85%61.13%66.24%P/L with gearing74.24%114.88%113.04%Total trades404052Profitable trades80.00%71.43%80.77%Average gearing1.281.621.434 Threshold = 2.0
Out-of-sample(1330-1610)(12/04/1999-09/05/2000)GRNNR(44-10-5-1) RNN(44-1-1) GARCH(1,1) COM1GR60.24%7.09%12.74%12.08% 8.33% 9.65%88.26%8.65%20.23%20.79% 10.53% 11.03%477895678.72%85.71%87.50%66.67% 80.00% 83.33%1.321.181.611.481.191.13
COM162.26%80.65%3183.87%1.24
Out-of-sample(1330-1610)(12/04/1999-09/05/2000)NNR(44-10-5-1) RNN(44-1-1) GARCH(1,1) COM1GRGR39.22%9.26%8.67%8.93% 10.36% 8.69%49.52%11.16%11.75%11.32% 11.00% 9.92%244663283.33%100.00%83.33%83.33% 100% 100.00%1.191.121.311.221.051.14
COM148.35%63.05%1694.12%1.25
Out-of-sample(1330-1610)(12/04/1999-09/05/2000)GRNNR(44-10-5-1) RNN(44-1-1) GARCH(1,1) COM120.97%4.39%7.80%10.54% 4.39%24.25%5.37%8.71%11.29% 4.70%8144188.89%100.00%100.00%100.00% 100%1.161.221.091.061.06
Trading days = 21
In-sampleSample(1-1329)ObservationPeriod(31/12/1993-09/04/1999)ModelsNNR(44-10-5-1) RNN(44-1-1) GARCH(1,1)69.03%63.04%60.57%P/L without gearingP/L with gearing103.33%98.22%85.19%Total trades243331Profitable trades82.76%78.57%79.49%Average gearing1.351.401.24
GR0-
Note: Cumulative P/L figures are expressed in volatility points.
Secondly, we evaluate the performance of model combinations. It is quite disappointingas, for both monthly volatility trading strategies, model combinations produce on averagemuch lower cumulative returns than alternative strategies based on NNR/RNN models forthe USD/JPY volatility and on either the GARCH (1,1) or the RNN (44-1-1) model forthe GBP/USD volatility. As a general rule, the GR combination model fails to clearlyoutperform the simple average model combination COM1 during the out-of-sample period,something already noted by Dunis et al. (2001b).Overall, with the monthly holding period, RNN model-based strategies show thestrongest out-of-sample trading performance: in terms of geared cumulative proﬁt, theycome ﬁrst in four out of the eight monthly strategies analysed, and second best in theremaining four cases. The strategy with the highest return yields a 106.17% cumulativeproﬁt over the out-of-sample period and is achieved for the USD/JPY volatility with theRNN (44-5-1) model and a ﬁlter equal to 0.5.The results of the weekly trading strategy are presented in Tables G4.1 and G4.2 inAppendix G. They basically conﬁrm the superior performance achieved through the useof RNN model-based strategies and the comparatively weak results obtained through theuse of model combination.
Forecasting and Trading Currency Volatility
149
Table 4.4 USD/JPY monthly volatility trading strategy1 Threshold = 0.5SampleObservationPeriodModelsP/L without gearingP/L with gearingTotal tradesProfitable tradesAverage gearing2 Threshold = 1.0SampleObservationPeriodModelsP/L without gearingP/L with gearingTotal tradesProfitable tradesAverage gearing3 Threshold = 1.5SampleObservationPeriodModelsP/L without gearingP/L with gearingTotal tradesProfitable tradesAverage gearing4 Threshold = 2.0SampleObservationPeriodModelsP/L without gearingP/L with gearingTotal tradesProfitable tradesAverage gearing
Trading days = 21In-sample(1-1329)(31/12/1993-09/04/1999)NNR(44-1-1) RNN(44-5-1) GARCH(1,1)COM131.35%16.42%−26.42% 19.15%151.79%144.52%6.93% 152.36%6262606054.84%51.61%38.33% 53.33%3.893.983.772.41
Out-of-sample(1330-1610)(12/04/1999-09/05/2000)GRNNR(44-1-1) RNN(44-5-1) GARCH(1,1)19.73%16.21%20.71%−8.57%82.92%63.61%106.17%−4.42%6113131260.66%76.92%84.62%50.00%2.362.863.915.73
COM1GR16.19% 3.21%75.78% 11.50%121366.67% 61.54%2.631.86
Out-of-sample(1330-1610)(12/04/1999-09/05/2000)COM1GRNNR(44-1-1) RNN(44-5-1) GARCH(1,1)40.60% 64.35%20.70%21.32%−1.94%76.72% 122.80%52.81%45.81%42.14%515212121258.82% 69.23%83.33%83.33%41.67%1.581.551.971.943.03
COM1GR13.53% 26.96%21.41% 46.53%111263.64% 83.33%1.411.66
