Interest Rate Modelling
Rancy_Chepchirchir
library(yuima)
# Brownian Motion
m1=setModel(drift="0",diffusion="1", state.var="x",time.var="t",solve.var="x",xinit=100)
X=simulate(m1)
YUIMA: 'delta' (re)defined.plot(X)
d1=X@data@original.data
d1Time Series:
Start = c(0, 1)
End = c(1, 1)
Frequency = 100
Series 1
[1,] 100.00000
[2,] 100.07864
[3,] 100.14744
[4,] 100.26101
[5,] 100.19600
[6,] 100.32525
[7,] 100.36930
[8,] 100.40041
[9,] 100.37561
[10,] 100.40049
[11,] 100.37754
[12,] 100.34133
[13,] 100.33243
[14,] 100.29706
[15,] 100.10629
[16,] 99.96946
[17,] 99.97327
[18,] 99.78895
[19,] 99.95958
[20,] 99.93938
[21,] 99.91650
[22,] 99.74287
[23,] 99.63650
[24,] 99.73448
[25,] 99.79344
[26,] 99.79658
[27,] 99.76694
[28,] 99.82420
[29,] 99.73991
[30,] 99.70667
[31,] 99.75231
[32,] 99.75777
[33,] 99.73275
[34,] 99.79234
[35,] 99.96229
[36,] 100.01699
[37,] 99.94456
[38,] 100.04605
[39,] 100.12503
[40,] 99.83230
[41,] 99.66571
[42,] 99.73146
[43,] 99.67869
[44,] 99.82062
[45,] 99.85868
[46,] 99.90330
[47,] 99.86971
[48,] 99.92948
[49,] 99.92796
[50,] 99.97979
[51,] 100.00060
[52,] 99.84626
[53,] 99.93828
[54,] 100.02102
[55,] 100.10759
[56,] 100.05319
[57,] 100.03747
[58,] 100.01580
[59,] 99.98291
[60,] 100.21446
[61,] 100.06334
[62,] 100.02742
[63,] 100.01447
[64,] 100.06580
[65,] 100.12965
[66,] 100.11220
[67,] 100.19206
[68,] 100.05881
[69,] 100.09331
[70,] 99.98243
[71,] 99.84947
[72,] 99.76524
[73,] 99.76862
[74,] 99.84772
[75,] 99.78450
[76,] 99.76108
[77,] 99.76783
[78,] 99.66762
[79,] 99.68972
[80,] 99.75127
[81,] 99.90197
[82,] 99.71865
[83,] 99.71770
[84,] 99.79685
[85,] 99.83034
[86,] 99.79276
[87,] 99.75700
[88,] 99.84208
[89,] 99.73915
[90,] 100.04932
[91,] 99.96380
[92,] 100.20145
[93,] 100.37708
[94,] 100.58153
[95,] 100.50820
[96,] 100.52103
[97,] 100.57906
[98,] 100.58203
[99,] 100.57689
[100,] 100.58789
[101,] 100.43905grid=setSampling(Terminal=1,n=1000)
YUIMA: 'delta' (re)defined.X=simulate(m1,sampling=grid)
plot(X)
# Geometric BM
m1=setModel(drift="mu*s",diffusion="sigma*s", state.var="s",time.var="t",solve.var="s",xinit=100)
X=simulate(m1,true.param=list(mu=0.1,sigma=0.2))
YUIMA: 'delta' (re)defined.plot(X)
# Prob distr associated with the GBM
m1=setModel(drift="mu*s",diffusion="sigma*s", state.var="s",time.var="t",solve.var="s",xinit=100)
simnum=100
dist=c(.31, .52,0.6,0.7, .95)
newsim=function(i){simulate(m1,true.param=list(mu=0.1,sigma=0.2))@data@original.data}
newsim(1)
YUIMA: 'delta' (re)defined.Time Series:
Start = c(0, 1)
End = c(1, 1)
Frequency = 100
Series 1
[1,] 100.00000
[2,] 97.19203
[3,] 97.84168
[4,] 97.07919
[5,] 100.65220
[6,] 103.98439
[7,] 107.48986
[8,] 106.58158
[9,] 107.81131
[10,] 109.20865
[11,] 105.70226
[12,] 103.38126
[13,] 103.35130
[14,] 103.81394
[15,] 103.37128
[16,] 102.28601
[17,] 100.40426
[18,] 102.63987
[19,] 103.72516
[20,] 106.72968
[21,] 105.87837
[22,] 105.21644
[23,] 105.03845
[24,] 103.82663
[25,] 105.18940
[26,] 103.71588
[27,] 105.47425
[28,] 107.79912
[29,] 108.36675
[30,] 107.39034
[31,] 104.99071
[32,] 106.53712
[33,] 113.09387
[34,] 111.67281
[35,] 111.76251
[36,] 113.48091
[37,] 113.86456
[38,] 118.79748
[39,] 119.94318
[40,] 124.66802
[41,] 126.76182
[42,] 122.36396
[43,] 120.75158
[44,] 120.37536
