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Finally, allowing for transaction costs, it is worth noting that all the trading strategies
retained produce positive returns, except some based on the GARCH (1,1) benchmark
model for the USD/JPY volatility. RNN models appear as the best single modelling
approach for short-term volatility trading. Somewhat surprisingly, model combination,
the overall best performing approach in terms of forecasting accuracy, fails to improve
the RNN-based volatility trading results.

4.7 CONCLUDING REMARKS AND FURTHER WORK
The rationale for this chapter was to develop a nonlinear nonparametric approach to
forecast FX volatility, identify mispriced options and subsequently develop a trading
strategy based upon this modelling procedure.
Using daily data from December 1993 through April 1999, we examined the use of
NNR and RNN regression models for forecasting and subsequently trading currency
volatility, with an application to the GBP/USD and USD/JPY exchange rates.
These models were then tested out-of-sample over the period April 1999“May 2000,
not only in terms of forecasting accuracy, but also in terms of trading performance. In
150 Applied Quantitative Methods for Trading and Investment

order to do so, we applied a realistic volatility trading strategy using FX option straddles
once mispriced options had been identi¬ed.
Allowing for transaction costs, most of the trading strategies retained produced positive
returns. RNN models appeared as the best single modelling approach in a short-term
trading context.
Model combination, despite its superior performance in terms of forecasting accuracy,
failed to produce superior trading strategies. Admittedly, other combination procedures
such as decision trees, neural networks, as in Donaldson and Kamstra (1996) or Shadbolt
and Taylor (2002), and unanimity or majority voting schemes as applied by Albanis and
Batchelor (2001) should be investigated.
Further work is also needed to compare the results from NNR and RNN models with
those from more “re¬ned” parametric or semiparametric approaches than our GARCH
(1,1) benchmark model, such as Donaldson and Kamstra (1997), So et al. (1999), Bollen
et al. (2000), Flˆ res and Roche (2000) and Beine and Laurent (2001).
o
Finally, applying dynamic risk management, the trading strategy retained could also
be re¬ned to integrate more realistic trading assumptions than those of a ¬xed holding
period of either 5 or 21 trading days.
However, despite the limitations of this chapter, we were clearly able to develop rea-
sonably accurate FX volatility forecasts, identify mispriced options and subsequently
simulate a pro¬table trading strategy. In the circumstances, the unambiguous implication
from our results is that, for the period and currencies considered, the currency option
market was inef¬cient and/or the pricing formulae applied by market participants were
inadequate.

ACKNOWLEDGEMENTS
We are grateful to Professor Ken Holden and Professor John Thompson of Liverpool
Business School and to an anonymous referee for helpful comments on an earlier version
of this chapter. The usual disclaimer applies.

APPENDIX A
Table A4.1 Summary statistics “ GBP/USD realised and implied
1-month volatility (31/12/1993“09/04/1999)

Sample observations : 1 to 1329
Variable(s) : Historical vol. Implied vol.
Maximum : 15.3644 15.0000
Minimum : 2.9448 3.2500
Mean : 6.8560 8.2575
Std. Dev. : 1.9928 1.6869
Skewness : 0.69788 0.38498
Kurtosis “ 3 : 0.81838 1.1778
Coeff. of variation : 0.29067 0.20429
Forecasting and Trading Currency Volatility 151
Table A4.2 Correlation matrix of realised and
implied volatility (GBP/USD)

Realised vol. Implied vol.
Realised vol. 1.0000 0.79174
Implied vol. 0.79174 1.0000




Implied Vol. Realised Vol.
16 16
14 14
12 12
10 10
8 8
.
6 6
4 4
2 2
0 0
05/01/94
05/04/94
05/07/94
05/10/94
05/01/95
05/04/95
05/07/95
05/10/95
05/01/96
05/04/96
05/07/96
05/10/96
05/01/97
05/04/97
05/07/97
05/10/97
05/01/98
05/04/98
05/07/98
05/10/98
05/01/99
05/04/99
Figure A4.1 GBP/USD realised and implied volatility (%) from 31/12/1993 to 09/04/1999

Table A4.3 Summary statistics “ USD/JPY realised and implied
1-month volatility (31/12/1993“09/04/1999)

Sample observations : 1 to 1329
Variable(s) : Historical vol. Implied vol.
Maximum : 33.0446 35.0000
Minimum : 4.5446 6.1500
Mean : 11.6584 12.2492
Std. Dev. : 5.2638 3.6212
Skewness : 1.4675 0.91397
Kurtosis “ 3 : 2.6061 2.1571
Coeff. of variation : 0.45150 0.29563




Table A4.4 Correlation matrix of realised and
implied volatility (USD/JPY)

Realised vol. Implied vol.
Realised vol. 1.0000 0.80851
Implied vol. 0.80851 1.0000
152 Applied Quantitative Methods for Trading and Investment

Implied Vol. Realised Vol.
40 40

35 35

30 30

25 25

20 20

15 15

10 10

5
5

0
0
05/01/94
05/04/94
05/07/94
05/10/94
05/01/95
05/04/95
05/07/95
05/10/95
05/01/96
05/04/96
05/07/96
05/10/96
05/01/97
05/04/97
05/07/97
05/10/97
05/01/98
05/04/98
05/07/98
05/10/98
05/01/99
05/04/99
Figure A4.2 USD/JPY realised and implied volatility (%) from 31/12/1993 to 09/04/1999