Trading days = 21In-sample(1-1329)(31/12/1993-09/04/1999)NNR(44-1-1) RNN(44-5-1) GARCH(1,1)25.83%44.59%−9.71%67.66%105.01%43.09%58585762.07%56.90%36.84%2.012.161.97Trading days = 21In-sample(1-1329)(31/12/1993-09/04/1999)NNR(44-1-1) RNN(44-5-1) GARCH(1,1)47.07%23.62%40.93%92.67%75.69%86.60%51514664.71%55.77%52.17%1.711.721.60
Out-of-sample(1330-1610)(12/04/1999-09/05/2000)COM1GRNNR(44-1-1) RNN(44-5-1) GARCH(1,1) COM1GR46.63% 84.17%19.65%25.54%1.87% 5.09% 23.80%75.70% 109.49%40.49%73.19%31.31% 10.65% 32.91%374210101261059.46% 73.81%80.00%80.00%33.33%50% 90.00%1.331.351.541.942.211.371.41
Trading days = 21In-sample(1-1329)(31/12/1993-09/04/1999)NNR(44-1-1) RNN(44-5-1) GARCH(1,1)52.52%28.98%39.69%202.23%72.21%49.77%37374159.46%61.54%56.10%1.671.591.39
COM127.89%35.77%2161.90%1.25
Out-of-sample(1330-1610)(12/04/1999-09/05/2000)GRNNR(44-1-1) RNN(44-5-1) GARCH(1,1) COM175.99%34.35%33.70%0.11% 7.74%94.30%60.33%64.76%−5.11% 12.35%26101012380.77%90.00%90.00%33%100%1.261.541.681.541.54
GR3.09%5.48%771%1.23
Note: Cumulative P/L figures are expressed in volatility points.
Finally, allowing for transaction costs, it is worth noting that all the trading strategiesretained produce positive returns, except some based on the GARCH (1,1) benchmarkmodel for the USD/JPY volatility. RNN models appear as the best single modellingapproach for short-term volatility trading. Somewhat surprisingly, model combination,the overall best performing approach in terms of forecasting accuracy, fails to improvethe RNN-based volatility trading results.
4.7 CONCLUDING REMARKS AND FURTHER WORKThe rationale for this chapter was to develop a nonlinear nonparametric approach toforecast FX volatility, identify mispriced options and subsequently develop a tradingstrategy based upon this modelling procedure.Using daily data from December 1993 through April 1999, we examined the use ofNNR and RNN regression models for forecasting and subsequently trading currencyvolatility, 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 straddlesonce mispriced options had been identiﬁed.Allowing for transaction costs, most of the trading strategies retained produced positivereturns. RNN models appeared as the best single modelling approach in a short-termtrading context.Model combination, despite its superior performance in terms of forecasting accuracy,failed to produce superior trading strategies. Admittedly, other combination proceduressuch as decision trees, neural networks, as in Donaldson and Kamstra (1996) or Shadboltand Taylor (2002), and unanimity or majority voting schemes as applied by Albanis andBatchelor (2001) should be investigated.Further work is also needed to compare the results from NNR and RNN models withthose from more “reﬁned” parametric or semiparametric approaches than our GARCH(1,1) benchmark model, such as Donaldson and Kamstra (1997), So et al. (1999), Bollenet al. (2000), Flôres and Roche (2000) and Beine and Laurent (2001).Finally, applying dynamic risk management, the trading strategy retained could alsobe reﬁned to integrate more realistic trading assumptions than those of a ﬁxed holdingperiod of either 5 or 21 trading days.However, despite the limitations of this chapter, we were clearly able to develop reasonably accurate FX volatility forecasts, identify mispriced options and subsequentlysimulate a proﬁtable trading strategy. In the circumstances, the unambiguous implicationfrom our results is that, for the period and currencies considered, the currency optionmarket was inefﬁcient and/or the pricing formulae applied by market participants wereinadequate.
ACKNOWLEDGEMENTSWe are grateful to Professor Ken Holden and Professor John Thompson of LiverpoolBusiness School and to an anonymous referee for helpful comments on an earlier versionof this chapter. The usual disclaimer applies.