[45,] 116.69196
[46,] 118.17884
[47,] 117.23031
[48,] 115.21940
[49,] 117.96167
[50,] 118.88724
[51,] 119.81965
[52,] 118.50062
[53,] 115.94151
[54,] 115.71680
[55,] 115.19419
[56,] 116.97677
[57,] 120.20351
[58,] 121.30979
[59,] 125.39372
[60,] 125.58006
[61,] 122.73358
[62,] 123.37971
[63,] 125.88543
[64,] 128.30894
[65,] 130.74847
[66,] 130.50574
[67,] 133.75711
[68,] 129.50016
[69,] 128.29090
[70,] 128.72886
[71,] 131.67811
[72,] 129.90731
[73,] 129.93822
[74,] 130.54475
[75,] 128.52221
[76,] 128.90325
[77,] 130.93232
[78,] 129.53575
[79,] 129.44351
[80,] 131.49127
[81,] 129.80621
[82,] 129.13792
[83,] 131.60556
[84,] 131.81339
[85,] 128.62381
[86,] 126.97143
[87,] 124.64330
[88,] 120.96342
[89,] 121.57048
[90,] 121.84470
[91,] 122.47573
[92,] 119.08831
[93,] 121.56533
[94,] 124.15594
[95,] 121.84537
[96,] 121.79410
[97,] 120.42035
[98,] 120.56409
[99,] 116.88969
[100,] 115.88171
[101,] 118.40706sim=sapply(1:simnum,function(x)newsim(x))
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YUIMA: 'delta' (re)defined.
YUIMA: 'delta' (re)defined.m2=t(sim)
m2 [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
[1,] 100 101.18518 101.01740 100.00626 102.09068 100.42025 97.49887 97.42893
[2,] 100 101.79164 101.56852 101.91625 102.89643 99.64953 101.31443 103.74913
[3,] 100 102.27903 100.83225 104.01435 101.40922 104.19604 105.55898 106.25488
[4,] 100 98.94275 99.39266 101.20928 101.82924 103.62218 104.30849 100.86677
[5,] 100 101.20447 99.50208 97.27477 96.40061 94.28205 93.71722 92.26480
[6,] 100 100.09522 96.91053 98.02488 99.78116 99.96834 101.80619 103.72681
[7,] 100 95.42716 97.86469 99.48474 98.16090 100.68950 102.03226 103.25876
[8,] 100 97.08198 98.17092 98.73457 98.56117 97.13604 97.80148 95.78353
[9,] 100 97.88494 99.45885 97.14966 99.85362 98.96915 100.04083 98.18730
[,9] [,10] [,11] [,12] [,13] [,14] [,15] [,16]
[1,] 96.13374 94.88593 93.86602 94.22044 92.16776 90.90956 88.09168 87.98965
[2,] 102.92854 103.96121 106.34364 108.91433 108.28692 107.89249 106.02505 106.20248
[3,] 105.23986 107.63184 108.01184 108.88789 112.33791 110.23246 112.86386 112.64341
[4,] 96.87397 97.51671 98.63526 97.90134 98.45394 97.05837 98.17057 103.17369
[5,] 91.68878 91.47361 89.78319 91.82614 90.05582 91.18509 92.52184 92.45454
[6,] 101.98210 104.20271 102.32392 104.13961 104.17284 103.19867 102.74649 104.50907
[7,] 103.04912 103.97924 102.48069 103.05603 102.38367 102.89473 100.79696 105.71145
[8,] 96.99624 97.93465 98.64537 99.21477 100.30038 97.58281 96.82598 96.27717
[9,] 99.53211 96.51336 98.34056 96.43378 95.77150 94.69811 92.68397 93.69433
[,17] [,18] [,19] [,20] [,21] [,22] [,23] [,24]
[1,] 87.71587 87.54880 90.21775 91.73217 90.96146 85.64129 86.23837 82.62968
[2,] 104.92603 105.29076 105.95691 103.99468 108.74303 105.52038 111.00715 108.70090
[3,] 112.15293 111.71072 113.08462 114.08212 112.13110 110.39470 109.15638 110.54663
[4,] 103.08199 102.97058 103.76647 103.82996 103.09043 100.87899 97.83767 95.07999
[5,] 92.41448 93.25877 96.76708 99.78606 99.69639 98.37958 98.79946 101.18628
[6,] 106.36821 105.85140 108.62851 110.13732 112.71661 108.68148 102.70255 105.38811
[7,] 104.20700 107.21274 103.85837 99.42938 100.65778 104.73286 105.69821 103.20512