APPENDIX B
B.4.1 GBP/USD GARCH (1,1) assuming a t-distribution and Wald test
GBP/USD GARCH (1,1) assuming a t-distribution
converged after 30 iterations

Dependent variable is DLUSD
1327 observations used for estimation from 3 to 1329

Regressor Coef¬cient Standard error T-Ratio[Prob]

ONE 0.0058756 0.010593 0.55466[0.579]
DLUSD(’1) 0.024310 0.027389 0.88760[0.375]

’0.0011330
R-Squared R-Squared
0.0011320
F -stat. F (3,1323)
S.E. of regression 0.45031 0.49977[0.682]
Mean of dependent 0.0037569 S.D. of dependent variable 0.45006
variable
’749.6257
Residual sum of 268.2795 Equation log-likelihood
squares
’754.6257 ’767.6024
Akaike info. Schwarz Bayesian criterion
criterion
DW-statistic 1.9739
Forecasting and Trading Currency Volatility 153

Parameters of the Conditional Heteroskedastic Model
Explaining H-SQ, the Conditional Variance of the Error Term

Coef¬cient Asymptotic standard error

Constant 0.0021625 0.0015222
E-SQ(’1) 0.032119 0.010135
H-SQ(’1) 0.95864 0.013969
D.F. of t-dist. 5.1209 0.72992

H-SQ stands for the conditional variance of the error term.
E-SQ stands for the square of the error term.

Wald test of restriction(s) imposed on parameters

Based on GARCH regression of DLUSD on:
ONE DLUSD(’1)
1327 observations used for estimation from 3 to 1329

Coef¬cients A1 to A2 are assigned to the above regressors respectively.
Coef¬cients B1 to B4 are assigned to ARCH parameters respectively.
List of restriction(s) for the Wald test:
b2 = 0; b3 = 0

CHSQ(2) = 16951.2[0.000]
Wald statistic

B.4.2 GBP/USD GARCH (1,1) assuming a normal distribution and Wald test
GBP/USD GARCH (1,1) assuming a normal distribution
converged after 35 iterations

Dependent variable is DLUSD
1327 observations used for estimation from 3 to 1329

Regressor Coef¬cient Standard error T-Ratio[Prob]

ONE 0.0046751 0.011789 0.39657[0.692]
DLUSD(’1) 0.047043 0.028546 1.6480[0.100]

’0.0010999
R-Squared R-Squared
0.0011651
F -stat. F (3,1323)
S.E. of regression 0.45030 0.51439[0.672]
Mean of dependent 0.0037569 S.D. of dependent variable 0.45006
variable
’796.3501
Residual sum of 268.2707 Equation log-likelihood
squares
’800.3501 ’810.7315
Akaike info. Schwarz Bayesian criterion
criterion
DW-statistic 2.0199
154 Applied Quantitative Methods for Trading and Investment

Parameters of the Conditional Heteroskedastic Model
Explaining H-SQ, the Conditional Variance of the Error Term

Coef¬cient Asymptotic standard error

Constant 0.0033874 0.0016061
E-SQ(’1) 0.028396 0.0074743
H-SQ(’1) 0.95513 0.012932

H-SQ stands for the conditional variance of the error term.
E-SQ stands for the square of the error term.

Wald test of restriction(s) imposed on parameters

Based on GARCH regression of DLUSD on:
ONE DLUSD(’1)
1327 observations used for estimation from 3 to 1329

Coef¬cients A1 to A2 are assigned to the above regressors respectively.
Coef¬cients B1 to B3 are assigned to ARCH parameters respectively.
List of restriction(s) for the Wald test:
b2 = 0; b3 = 0

CHSQ(2) = 16941.1[0.000]
Wald statistic


B.4.3 USD/JPY GARCH (1,1) assuming a t-distribution and Wald test
USD/JPY GARCH (1,1) assuming a t-distribution
converged after 26 iterations

Dependent variable is DLYUSD
1327 observations used for estimation from 3 to 1329

Regressor Coef¬cient Standard error T-Ratio[Prob]

ONE 0.037339 0.015902 2.3480[0.019]
DLYUSD(’1) 0.022399 0.026976 0.83032[0.407]

’0.0011874
R-Squared R-Squared
0.0010778
F -stat. F (3,1323)
S.E. of regression 0.79922 0.47580[0.699]
Mean of dependent 0.0056665 S.D. of dependent variable 0.79875
variable
’1385.3
Residual sum of 845.0744 Equation log-likelihood
squares
’1390.3 ’1403.3
Akaike info. Schwarz Bayesian criterion
criterion
DW-statistic 1.8999
Forecasting and Trading Currency Volatility 155

Parameters of the Conditional Heteroskedastic Model
Explaining H-SQ, the Conditional Variance of the Error Term

Coef¬cient Asymptotic standard error

Constant 0.0078293 0.0045717
E-SQ(’1) 0.068118 0.021094
H-SQ(’1) 0.92447 0.023505
D.F. of t-dist. 4.3764 0.54777

H-SQ stands for the conditional variance of the error term.
E-SQ stands for the square of the error term.

