library(AER)
library(urca)
library(fUnitRoots)
library(car)
library(rugarch)
library(rmgarch)
library(vars)
library(forecast)spec = ugarchspec (mean.model = list(armaOrder =c(0,0)),variance.model = list(garchOrder =c(1,1),model ="sGARCH"))
ugarchfit(spec, data = currencies$rjpy)##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.006403 0.004903 1.3059 0.19158
## omega 0.001640 0.000216 7.6027 0.00000
## alpha1 0.036849 0.001690 21.8038 0.00000
## beta1 0.956418 0.001184 808.0811 0.00000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.006403 0.005947 1.0768 0.28157
## omega 0.001640 0.000608 2.6964 0.00701
## alpha1 0.036849 0.003673 10.0336 0.00000
## beta1 0.956418 0.000656 1457.4040 0.00000
##
## LogLikelihood : -4344.782
##
## Information Criteria
## ------------------------------------
##
## Akaike 1.2180
## Bayes 1.2218
## Shibata 1.2180
## Hannan-Quinn 1.2193
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 181.0 0
## Lag[2*(p+q)+(p+q)-1][2] 184.5 0
## Lag[4*(p+q)+(p+q)-1][5] 189.1 0
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 20.79 5.114e-06
## Lag[2*(p+q)+(p+q)-1][5] 25.75 4.923e-07
## Lag[4*(p+q)+(p+q)-1][9] 31.56 1.634e-07
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 3.959 0.500 2.000 0.046614
## ARCH Lag[5] 8.886 1.440 1.667 0.012264
## ARCH Lag[7] 13.405 2.315 1.543 0.002842
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 1.1106
## Individual Statistics:
## mu 0.1077
## omega 0.3804
## alpha1 0.4109
## beta1 0.5065
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.07 1.24 1.6
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 1.1641 2.444e-01
## Negative Sign Bias 6.0959 1.145e-09 ***
## Positive Sign Bias 0.9241 3.555e-01
## Joint Effect 39.5060 1.356e-08 ***
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 1911 0
## 2 30 2522 0
## 3 40 3038 0
## 4 50 3460 0
##
##
## Elapsed time : 0.663445LjungBox Test: no auto correlated to itseft
spec = ugarchspec (mean.model = list(armaOrder =c(0,0)),variance.model = list(garchOrder =c(1,1),model ="eGARCH"))
ugarchfit(spec, data = currencies$rjpy)##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : eGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.003186 0.004741 0.67207 0.50154
## omega -0.011004 0.003004 -3.66269 0.00025
## alpha1 -0.028417 0.001212 -23.43837 0.00000
## beta1 0.986437 0.002677 368.53083 0.00000
## gamma1 0.102903 0.007283 14.12952 0.00000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.003186 0.006297 0.50595 0.612894
## omega -0.011004 0.009325 -1.18000 0.238001
## alpha1 -0.028417 0.009386 -3.02763 0.002465
## beta1 0.986437 0.007030 140.31135 0.000000
## gamma1 0.102903 0.026863 3.83062 0.000128
##
## LogLikelihood : -4323
##
## Information Criteria
## ------------------------------------
##
## Akaike 1.2122
## Bayes 1.2170
## Shibata 1.2122
## Hannan-Quinn 1.2138
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 182.6 0
## Lag[2*(p+q)+(p+q)-1][2] 185.7 0
## Lag[4*(p+q)+(p+q)-1][5] 189.8 0
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 16.41 5.109e-05
## Lag[2*(p+q)+(p+q)-1][5] 22.55 3.845e-06
## Lag[4*(p+q)+(p+q)-1][9] 29.87 4.736e-07
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 4.453 0.500 2.000 0.0348425
## ARCH Lag[5] 10.646 1.440 1.667 0.0045694
## ARCH Lag[7] 16.477 2.315 1.543 0.0004957
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 1.8241
## Individual Statistics:
## mu 0.09764
## omega 0.70287
## alpha1 0.17568
## beta1 0.57183
## gamma1 0.68365
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.28 1.47 1.88
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 1.350 1.769e-01
## Negative Sign Bias 5.095 3.575e-07 ***
## Positive Sign Bias 1.124 2.610e-01
## Joint Effect 27.335 5.008e-06 ***
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 1749 0
## 2 30 2308 0
## 3 40 2827 0
## 4 50 3225 0
##
##
## Elapsed time : 1.107285gamma represent for leverage effect
spec = ugarchspec (mean.model = list(armaOrder =c(0,0)),variance.model = list(garchOrder =c(1,1),model ="gjrGARCH"))
ugarchfit(spec, data = currencies$rjpy)##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.003522 0.004913 0.7169 0.473438
## omega 0.002037 0.000333 6.1231 0.000000
## alpha1 0.027579 0.003291 8.3791 0.000000
## beta1 0.950473 0.004008 237.1241 0.000000
## gamma1 0.026850 0.005349 5.0195 0.000001
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.003522 0.005938 0.59322 0.553036
## omega 0.002037 0.000764 2.66594 0.007677
## alpha1 0.027579 0.006218 4.43533 0.000009
## beta1 0.950473 0.006388 148.79911 0.000000
## gamma1 0.026850 0.010414 2.57816 0.009933
##
## LogLikelihood : -4329.648
##
## Information Criteria
## ------------------------------------
##
## Akaike 1.2140
## Bayes 1.2188
## Shibata 1.2140
## Hannan-Quinn 1.2157
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 185.1 0
## Lag[2*(p+q)+(p+q)-1][2] 188.2 0
## Lag[4*(p+q)+(p+q)-1][5] 192.4 0
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 15.96 6.468e-05
## Lag[2*(p+q)+(p+q)-1][5] 23.44 2.163e-06
## Lag[4*(p+q)+(p+q)-1][9] 30.42 3.356e-07
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 5.856 0.500 2.000 0.0155276
## ARCH Lag[5] 12.548 1.440 1.667 0.0015551
## ARCH Lag[7] 17.377 2.315 1.543 0.0002948
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 1.5211
## Individual Statistics:
## mu 0.05841
## omega 0.47122
## alpha1 0.47883
## beta1 0.63269
## gamma1 0.35973
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.28 1.47 1.88
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.9567 3.388e-01
## Negative Sign Bias 5.0233 5.202e-07 ***
## Positive Sign Bias 1.3087 1.907e-01
## Joint Effect 27.4114 4.827e-06 ***
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 1795 0
## 2 30 2328 0
## 3 40 2892 0
## 4 50 3214 0
##
##
## Elapsed time : 0.7462869archm = T –> GARCH in mean
Using higher order for ARMA –> arma(2,3) Runing auto ARMA –> arma(5,3)
best_model <- auto.arima(currencies$rjpy)
# Hiển thị thông tin về mô hình tốt nhất
summary(best_model)## Series: currencies$rjpy