[8,] 93.77103 93.31365 92.64741 88.64081 88.37145 88.53658 89.28601 88.83309
[9,] 92.12598 92.65853 95.49348 96.49286 94.53750 94.29583 93.33238 92.48116
[,25] [,26] [,27] [,28] [,29] [,30] [,31] [,32]
[1,] 80.86905 79.11802 76.42300 77.21622 78.88690 79.33634 79.97334 80.16589
[2,] 111.17302 112.52817 116.04167 116.52849 115.03073 114.48509 116.86538 117.51740
[3,] 111.63194 112.45692 113.03887 112.87563 114.34910 116.19536 114.14559 115.96109
[4,] 97.10991 96.92928 93.22291 96.19572 94.91805 92.55768 95.34292 95.17612
[5,] 104.54498 102.18234 105.05891 105.10206 107.46042 110.81676 109.12960 111.54662
[6,] 106.25145 107.57945 106.99444 108.82045 106.30204 105.08236 106.19741 106.35881
[7,] 102.43459 103.66009 102.29248 104.74016 103.88068 103.57443 106.77347 103.88210
[8,] 87.86535 84.97344 84.78152 85.10881 87.72917 90.82600 89.76171 89.95942
[9,] 90.02294 88.95922 86.94611 89.02963 88.38918 89.51134 89.82846 90.19989
[,33] [,34] [,35] [,36] [,37] [,38] [,39] [,40]
[1,] 79.68328 78.96362 77.47333 79.56490 80.29526 77.53085 78.39410 80.87233
[2,] 117.22061 115.05667 114.02739 120.12604 117.56120 122.12786 125.92693 129.10149
[3,] 112.91256 111.33058 112.17216 112.86121 114.44146 116.22868 117.36331 118.06420
[4,] 94.82195 95.04297 94.75987 93.91254 94.42077 93.48605 93.13330 94.53638
[5,] 117.06410 121.68550 124.51762 125.57626 124.73550 128.60922 126.29179 121.67458
[6,] 106.65643 107.04308 104.73275 102.55150 101.26121 103.81158 102.91833 101.24518
[7,] 102.92706 104.51808 103.88327 104.13824 105.34348 105.06900 105.57212 101.77549
[8,] 88.55845 87.94231 87.96000 85.80866 83.85843 84.00288 82.91146 83.23881
[9,] 93.93408 92.60186 94.35428 90.61188 90.02926 91.67766 92.32661 94.35029
[,41] [,42] [,43] [,44] [,45] [,46] [,47] [,48]
[1,] 84.00131 85.40679 80.53166 81.64590 79.76962 82.61344 81.90990 84.04507
[2,] 130.22111 127.83372 125.91396 125.60989 127.01733 125.38759 123.42356 122.05001
[3,] 120.58374 121.10794 121.09319 120.20835 128.39959 125.65456 126.34814 122.83976
[4,] 95.64372 98.31938 97.25195 100.40020 98.13732 99.80515 100.88505 95.93896
[5,] 122.99029 122.29783 125.76466 124.89282 124.30083 126.71639 126.58566 122.56280
[6,] 100.93598 99.91595 96.88511 100.09787 100.97975 100.75949 101.34632 98.37517
[7,] 101.04086 99.14493 96.31173 98.23419 98.57118 98.38213 97.55486 95.21529
[8,] 83.33404 82.02975 83.66097 84.22369 79.66820 79.33571 78.07533 77.95602
[9,] 94.34700 96.39383 94.50575 94.66385 96.12558 95.34402 93.27418 90.88412
[,49] [,50] [,51] [,52] [,53] [,54] [,55] [,56]
[1,] 82.38154 83.43668 83.16552 81.89154 80.96764 80.43920 80.72408 82.66145
[2,] 119.64807 117.67786 117.80392 118.53886 118.39049 116.92984 118.09133 116.32236
[3,] 122.25321 120.62252 121.23976 119.76817 117.56043 116.90650 115.14133 117.66082
[4,] 95.50511 96.80111 97.46099 99.11210 97.66308 96.27288 95.67262 93.27242
[5,] 125.33462 126.48994 128.98012 133.00796 133.03765 132.08129 135.75581 134.72516
[6,] 101.96616 99.68871 99.26167 98.08559 100.59990 100.08266 100.49105 101.20165
[7,] 94.62024 90.63342 89.74187 93.06646 96.08388 97.42283 100.03356 102.77101
[8,] 78.30103 76.92115 80.71203 81.54998 81.66691 83.61994 84.02204 84.25724
