
mean(z)
0.6342
std(z)
1.3569
jump days
166
Jump %
3.68%
Semivariance, Realized upVariance
S&P500, 1990-2007
Statistic
Value
mean(RS)
4.0894e-05
std(RS)
7.1114e-05
mean(upRV)
4.0405e-05
std(upRV)
6.3970e-05
Bipower Downward Variation
Statistic
Value
mean(BV)
7.6804e-05
std(BV)
1.1303e-04
mean(BPDV) 2.4916e-06
std(BPDV)
S&P500, 1990-2007
2.7787e-05
Correlogram – Realized Variance
S&P500, 1990-2007
Correlogram – Realized Semivariance
S&P500, 1990-2007
Correlogram – Realized upVariance
S&P500, 1990-2007