## ARIMA(5,0,3) with non-zero mean
##
## Coefficients:
## ar1 ar2 ar3 ar4 ar5 ma1 ma2 ma3
## -0.5434 0.0240 -0.1596 -0.0172 -0.0381 0.6758 0.0839 0.1691
## s.e. 0.3383 0.3039 0.4597 0.0695 0.0193 0.3383 0.3068 0.4669
## mean
## -0.0011
## s.e. 0.0061
##
## sigma^2 = 0.2179: log likelihood = -4687.96
## AIC=9395.92 AICc=9395.95 BIC=9464.65
##
## Training set error measures:
## ME RMSE MAE MPE MAPE MASE ACF1
## Training set 0.000319364 0.4665194 0.315304 NaN Inf 0.7222998 2.462168e-05spec = ugarchspec (mean.model = list(armaOrder =c(5,3)),variance.model = list(garchOrder =c(1,1),model ="eGARCH"))
ugarchfit(spec, data = currencies$rjpy)##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : eGARCH(1,1)
## Mean Model : ARFIMA(5,0,3)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.002043 0.003182 6.4186e-01 0.520962
## ar1 -0.674212 0.002508 -2.6886e+02 0.000000
## ar2 0.096606 0.002895 3.3375e+01 0.000000
## ar3 0.612132 0.007636 8.0162e+01 0.000000
## ar4 -0.136416 0.005301 -2.5734e+01 0.000000
## ar5 -0.012144 0.000938 -1.2942e+01 0.000000
## ma1 0.865850 0.000183 4.7393e+03 0.000000
## ma2 0.071820 0.000120 5.9708e+02 0.000000
## ma3 -0.610346 0.000001 -1.0401e+06 0.000000
## omega -0.012855 0.006328 -2.0316e+00 0.042195
## alpha1 -0.036833 0.006945 -5.3039e+00 0.000000
## beta1 0.986161 0.004814 2.0485e+02 0.000000
## gamma1 0.107242 0.014518 7.3870e+00 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.002043 0.002628 7.7735e-01 0.43695
## ar1 -0.674212 0.003018 -2.2339e+02 0.00000
## ar2 0.096606 0.003479 2.7766e+01 0.00000
## ar3 0.612132 0.005255 1.1648e+02 0.00000
## ar4 -0.136416 0.003990 -3.4192e+01 0.00000
## ar5 -0.012144 0.000848 -1.4324e+01 0.00000
## ma1 0.865850 0.000115 7.5282e+03 0.00000
## ma2 0.071820 0.000620 1.1580e+02 0.00000
## ma3 -0.610346 0.000000 -1.9365e+06 0.00000
## omega -0.012855 0.032297 -3.9804e-01 0.69061
## alpha1 -0.036833 0.037784 -9.7483e-01 0.32964
## beta1 0.986161 0.023714 4.1586e+01 0.00000
## gamma1 0.107242 0.079550 1.3481e+00 0.17762
##
## LogLikelihood : -4210.115
##
## Information Criteria
## ------------------------------------
##
## Akaike 1.1828
## Bayes 1.1953
## Shibata 1.1828
## Hannan-Quinn 1.1871
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.7912 0.3737
## Lag[2*(p+q)+(p+q)-1][23] 8.3292 1.0000
## Lag[4*(p+q)+(p+q)-1][39] 17.0633 0.8096
## d.o.f=8
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 12.22 4.721e-04
## Lag[2*(p+q)+(p+q)-1][5] 17.39 1.002e-04
## Lag[4*(p+q)+(p+q)-1][9] 23.14 3.054e-05
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 4.119 0.500 2.000 0.042395
## ARCH Lag[5] 8.802 1.440 1.667 0.012856
## ARCH Lag[7] 13.312 2.315 1.543 0.002993
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 13.9079
## Individual Statistics:
## mu 0.10992
## ar1 0.06646
## ar2 0.06467
## ar3 0.12690
## ar4 0.29472
## ar5 0.23116
## ma1 0.06338
## ma2 0.06007
## ma3 0.12056
## omega 0.73560
## alpha1 0.20066
## beta1 0.55089
## gamma1 0.50904
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.89 3.15 3.69
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.3538 7.235e-01
## Negative Sign Bias 4.0134 6.046e-05 ***
## Positive Sign Bias 1.5270 1.268e-01
## Joint Effect 19.5902 2.064e-04 ***
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 837.9 1.868e-165
## 2 30 900.7 7.692e-171
## 3 40 923.0 2.728e-168
## 4 50 956.2 1.825e-168
##
##
## Elapsed time : 3.423373meanmodel = list(armaOrder =c(0,0),archm =T, archpow =2)
varmodel = list(garchOrder =c(1,1),model ="sGARCH")
spec = ugarchspec(mean.model = meanmodel , variance.model = varmodel)
ugarchfit(spec , data = currencies$rjpy)##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.011268 0.010089 1.11679 0.26409
## archm -0.027702 0.050193 -0.55192 0.58101
## omega 0.001626 0.000156 10.44456 0.00000
## alpha1 0.036768 0.000379 96.99834 0.00000
## beta1 0.956565 0.001895 504.80735 0.00000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.011268 0.012151 0.92732 0.353762
## archm -0.027702 0.053756 -0.51533 0.606322
## omega 0.001626 0.000600 2.70965 0.006735
## alpha1 0.036768 0.003722 9.87807 0.000000
## beta1 0.956565 0.001787 535.35329 0.000000
##
## LogLikelihood : -4344.631
##
## Information Criteria
## ------------------------------------
##
## Akaike 1.2182
## Bayes 1.2230
## Shibata 1.2182
## Hannan-Quinn 1.2199
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 181.0 0
## Lag[2*(p+q)+(p+q)-1][2] 184.5 0
## Lag[4*(p+q)+(p+q)-1][5] 189.1 0
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 21.08 4.414e-06
## Lag[2*(p+q)+(p+q)-1][5] 26.01 4.178e-07
## Lag[4*(p+q)+(p+q)-1][9] 31.82 1.382e-07
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 3.933 0.500 2.000 0.047344
## ARCH Lag[5] 8.835 1.440 1.667 0.012622
## ARCH Lag[7] 13.384 2.315 1.543 0.002875
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 1.2634
## Individual Statistics:
## mu 0.08123
## archm 0.06069
## omega 0.38441
## alpha1 0.40883
## beta1 0.50746
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.28 1.47 1.88
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 1.065 2.870e-01
## Negative Sign Bias 6.068 1.358e-09 ***
## Positive Sign Bias 1.001 3.169e-01
## Joint Effect 39.673 1.250e-08 ***
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 1799 0
## 2 30 2339 0
## 3 40 2826 0
## 4 50 3161 0
##
##
## Elapsed time : 1.277665par ( lwd =2, cex.axis = 2)
x_axis = currencies$Date [ currencies$Date >= "2016-08-03"]
plot ( x_axis , static_fc@forecast$sigmaFor , type ="l", xlab ="", ylab ="", col ="blue3"
)
lines ( x_axis , dynamic_fc@forecast$sigmaFor , col ="brown1")
legend ("bottomleft", legend = c ("Static", "Dynamic") , col = c("blue3","brown1") ,
lty = 1)