[9,] 87.14756 87.10963 89.36855 91.45665 91.20791 89.87492 92.51004 94.32829
[,57] [,58] [,59] [,60] [,61] [,62] [,63] [,64]
[1,] 82.12077 83.67837 84.81809 84.30494 84.36391 84.72429 81.98822 81.81608
[2,] 111.87459 110.62789 112.08722 112.53609 113.47191 112.56156 113.05134 113.95784
[3,] 117.91973 119.83590 118.81346 120.73045 119.56476 119.09231 116.60675 113.63403
[4,] 92.76377 90.37599 90.78838 92.21722 89.72355 86.74002 86.88406 89.43915
[5,] 135.22989 137.32651 137.03859 138.82579 135.99098 138.69199 133.39330 130.73849
[6,] 102.08107 102.39923 99.56641 102.56183 103.47466 102.89751 106.69369 109.60487
[7,] 101.19769 98.67952 97.93395 97.91563 99.92062 98.26021 95.80348 96.46950
[8,] 83.63939 82.05870 81.70141 79.80633 78.91245 78.33525 76.17332 75.44906
[9,] 94.11265 91.94127 91.63312 91.05862 92.10544 94.26182 91.15541 89.85279
[,65] [,66] [,67] [,68] [,69] [,70] [,71] [,72]
[1,] 84.33586 81.75353 81.67648 79.68473 79.22878 79.01242 76.99730 77.72888
[2,] 115.27397 115.99857 117.55905 115.88900 116.71113 117.44107 116.54929 120.07496
[3,] 117.16753 113.13086 110.56153 113.44642 114.03266 113.04336 119.74437 118.55082
[4,] 87.20412 89.55599 88.07676 89.07696 89.37147 88.45108 89.49639 86.87992
[5,] 126.89338 124.25692 130.29274 134.25944 133.34951 134.24563 131.62206 133.01799
[6,] 110.26430 111.55091 111.56581 109.50475 108.96209 109.32598 113.12000 113.68848
[7,] 97.95236 97.32589 99.43559 102.07753 105.77990 107.95801 108.07936 110.07331
[8,] 75.99967 74.88968 75.45625 76.06054 76.72160 73.74859 73.41436 74.22759
[9,] 90.17319 88.70373 90.62586 88.80590 88.37555 86.29057 86.12667 87.59822
[,73] [,74] [,75] [,76] [,77] [,78] [,79] [,80]
[1,] 76.80002 74.28278 74.43355 72.53441 74.21714 74.64116 73.60695 70.91158
[2,] 125.35987 125.52400 130.58134 128.63456 122.99145 128.43137 129.77670 129.46597
[3,] 116.51993 113.96109 109.55904 111.36560 111.07120 108.28248 108.43999 105.60170
[4,] 86.53363 85.96563 87.68659 87.66982 87.13324 88.99503 86.92195 87.13903
[5,] 131.03256 130.71045 128.82157 128.31523 131.56021 132.18075 131.14726 134.03302
[6,] 112.77043 115.12565 114.52101 119.17634 119.77591 123.46916 122.68402 120.46702
[7,] 111.43026 106.47862 107.78756 105.60856 108.02096 105.95915 106.06018 107.60302
[8,] 73.46775 72.06090 71.83678 71.91375 73.00912 76.51743 78.09945 77.71747
[9,] 88.66398 88.39820 88.84248 87.07835 86.22255 84.64580 84.42685 85.57572
[,81] [,82] [,83] [,84] [,85] [,86] [,87] [,88]
[1,] 69.75271 69.83285 69.50332 69.13801 72.76024 73.27411 72.01226 69.91126
[2,] 129.81745 128.38337 124.79154 125.48919 122.75351 124.34735 125.30241 127.52659
[3,] 107.29262 107.73074 104.41115 106.92891 110.98966 112.48103 114.04248 115.13259
[4,] 87.84313 87.80368 90.14486 91.03151 93.37053 90.59564 90.06220 89.29830
[5,] 133.72609 134.08247 134.76396 133.18726 133.95088 128.39734 129.44021 125.37207
[6,] 121.10702 117.08480 113.74663 110.40687 112.92205 110.58111 111.64979 112.45637
[7,] 105.27331 107.63062 109.94097 112.76492 109.49888 109.47652 109.29503 108.30725
[8,] 78.81293 77.96325 79.62576 78.27693 78.80453 81.22292 82.22821 80.16564
[9,] 86.72456 88.52851 88.62094 91.51423 91.21652 91.39549 90.61611 92.44170
[,89] [,90] [,91] [,92] [,93] [,94] [,95] [,96]