APPENDIX ATable A4.1 Summary statistics – GBP/USD realised and implied1-month volatility (31/12/1993–09/04/1999)Sample observationsVariable(s)MaximumMinimumMeanStd. Dev.SkewnessKurtosis – 3Coeff. of variation
:::::::::
1 to 1329Historical vol.15.36442.94486.85601.99280.697880.818380.29067
Implied vol.15.00003.25008.25751.68690.384981.17780.20429
Forecasting and Trading Currency Volatility
151
Table A4.2 Correlation matrix of realised andimplied volatility (GBP/USD)Realised vol.
Implied vol.
1.00000.79174
0.791741.0000
Realised vol.Implied vol.
Implied Vol.
Realised Vol.
16
16
14
14
12
12
10
10
8
8.
6
6
05/04/99
05/01/99
05/10/98
05/07/98
05/04/98
05/01/98
05/10/97
05/07/97
05/04/97
05/01/97
05/10/96
05/07/96
05/04/96
05/01/96
05/10/95
05/07/95
05/04/95
05/01/95
05/10/94
005/07/94
2
005/04/94
4
205/01/94
4
Figure A4.1 GBP/USD realised and implied volatility (%) from 31/12/1993 to 09/04/1999Table A4.3 Summary statistics – USD/JPY realised and implied1-month volatility (31/12/1993–09/04/1999)Sample observationsVariable(s)MaximumMinimumMeanStd. Dev.SkewnessKurtosis – 3Coeff. of variation
:::::::::
1 to 1329Historical vol.33.04464.544611.65845.26381.46752.60610.45150
Implied vol.35.00006.150012.24923.62120.913972.15710.29563
Table A4.4 Correlation matrix of realised andimplied volatility (USD/JPY)
Realised vol.Implied vol.
Realised vol.
Implied vol.
1.00000.80851
0.808511.0000
152
Applied Quantitative Methods for Trading and InvestmentImplied Vol.
Realised Vol.
05/04/99
05/01/99
05/10/98
05/07/98
05/04/98
005/01/98
005/10/97
5
05/07/97
5
05/04/97
10
05/01/97
10
05/10/96
15
05/07/96
15
05/04/96
20
05/01/96
20
05/10/95
25
05/07/95
25
05/04/95
30
05/01/95
30
05/10/94
35
05/07/94
35
05/04/94
40
05/01/94
40
Figure A4.2 USD/JPY realised and implied volatility (%) from 31/12/1993 to 09/04/1999
APPENDIX BB.4.1 GBP/USD GARCH (1,1) assuming a t-distribution and Wald testGBP/USD GARCH (1,1) assuming a t-distributionconverged after 30 iterationsDependent variable is DLUSD1327 observations used for estimation from 3 to 1329Regressor
Coefﬁcient
Standard error
ONEDLUSD(−1)
0.00587560.024310
0.0105930.027389
R-SquaredS.E. of regressionMean of dependentvariableResidual sum ofsquaresAkaike info.criterionDW-statistic
0.00113200.450310.0037569
R-SquaredF -stat. F (3,1323)S.D. of dependent variable
268.2795−754.62571.9739
T-Ratio[Prob]0.55466[0.579]0.88760[0.375]−0.00113300.49977[0.682]0.45006
Equation log-likelihood
−749.6257
Schwarz Bayesian criterion
−767.6024
Forecasting and Trading Currency Volatility
153
Parameters of the Conditional Heteroskedastic ModelExplaining H-SQ, the Conditional Variance of the Error Term
ConstantE-SQ(−1)H-SQ(−1)D.F. of t-dist.