## 1987-08-21 1987-08-22 1987-08-23 1987-08-24 1987-08-25 1987-08-26
## T+1 0.6826303 0.678263 0.6641801 0.6505759 0.6434195 0.6299555
## 1987-08-27 1987-08-28 1987-08-29 1987-08-30 1987-08-31 1987-09-01
## T+1 0.6231143 0.6102896 0.6184922 0.6056829 0.5962448 0.589531
## 1987-09-02 1987-09-03 1987-09-04 1987-09-05 1987-09-06 1987-09-07
## T+1 0.5774568 0.5661485 0.5720892 0.5605903 0.5569587 0.5463362
## 1987-09-08 1987-09-09 1987-09-10 1987-09-11 1987-09-12 1987-09-13
## T+1 0.5354129 0.5247622 0.5184104 0.5132586 0.5041226 0.494981
## 1987-09-14 1987-09-15 1987-09-16 1987-09-17 1987-09-18 1987-09-19
## T+1 0.487564 0.5241105 0.5137614 0.5068199 0.4999974 0.5127762
## 1987-09-20 1987-09-21 1987-09-22 1987-09-23 1987-09-24 1987-09-25
## T+1 0.5039947 0.4956527 0.4909525 0.4814986 0.4791588 0.4782978
## 1987-09-26 1987-09-27 1987-09-28 1987-09-29 1987-09-30 1987-10-01
## T+1 0.5465182 0.5370277 0.5393924 0.5306867 0.5201619 0.5187172
## 1987-10-02 1987-10-03 1987-10-04 1987-10-05 1987-10-06 1987-10-07
## T+1 0.5093211 0.5155222 0.5112841 0.5034367 0.4946633 0.4851085
## 1987-10-08 1987-10-09 1987-10-10 1987-10-11 1987-10-12 1987-10-13
## T+1 0.4794534 0.4712405 0.4699986 0.4877799 0.4844847 0.4752462
## 1987-10-14 1987-10-15 1987-10-16 1987-10-17 1987-10-18 1987-10-19
## T+1 0.4662225 0.4632573 0.4557307 0.4478942 0.4601717 0.452094
## 1987-10-20 1987-10-21 1987-10-22 1987-10-23 1987-10-24 1987-10-25
## T+1 0.4447112 0.4365367 0.4288566 0.457448 0.4696097 0.4747271
## 1987-10-26 1987-10-27 1987-10-28 1987-10-29 1987-10-30 1987-10-31
## T+1 0.4663039 0.4754378 0.4664088 0.463269 0.4616188 0.453062
## 1987-11-01 1987-11-02 1987-11-03 1987-11-04 1987-11-05 1987-11-06
## T+1 0.4448626 0.4382886 0.4307612 0.4229812 0.4157362 0.4099633
## 1987-11-07 1987-11-08 1987-11-09 1987-11-10 1987-11-11 1987-11-12
## T+1 0.4083028 0.4021054 0.3968392 0.3903962 0.3837859 0.3787619
## 1987-11-13 1987-11-14 1987-11-15 1987-11-16 1987-11-17 1987-11-18
## T+1 0.3797603 0.3741547 0.3781163 0.3794811 0.383203 0.3768062
## 1987-11-19 1987-11-20 1987-11-21 1987-11-22 1987-11-23 1987-11-24
## T+1 0.3715973 0.3672546 0.417289 0.42419 0.4166041 0.4093869
## 1987-11-25 1987-11-26 1987-11-27 1987-11-28 1987-11-29 1987-11-30
## T+1 0.4022206 0.454197 0.4500036 0.4620435 0.6068672 0.6015014
## 1987-12-01 1987-12-02 1987-12-03 1987-12-04 1987-12-05 1987-12-06
## T+1 0.589392 0.5773214 0.595795 0.5978751 0.6044845 0.5920351
## 1987-12-07 1987-12-08 1987-12-09 1987-12-10 1987-12-11 1987-12-12
## T+1 0.6216099 0.6132946 0.6006045 0.5882856 0.5763394 0.5759664
## 1987-12-13 1987-12-14 1987-12-15 1987-12-16 1987-12-17 1987-12-18
## T+1 0.6196728 0.6082048 0.5956909 0.5834577 0.5956134 0.5835285
## 1987-12-19 1987-12-20 1987-12-21 1987-12-22 1987-12-23 1987-12-24
## T+1 0.5860833 0.6104056 0.6027306 0.5932877 0.5811158 0.5721012
## 1987-12-25 1987-12-26 1987-12-27 1987-12-28 1987-12-29 1987-12-30
## T+1 0.5605248 0.5497197 0.5410226 0.5532263 0.5538819 0.5427417
## 1987-12-31 1988-01-01 1988-01-02 1988-01-03 1988-01-04 1988-01-05
## T+1 0.5325126 0.5244241 0.514118 0.6629531 0.6514462 0.6385486
## 1988-01-06 1988-01-07 1988-01-08 1988-01-09 1988-01-10 1988-01-11
## T+1 0.6252091 0.6196652 0.6097664 0.5973745 0.5850986 0.5740843
## 1988-01-12 1988-01-13 1988-01-14 1988-01-15 1988-01-16 1988-01-17
## T+1 0.5625122 0.5511447 0.5400768 0.5293421 0.5198033 0.5317406
## 1988-01-18 1988-01-19 1988-01-20 1988-01-21 1988-01-22 1988-01-23
## T+1 0.5213064 0.5131527 0.5030973 0.4933136 0.4962347 0.4866364
## 1988-01-24 1988-01-25 1988-01-26 1988-01-27 1988-01-28 1988-01-29
## T+1 0.5324183 0.5225299 0.531903 0.5213459 0.5114069 0.5330815
## 1988-01-30 1988-01-31 1988-02-01 1988-02-02 1988-02-03 1988-02-04
## T+1 0.5230863 0.5694731 0.5599628 0.550732 0.5396751 0.5323597
## 1988-02-05 1988-02-06 1988-02-07 1988-02-08 1988-02-09 1988-02-10
## T+1 0.5369777 0.527452 0.5829762 0.5712142 0.561938 0.5505855
## 1988-02-11 1988-02-12 1988-02-13 1988-02-14 1988-02-15 1988-02-16
## T+1 0.5681389 0.560521 0.5548771 0.5452472 0.5670394 0.5559006
## 1988-02-17 1988-02-18 1988-02-19 1988-02-20 1988-02-21 1988-02-22
## T+1 0.5447071 0.5467009 0.5647557 0.5538555 0.550079 0.5396708
## 1988-02-23 1988-02-24 1988-02-25 1988-02-26 1988-02-27 1988-02-28
## T+1 0.5313057 0.5207646 0.5118213 0.5053268 0.4958756 0.4883479
## 1988-02-29 1988-03-01 1988-03-02 1988-03-03 1988-03-04 1988-03-05
## T+1 0.5159902 0.5086768 0.4987432 0.4978179 0.4882432 0.4925098
## 1988-03-06 1988-03-07 1988-03-08 1988-03-09 1988-03-10 1988-03-11
## T+1 0.4971039 0.4993747 0.4927152 0.4832148 0.476444 0.4739222
## 1988-03-12 1988-03-13 1988-03-14 1988-03-15 1988-03-16 1988-03-17
## T+1 0.4659558 0.4602569 0.4617943 0.4585968 0.450034 0.4423259
## 1988-03-18 1988-03-19 1988-03-20 1988-03-21 1988-03-22 1988-03-23
## T+1 0.4345447 0.4681733 0.4756273 0.4670454 0.4615089 0.4528644
## 1988-03-24 1988-03-25 1988-03-26 1988-03-27 1988-03-28 1988-03-29
## T+1 0.4453835 0.4376055 0.4302198 0.431659 0.4341363 0.4313294
## 1988-03-30 1988-03-31 1988-04-01 1988-04-02 1988-04-03 1988-04-04
## T+1 0.4235383 0.4161653 0.4092982 0.405272 0.4529917 0.4458293
## 1988-04-05 1988-04-06 1988-04-07 1988-04-08 1988-04-09 1988-04-10
## T+1 0.4442564 0.4360947 0.4283368 0.4227297 0.4557239 0.4485474
## 1988-04-11 1988-04-12 1988-04-13 1988-04-14 1988-04-15 1988-04-16
## T+1 0.4403067 0.4333807 0.4255268 0.4474445 0.4408751 0.4391777
## 1988-04-17 1988-04-18 1988-04-19 1988-04-20 1988-04-21 1988-04-22
## T+1 0.432446 0.4329429 0.43026 0.4224943 0.415633 0.4246301
## 1988-04-23 1988-04-24 1988-04-25 1988-04-26 1988-04-27 1988-04-28
## T+1 0.4196516 0.4138938 0.4067772 0.4034055 0.3964126 0.3905493
## 1988-04-29 1988-04-30 1988-05-01 1988-05-02 1988-05-03 1988-05-04
## T+1 0.4116402 0.4317127 0.4348856 0.4279216 0.422645 0.4150967
## 1988-05-05 1988-05-06 1988-05-07 1988-05-08 1988-05-09 1988-05-10
## T+1 0.4092702 0.4074795 0.4006308 0.3974821 0.3911792 0.385262
## 1988-05-11 1988-05-12 1988-05-13 1988-05-14 1988-05-15 1988-05-16
## T+1 0.3788039 0.4105523 0.4084922 0.431604 0.4238885 0.4164455
## 1988-05-17 1988-05-18 1988-05-19 1988-05-20 1988-05-21 1988-05-22
## T+1 0.4103062 0.4031133 0.3968271 0.3954502 0.3898469 0.393243
## 1988-05-23 1988-05-24 1988-05-25 1988-05-26 1988-05-27 1988-05-28
## T+1 0.3892493 0.3842738 0.3778451 0.3719643 0.3964503 0.3920499
## 1988-05-29 1988-05-30 1988-05-31 1988-06-01 1988-06-02 1988-06-03
## T+1 0.385924 0.3875727 0.3842706 0.377842 0.37244 0.3665433
## 1988-06-04 1988-06-05 1988-06-06 1988-06-07 1988-06-08 1988-06-09
## T+1 0.4290777 0.4648807 0.4603475 0.4524429 0.4440516 0.436135
## 1988-06-10 1988-06-11 1988-06-12 1988-06-13 1988-06-14 1988-06-15
## T+1 0.4283055 0.4324571 0.4252689 0.422179 0.4148964 0.4075708
## 1988-06-16 1988-06-17 1988-06-18 1988-06-19 1988-06-20 1988-06-21
## T+1 0.4004555 0.398861 0.3926951 0.3888089 0.3823703 0.4010424
## 1988-06-22 1988-06-23 1988-06-24 1988-06-25 1988-06-26 1988-06-27
## T+1 0.3941191 