[1,] 70.93617 72.74206 72.05301 72.82859 72.89377 70.92799 70.26099 68.54992
[2,] 126.87092 124.63214 126.56333 129.43634 129.67169 126.81656 124.57240 126.29234
[3,] 117.79594 117.12171 116.14400 118.17053 119.57393 118.64975 123.57737 124.94866
[4,] 89.88999 89.25424 88.51117 91.27233 89.96276 90.34926 88.26589 90.79435
[5,] 126.26223 127.36304 127.49978 122.10623 124.01718 126.05856 128.21195 125.52760
[6,] 115.40980 113.88486 110.17583 113.91078 115.29197 116.08859 118.28677 124.76476
[7,] 106.31407 108.01879 106.75536 101.65070 98.69019 99.68984 102.71400 101.93185
[8,] 78.19232 80.19172 77.86292 77.43780 77.92506 78.29078 77.24671 76.61510
[9,] 90.85599 90.67343 89.07628 86.43119 88.56912 90.02083 88.52346 91.14222
[,97] [,98] [,99] [,100] [,101]
[1,] 68.14642 67.66693 65.68832 64.96944 64.34718
[2,] 128.42334 127.91077 131.18461 122.27445 124.60646
[3,] 127.51729 128.75298 127.48548 130.52949 129.45207
[4,] 92.77267 93.64012 95.48032 94.39380 91.72619
[5,] 123.99732 124.12082 123.15389 122.57698 119.25667
[6,] 124.52909 121.64920 124.28928 114.67941 117.20822
[7,] 106.47141 103.99241 105.31273 105.57035 105.48666
[8,] 74.44297 75.88571 75.31025 73.94872 73.49585
[9,] 94.21307 94.92788 93.41056 89.89473 90.12307
[ reached getOption("max.print") -- omitted 91 rows ]apply(m2,2,mean) [1] 100.00000 99.80552 99.85919 100.23874 100.58012 100.86709 101.28876 101.75329
[9] 101.93259 101.80620 101.80784 102.06232 101.85442 101.84322 102.17194 102.27718
[17] 102.41693 102.43512 102.52022 102.75309 102.96643 103.01598 102.80205 103.08381
[25] 103.04718 103.07741 103.09164 103.55296 103.43602 103.37619 103.50430 103.89953
[33] 104.22227 104.06543 104.31056 104.42726 104.51560 104.82578 104.98856 105.23395
[41] 105.33602 105.55647 105.69421 106.10316 105.99078 106.16826 106.05415 105.96438
[49] 106.03450 106.36139 106.20341 106.43528 106.74391 106.70582 106.98090 107.52136
[57] 107.76003 108.21637 108.49715 108.20700 108.38414 108.61068 108.62799 108.75060
[65] 109.16628 109.30056 109.42836 109.78341 109.93273 110.05809 110.54485 110.58351
[73] 111.09726 111.38695 111.56280 111.74869 111.71078 111.66241 111.92437 111.60229
[81] 111.66917 111.45854 111.16454 111.32642 111.79965 111.84782 112.19376 112.10945
[89] 112.20600 112.20717 112.58884 112.64552 113.10220 113.05640 113.08391 113.26891
[97] 113.83565 113.83061 113.91001 113.71467 113.41924tile=sapply(1:100,function(x)quantile(m2[,x], dist) )
tile [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
31% 100 98.99673 98.74636 98.72704 98.85074 99.07565 99.3419 99.98513 99.95754
52% 100 99.85474 100.04089 100.50650 100.26871 101.12311 101.4541 102.28166 102.34548
60% 100 100.28371 100.43848 101.22317 101.37107 101.77188 102.0551 103.18562 103.04486
70% 100 100.80526 101.09232 102.13361 102.54195 102.73820 103.0292 104.02751 104.51765
95% 100 103.14850 103.96020 104.46127 105.69184 106.39651 107.6843 108.28933 109.71761
[,10] [,11] [,12] [,13] [,14] [,15] [,16] [,17]
31% 99.5379 99.23588 99.28597 99.59219 98.63421 98.92668 98.91951 98.3498
52% 102.4262 101.98294 102.34137 102.08433 102.34780 102.51612 101.96949 103.0058
60% 103.4922 102.94531 103.23186 103.18372 103.55971 103.11338 103.54649 103.9515