Wald test of restriction(s) imposed on parameters

Based on GARCH regression of DLYUSD on:
ONE DLYUSD(’1)
1327 observations used for estimation from 3 to 1329

Coef¬cients A1 to A2 are assigned to the above regressors respectively.
Coef¬cients B1 to B4 are assigned to ARCH parameters respectively.
List of restriction(s) for the Wald test:
b2 = 0; b3 = 0

CHSQ(2) = 11921.3[0.000]
Wald Statistic


APPENDIX C
GARCH Vol. Forecast Realised Vol.
40 40

35 35

30 30

25 25

20 20
.
15 15

10 10

5 5

0 0
05/01/94

05/05/94

05/09/94

05/01/95

05/05/95

05/09/95

05/01/96

05/05/96

05/09/96

05/01/97

05/05/97

05/09/97

05/01/98

05/05/98

05/09/98

05/01/99

05/05/99

05/09/99

05/01/00

05/05/00




Figure C4.1 USD/JPY GARCH (1,1) volatility forecast (%)
156 Applied Quantitative Methods for Trading and Investment

APPENDIX D

Table D4.1a GBP/USD NNR test results for the validation data set

NNR NNR NNR NNR NNR
(44-1-1) (44-5-1) (44-10-1) (44-10-5-1) (44-15-10-1)

Explained variance 1.4% 5.9% 8.1% 12.5% 15.6%
Average relative error 0.20 0.20 0.20 0.20 0.21
Average absolute error 1.37 1.39 1.40 1.41 1.44
Average direction error 33.3% 32.3% 31.5% 30.5% 31.5%


Table D4.1b GBP/USD RNN test results for the validation data set

RNN RNN RNN RNN RNN
(44-1-1) (44-5-1) (44-10-1) (44-10-5-1) (44-15-10-1)

Explained variance 13.3% 9.3% 5.5% 6.8% 12.0%
Average relative error 0.18 0.19 0.19 0.20 0.20
Average absolute error 1.25 1.29 1.33 1.38 1.42
Average direction error 30.1% 32.6% 32.6% 30.8% 32.3%

NNR/RNN (a-b-c) represents different neural network models, where:
a = number of input variables;
b = number of hidden nodes;
c = number of output nodes.
Realised Vol(t) = f [lVol(t ’ 21), Realised Vol(t ’ 21), |r|(t ’ 21, . . . , t ’ 41), DLGOLD(t ’ 21, . . . , t ’ 41)].


Table D4.2a USD/JPY NNR test results for the validation data set

NNR NNR NNR NNR NNR
(44-1-1) (44-5-1) (44-10-1) (44-10-5-1) (44-15-10-1)

Explained variance 5.1% 5.4% 5.4% 2.5% 2.9%
Average relative error 0.16 0.16 0.16 0.16 0.16
Average absolute error 1.88 1.87 1.86 1.85 1.84
Average direction error 30.1% 30.8% 30.8% 32.6% 32.3%


Table D4.2b USD/JPY RNN test results for the validation data set

RNN RNN RNN RNN RNN
(44-1-1) (44-5-1) (44-10-1) (44-10-5-1) (44-15-10-1)

Explained variance 8.4% 8.5% 8.0% 3.2% 3.0%
Average relative error 0.16 0.16 0.16 0.16 0.16
Average absolute error 1.86 1.85 1.84 1.85 1.85
Average direction error 30.1% 29.4% 29.4% 31.5% 30.1%

NNR/RNN (a-b-c) represents different neural network models, where:
a = number of input variables;
b = number of hidden nodes;
c = number of output nodes.
Realised Vol(t) = f [lVol(t ’ 21), Realised Vol(t ’ 21), |r|(t ’ 21, . . . , t ’ 41), DLOIL(t ’ 21, . . . , t ’ 41)].
0
5
10
15
20
25
30
35
0
2
4
6
8
10
12
14
16
18
05/01/94
05/01/94
05/05/94
05/05/94
05/09/94
05/09/94
05/01/95
05/01/95
05/05/95
05/05/95




Figure E4.2
Figure E4.1
05/09/95
05/09/95
05/01/96
05/01/96
05/05/96
05/05/96
05/09/96
Forecasting and Trading Currency Volatility




05/09/96
RNN Vol. Forecast




05/01/97 05/01/97




.
05/05/97




RNN Vol. Forecast
05/05/97
05/09/97 05/09/97
APPENDIX E




05/01/98 05/01/98
05/05/98 05/05/98
05/09/98
Realised Vol.




05/09/98

Realised Vol.
05/01/99 05/01/99
05/05/99 05/05/99




USD/JPY RNN (44-5-1) volatility forecast (%)
GBP/USD RNN (44-1-1) volatility forecast (%)




05/09/99 05/09/99
05/01/00
05/01/00

05/05/00
05/05/00
0
2
4
6
8




0
5
10
15
20
25
30
35
10
12
14
16
18
157
158




0
5
10
15
20
25
30
35
0
2
4
6
8
10
12
14
16
18
05/01/94 05/01/94
05/05/94 05/05/94
05/09/94 05/09/94
05/01/95 05/01/95
05/05/95 05/05/95
05/09/95




Figure F4.1
05/09/95
05/01/96 05/01/96




GR Comb.
GR Comb.




05/05/96 05/05/96
05/09/96 05/09/96
05/01/97 05/01/97




.
05/05/97 05/05/97
05/09/97 05/09/97




COM1 Comb.
APPENDIX F
COM1 Comb.




05/01/98 05/01/98
05/05/98 05/05/98
05/09/98 05/09/98
05/01/99 05/01/99
05/05/99 05/05/99
Realised Vol.




Figure F4.2 USD/JPY volatility forecast combinations (%)
GBP/USD volatility forecast combinations (%)
Realised Vol.




