Black-Scholes-Merton formula implementation, pricing european options.
Arguments
- type
'call' for call option, any other value for put.
- spot
current stock price.
- strike
the strike price.
- time
time to option expiration in years.
- rate
the risk-free interest rate.
- yield
the dividends that are expected to be paid.
- sigma
volatility of the stock price.
Recycle rule
These arguments handle the recycle rule so vectors can be provided and once those vectors differs in length the recycle rule is applied.
Examples
bsmprice(c("call", "put"), c(50, 49, 48, 47), 45, 0.25, 0.13, 0.01, 0.2)
#> [1] 6.5031266 0.2854164 4.7405597 0.6079959
bsmprice("call", 50, 45, 0.25, 0.13, 0.01, 1:1000 / 1000)
#> [1] 6.314146 6.314146 6.314146 6.314146 6.314146 6.314146 6.314146
#> [8] 6.314146 6.314146 6.314146 6.314146 6.314146 6.314146 6.314146
#> [15] 6.314146 6.314146 6.314146 6.314146 6.314146 6.314146 6.314146
#> [22] 6.314146 6.314146 6.314146 6.314146 6.314146 6.314146 6.314146
#> [29] 6.314146 6.314146 6.314146 6.314146 6.314146 6.314146 6.314146
#> [36] 6.314146 6.314146 6.314146 6.314146 6.314146 6.314146 6.314146
#> [43] 6.314146 6.314146 6.314146 6.314146 6.314146 6.314146 6.314146
#> [50] 6.314146 6.314146 6.314146 6.314146 6.314146 6.314146 6.314146
#> [57] 6.314146 6.314146 6.314146 6.314147 6.314147 6.314148 6.314149
#> [64] 6.314150 6.314151 6.314153 6.314155 6.314158 6.314162 6.314167
#> [71] 6.314173 6.314180 6.314188 6.314199 6.314212 6.314227 6.314244
#> [78] 6.314265 6.314290 6.314318 6.314350 6.314388 6.314430 6.314479
#> [85] 6.314534 6.314596 6.314665 6.314743 6.314829 6.314925 6.315031
#> [92] 6.315148 6.315276 6.315417 6.315570 6.315737 6.315919 6.316116
#> [99] 6.316329 6.316558 6.316805 6.317070 6.317354 6.317658 6.317983
#> [106] 6.318328 6.318695 6.319085 6.319499 6.319936 6.320399 6.320886
#> [113] 6.321400 6.321940 6.322508 6.323104 6.323729 6.324383 6.325067
#> [120] 6.325781 6.326526 6.327302 6.328110 6.328951 6.329825 6.330732
#> [127] 6.331674 6.332649 6.333659 6.334704 6.335785 6.336901 6.338053
#> [134] 6.339242 6.340468 6.341731 6.343031 6.344369 6.345744 6.347158
#> [141] 6.348609 6.350100 6.351629 6.353196 6.354803 6.356448 6.358133
#> [148] 6.359857 6.361621 6.363424 6.365267 6.367149 6.369071 6.371032
#> [155] 6.373034 6.375075 6.377156 6.379277 6.381437 6.383637 6.385877
#> [162] 6.388157 6.390476 6.392834 6.395233 6.397670 6.400147 6.402663
#> [169] 6.405219 6.407813 6.410447 6.413119 6.415830 6.418580 6.421368
#> [176] 6.424194 6.427059 6.429962 6.432902 6.435881 6.438897 6.441951
#> [183] 6.445042 6.448170 6.451335 6.454537 6.457775 6.461050 6.464361
#> [190] 6.467709 6.471092 6.474511 6.477966 6.481455 6.484981 6.488541
#> [197] 6.492135 6.495765 6.499429 6.503127 6.506859 6.510624 6.514424
#> [204] 6.518256 6.522122 6.526021 6.529953 6.533917 6.537914 6.541943
#> [211] 6.546003 6.550096 6.554220 6.558375 6.562562 6.566779 6.571027