70% 104.2121 105.12443 104.87608 104.93474 104.93510 105.32227 106.12132 106.3082
95% 110.2481 111.63925 112.52884 112.22625 113.78326 114.24616 113.47761 113.7814
[,18] [,19] [,20] [,21] [,22] [,23] [,24] [,25]
31% 98.12382 98.2706 98.29303 97.54254 97.42907 97.46337 97.98474 97.71178
52% 102.31756 102.7178 103.08808 102.76256 102.41511 102.81371 102.51275 102.75315
60% 104.13703 104.3424 103.86069 104.43518 104.89765 104.16129 104.82088 105.42032
70% 106.16086 105.7390 105.25751 106.76681 106.98264 106.18725 106.67308 107.96580
95% 113.74492 115.4237 117.94778 120.00633 120.39006 119.07846 118.50242 118.90954
[,26] [,27] [,28] [,29] [,30] [,31] [,32] [,33]
31% 96.73035 96.66091 97.16712 96.94656 96.13563 96.31956 96.79762 97.29064
52% 103.07187 103.88861 104.62334 103.89583 103.93202 104.10511 103.11711 104.18791
60% 105.99786 106.53404 106.93371 107.53978 107.60647 107.58486 106.42533 107.60957
70% 109.26486 109.20922 109.14067 109.71281 110.43939 110.16994 111.14815 112.56212
95% 120.00813 121.40782 121.97870 121.10883 121.29943 121.00432 122.25278 123.38989
[,34] [,35] [,36] [,37] [,38] [,39] [,40] [,41]
31% 96.35997 95.59557 96.32017 96.54817 96.5442 95.89276 96.36273 97.19844
52% 104.34923 105.06877 104.21524 105.22656 106.7563 106.23732 105.84264 104.90867
60% 107.21451 107.67009 108.41471 107.92688 108.3539 108.61054 109.40692 109.12363
70% 110.71150 111.49017 111.76062 111.68966 111.8326 111.88276 112.15029 111.79010
95% 124.70529 125.01808 125.61432 124.94037 127.5558 128.50076 130.98688 131.03350
[,42] [,43] [,44] [,45] [,46] [,47] [,48] [,49]
31% 98.34004 97.13823 98.17252 98.47929 99.3103 98.85825 97.65287 96.52426
52% 105.68685 104.93697 106.09422 105.43938 104.4835 105.23687 106.06270 104.85968
60% 108.43472 108.68759 108.76560 108.39756 109.2144 109.76262 109.32604 109.07555
70% 111.26743 112.81870 112.76692 113.28812 113.2218 112.09203 112.61376 112.86195
95% 130.95486 130.55046 130.99516 130.46893 128.0444 128.86900 129.93635 132.09056
[,50] [,51] [,52] [,53] [,54] [,55] [,56] [,57]
31% 97.17117 96.88472 97.50245 97.54276 97.54069 99.18602 100.0202 98.13187
52% 105.72288 104.38699 103.82652 104.40721 105.46884 105.60206 106.2461 106.27073
60% 107.24389 107.22227 107.88611 108.85063 108.36705 108.29080 107.9935 109.20103
70% 113.07533 113.86990 113.10393 114.28017 113.64119 112.93052 114.5878 113.13434
95% 134.81547 135.63538 135.75579 136.80685 134.23760 135.02899 134.9627 133.86049
[,58] [,59] [,60] [,61] [,62] [,63] [,64] [,65] [,66]
31% 99.34221 100.3937 98.92933 99.06021 98.95314 99.46912 99.48747 100.3613 100.4786
52% 107.49230 107.4687 106.18375 107.25078 107.16131 106.81264 108.20067 108.5328 108.9625
60% 110.44058 110.1791 109.57066 109.77740 110.50219 109.77962 110.32095 111.4066 111.1358
70% 114.42867 114.4202 115.71404 113.99504 115.68819 115.58023 115.31263 115.8420 116.6111
95% 137.07576 138.0620 137.02328 136.96121 137.27444 138.11767 138.30217 140.8062 139.7156
[,67] [,68] [,69] [,70] [,71] [,72] [,73] [,74] [,75]
31% 99.71205 100.1377 99.73075 99.28164 99.90634 100.3721 100.8084 101.7464 100.1480
52% 108.38356 108.8998 107.93520 108.39572 109.32228 111.4504 112.0614 111.9760 112.0489
60% 113.45704 113.8198 114.84702 115.34146 114.99878 115.2251 115.2233 116.1485 116.6311