05/09/99 05/09/99
05/01/00 05/01/00
05/05/00 05/05/00




0
5
0
2
4
6
8




10
15
20
25
30
35
10
12
14
16
18
Applied Quantitative Methods for Trading and Investment
Forecasting and Trading Currency Volatility 159

APPENDIX G
Table G4.1 GBP/USD weekly volatility trading strategy

1 Threshold = 0.5 Trading days = 5
In-sample Out-of-sample
Sample
Observation (1-1329) (1330-1610)
Period (31/12/1993-09/04/1999) (12/04/1999-09/05/2000)
Models NNR(44-10-5-1) RNN(44-1-1) GARCH(1,1) COM1 GR NNR(44-10-5-1) RNN(44-1-1) GARCH(1,1) COM1 GR
’12.85% 8.30% 30.08%
P/L without gearing 162.15% 160.00% 262.41% 189.82% 313.55% 15.37% 13.30%
’2.39% 20.92% 53.64%
P/L with gearing 626.62% 852.70% 889.04% 597.16% 792.87% 36.46% 58.98%
Total trades 221 221 224 210 232 39 43 40 35 33
Profitable trades 66.97% 68.07% 81.70% 72.86% 91.38% 53.85% 60.47% 30.00% 45.71% 72.73%
Average gearing 2.86 4.20 2.71 2.54 2.19 1.72 2.58 1.82 1.62 1.63

2 Threshold = 1.0 Trading days = 5
In-sample Out-of-sample
Sample
Observation (1-1329) (1330-1610)
Period (31/12/1993-09/04/1999) (12/04/1999-09/05/2000)
Models NNR(44-10-5-1) RNN(44-1-1) GARCH(1,1) COM1 GR NNR(44-10-5-1) RNN(44-1-1) GARCH(1,1) COM1 GR
P/L without gearing 166.15% 147.13% 213.13% 167.03% 243.71% 20.09% 12.02% 7.87% 11.11% 18.57%
P/L with gearing 333.20% 420.39% 300.56% 366.44% 27.01% 20.87% 12.80% 14.27% 23.42%
434.96%
Total trades 166 166 147 138 21 27 17 13 14
153
Profitable trades 73.49% 72.00% 77.55% 94.93% 76.19% 55.56% 58.82% 76.92% 85.71%
84.31%
Average gearing 1.82 2.35 1.73 1.61 1.42 1.24 1.64 1.31 1.16 1.21

3 Threshold = 1.5 Trading days = 5
In-sample Out-of-sample
Sample
Observation (1-1329) (1330-1610)
Period (31/12/1993-09/04/1999) (12/04/1999-09/05/2000)
Models NNR(44-10-5-1) RNN(44-1-1) GARCH(1,1) COM1 GR NNR(44-10-5-1) RNN(44-1-1) GARCH(1,1) COM1 GR
P/L without gearing 128.90% 137.62% 176.30% 122.22% 129.73% 16.49% 7.06% 7.08% 7.08% 4.39%
P/L with gearing 198.76% 278.91% 268.10% 173.28% 158.83% 19.49% 10.52% 8.56% 7.61% 5.21%
Total trades 113 113 94 82 61 10 18 4 4 2
Profitable trades 74.34% 69.70% 93.62% 82.93% 95.08% 90.00% 50.00% 100.00% 100% 100.00%
Average gearing 1.49 1.81 1.45 1.38 1.21 1.12 1.27 1.19 1.07 1.14

4 Threshold = 2.0 Trading days = 5
In-sample Out-of-sample
Sample
Observation (1-1329) (1330-1610)
Period (31/12/1993-09/04/1999) (12/04/1999-09/05/2000)
Models NNR(44-10-5-1) RNN(44-1-1) GARCH(1,1) COM1 GR NNR(44-10-5-1) RNN(44-1-1) GARCH(1,1) COM1 GR
P/L without gearing 95.08% 115.09% 118.43% 62.47% 45.67% 3.26% 7.25% 4.67% 3.26% -
P/L with gearing 131.34% 194.33% 158.35% 79.39% 52.97% 3.99% 9.27% 5.65% 3.48% -
Total trades 68 68 56 36 18 1 10 2 1 0
Profitable trades 82.35% 72.44% 94.64% 88.89% 94.44% 100.00% 80.00% 100% 100% -
Average gearing 1.34 1.55 1.31 1.29 1.14 1.22 1.14 1.16 1.07 -
Note: Cumulative P/L figures are expressed in volatility points.
160 Applied Quantitative Methods for Trading and Investment
Table G4.2 USD/JPY weekly volatility trading strategy

1 Threshold = 0.5 Trading days = 5
In-sample Out-of-sample
Sample
Observation (1-1329) (1330-1610)
Period (31/12/1993-09/04/1999) (12/04/1999-09/05/2000)
Models NNR(44-1-1) RNN(44-5-1) GARCH(1,1) COM1 GR NNR(44-1-1) RNN(44-5-1) GARCH(1,1) COM1 GR
’10.17% ’11.66% ’82.07% 39.23% 69.31%
P/L without gearing 208.63% 77.06% 387.38% 63.95% 78.77%
’79.31% ’32.08% ’359.44% 111.86% 206.43%
P/L with gearing 1675.67% 517.50% 1095.73% 251.14% 372.70%
Total trades 251 251 242 232 229 50 50 53 48 50
Profitable trades 51.00% 50.60% 54.13% 55.17% 78.60% 76.00% 86.00% 18.87% 60.42% 78.00%
Average gearing 3.76 3.88 3.67 2.67 2.42 3.22 3.73 5.92 2.18 2.35