Coefﬁcient
Asymptotic standard error
0.00216250.0321190.958645.1209
0.00152220.0101350.0139690.72992
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 parametersBased on GARCH regression of DLUSD on:ONEDLUSD(−1)1327 observations used for estimation from 3 to 1329Coefﬁcients A1 to A2 are assigned to the above regressors respectively.Coefﬁcients B1 to B4 are assigned to ARCH parameters respectively.List of restriction(s) for the Wald test:b2 = 0; b3 = 0Wald statistic
CHSQ(2) = 16951.2[0.000]
B.4.2 GBP/USD GARCH (1,1) assuming a normal distribution and Wald testGBP/USD GARCH (1,1) assuming a normal distributionconverged after 35 iterationsDependent variable is DLUSD1327 observations used for estimation from 3 to 1329Regressor
Coefﬁcient
Standard error
ONEDLUSD(−1)
0.00467510.047043
0.0117890.028546
R-SquaredS.E. of regressionMean of dependentvariableResidual sum ofsquaresAkaike info.criterionDW-statistic
0.00116510.450300.0037569
R-SquaredF -stat. F (3,1323)S.D. of dependent variable
268.2707−800.35012.0199
T-Ratio[Prob]0.39657[0.692]1.6480[0.100]−0.00109990.51439[0.672]0.45006
Equation log-likelihood
−796.3501
Schwarz Bayesian criterion
−810.7315
154
Applied Quantitative Methods for Trading and Investment
Parameters of the Conditional Heteroskedastic ModelExplaining H-SQ, the Conditional Variance of the Error Term
ConstantE-SQ(−1)H-SQ(−1)
Coefﬁcient
Asymptotic standard error
0.00338740.0283960.95513
0.00160610.00747430.012932
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 parametersBased on GARCH regression of DLUSD on:ONEDLUSD(−1)1327 observations used for estimation from 3 to 1329Coefﬁcients A1 to A2 are assigned to the above regressors respectively.Coefﬁcients B1 to B3 are assigned to ARCH parameters respectively.List of restriction(s) for the Wald test:b2 = 0; b3 = 0Wald statistic
CHSQ(2) = 16941.1[0.000]
B.4.3 USD/JPY GARCH (1,1) assuming a t-distribution and Wald testUSD/JPY GARCH (1,1) assuming a t-distributionconverged after 26 iterationsDependent variable is DLYUSD1327 observations used for estimation from 3 to 1329RegressorONEDLYUSD(−1)
Coefﬁcient0.0373390.022399
R-Squared0.0010778S.E. of regression0.79922Mean of dependent0.0056665variableResidual sum of845.0744squaresAkaike info.−1390.3criterionDW-statistic1.8999
Standard error0.0159020.026976
T-Ratio[Prob]2.3480[0.019]0.83032[0.407]
R-SquaredF -stat. F (3,1323)S.D. of dependent variable
−0.00118740.47580[0.699]0.79875
Equation log-likelihood
−1385.3
Schwarz Bayesian criterion
−1403.3
Forecasting and Trading Currency Volatility
155
Parameters of the Conditional Heteroskedastic ModelExplaining H-SQ, the Conditional Variance of the Error Term
ConstantE-SQ(−1)H-SQ(−1)D.F. of t-dist.
Coefﬁcient
Asymptotic standard error
0.00782930.0681180.924474.3764
0.00457170.0210940.0235050.54777
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 parametersBased on GARCH regression of DLYUSD on:ONEDLYUSD(−1)1327 observations used for estimation from 3 to 1329Coefﬁcients A1 to A2 are assigned to the above regressors respectively.Coefﬁcients B1 to B4 are assigned to ARCH parameters respectively.List of restriction(s) for the Wald test:b2 = 0; b3 = 0CHSQ(2) = 11921.3[0.000]
Wald Statistic
APPENDIX CGARCH Vol. Forecast
Realised Vol.
40
40
35
35
30
30
25
25
20
20.
15
15
Figure C4.1
USD/JPY GARCH (1,1) volatility forecast (%)
05/05/00
05/01/00
05/09/99
05/05/99
05/01/99
05/09/98
05/05/98
05/01/98
05/09/97
05/05/97
05/01/97
05/09/96
05/05/96
05/01/96
05/09/95
005/05/95
005/01/95
5
05/09/94
5
05/05/94
10
05/01/94
10
156
Applied Quantitative Methods for Trading and Investment
APPENDIX DTable D4.1a GBP/USD NNR test results for the validation data setNNR(44-1-1)
NNR(44-5-1)
NNR(44-10-1)
NNR(44-10-5-1)
NNR(44-15-10-1)
1.4%0.201.3733.3%
5.9%0.201.3932.3%
8.1%0.201.4031.5%
12.5%0.201.4130.5%
15.6%0.211.4431.5%
Explained varianceAverage relative errorAverage absolute errorAverage direction error
Table D4.1b GBP/USD RNN test results for the validation data set
Explained varianceAverage relative errorAverage absolute errorAverage direction error
RNN(44-1-1)
RNN(44-5-1)
RNN(44-10-1)
RNN(44-10-5-1)
RNN(44-15-10-1)
13.3%0.181.2530.1%
9.3%0.191.2932.6%
5.5%0.191.3332.6%
6.8%0.201.3830.8%
12.0%0.201.4232.3%
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
Explained varianceAverage relative errorAverage absolute errorAverage direction error
Table D4.2b
Explained varianceAverage relative errorAverage absolute errorAverage direction error
USD/JPY NNR test results for the validation data setNNR(44-1-1)
NNR(44-5-1)
NNR(44-10-1)
NNR(44-10-5-1)
NNR(44-15-10-1)
5.1%0.161.8830.1%
5.4%0.161.8730.8%
5.4%0.161.8630.8%
2.5%0.161.8532.6%
2.9%0