70% 118.41943 119.0384 118.75436 118.12041 119.52286 120.0941 120.8456 122.6610 121.1654
95% 137.24391 138.8241 141.66989 142.24964 139.47579 136.6359 137.4837 138.7181 136.7132
[,76] [,77] [,78] [,79] [,80] [,81] [,82] [,83] [,84]
31% 101.6936 100.5634 100.7272 101.8074 101.6448 100.4757 100.5536 100.3384 100.5341
52% 111.1466 111.2652 111.9814 112.8038 113.7169 113.5450 113.1684 112.4759 111.7671
60% 117.0908 119.1298 118.2874 117.8290 116.9933 116.7438 117.5530 116.5994 116.7624
70% 122.3774 122.4759 123.3194 123.1594 122.0148 121.3681 121.3980 121.7900 122.9229
95% 140.4041 140.7956 139.7178 139.9140 138.1337 139.6609 138.0458 138.3475 140.1224
[,85] [,86] [,87] [,88] [,89] [,90] [,91] [,92] [,93]
31% 100.1791 100.4125 101.3379 101.7925 102.9354 101.8745 101.7453 101.2008 101.4298
52% 111.7795 111.4196 112.1865 112.4036 112.8674 112.3069 111.7972 113.8525 113.1339
60% 116.2596 115.6021 118.6767 118.3062 117.1908 117.3566 116.4925 117.6598 118.4068
70% 123.3044 122.3407 123.8944 124.5977 124.4845 125.0502 125.5289 123.6484 124.1547
95% 143.3990 144.5335 140.8974 138.7456 141.8494 141.6551 144.3496 145.3608 148.5418
[,94] [,95] [,96] [,97] [,98] [,99] [,100]
31% 101.0671 102.9134 101.6505 102.1901 103.8117 102.9959 101.8777
52% 112.5806 113.2749 113.8768 113.6231 113.3654 114.3569 114.5950
60% 118.7495 120.4766 120.6747 120.7567 118.3861 118.9517 118.3056
70% 124.1514 124.2212 124.8199 125.2985 125.4502 124.9098 124.9882
95% 148.3806 148.1281 148.0997 147.0227 149.2329 146.9591 148.8924# Vasicek Model
m1 = setModel(drift = "theta*(mu-x)", diffusion = "sigma", state.var = "x",
time.var = "t", solve.var = "x", xinit = 0.5)
X = simulate(m1, true.param = list(mu = 0.5, sigma = 0.2,
theta = 2))
YUIMA: 'delta' (re)defined.plot(X)
# CIR Model
m1 = setModel(drift = "theta*(mu-x)", diffusion = "sigma*(x^0.5)", state.var = "x",
time.var = "t", solve.var = "x", xinit = 0.5)
X = simulate(m1, true.param = list(mu = 0.5, sigma = 0.2,
theta = 2))
YUIMA: 'delta' (re)defined.plot(X)
# CKLS MOdel
grid=setSampling(Terminal=1, n=1000)
YUIMA: 'delta' (re)defined.m1 = setModel(drift = "alpha1+(alpha2*x)", diffusion = "alpha3*(x^alpha4)", state.var = "x",
time.var = "t", solve.var = "x", xinit = 0.5)
X = simulate(m1, true.param = list(alpha1 = 0.1, alpha2 = 0.2, alpha3 = 0.3,
alpha4 = 0.4), sampling=grid)
plot(X)
# Hyperbolic Processes
m1 = setModel(drift = "-theta*x/((1+x^2)^0.5)", diffusion = "1", state.var = "x",
time.var = "t", solve.var = "x", xinit = 0.5)
X = simulate(m1, true.param = list(theta = 1))
YUIMA: 'delta' (re)defined.plot(X)
# Fitting SDE to given data
library(Ecdat)package 㤼㸱Ecdat㤼㸲 was built under R version 3.6.3Loading required package: Ecfun
package 㤼㸱Ecfun㤼㸲 was built under R version 3.6.3
Attaching package: 㤼㸱Ecfun㤼㸲
The following object is masked from 㤼㸱package:base㤼㸲:
sign
Attaching package: 㤼㸱Ecdat㤼㸲
The following object is masked from 㤼㸱package:datasets㤼㸲:
Orangedata (Irates)
rates <-Irates[,'r1']
plot(rates)
X<-window(rates, start=1964.471, end=1989.333)
m1 = setModel(drift = "theta*(mu-x)", diffusion = "sigma", state.var = "x",
time.var = "t", solve.var = "x", xinit = 0.5)
X = simulate(m1, true.param = list(mu=0.1, sigma=0.2, theta=2))