2 Threshold = 1.0 Trading days = 5
In-sample Out-of-sample
Sample
Observation (1-1329) (1330-1610)
Period (31/12/1993-09/04/1999) (12/04/1999-09/05/2000)
Models NNR(44-1-1) RNN(44-5-1) GARCH(1,1) COM1 GR NNR(44-1-1) RNN(44-5-1) GARCH(1,1) COM1 GR
’75.61%
P/L without gearing 18.73% 30.43% 234.66% 142.37% 308.50% 52.70% 63.01% 31.71% 76.88%
’162.97%
P/L with gearing 0.43% 102.50% 331.50% 578.10% 88.19% 135.49% 49.88% 124.63%
897.30%
Total trades 213 213 168 165 39 43 53 30 37
201
Profitable trades 52.11% 54.17% 58.93% 82.42% 76.92% 81.40% 24.53% 66.67% 94.59%
54.23%
Average gearing 2.16 2.16 2.15 1.68 1.65 1.77 1.98 2.96 1.52 1.56

3 Threshold = 1.5 Trading days = 5
In-sample Out-of-sample
Sample
Observation (1-1329) (1330-1610)
Period (31/12/1993-09/04/1999) (12/04/1999-09/05/2000)
Models NNR(44-1-1) RNN(44-5-1) GARCH(1,1) COM1 GR NNR(44-1-1) RNN(44-5-1) GARCH(1,1) COM1 GR
’65.02% 5.33% 52.59%
P/L without gearing 34.15% 54.46% 251.07% 104.67% 282.96% 64.00% 55.83%
’102.88% 11.99% 77.99%
P/L with gearing 60.28% 164.86% 643.47% 203.58% 399.48% 108.52% 81.09%
Total trades 157 157 144 98 104 33 35 50 13 22
Profitable trades 54.78% 56.21% 63.89% 60.20% 86.54% 81.82% 80.00% 24.00% 69% 100.00%
Average gearing 1.83 1.75 1.78 1.45 1.35 1.56 1.51 2.00 1.38 1.42

4 Threshold = 2.0 Trading days = 5
In-sample Out-of-sample
Sample
Observation (1-1329) (1330-1610)
Period (31/12/1993-09/04/1999) (12/04/1999-09/05/2000)
Models NNR(44-1-1) RNN(44-5-1) GARCH(1,1) COM1 GR NNR(44-1-1) RNN(44-5-1) GARCH(1,1) COM1 GR
’6.13% ’43.71% 9.23%
P/L without gearing 14.08% 247.22% 71.30% 154.88% 49.61% 53.90% 30.33%
’35.86% ’52.64% 14.94%
P/L with gearing 63.43% 514.30% 105.49% 202.86% 71.66% 86.20% 40.28%
Total trades 117 117 113 47 50 21 23 47 6 11
Profitable trades 51.28% 52.94% 70.80% 65.96% 86.00% 90.48% 86.96% 28% 83% 100%
Average gearing 1.58 1.66 1.56 1.28 1.25 1.46 1.52 1.74 1.55 1.30




REFERENCES
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in Forecast Combination and Portfolio Choice, Dunis C, Moody J, Timmermann A (eds); John
Wiley: Chichester.
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Analysis of Financial Time Series. Ruey S. Tsay
Copyright ™ 2002 John Wiley & Sons, Inc.
ISBN: 0-471-41544-8



CHAPTER 5




High-Frequency Data Analysis
and Market Microstructure


High-frequency data are observations taken at ¬ne time intervals. In ¬nance, they
often mean observations taken daily or at a ¬ner time scale. These data have become
available primarily due to advances in data acquisition and processing techniques,
and they have attracted much attention because they are important in empirical
study of market microstructure. The ultimate high-frequency data in ¬nance are the
transaction-by-transaction or trade-by-trade data in security markets. Here time is
often measured in seconds. The Trades and Quotes (TAQ) database of the New York
Stock Exchange (NYSE) contains all equity transactions reported on the Consoli-
dated Tape from 1992 to present, which includes transactions on NYSE, AMEX,
NASDAQ, and the regional exchanges. The Berkeley Options Data Base provides
similar data for options transactions from August 1976 to December 1996. Trans-
actions data for many other securities and markets, both domestic and foreign,
are continuously collected and processed. Wood (2000) provides some historical
perspective of high-frequency ¬nancial study.
High-frequency ¬nancial data are important in studying a variety of issues related
to trading process and market microstructure. They can be used to compare the ef¬-
ciency of different trading systems in price discovery (e.g., the open out-cry system
of NYSE and the computer trading system of NASDAQ). They can also be used to
study the dynamics of bid and ask quotes of a particular stock (e.g., Hasbrouck, 1999;
Zhang, Russell, and Tsay, 2001b). In an order-driven stock market (e.g., the Taiwan
Stock Exchange), high-frequency data can be used to study the order dynamic and,
more interesting, to investigate the question “who provides the market liquidity.”
Cho, Russell, Tiao, and Tsay (2000) use intraday 5-minute returns of more than 340
stocks traded in the Taiwan Stock Exchange to study the impact of daily stock price
limits and ¬nd signi¬cant evidence of magnet effects toward the price ceiling.
However, high-frequency data have some unique characteristics that do not appear
in lower frequencies. Analysis of these data thus introduces new challenges to ¬nan-
cial economists and statisticians. In this chapter, we study these special characteris-
tics, consider methods for analyzing high-frequency data, and discuss implications

175
176 HIGH-FREQUENCY DATA


of the results obtained. In particular, we discuss nonsynchronous trading, bid-ask
spread, duration models, price movements that are in multiples of tick size, and
bivariate models for price changes and time durations between transactions associ-
ated with price changes. The models discussed are also applicable to other scienti¬c
areas such as telecommunications and environmental studies.