0.3878806 0.4071112 0.402717 0.4043487 0.4023987
## 1988-06-28 1988-06-29 1988-06-30 1988-07-01 1988-07-02 1988-07-03
## T+1 0.395446 0.3886868 0.3853564 0.3789581 0.3745541 0.3698783
## 1988-07-04 1988-07-05 1988-07-06 1988-07-07 1988-07-08 1988-07-09
## T+1 0.4063803 0.400408 0.3935032 0.3892863 0.3900911 0.3858324
## 1988-07-10 1988-07-11 1988-07-12 1988-07-13 1988-07-14 1988-07-15
## T+1 0.3801215 0.3739182 0.3677987 0.3618651 0.3575606 0.3605399
## 1988-07-16 1988-07-17 1988-07-18 1988-07-19 1988-07-20 1988-07-21
## T+1 0.3569844 0.3524432 0.3497508 0.3477913 0.342473 0.3452645
## 1988-07-22 1988-07-23 1988-07-24 1988-07-25 1988-07-26 1988-07-27
## T+1 0.3461796 0.3410506 0.3362145 0.3416815 0.3383419 0.3333204
## 1988-07-28 1988-07-29 1988-07-30 1988-07-31 1988-08-01 1988-08-02
## T+1 0.3305042 0.3257462 0.3506338 0.3466331 0.3441676 0.3483502
## 1988-08-03 1988-08-04 1988-08-05 1988-08-06 1988-08-07 1988-08-08
## T+1 0.3430126 0.3380803 0.3441849 0.3406681 0.3359323 0.3440646
## 1988-08-09 1988-08-10 1988-08-11 1988-08-12 1988-08-13 1988-08-14
## T+1 0.344774 0.3396526 0.3353392 0.3395619 0.3383181 0.3466603
## 1988-08-15 1988-08-16 1988-08-17 1988-08-18 1988-08-19 1988-08-20
## T+1 0.344628 0.3431209 0.3382964 0.3342881 0.3311482 0.3323302
## 1988-08-21 1988-08-22 1988-08-23 1988-08-24 1988-08-25 1988-08-26
## T+1 0.3312552 0.3264538 0.326121 0.3216611 0.317183 0.3167578
## 1988-08-27 1988-08-28 1988-08-29 1988-08-30 1988-08-31 1988-09-01
## T+1 0.3302413 0.3289446 0.341153 0.3365259 0.3316508 0.3356656
## 1988-09-02 1988-09-03 1988-09-04 1988-09-05 1988-09-06 1988-09-07
## T+1 0.3629771 0.3583138 0.3785949 0.3902975 0.3836899 0.3777038
## 1988-09-08 1988-09-09 1988-09-10 1988-09-11 1988-09-12 1988-09-13
## T+1 0.3756565 0.3748537 0.3694672 0.3636288 0.358945 0.3534871
## 1988-09-14 1988-09-15 1988-09-16 1988-09-17 1988-09-18 1988-09-19
## T+1 0.3495565 0.3450496 0.3413106 0.3900043 0.3858789 0.3809718
## 1988-09-20 1988-09-21 1988-09-22 1988-09-23 1988-09-24 1988-09-25
## T+1 0.3750732 0.3801934 0.374265 0.3812101 0.3769569 0.3718206
## 1988-09-26 1988-09-27 1988-09-28 1988-09-29 1988-09-30 1988-10-01
## T+1 0.4027266 0.3957926 0.3972331 0.4017237 0.4324378 0.4328393
## 1988-10-02 1988-10-03 1988-10-04 1988-10-05 1988-10-06 1988-10-07
## T+1 0.4265198 0.4211777 0.414014 0.4089782 0.4048296 0.3991335
## 1988-10-08 1988-10-09 1988-10-10 1988-10-11 1988-10-12 1988-10-13
## T+1 0.3924822 0.4108811 0.41176 0.4045493 0.4006333 0.3964465
## 1988-10-14 1988-10-15 1988-10-16 1988-10-17 1988-10-18 1988-10-19
## T+1 0.3900571 0.4066039 0.399588 0.3934826 0.3868256 0.380318
## 1988-10-20 1988-10-21 1988-10-22 1988-10-23 1988-10-24 1988-10-25
## T+1 0.3761515 0.3713027 0.3692452 0.3633835 0.3585138 0.3552363
## 1988-10-26 1988-10-27 1988-10-28 1988-10-29 1988-10-30 1988-10-31
## T+1 0.3497216 0.3443395 0.3412192 0.3364395 0.3316234 0.3321103
## 1988-11-01 1988-11-02 1988-11-03 1988-11-04 1988-11-05 1988-11-06
## T+1 0.3287848 0.3240624 0.3198268 0.3189177 0.3241338 0.3195856
## 1988-11-07 1988-11-08 1988-11-09 1988-11-10 1988-11-11 1988-11-12
## T+1 0.3320613 0.3291294 0.3244179 0.3209376 0.3165117 0.3130751
## 1988-11-13 1988-11-14 1988-11-15 1988-11-16 1988-11-17 1988-11-18
## T+1 0.3094115 0.3077153 0.3036875 0.3043787 0.3037034 0.3001741
## 1988-11-19 1988-11-20 1988-11-21 1988-11-22 1988-11-23 1988-11-24
## T+1 0.3146359 0.3103728 0.3064479 0.3025648 0.2987041 0.2950533
## 1988-11-25 1988-11-26 1988-11-27 1988-11-28 1988-11-29 1988-11-30
## T+1 0.2919979 0.2927157 0.2912954 0.2884125 0.2852257 0.2819596
## 1988-12-01 1988-12-02 1988-12-03 1988-12-04 1988-12-05 1988-12-06
## T+1 0.278825 0.27585 0.2926611 0.2892503 0.3064438 0.3087497
## 1988-12-07 1988-12-08 1988-12-09 1988-12-10 1988-12-11 1988-12-12
## T+1 0.3046869 0.3025953 0.300729 0.3172102 0.3324874 0.3299682
## 1988-12-13 1988-12-14 1988-12-15 1988-12-16 1988-12-17 1988-12-18
## T+1 0.3252133 0.3207779 0.321729 0.3174081 0.3210697 0.3319153
## 1988-12-19 1988-12-20 1988-12-21 1988-12-22 1988-12-23 1988-12-24
## T+1 0.3275395 0.3263092 0.3217497 0.3336278 0.3304344 0.3321224
## 1988-12-25 1988-12-26 1988-12-27 1988-12-28 1988-12-29 1988-12-30
## T+1 0.3367802 0.3553276 0.3497756 0.3444566 0.3392683 0.3342508
## 1988-12-31 1989-01-01 1989-01-02 1989-01-03 1989-01-04 1989-01-05
## T+1 0.335312 0.3463348 0.3429069 0.3377415 0.3352517 0.3304592
## 1989-01-06 1989-01-07 1989-01-08 1989-01-09 1989-01-10 1989-01-11
## T+1 0.3266338 0.3300001 0.3279808 0.3236734 0.3191213 0.3147172
## 1989-01-12 1989-01-13 1989-01-14 1989-01-15 1989-01-16 1989-01-17
## T+1 0.3106028 0.3065645 0.3025914 0.3046414 0.3045044 0.300584
## 1989-01-18 1989-01-19 1989-01-20 1989-01-21 1989-01-22 1989-01-23
## T+1 0.2967958 0.2931565 0.2943833 0.291006 0.2934429 0.2969793
## 1989-01-24 1989-01-25 1989-01-26 1989-01-27 1989-01-28 1989-01-29
## T+1 0.2933134 0.2897825 0.2864406 0.2933427 0.3346778 0.3330572
## 1989-01-30 1989-01-31 1989-02-01 1989-02-02 1989-02-03 1989-02-04
## T+1 0.3328588 0.3280114 0.3244131 0.3291519 0.3245634 0.3204363
## 1989-02-05 1989-02-06 1989-02-07 1989-02-08 1989-02-09 1989-02-10
## T+1 0.321937 0.3261578 0.3216345 0.3171692 0.3129181 0.3123861
## 1989-02-11 1989-02-12 1989-02-13 1989-02-14 1989-02-15 1989-02-16
## T+1 0.3600856 0.3656425 0.3599927 0.3620209 0.3562593 0.3533836
## 1989-02-17 1989-02-18 1989-02-19 1989-02-20 1989-02-21 1989-02-22
## T+1 0.3485466 0.3448695 0.3505141 0.3555523 0.3522391 0.3467816
## 1989-02-23 1989-02-24 1989-02-25 1989-02-26 1989-02-27 1989-02-28
## T+1 0.3478306 0.3626271 0.3576178 0.3519933 0.3562939 0.3508539
## 1989-03-01 1989-03-02 1989-03-03 1989-03-04 1989-03-05 1989-03-06
## T+1 0.345439 0.3412971 0.3623612 0.3807218 0.400955 0.3992992
## 1989-03-07 1989-03-08 1989-03-09 1989-03-10 1989-03-11 1989-03-12
## T+1 0.3939642 0.3872495 0.3825066 0.3899231 0.3964793 0.4011643
## 1989-03-13 1989-03-14 1989-03-15 1989-03-16 1989-03-17 1989-03-18
## T+1 0.3981003 0.3913425 0.384703 0.3789032 0.3777692 0.3725334
## 1989-03-19 1989-03-20 1989-03-21 1989-03-22 1989-03-23 1989-03-24
## T+1 0.3710022 0.4066173 0.3995319 0.3926687 0.3860263 0.3858727
## 1989-03-25 1989-03-26 1989-03-27 1989-03-28 1989-03-29 1989-03-30
## T+1 0.3844789 0.3818592 