YUIMA: 'delta' (re)defined.initialise=list(mu=0.05, sigma=0.5, theta=1)
lowbound=list(mu=0, sigma=0, theta=0)
upbound=list(mu=0.2, sigma=2, theta=3)
mle=qmle(X, start=initialise, lower=lowbound, upper=upbound)bounds can only be used with method L-BFGS-B (or Brent)summary(mle)Quasi-Maximum likelihood estimation
Call:
qmle(yuima = X, start = initialise, lower = lowbound, upper = upbound)
Coefficients:
Estimate Std. Error
sigma 0.2050689 0.01463308
theta 3.0000000 2.15053932
mu 0.1997730 0.10319003
-2 log L: -494.9961 # Incorporating Jumps into the SDE
library(yuima)
grid=setSampling(Terminal=1, n=1000)
YUIMA: 'delta' (re)defined.jump=list(intensity="7", df=list("dnorm(z, 0, 0.2)"))
m1 = setModel(drift = "theta*(mu-x)", diffusion = "sigma", state.var = "x",
time.var = "t", solve.var = "x", xinit = 0.2, jump.coeff="1",
measure=jump, measure.type="CP")
X = simulate(m1, true.param = list(mu=0.1, sigma=0.2, theta=2), sampling=grid)
plot(X)
# Fractional Brownian Motion
grid=setSampling(Terminal=1, n=1000)
YUIMA: 'delta' (re)defined.m1 = setModel(drift = "theta*(mu-x)", diffusion = "sigma", state.var = "x",
time.var = "t", solve.var = "x", xinit = 0.2, hurst= 0.3)
m2 = setModel(drift = "theta*(mu-x)", diffusion = "sigma", state.var = "x",
time.var = "t", solve.var = "x", xinit = 0.2, hurst= 0.7)
X = simulate(m1, true.param = list(mu=0.1, sigma=0.2, theta=2), sampling=grid)
Y = simulate(m2, true.param = list(mu=0.1, sigma=0.2, theta=2), sampling=grid)
par(mfrow=c(2, 1)); plot(X); plot(Y)
# Correlated BMs
solution=c("x1", "x2", "x3")
drift=c("b1*x1", "b2*x2", "b3*x3")
c1=c(2, 1, 3, 1, 4, 2, 3, 2, 5)
cov=matrix(c1, 3, 3)
cov==t(cov); chol=chol(cov); diff=chol; diff [,1] [,2] [,3]
[1,] TRUE TRUE TRUE
[2,] TRUE TRUE TRUE
[3,] TRUE TRUE TRUE
[,1] [,2] [,3]
[1,] 1.414214 0.7071068 2.1213203
[2,] 0.000000 1.8708287 0.2672612
[3,] 0.000000 0.0000000 0.6546537m1 = setModel(drift = drift, diffusion = diff, solve.variable = solution,
xinit = c(1, 2, 3))
X = simulate(m1, true.param = list(b1=0.5, b2=0.6, b3=0.7))
YUIMA: 'delta' (re)defined.plot(X)# Multidimensional BM (Hull-White 2-factor Model)
solution =c("r","u")
drift=c("theta-(alpha*r)+u","-b*u")
c1=c("sigma1","0","0","sigma2")
diff=matrix(c1,2,2)
m1=setModel(drift=drift,diffusion=diff,
solve.variable=solution,xinit=c(0.1,0.2))
X=simulate(m1,true.param=list(theta=1,alpha=1,b=1,sigma1=2,sigma2=2))
YUIMA: 'delta' (re)defined.plot(X)
# Heston Model
solution =c("s1","s2")
drift=c("mu*s1","k*(theta-s2)")
d2=c("c1*s1*(s2^{0.5})","c2*s1*(s2^(0.5))","0","c3*eta*(s2^(0.5))")
diff=matrix(d2,byrow=T,2)
cov=matrix(c(2,0.7,0.7,5),2,2)
cov;chol(cov); [,1] [,2]
[1,] 2.0 0.7
[2,] 0.7 5.0
[,1] [,2]
[1,] 1.414214 0.4949747
[2,] 0.000000 2.1805962m1=setModel(drift=drift,diffusion=diff,
solve.variable=solution,xinit=c(50,5))
X=simulate(m1,true.param=list(theta=1,eta=1,mu=1,k =2,
c1=chol(cov)[1,1],c2=chol(cov)[1,2],c3=chol(cov)[2,2]))
YUIMA: 'delta' (re)defined.plot(X)
References: Maria, S I. (2012) The yuima package: an R framework for simulation and inference of stochastic differential equations: Department of Economics, Business and Statistics University of Milan, Milan, Italy
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