5.1 NONSYNCHRONOUS TRADING

We begin with nonsynchronous trading. Stock tradings such as those on the NYSE
do not occur in a synchronous manner; different stocks have different trading fre-
quencies, and even for a single stock the trading intensity varies from hour to hour
and from day to day. Yet we often analyze a return series in a ¬xed time interval such
as daily, weekly, or monthly. For daily series, price of a stock is its closing price,
which is the last transaction price of the stock in a trading day. The actual time of the
last transaction of the stock varies from day to day. As such we incorrectly assume
daily returns as an equally-spaced time series with a 24-hour interval. It turns out
that such an assumption can lead to erroneous conclusions about the predictability
of stock returns even if the true return series are serially independent.
For daily stock returns, nonsynchronous trading can introduce (a) lag-1 cross-
correlation between stock returns, (b) lag-1 serial correlation in a portfolio return,
and (c) in some situations negative serial correlations of the return series of a single
stock. Consider stocks A and B. Assume that the two stocks are independent and
stock A is traded more frequently than stock B. For special news affecting the market
that arrives near the closing hour on one day, stock A is more likely than B to show
the effect of the news on the same day simply because A is traded more frequently.
The effect of the news on B will eventually appear, but it may be delayed until the
following trading day. If this situation indeed happens, return of stock A appears
to lead that of stock B. Consequently, the return series may show a signi¬cant lag-
1 cross-correlation from A to B even though the two stocks are independent. For
a portfolio that holds stocks A and B, the prior cross-correlation would become a
signi¬cant lag-1 serial correlation.
In a more complicated manner, nonsynchronous trading can also induce erroneous
negative serial correlations for a single stock. There are several models available in
the literature to study this phenomenon; see Campbell, Lo, and MacKinlay (1997)
and the references therein. Here we adopt a simpli¬ed version of the model pro-
posed in Lo and MacKinlay (1990). Let rt be the continuously compounded return
of a security at the time index t. For simplicity, assume that {rt } is a sequence of
independent and identically distributed random variables with mean E(rt ) = µ and
variance Var(rt ) = σ 2 . For each time period, the probability that the security is not
traded is π, which is time-invariant and independent of rt . Let rto be the observed
return. When there is no trade at time index t, we have rto = 0 because there is
no information available. Yet when there is a trade at time index t, we de¬ne rto as
the cumulative return from the previous trade (i.e., rto = rt + rt’1 + · · · + rt’kt ,
where kt is the largest non-negative integer such that no trade occurred in the periods
177
NONSYNCHRONOUS TRADING


t ’ kt , t ’ kt + 1, . . . , t ’ 1). Mathematically, the relationship between rt and rto is
±
with probability π
0

r
t with probability (1 ’ π)2


r + r
t with probability (1 ’ π)2 π

 t’1
r + r
t’1 + rt’2 with probability (1 ’ π)2 π 2
t
rto = . (5.1)
.
. .
. .


k

 i=0 rt’i with probability (1 ’ π)2 π k’1


. .
. .
. .

These probabilities are easy to understand. For example, rto = rt if and only if there
are trades at both t and t ’ 1, rto = rt + rt’1 if and only if there are trades at t and
t ’ 2, but no trade at t ’ 1, and rto = rt + rt’1 + rt’2 if and only if there are trades
at t and t ’ 3, but no trades at t ’ 1 and t ’ 2, and so on. As expected, the total
probability is 1 given by

1
π + (1 ’ π)2 [1 + π + π 2 + · · ·] = π + (1 ’ π)2 = π + 1 ’ π = 1.
1’π
We are ready to consider the moment equations of the observed return series {rto }.
First, the expectation of rto is

E(rto ) = (1 ’ π)2 E(rt ) + (1 ’ π)2 π E(rt + rt’1 ) + · · ·
= (1 ’ π)2 µ + (1 ’ π)2 π2µ + (1 ’ π)2 π 2 3µ + · · ·
= (1 ’ π)2 µ[1 + 2π + 3π 2 + 4π 3 + · · ·]
1
= (1 ’ π)2 µ = µ. (5.2)
(1 ’ π)2

In the prior derivation, we use the result 1 + 2π + 3π 2 + 4π 3 + · · · = 1
. Next,
(1’π)2
for the variance of rto , we use Var(rto ) = E[(rto )2 ] ’ [E(rto )]2 and

E(rto )2 = (1 ’ π)2 E[(rt )2 ] + (1 ’ π)2 π E[(rt + rt’1 )2 ] + · · ·
= (1 ’ π)2 [(σ 2 + µ2 ) + π(2σ 2 + 4µ2 ) + π 2 (3σ 2 + 9µ2 ) + ·] (5.3)
= (1 ’ π)2 {σ 2 [1 + 2π + 3π 2 + · · ·] + µ2 [1 + 4π + 9π 2 + · · ·]} (5.4)
2
= σ 2 + µ2 ’1 . (5.5)
1’π