0.3940842 0.3874274 0.3809029 0.3783165
## 1989-03-31 1989-04-01 1989-04-02 1989-04-03 1989-04-04 1989-04-05
## T+1 0.3732744 0.3712316 0.3712482 0.3660762 0.3602515 0.354549
## 1989-04-06 1989-04-07 1989-04-08 1989-04-09 1989-04-10 1989-04-11
## T+1 0.3491106 0.3521371 0.3472017 0.3680228 0.3831051 0.3781751
## 1989-04-12 1989-04-13 1989-04-14 1989-04-15 1989-04-16 1989-04-17
## T+1 0.3719271 0.3726788 0.3769358 0.3816742 0.3831725 0.3807561
## 1989-04-18 1989-04-19 1989-04-20 1989-04-21 1989-04-22 1989-04-23
## T+1 0.3744988 0.3683609 0.3632452 0.3577782 0.3546749 0.3688651
## 1989-04-24 1989-04-25 1989-04-26 1989-04-27 1989-04-28 1989-04-29
## T+1 0.3628957 0.3596968 0.3540088 0.3485011 0.3439977 0.3403224
## 1989-04-30 1989-05-01 1989-05-02 1989-05-03 1989-05-04 1989-05-05
## T+1 0.3363927 0.3386149 0.3344762 0.3295779 0.3262822 0.3235796
## 1989-05-06 1989-05-07 1989-05-08 1989-05-09 1989-05-10 1989-05-11
## T+1 0.3211389 0.3175495 0.3157392 0.3114633 0.3073065 0.3206513
## 1989-05-12 1989-05-13 1989-05-14 1989-05-15 1989-05-16 1989-05-17
## T+1 0.333282 0.3339082 0.3296325 0.3255422 0.3222394 0.3177283
## 1989-05-18 1989-05-19 1989-05-20 1989-05-21 1989-05-22 1989-05-23
## T+1 0.3149243 0.316982 0.314733 0.3196829 0.3211028 0.3166402
## 1989-05-24 1989-05-25 1989-05-26 1989-05-27 1989-05-28 1989-05-29
## T+1 0.3123157 0.308155 0.3046181 0.3185786 0.3142945 0.313786
## 1989-05-30 1989-05-31 1989-06-01 1989-06-02 1989-06-03 1989-06-04
## T+1 0.3095859 0.3055021 0.3022181 0.3145564 0.3114712 0.3136584
## 1989-06-05 1989-06-06 1989-06-07 1989-06-08 1989-06-09 1989-06-10
## T+1 0.3122997 0.3086259 0.304562 0.3077512 0.3062141 0.334046
## 1989-06-11 1989-06-12 1989-06-13 1989-06-14 1989-06-15 1989-06-16
## T+1 0.3537448 0.3482463 0.3430511 0.3379105 0.3329464 0.3464595
## 1989-06-17 1989-06-18 1989-06-19 1989-06-20 1989-06-21 1989-06-22
## T+1 0.3412223 0.3362729 0.3475693 0.3438123 0.3386198 0.3343404
## 1989-06-23 1989-06-24 1989-06-25 1989-06-26 1989-06-27 1989-06-28
## T+1 0.3302292 0.3285887 0.3252596 0.3279081 0.3232163 0.3187117
## 1989-06-29 1989-06-30 1989-07-01 1989-07-02 1989-07-03 1989-07-04
## T+1 0.3215971 0.3246116 0.321513 0.3191691 0.3195368 0.3151492
## 1989-07-05 1989-07-06 1989-07-07 1989-07-08 1989-07-09 1989-07-10
## T+1 0.3108757 0.308352 0.3222194 0.321155 0.3170618 0.3165078
## 1989-07-11 1989-07-12 1989-07-13 1989-07-14 1989-07-15 1989-07-16
## T+1 0.3123068 0.3081416 0.3129114 0.3107968 0.3112928 0.309541
## 1989-07-17 1989-07-18 1989-07-19 1989-07-20
## T+1 0.3118025 0.307633 0.3036066 0.3003993## 1987-08-21 1987-08-22 1987-08-23 1987-08-24 1987-08-25
## T+1 -0.004654263 -0.004418605 -0.003669009 -0.002959826 -0.002592656
## 1987-08-26 1987-08-27 1987-08-28 1987-08-29 1987-08-30
## T+1 -0.001912876 -0.001572978 -0.0009457949 -0.001345435 -0.0007236954
## 1987-08-31 1987-09-01 1987-09-02 1987-09-03 1987-09-04
## T+1 -0.0002739087 4.174126e-05 0.0006004269 0.001113186 0.0008450733
## 1987-09-05 1987-09-06 1987-09-07 1987-09-08 1987-09-09 1987-09-10
## T+1 0.001361497 0.001522415 0.001987096 0.00245561 0.00290332 0.003166037
## 1987-09-11 1987-09-12 1987-09-13 1987-09-14 1987-09-15 1987-09-16
## T+1 0.003376776 0.003745309 0.004107446 0.004396396 0.002930422 0.0033563
## 1987-09-17 1987-09-18 1987-09-19 1987-09-20 1987-09-21 1987-09-22
## T+1 0.003637195 0.003909548 0.003396399 0.003750424 0.004081064 0.004264931
## 1987-09-23 1987-09-24 1987-09-25 1987-09-26 1987-09-27 1987-09-28
## T+1 0.004629447 0.004718571 0.004751257 0.001979214 0.002386946 0.002286019
## 1987-09-29 1987-09-30 1987-10-01 1987-10-02 1987-10-03 1987-10-04
## T+1 0.002655388 0.003093912 0.003153422 0.003536422 0.003284442 0.003456985
## 1987-10-05 1987-10-06 1987-10-07 1987-10-08 1987-10-09 1987-10-10
## T+1 0.003772711 0.004119912 0.004491096 0.004707374 0.005016957 0.005063302
## 1987-10-11 1987-10-12 1987-10-13 1987-10-14 1987-10-15 1987-10-16
## T+1 0.004388048 0.004515077 0.004866629 0.005203475 0.005312755 0.005587003
## 1987-10-17 1987-10-18 1987-10-19 1987-10-20 1987-10-21 1987-10-22
## T+1 0.005867773 0.005425727 0.005717906 0.005980427 0.006266053 0.006529577
## 1987-10-23 1987-10-24 1987-10-25 1987-10-26 1987-10-27 1987-10-28
## T+1 0.005524825 0.005077792 0.004886183 0.005200464 0.004859409 0.005196587
## 1987-10-29 1987-10-30 1987-10-31 1987-11-01 1987-11-02 1987-11-03
## T+1 0.005312323 0.005372838 0.005683166 0.005975087 0.006205287 0.006464662
## 1987-11-04 1987-11-05 1987-11-06 1987-11-07 1987-11-08 1987-11-09
## T+1 0.006728018 0.006968952 0.007157947 0.007211822 0.007410959 0.007577781
## 1987-11-10 1987-11-11 1987-11-12 1987-11-13 1987-11-14 1987-11-15
## T+1 0.00777889 0.007981801 0.008133701 0.008103674 0.008271239 0.008153075
## 1987-11-16 1987-11-17 1987-11-18 1987-11-19 1987-11-20 1987-11-21
## T+1 0.008112077 0.007999527 0.00819229 0.008346859 0.008474078 0.006917664
## 1987-11-22 1987-11-23 1987-11-24 1987-11-25 1987-11-26 1987-11-27
## T+1 0.006687415 0.006940307 0.007176675 0.007407284 0.005642337 0.005792676
## 1987-11-28 1987-11-29 1987-11-30 1987-12-01 1987-12-02
## T+1 0.005357285 -0.000780631 -0.0005235497 4.823872e-05 0.0006066252
## 1987-12-03 1987-12-04 1987-12-05 1987-12-06 1987-12-07
## T+1 -0.0002526517 -0.0003510986 -0.0006661915 -7.557339e-05 -0.001498729
## 1987-12-08 1987-12-09 1987-12-10 1987-12-11 1987-12-12
## T+1 -0.001091583 -0.0004807986 9.990291e-05 0.0006515416 0.0006685871
## 1987-12-13 1987-12-14 1987-12-15 1987-12-16 1987-12-17
## T+1 -0.001403391 -0.0008450735 -0.0002477342 0.0003242036 -0.0002440742
## 1987-12-18 1987-12-19 1987-12-20 1987-12-21 1987-12-22
## T+1 0.0003209299 0.0002024503 -0.00095141 -0.0005822377 -0.0001344397
## 1987-12-23 1987-12-24 1987-12-25 1987-12-26 1987-12-27 1987-12-28
## T+1 0.0004323429 0.0008445298 0.001364408 0.001840059 0.002216188 0.001686709
## 1987-12-29 1987-12-30 1987-12-31 1988-01-01 1988-01-02 1988-01-03
## T+1 0.001657928 0.002142316 0.002578418 0.002917383 0.003341765 -0.003604445
## 1988-01-04 1988-01-05 1988-01-06 1988-01-07 1988-01-08
## T+1 -0.003004757 -0.002345068 -0.001676662 -0.001403018 -0.0009204868
## 1988-01-09 1988-01-10 1988-01-11 1988-01-12 1988-01-13 1988-01-14
## T+1 -0.0003273764 0.0002481748 0.0007544066 