#> [218] 6.575306 6.579615 6.583954 6.588323 6.592722 6.597150 6.601608
#> [225] 6.606094 6.610610 6.615154 6.619727 6.624328 6.628957 6.633614
#> [232] 6.638299 6.643011 6.647750 6.652517 6.657311 6.662131 6.666978
#> [239] 6.671851 6.676750 6.681676 6.686627 6.691604 6.696606 6.701633
#> [246] 6.706685 6.711763 6.716865 6.721991 6.727142 6.732317 6.737515
#> [253] 6.742738 6.747984 6.753254 6.758547 6.763863 6.769202 6.774563
#> [260] 6.779948 6.785354 6.790783 6.796234 6.801707 6.807202 6.812718
#> [267] 6.818256 6.823814 6.829395 6.834996 6.840617 6.846260 6.851923
#> [274] 6.857606 6.863309 6.869033 6.874776 6.880539 6.886322 6.892124
#> [281] 6.897946 6.903786 6.909646 6.915525 6.921422 6.927338 6.933272
#> [288] 6.939225 6.945195 6.951184 6.957191 6.963216 6.969258 6.975317
#> [295] 6.981395 6.987489 6.993600 6.999729 7.005874 7.012037 7.018215
#> [302] 7.024411 7.030622 7.036850 7.043094 7.049355 7.055631 7.061922
#> [309] 7.068230 7.074553 7.080892 7.087245 7.093615 7.099999 7.106398
#> [316] 7.112812 7.119241 7.125685 7.132143 7.138616 7.145103 7.151604
#> [323] 7.158119 7.164649 7.171192 7.177749 7.184320 7.190905 7.197503
#> [330] 7.204115 7.210740 7.217378 7.224030 7.230695 7.237372 7.244063
#> [337] 7.250766 7.257482 7.264210 7.270951 7.277705 7.284471 7.291249
#> [344] 7.298039 7.304842 7.311656 7.318482 7.325321 7.332171 7.339032
#> [351] 7.345905 7.352790 7.359686 7.366594 7.373512 7.380442 7.387383
#> [358] 7.394335 7.401298 7.408272 7.415257 7.422253 7.429259 7.436275
#> [365] 7.443303 7.450340 7.457388 7.464447 7.471515 7.478594 7.485683
#> [372] 7.492781 7.499890 7.507009 7.514137 7.521275 7.528423 7.535581
#> [379] 7.542748 7.549924 7.557110 7.564305 7.571510 7.578724 7.585947
#> [386] 7.593179 7.600420 7.607670 7.614929 7.622197 7.629474 7.636759
#> [393] 7.644053 7.651356 7.658668 7.665988 7.673316 7.680653 7.687998
#> [400] 7.695351 7.702713 7.710083 7.717461 7.724847 7.732241 7.739643
#> [407] 7.747053 7.754471 7.761896 7.769330 7.776771 7.784220 7.791676
#> [414] 7.799140 7.806612 7.814091 7.821577 7.829071 7.836572 7.844081
#> [421] 7.851596 7.859119 7.866649 7.874186 7.881730 7.889281 7.896839
#> [428] 7.904404 7.911976 7.919555 7.927141 7.934733 7.942332 7.949937
#> [435] 7.957549 7.965168 7.972794 7.980425 7.988064 7.995708 8.003359
#> [442] 8.011017 8.018680 8.026350 8.034026 8.041709 8.049397 8.057092
#> [449] 8.064792 8.072499 8.080211 8.087930 8.095654 8.103385 8.111121
#> [456] 8.118863 8.126610 8.134364 8.142123 8.149888 8.157658 8.165434
#> [463] 8.173216 8.181003 8.188796 8.196594 8.204397 8.212206 8.220020
#> [470] 8.227840 8.235665 8.243495 8.251330 8.259171 8.267016 8.274867
#> [477] 8.282723 8.290584 8.298450 8.306321 8.314197 8.322078 8.329964
#> [484] 8.337855 8.345750 8.353651 8.361556 8.369466 8.377381 8.385301
#> [491] 8.393225 8.401154 8.409087 8.417025 8.424968 8.432915 8.440867