In Eq. (5.3), we use
2 2
k k k
= Var +E = (k + 1)σ 2 + [(k + 1)µ]2
E rt’i rt’i rt’i
i=0 i=0 i=0
178 HIGH-FREQUENCY DATA


under the serial independence assumption of rt . Using techniques similar to that of
Eq. (5.2), we can show that the ¬rst term of Eq. (5.4) reduces to σ 2 . For the second
term of Eq. (5.4), we use the identity
2 1
1 + 4π + 9π 2 + 16π 3 + · · · = ’ ,
(1 ’ π)3 (1 ’ π)2
which can be obtained as follows: Let

H = 1 + 4π + 9π 2 + 16π 3 + · · · G = 1 + 3π + 5π 2 + 7π 3 + · · · .
and

Then (1 ’ π)H = G and

(1 ’ π)G = 1 + 2π + 2π 2 + 2π 3 + · · ·
2
= 2(1 + π + π 2 + · · ·) ’ 1 = ’ 1.
(1 ’ π)
Consequently, from Eqs. (5.2) and (5.5), we have

2πµ2
2
Var(rto ) =σ +µ ’1 ’µ =σ + .
2 2 2 2
(5.6)
1’π 1’π
Consider next the lag-1 autocovariance of {rto }. Here we use Cov(rto , rt’1 ) = o

E(rtort’1 ) ’ E(rt0 )E(rt’1 ) = E(rtort’1 ) ’ µ2 . The question then reduces to ¬nding
o o o

E(rtort’1 ). Notice that rto rt’1 is zero if there is no trade at t, no trade at t ’ 1, or no
o o

trade at both t and t ’ 1. Therefore, we have
±
with probability 2π ’ π 2
0

r r
 t t’1 with probability (1 ’ π)3


r (r
 t t’1 + rt’2 ) with probability (1 ’ π)3 π


r (r
t t’1 + rt’2 + rt’3 ) with probability (1 ’ π) π
32
rt rt’1 = .
oo
(5.7)
.
. .
. .



r (
t k
i=1 rt’i ) with probability (1 ’ π)3 π k’1


. .
. .
. .

Again the total probability is unity. To understand the prior result, notice that
rto rt’1 = rt rt’1 if and only if there are three consecutive trades at t ’ 2, t ’ 1,
o

and t. Using Eq. (5.7) and the fact that E(rt rt’ j ) = E(rt )E(rt’ j ) = µ2 for j > 0,
we have

E(rtort’1 ) = (1 ’ π)3 {E(rt rt’1 ) + π E[rt (rt’1 + rt’2 )]
o

3
+ π E rt + · · ·}
2
rt’i
i=1

= (1 ’ π)3 µ2 [1 + 2π + 3π 2 + · · ·] = (1 ’ π)µ2 .
179
BID-ASK SPREAD


The lag-1 autocovariance of {rto } is then

Cov(rto , rt’1 ) = ’πµ2 .
o
(5.8)

Provided that µ is not zero, the nonsynchronous trading induces a negative lag-1
autocorrelation in rto given by

’(1 ’ π)πµ2
ρ1 (rto ) = .
(1 ’ π)σ 2 + 2πµ2

In general, we can extend the prior result and show that

Cov(rto , rt’ j ) = ’µ2 π j , j ≥ 1.
o


The magnitude of the lag-1 ACF depends on the choices of µ, π, and σ and can
be substantial. Thus, when µ = 0, the nonsynchronous trading induces negative
autocorrelations in an observed security return series.
The previous discussion can be generalized to the return series of a portfolio that
consists of N securities; see Campbell, Lo, and MacKinlay (1997, Chapter 3). In
the time series literature, effects of nonsynchronous trading on the return of a single
security are equivalent to that of random temporal aggregation on a time series, with
the trading probability π governing the mechanism of aggregation.


5.2 BID-ASK SPREAD

In some stock exchanges (e.g., NYSE) market makers play an important role in facil-
itating trades. They provide market liquidity by standing ready to buy or sell when-
ever the public wishes to sell or buy. By market liquidity, we mean the ability to buy
or sell signi¬cant quantities of a security quickly, anonymously, and with little price
impact. In return for providing liquidity, market makers are granted monopoly rights
by the exchange to post different prices for purchases and sales of a security. They
buy at the bid price Pb and sell at a higher ask price Pa . (For the public, Pb is the
sale price and Pa is the purchase price.) The difference Pa ’ Pb is call the bid-ask
spread, which is the primary source of compensation for market makers. Typically,
the bid-ask spread is small”namely, one or two ticks.
The existence of bid-ask spread, although small in magnitude, has several impor-
tant consequences in time series properties of asset returns. We brie¬‚y discuss the
bid-ask bounce”namely, the bid-ask spread introduces negative lag-1 serial corre-
lation in an asset return. Consider the simple model of Roll (1984). The observed
market price Pt of an asset is assumed to satisfy

S
Pt = Pt— + It , (5.9)
2
180 HIGH-FREQUENCY DATA


where S = Pa ’ Pb is the bid-ask spread, Pt— is the time-t fundamental value of the
asset in a frictionless market, and {It } is a sequence of independent binary random
variables with equal probabilities (i.e., It = 1 with probability 0.5 and = ’1 with
probability 0.5). The It can be interpreted as an order-type indicator, with 1 signify-
ing buyer-initiated transaction and ’1 seller-initiated transaction. Alternatively, the
model can be written as

+S/2 with probability 0.5,
Pt = Pt— +
’S/2 with probability 0.5.