0.001275911 0.001777859 0.002256729
## 1988-01-15 1988-01-16 1988-01-17 1988-01-18 1988-01-19 1988-01-20
## T+1 0.002711904 0.003108701 0.002610992 0.003046652 0.003381084 0.003786258
## 1988-01-21 1988-01-22 1988-01-23 1988-01-24 1988-01-25 1988-01-26
## T+1 0.004172784 0.004058177 0.004432225 0.002582399 0.002996015 0.002604143
## 1988-01-27 1988-01-28 1988-01-29 1988-01-30 1988-01-31 1988-02-01
## T+1 0.003045019 0.003452006 0.002554381 0.002972947 0.0009634875 0.001389377
## 1988-02-02 1988-02-03 1988-02-04 1988-02-05 1988-02-06 1988-02-07
## T+1 0.001795887 0.002273926 0.002584873 0.002389075 0.002791102 0.0003464703
## 1988-02-08 1988-02-09 1988-02-10 1988-02-11 1988-02-12 1988-02-13
## T+1 0.0008847386 0.001301512 0.001802285 0.001023667 0.001364576 0.001614177
## 1988-02-14 1988-02-15 1988-02-16 1988-02-17 1988-02-18 1988-02-19
## T+1 0.002034232 0.001073153 0.001569104 0.002057574 0.001971293 0.001175637
## 1988-02-20 1988-02-21 1988-02-22 1988-02-23 1988-02-24 1988-02-25
## T+1 0.001659089 0.001824389 0.00227411 0.002629324 0.003069038 0.003435192
## 1988-02-26 1988-02-27 1988-02-28 1988-02-29 1988-03-01 1988-03-02
## T+1 0.003697115 0.004072301 0.004366064 0.0032653 0.003562428 0.003959214
## 1988-03-03 1988-03-04 1988-03-05 1988-03-06 1988-03-07 1988-03-08
## T+1 0.003995777 0.004370116 0.004204204 0.004023942 0.003934223 0.004196179
## 1988-03-09 1988-03-10 1988-03-11 1988-03-12 1988-03-13 1988-03-14
## T+1 0.004563801 0.004821431 0.004916457 0.005213333 0.005422621 0.005366413
## 1988-03-15 1988-03-16 1988-03-17 1988-03-18 1988-03-19 1988-03-20
## T+1 0.005483101 0.005791591 0.006064318 0.006334851 0.0051312 0.004852262
## 1988-03-21 1988-03-22 1988-03-23 1988-03-24 1988-03-25 1988-03-26
## T+1 0.005173025 0.005376862 0.005690266 0.005956698 0.006229007 0.006483142
## 1988-03-27 1988-03-28 1988-03-29 1988-03-30 1988-03-31 1988-04-01
## T+1 0.006433961 0.006348919 0.00644524 0.006709319 0.006954797 0.007179551
## 1988-04-02 1988-04-03 1988-04-04 1988-04-05 1988-04-06 1988-04-07
## T+1 0.007309589 0.005685693 0.005940944 0.005996456 0.006281345 0.006547244
## 1988-04-08 1988-04-09 1988-04-10 1988-04-11 1988-04-12 1988-04-13
## T+1 0.006736452 0.005587252 0.005844559 0.006134984 0.006374911 0.006642376
## 1988-04-14 1988-04-15 1988-04-16 1988-04-17 1988-04-18 1988-04-19
## T+1 0.005883739 0.006115124 0.006174351 0.006406996 0.006389947 0.006481772
## 1988-04-20 1988-04-21 1988-04-22 1988-04-23 1988-04-24 1988-04-25
## T+1 0.006744341 0.006972351 0.006672605 0.00683926 0.007029557 0.007261125
## 1988-04-26 1988-04-27 1988-04-28 1988-04-29 1988-04-30 1988-05-01
## T+1 0.007369437 0.0075912 0.00777415 0.00710332 0.006432122 0.0063231
## 1988-05-02 1988-05-03 1988-05-04 1988-05-05 1988-05-06 1988-05-07
## T+1 0.00656134 0.006739293 0.006990019 0.007180461 0.007238453 0.007457891
## 1988-05-08 1988-05-09 1988-05-10 1988-05-11 1988-05-12 1988-05-13
## T+1 0.007557533 0.007754627 0.00793679 0.008132439 0.007138787 0.007205688
## 1988-05-14 1988-05-15 1988-05-16 1988-05-17 1988-05-18 1988-05-19
## T+1 0.006435845 0.006697552 0.006945548 0.007146795 0.007378779 0.007578163
## 1988-05-20 1988-05-21 1988-05-22 1988-05-23 1988-05-24 1988-05-25
## T+1 0.007621416 0.007795885 0.007690438 0.007814344 0.007966942 0.008161204
## 1988-05-26 1988-05-27 1988-05-28 1988-05-29 1988-05-30 1988-05-31
## T+1 0.008336038 0.007590013 0.007727587 0.00791655 0.007865985 0.00796704
## 1988-06-01 1988-06-02 1988-06-03 1988-06-04 1988-06-05 1988-06-06
## T+1 0.008161298 0.008321998 0.008494775 0.006522054 0.005253013 0.005419312
## 1988-06-07 1988-06-08 1988-06-09 1988-06-10 1988-06-11 1988-06-12
## T+1 0.005705395 0.006003669 0.006279951 0.006548307 0.006406618 0.006651077
## 1988-06-13 1988-06-14 1988-06-15 1988-06-16 1988-06-17 1988-06-18
## T+1 0.006754903 0.006996609 0.0072355 0.007463462 0.007513996 0.007707511
## 1988-06-19 1988-06-20 1988-06-21 1988-06-22 1988-06-23 1988-06-24
## T+1 0.00782793 0.008024805 0.007444809 0.007663086 0.007856519 0.007250347
## 1988-06-25 1988-06-26 1988-06-27 1988-06-28 1988-06-29 1988-06-30
## T+1 0.007391444 0.007339227 0.007401604 0.007621547 0.007831696 0.007933908
## 1988-07-01 1988-07-02 1988-07-03 1988-07-04 1988-07-05 1988-07-06
## T+1 0.008127805 0.008259384 0.008397395 0.00727392 0.00746497 0.00768232
## 1988-07-07 1988-07-08 1988-07-09 1988-07-10 1988-07-11 1988-07-12
## T+1 0.007813204 0.007788332 0.007919352 0.008092792 0.008278253 0.00845822
## 1988-07-13 1988-07-14 1988-07-15 1988-07-16 1988-07-17 1988-07-18
## T+1 0.008629886 0.008752673 0.008667844 0.008768996 0.008896733 0.008971695
## 1988-07-19 1988-07-20 1988-07-21 1988-07-22 1988-07-23 1988-07-24
## T+1 0.00902589 0.009171445 0.009095326 0.009070237 0.009209994 0.00933986
## 1988-07-25 1988-07-26 1988-07-27 1988-07-28 1988-07-29 1988-07-30
## T+1 0.009192917 0.00928296 0.00941669 0.009490814 0.009614616 0.008947174
## 1988-07-31 1988-08-01 1988-08-02 1988-08-03 1988-08-04 1988-08-05
## T+1 0.009057781 0.00912531 0.009010463 0.00915678 0.009289976 0.009124838
## 1988-08-06 1988-08-07 1988-08-08 1988-08-09 1988-08-10 1988-08-11
## T+1 0.009220334 0.009347382 0.009128119 0.009108744 0.009247725 0.009363168
## 1988-08-12 1988-08-13 1988-08-14 1988-08-15 1988-08-16 1988-08-17
## T+1 0.009250169 0.009283599 0.009057032 0.009112735 0.009153832 0.009284182
## 1988-08-18 1988-08-19 1988-08-20 1988-08-21 1988-08-22 1988-08-23
## T+1 0.009391074 0.009473918 0.009442823 0.009471107 0.009596318 0.009604931
## 1988-08-24 1988-08-25 1988-08-26 1988-08-27 1988-08-28 1988-08-29
## T+1 0.009719478 0.009832909 0.009843598 0.009497701 0.009531591 0.009207224
## 1988-08-30 1988-08-31 1988-09-01 1988-09-02 1988-09-03 1988-09-04
## T+1 0.009331554 0.00946071 0.009354483 0.008597927 0.008731294 0.008138717
## 1988-09-05 1988-09-06 1988-09-07 1988-09-08 1988-09-09 1988-09-10
## T+1 0.007781947 0.007984722 0.008165436 0.008226592 0.008250481 0.008409447
## 1988-09-11 1988-09-12 1988-09-13 1988-09-14 1988-09-15 1988-09-16
## T+1 0.008579153 0.008713342 0.008867514 0.008977082 0.009101206 0.009202961
## 1988-09-17 1988-09-18 1988-09-19 1988-09-20 1988-09-21 1988-09-22
## T+1 0.007791018 0.007917929 0.008067131 0.008243953 0.008090623 0.008267967
## 