#> [498] 8.448824 8.456784 8.464750 8.472719 8.480693 8.488672 8.496654
#> [505] 8.504642 8.512633 8.520629 8.528628 8.536633 8.544641 8.552653
#> [512] 8.560670 8.568691 8.576716 8.584745 8.592778 8.600815 8.608856
#> [519] 8.616901 8.624949 8.633002 8.641059 8.649120 8.657184 8.665253
#> [526] 8.673325 8.681401 8.689481 8.697564 8.705652 8.713743 8.721837
#> [533] 8.729936 8.738038 8.746143 8.754253 8.762365 8.770482 8.778602
#> [540] 8.786725 8.794852 8.802983 8.811117 8.819254 8.827395 8.835539
#> [547] 8.843687 8.851838 8.859992 8.868150 8.876311 8.884475 8.892643
#> [554] 8.900814 8.908988 8.917165 8.925345 8.933529 8.941716 8.949906
#> [561] 8.958099 8.966295 8.974495 8.982697 8.990903 8.999111 9.007323
#> [568] 9.015537 9.023755 9.031975 9.040199 9.048425 9.056655 9.064887
#> [575] 9.073122 9.081360 9.089601 9.097845 9.106091 9.114341 9.122593
#> [582] 9.130848 9.139106 9.147366 9.155629 9.163895 9.172164 9.180436
#> [589] 9.188710 9.196986 9.205266 9.213548 9.221832 9.230120 9.238409
#> [596] 9.246702 9.254997 9.263294 9.271594 9.279897 9.288202 9.296509
#> [603] 9.304820 9.313132 9.321447 9.329764 9.338084 9.346406 9.354731
#> [610] 9.363058 9.371387 9.379719 9.388053 9.396390 9.404728 9.413070
#> [617] 9.421413 9.429758 9.438106 9.446457 9.454809 9.463164 9.471520
#> [624] 9.479879 9.488241 9.496604 9.504970 9.513338 9.521707 9.530079
#> [631] 9.538454 9.546830 9.555208 9.563589 9.571971 9.580356 9.588743
#> [638] 9.597131 9.605522 9.613915 9.622310 9.630707 9.639105 9.647506
#> [645] 9.655909 9.664313 9.672720 9.681129 9.689539 9.697952 9.706366
#> [652] 9.714782 9.723200 9.731620 9.740042 9.748466 9.756891 9.765318
#> [659] 9.773748 9.782179 9.790611 9.799046 9.807482 9.815920 9.824360
#> [666] 9.832802 9.841245 9.849690 9.858137 9.866586 9.875036 9.883488
#> [673] 9.891942 9.900397 9.908854 9.917313 9.925773 9.934235 9.942698
#> [680] 9.951164 9.959630 9.968099 9.976569 9.985040 9.993514 10.001988
#> [687] 10.010465 10.018942 10.027422 10.035903 10.044385 10.052869 10.061355
#> [694] 10.069842 10.078330 10.086820 10.095311 10.103804 10.112299 10.120795
#> [701] 10.129292 10.137791 10.146291 10.154792 10.163295 10.171799 10.180305
#> [708] 10.188812 10.197321 10.205831 10.214342 10.222855 10.231369 10.239884
#> [715] 10.248401 10.256919 10.265438 10.273958 10.282480 10.291004 10.299528
#> [722] 10.308054 10.316581 10.325109 10.333639 10.342170 10.350702 10.359235
#> [729] 10.367770 10.376305 10.384842 10.393381 10.401920 10.410461 10.419002
#> [736] 10.427545 10.436090 10.444635 10.453182 10.461729 10.470278 10.478828
#> [743] 10.487379 10.495931 10.504485 10.513039 10.521595 10.530151 10.538709
#> [750] 10.547268 10.555828 10.564389 10.572951 10.581514 10.590079 10.598644
#> [757] 10.607210 10.615778 10.624346 10.632915 10.641486 10.650057 10.658630
#> [764] 10.667203 10.675778 10.684353 10.692930 10.701507 10.710086 10.718665