If there is no change in Pt— , then the observed process of price changes is

S
Pt = (It ’ It’1 ) . (5.10)
2

Under the assumption of It in Eq. (5.9), E(It ) = 0 and Var(It ) = 1, and we have
E( Pt ) = 0 and

Var( Pt ) = S 2 /2 (5.11)
Cov( Pt , Pt’1 ) = ’S 2 /4 (5.12)
Cov( Pt , Pt’ j ) = 0, j > 1. (5.13)

Therefore, the autocorrelation function of Pt is

’0.5 if j = 1,
ρ j ( Pt ) = (5.14)
if j > 1.
0

The bid-ask spread thus introduces a negative lag-1 serial correlation in the series
of observed price changes. This is referred to as the bid-ask bounce in the ¬nance
literature. Intuitively, the bounce can be seen as follows. Assume that the fundamen-
tal price Pt— is equal to (Pa + Pb )/2. Then Pt assumes the value Pa or Pb . If the
previously observed price is Pa (the higher value), then the current observed price is
either unchanged or lower at Pb . Thus, Pt is either 0 or ’S. However, if the pre-
vious observed price is Pb (the lower value), then Pt is either 0 or S. The negative
lag-1 correlation in Pt becomes apparent. The bid-ask spread does not introduce
any serial correlation beyond lag 1, however.
A more realistic formulation is to assume that Pt— follows a random walk so that

Pt— = Pt— ’ Pt’1 = t , which forms a sequence of independent and identically
distributed random variables with mean zero and variance σ 2 . In addition, { t } is
independent of {It }. In this case, Var( Pt ) = σ 2 + S 2 /2, but Cov( Pt , Pt’ j )
remains unchanged. Therefore,

’S 2 /4
ρ1 ( Pt ) = ¤ 0.
S 2 /2 + σ 2
181
EMPIRICAL CHARACTERISTICS


The magnitude of lag-1 autocorrelation of Pt is reduced, but the negative effect
remains when S = Pa ’ Pb > 0. In ¬nance, it might be of interest to study the
components of the bid-ask spread. Interested readers are referred to Campbell, Lo,
and MacKinlay (1997) and the references therein.
The effect of bid-ask spread continues to exist in portfolio returns and in multivari-
ate ¬nancial time series. Consider the bivariate case. Denote the bivariate order-type
indicator by It = (I1t , I2t ) , where I1t is for the ¬rst security and I2t for the second
security. If I1t and I2t are contemporaneously correlated, then the bid-ask spreads
can introduce negative lag-1 cross-correlations.



5.3 EMPIRICAL CHARACTERISTICS OF TRANSACTIONS DATA

Let ti be the calendar time, measured in seconds from midnight, at which the i-th
transaction of an asset takes place. Associated with the transaction are several vari-
ables such as the transaction price, the transaction volume, the prevailing bid and ask
quotes, and so on. The collection of ti and the associated measurements are referred
to as the transactions data. These data have several important characteristics that do
not exist when the observations are aggregated over time. Some of the characteristics
are given next.

1. Unequally spaced time intervals: Transactions such as stock tradings on an
exchange do not occur at equally spaced time intervals. As such the observed
transaction prices of an asset do not form an equally spaced time series. The
time duration between trades becomes important and might contain useful
information about market microstructure (e.g., trading intensity).
2. Discrete-valued prices: The price change of an asset from one transaction to
the next only occurs in multiples of tick size. In the NYSE, the tick size was
one eighth of a dollar before June 24, 1997, and was one sixteenth of a dollar
before January 29, 2001. All NYSE and AMEX stocks started to trade in dec-
imals on January 29, 2001. Therefore, the price is a discrete-valued variable in
transactions data. In some markets, price change may also be subject to limit
constraints set by regulators.
3. Existence of a daily periodic or diurnal pattern: Under the normal trading con-
ditions, transaction activity can exhibit periodic pattern. For instance, in the
NYSE, transactions are heavier at the beginning and closing of the trading
hours and thinner during the lunch hours, resulting in a “U-shape” transac-
tion intensity. Consequently, time durations between transactions also exhibit
a daily cyclical pattern.
4. Multiple transactions within a single second: It is possible that multiple trans-
actions, even with different prices, occur at the same time. This is partly due
to the fact that time is measured in seconds that may be too long a time scale
in periods of heavy tradings.
182 HIGH-FREQUENCY DATA


Table 5.1. Frequencies of Price Change in Multiples of Tick Size for IBM Stock from
November 1, 1990 to January 31, 1991.

¤ ’3 ’2 ’1 ≥3
Number (tick) 0 1 2
Percentage 0.66 1.33 14.53 67.06 14.53 1.27 0.63



To demonstrate these characteristics, we consider ¬rst the IBM transactions data
from November 1, 1990 to January 31, 1991. These data are from the Trades, Orders
Reports, and Quotes (TORQ) dataset; see Hasbrouck (1992). There are 63 trading
days and 60,328 transactions. To simplify the discussion, we ignore the price changes
between trading days and focus on the transactions that occurred in the normal trad-
ing hours from 9:30 am to 4:00 pm Eastern Time. It is well known that overnight
stock returns differ substantially from intraday returns; see Stoll and Whaley (1990)
and the references therein. Table 5.1 gives the frequencies in percentages of price
change measured in the tick size of $1/8 = $0.125. From the table, we make the
following observations:

1. About two-thirds of the intraday transactions were without price change.
2. The price changed in one tick approximately 29% of the intraday transactions.
3. Only 2.6% of the transactions were associated with two-tick price changes.
4. Only about 1.3% of the transactions resulted in price changes of three ticks or

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