1988-09-23 1988-09-24 1988-09-25 1988-09-26 1988-09-27 1988-09-28
## T+1 0.008059929 0.008187785 0.008340277 0.007391136 0.007610673 0.007565379
## 1988-09-29 1988-09-30 1988-10-01 1988-10-02 1988-10-03 1988-10-04
## T+1 0.007423125 0.006407277 0.006393505 0.006608831 0.006788383 0.007025609
## 1988-10-05 1988-10-06 1988-10-07 1988-10-08 1988-10-09 1988-10-10
## T+1 0.007189937 0.007323799 0.007505374 0.007714141 0.007128076 0.007099409
## 1988-10-11 1988-10-12 1988-10-13 1988-10-14 1988-10-15 1988-10-16
## T+1 0.007332794 0.007457813 0.007590132 0.007789386 0.007266715 0.00749098
## 1988-10-17 1988-10-18 1988-10-19 1988-10-20 1988-10-21 1988-10-22
## T+1 0.007682963 0.007888926 0.008086867 0.008211835 0.008355536 0.008415949
## 1988-10-23 1988-10-24 1988-10-25 1988-10-26 1988-10-27 1988-10-28
## T+1 0.008586224 0.008725609 0.008818362 0.008972506 0.009120618 0.009205433
## 1988-10-29 1988-10-30 1988-10-31 1988-11-01 1988-11-02 1988-11-03
## T+1 0.00933386 0.009461432 0.009448617 0.00953576 0.009657999 0.009766136
## 1988-11-04 1988-11-05 1988-11-06 1988-11-07 1988-11-08 1988-11-09
## T+1 0.009789159 0.009656163 0.009772249 0.009449908 0.009526769 0.00964886
## 1988-11-10 1988-11-11 1988-11-12 1988-11-13 1988-11-14 1988-11-15
## T+1 0.009737914 0.009849776 0.009935563 0.01002599 0.01006749 0.01016513
## 1988-11-16 1988-11-17 1988-11-18 1988-11-19 1988-11-20 1988-11-21
## T+1 0.01014847 0.01016475 0.01024925 0.009896719 0.01000236 0.01009835
## 1988-11-22 1988-11-23 1988-11-24 1988-11-25 1988-11-26 1988-11-27
## T+1 0.01019212 0.01028416 0.01037011 0.01044123 0.01042458 0.01045747
## 1988-11-28 1988-11-29 1988-11-30 1988-12-01 1988-12-02 1988-12-03
## T+1 0.01052374 0.01059622 0.01066967 0.01073937 0.0108048 0.01042585
## 1988-12-04 1988-12-05 1988-12-06 1988-12-07 1988-12-08 1988-12-09
## T+1 0.01050455 0.01009845 0.01004221 0.01014103 0.01019139 0.01023603
## 1988-12-10 1988-12-11 1988-12-12 1988-12-13 1988-12-14 1988-12-15
## T+1 0.009832225 0.00943868 0.009504849 0.009628371 0.009741977 0.009717747
## 1988-12-16 1988-12-17 1988-12-18 1988-12-19 1988-12-20 1988-12-21
## T+1 0.009827247 0.009734552 0.009453751 0.009568166 0.00960006 0.009717218
## 1988-12-22 1988-12-23 1988-12-24 1988-12-25 1988-12-26 1988-12-27
## T+1 0.00940856 0.009492642 0.009448298 0.009324764 0.008815789 0.008971007
## 1988-12-28 1988-12-29 1988-12-30 1988-12-31 1989-01-01 1989-01-02
## T+1 0.00911742 0.009258071 0.009392061 0.00936389 0.009065977 0.009159654
## 1989-01-03 1989-01-04 1989-01-05 1989-01-06 1989-01-07 1989-01-08
## T+1 0.009299055 0.009365493 0.009491993 0.009591659 0.009504016 0.009556696
## 1989-01-09 1989-01-10 1989-01-11 1989-01-12 1989-01-13 1989-01-14
## T+1 0.009667989 0.009784007 0.009894688 0.009996701 0.01009552 0.01019148
## 1989-01-15 1989-01-16 1989-01-17 1989-01-18 1989-01-19 1989-01-20
## T+1 0.01014213 0.01014543 0.01023949 0.01032922 0.01041434 0.01038577
## 1989-01-21 1989-01-22 1989-01-23 1989-01-24 1989-01-25 1989-01-26
## T+1 0.01046415 0.01040768 0.0103249 0.0104107 0.01049233 0.01056868
## 1989-01-27 1989-01-28 1989-01-29 1989-01-30 1989-01-31 1989-02-01
## T+1 0.01041002 0.009380737 0.009423643 0.009428883 0.0095559 0.009648983
## 1989-02-02 1989-02-03 1989-02-04 1989-02-05 1989-02-06 1989-02-07
## T+1 0.009526183 0.009645115 0.009750661 0.009712438 0.009603977 0.009720158
## 1989-02-08 1989-02-09 1989-02-10 1989-02-11 1989-02-12 1989-02-13
## T+1 0.009833257 0.009939462 0.009952652 0.008680824 0.008520927 0.008683477
## 1989-02-14 1989-02-15 1989-02-16 1989-02-17 1989-02-18 1989-02-19
## T+1 0.008625415 0.008789502 0.008870417 0.009005037 0.009106135 0.008950502
## 1989-02-20 1989-02-21 1989-02-22 1989-02-23 1989-02-24 1989-02-25
## T+1 0.008809456 0.008902437 0.009053696 0.009024807 0.008607996 0.00875105
## 1989-02-26 1989-02-27 1989-02-28 1989-03-01 1989-03-02 1989-03-03
## T+1 0.008909299 0.008788525 0.008941052 0.009090548 0.009203327 0.008615641
## 1989-03-04 1989-03-05 1989-03-06 1989-03-07 1989-03-08 1989-03-09
## T+1 0.008074682 0.00744759 0.007500128 0.007667928 0.007875915 0.00802067
## 1989-03-10 1989-03-11 1989-03-12 1989-03-13 1989-03-14 1989-03-15
## T+1 0.007793527 0.007589101 0.007440931 0.007538034 0.007749558 0.007953857
## 1989-03-16 1989-03-17 1989-03-18 1989-03-19 1989-03-20 1989-03-21
## T+1 0.008129455 0.008163477 0.008319239 0.008364382 0.007266283 0.007492756
## 1989-03-22 1989-03-23 1989-03-24 1989-03-25 1989-03-26 1989-03-27
## T+1 0.007708334 0.007913419 0.007918119 0.007960692 0.00804029 0.007664178
## 1989-03-28 1989-03-29 1989-03-30 1989-03-31 1989-04-01 1989-04-02
## T+1 0.00787045 0.008069213 0.00814707 0.008297327 0.00835763 0.008357141
## 1989-04-03 1989-04-04 1989-04-05 1989-04-06 1989-04-07 1989-04-08
## T+1 0.008508342 0.008676085 0.008837705 0.008989435 0.008905284 0.009042137
## 1989-04-09 1989-04-10 1989-04-11 1989-04-12 1989-04-13 1989-04-14
## T+1 0.008451682 0.008002502 0.008151312 0.008337135 0.008314943 0.008188416
## 1989-04-15 1989-04-16 1989-04-17 1989-04-18 1989-04-19 1989-04-20
## T+1 0.008045891 0.008000454 0.008073646 0.008261024 0.008441813 0.008590209
## 1989-04-21 1989-04-22 1989-04-23 1989-04-24 1989-04-25 1989-04-26
## T+1 0.0087465 0.008834164 0.008427074 0.008600271 0.008691922 0.008852881
## 1989-04-27 1989-04-28 1989-04-29 1989-04-30 1989-05-01 1989-05-02
## T+1 0.009006294 0.009129946 0.009229668 0.009335109 0.009275632 0.009386086
## 1989-05-03 1989-05-04 1989-05-05 1989-05-06 1989-05-07 1989-05-08
## T+1 0.009515055 0.009600759 0.009670398 0.00973279 0.009823686 0.009869143
## 1989-05-09 1989-05-10 1989-05-11 1989-05-12 1989-05-13 1989-05-14
## T+1 0.009975476 0.01007746 0.009745197 0.009417704 0.009401138 0.009513629
## 1989-05-15 1989-05-16 1989-05-17 1989-05-18 1989-05-19 1989-05-20
## T+1 0.009619885 0.009704716 0.009819183 0.009889518 0.009837965 0.009894295
## 1989-05-21 1989-05-22 1989-05-23 1989-05-24 1989-05-25 1989-05-26
## T+1 0.009769782 0.009733707 0.009846552 0.009954395 0.01005675 0.01014269
## 1989-05-27 1989-05-28 1989-05-29 1989-05-30 1989-05-31 1989-06-01
## T+1 0.00979773 0.009905231 0.009917895 0.01002171 0.0101213 0.01020043
## 1989-06-02 1989-06-03 1989-06-04 1989-06-05 1989-06-06 1989-06-07
## T+1 0.009898701 0.009975282 0.009921069 0.009954791 0.01004524 0.01014404
## 1989-06-08 1989-06-09 1989-06-10 1989-06-11 1989-06-12 1989-06-13
## T+1 0.01006662 0.01010403 0.00939749 0.00886029 0.009013333 0.009155733
## 1989-06-14 1989-06-15 1989-06-16 1989-06-17 1989-06-18 1989-06-19
## T+1 0.009294529 0.009426568 0.009062551 0.009205351 0.009338304 0.009032011
## 1989-06-20 1989-06-21 1989-06-22 1989-06-23 1989-06-24 1989-06-25
## T+1 0.009135003 0.009275501 0.009389687 0.009498017 0.009540871 0.009627177
## 1989-06-26 1989-06-27 1989-06-28 1989-06-29 1989-06-30 1989-07-01
## T+1 0.009558587 0.009679714 0.009794365 0.009721111 0.009643873 0.009723257
## 1989-07-02 1989-07-03 1989-07-04 1989-07-05 1989-07-06 1989-07-07
## T+1 0.009782799 0.009773486 0.009883899 0.009989975 0.01005194 0.009705225
## 1989-07-08 1989-07-09 1989-07-10 1989-07-11 1989-07-12 1989-07-13
## T+1 0.009732379 0.009835958 0.009849874 0.009954616 0.01005708 0.009939626
## 1989-07-14 1989-07-15 1989-07-16 1989-07-17 1989-07-18 1989-07-19
## T+1 0.009991921 0.009979687 0.01002281 0.009967094 0.0100695 0.01016708
## 1989-07-20
## T+1 0.01024389par ( lwd =2, cex.axis = 2)
x_axis = currencies$Date [ currencies$Date >= "2016-08-03"]
plot ( x_axis , static_fc@forecast$seriesFor , type ="l", xlab ="", ylab ="", col ="blue3"
)
lines ( x_axis , dynamic_fc@forecast$seriesFor , col ="brown1")
legend ("bottomleft", legend = c ("Static", "Dynamic") , col = c("blue3","brown1") ,
lty = 1)
=========================================================================================
0.1 Deriving Critical Values for a Dickey–Fuller Test Using Simulation
set.seed(123456)
tnone = tconst = ttrend = NULL
for (i in 1:500){
y = c(0, cumsum(rnorm(1199)))
dy = c(NA , diff ( y ) )
lagy = c(NA , y[1:1199])
# Model with no intercept (no drift)
lmnone = lm(dy[-(1:200)] ~ 0 + lagy[-(1:200)])
tnone = c(tnone , coef(summary(lmnone))[1]/coef(summary(lmnone))[2])
# Model with intercept, no trend
lmconst = lm(dy[-(1:200)] ~ 1 + lagy[-(1:200)])
tconst = c(tconst, coef(summary(lmconst))[2,1]/coef(summary(lmconst))[2,2])
# Model with intercept and trend
lmtrend = lm(dy[-(1:200)] ~ 1 + lagy[-(1:200)] + c(1:1000))
ttrend = c(ttrend, coef(summary(lmtrend))[2,1]/ coef(summary(lmtrend))[2,2])
}Explanation - ‘set.seed(123456)’: to replicate the exact t-values created with this program on any other computer at a later time and the command serves to define the starting value for draws from a standard normal distribution which are necessary in later stages to create variables that follow a standard normal distribution. - ‘tnone = tconst = ttrend = NULL’: variables to store the t-statistics. - N = 500 - ‘y = c(0, cumsum(rnorm(1199)))’: a random series of 1199 numbers with a standard normal distribution rnorm(1199) => a vector comprising 1200 entries that resembles a random walk series that follows a unit root process. - ‘dy = c(NA , diff ( y ) )’ and ‘lagy = c(NA , y[1:1199])’: the first differences and lagged values of y, adding an ’NA’ value at the start to keep the same length. - ‘lmnone’, ‘lmconst’, ‘lmtrend’: the linear model is estimated. - ‘tnone’, ‘tconst’, ‘ttrend’: computes the t-statistic and appends it to the corresponding vector. - Note that when generating random draws, it sometimes takes a while before the constructed series have the properties that they should, and therefore, when generating the regressions, we exclude the first 200 observations. => dy[-(1:200)] is used to remove first 200 observations. The independent variables change for the different models. - ‘dy[-(1:200)] ~ 0 + lagy[-(1:200)]’: only the lagged values of y, hence the values starting at 200, are used and also avoid an intercept. - ‘dy[-(1:200)] ~ 1 + lagy[-(1:200)]’: include an intercept and write 1 instead of 0. - ‘dy[-(1:200)] ~ 1 + lagy[-(1:200)] + c(1:1000)’: the model includes an intercept and a time trend. For each of these 50,000 repetitions, a new set of t-values will be generated and added to the vectors tnone, tconst and ttrend
## 1% 5% 10%
## -2.401095 -1.921597 -1.675099## 1% 5% 10%
## -3.651240 -2.870267 -2.571466## 1% 5% 10%
## -3.987071 -3.366447 -3.107267Explanation We compute the 1st, 5th and 10th percentile for the three variables ‘tnone’, ‘tconst’, ‘ttrend’ using the quantile function.
0.1.1 Pricing Asian Options
STEPS 1. Specify a data generating process for the underlying asset. A random walk with drift model is usually assumed. Specify also the assumed size of the drift component and the assumed size of the volatility parameter. Specify also a strike price K, and a time to maturity, T. 2. Draw a series of length T, the required number of observations for the life of the option, from a normal distribution. This will be the error series, so that ϵt ∼ N(0; 1). 3. Form a series of observations of length T on the underlying asset. 4. Observe the price of the underlying asset at maturity observation T. For a call option, if the value of the underlying asset on maturity date Pt ≤ K, the option expires worthless for this replication. If the value of the underlying asset on maturity date Pt > K, the option expires in the money, and has a value on that date equal to PT - K, which should be discounted back to the present using the risk-free rate. 5. Repeat steps 1 to 4 a total of N times, and take the average value of the option over N replications. This average will be the price of the option.
set.seed(123456)
# Initial values and derived constants
K = 6500; iv = 0.2652 # Simulation one
s0 = 6289.7; rf = 0.0624; dy = 0.0242; ttm = 0.5; obs = 125
dt = ttm /obs
drift = (rf -dy -iv^2/2)*dt
vsqrdt = iv*dt^0.5
putval = callval = NULL
for (i in 1:25000){
random = rnorm (obs ) # create cumulative sum of random numbers
# Spot price evolution for positive and negative random numbers
spot = s0* exp( drift *c(1: obs)+ vsqrdt * cumsum ( random ))
spot_neg = s0*exp( drift *c(1:obs)+ vsqrdt * cumsum (- random ))
# Compute call values
callval = c( callval , max ( mean ( spot )-K,0)*exp (-rf*ttm ))
callval = c( callval , max ( mean ( spot_neg )-K,0)*exp (-rf*ttm))
# Compute Put values
putval = c(putval , max (K- mean ( spot ),0)*exp (-rf*ttm ))
putval = c(putval , max (K- mean ( spot_neg ),0)*exp (-rf*ttm ))
}## [1] 204.6794## [1] 349.6241