#> [771] 10.727246 10.735827 10.744409 10.752993 10.761577 10.770162 10.778748
#> [778] 10.787335 10.795923 10.804512 10.813101 10.821692 10.830283 10.838876
#> [785] 10.847469 10.856063 10.864658 10.873254 10.881851 10.890449 10.899047
#> [792] 10.907647 10.916247 10.924848 10.933450 10.942052 10.950656 10.959260
#> [799] 10.967865 10.976471 10.985078 10.993685 11.002294 11.010903 11.019513
#> [806] 11.028123 11.036735 11.045347 11.053960 11.062574 11.071188 11.079804
#> [813] 11.088419 11.097036 11.105654 11.114272 11.122891 11.131510 11.140131
#> [820] 11.148752 11.157373 11.165996 11.174619 11.183243 11.191867 11.200492
#> [827] 11.209118 11.217745 11.226372 11.235000 11.243628 11.252258 11.260888
#> [834] 11.269518 11.278149 11.286781 11.295413 11.304046 11.312680 11.321314
#> [841] 11.329949 11.338584 11.347221 11.355857 11.364494 11.373132 11.381771
#> [848] 11.390410 11.399050 11.407690 11.416330 11.424972 11.433614 11.442256
#> [855] 11.450899 11.459543 11.468187 11.476832 11.485477 11.494123 11.502769
#> [862] 11.511416 11.520063 11.528711 11.537359 11.546008 11.554657 11.563307
#> [869] 11.571958 11.580609 11.589260 11.597912 11.606564 11.615217 11.623870
#> [876] 11.632524 11.641178 11.649833 11.658488 11.667144 11.675800 11.684456
#> [883] 11.693113 11.701771 11.710429 11.719087 11.727746 11.736405 11.745064
#> [890] 11.753724 11.762385 11.771046 11.779707 11.788369 11.797031 11.805693
#> [897] 11.814356 11.823019 11.831683 11.840347 11.849012 11.857676 11.866342
#> [904] 11.875007 11.883673 11.892339 11.901006 11.909673 11.918340 11.927008
#> [911] 11.935676 11.944345 11.953013 11.961682 11.970352 11.979022 11.987692
#> [918] 11.996362 12.005033 12.013704 12.022375 12.031047 12.039719 12.048392
#> [925] 12.057064 12.065737 12.074410 12.083084 12.091758 12.100432 12.109106
#> [932] 12.117781 12.126456 12.135131 12.143807 12.152482 12.161159 12.169835
#> [939] 12.178511 12.187188 12.195865 12.204543 12.213220 12.221898 12.230576
#> [946] 12.239255 12.247933 12.256612 12.265291 12.273970 12.282650 12.291329
#> [953] 12.300009 12.308690 12.317370 12.326050 12.334731 12.343412 12.352093
#> [960] 12.360775 12.369456 12.378138 12.386820 12.395502 12.404185 12.412867
#> [967] 12.421550 12.430233 12.438916 12.447599 12.456283 12.464966 12.473650
#> [974] 12.482334 12.491018 12.499702 12.508387 12.517071 12.525756 12.534441
#> [981] 12.543126 12.551811 12.560496 12.569182 12.577867 12.586553 12.595239
#> [988] 12.603925 12.612611 12.621297 12.629983 12.638670 12.647356 12.656043
#> [995] 12.664730 12.673416 12.682103 12.690790 12.699478 12.708165
bsmprice("put", 20, 30, 0:2000 / 1000, 0.15, 0, 0.25)
#> [1] 10.000000 9.995500 9.991001 9.986503 9.982005 9.977508 9.973012
#> [8] 9.968517 9.964022 9.959527 9.955034 9.950541 9.946049 9.941557
#> [15] 9.937066 9.932576 9.928086 9.923597 9.919109 9.914622 9.910135
#> [22] 9.905649 9.901163 9.896678 9.892194 9.887711 9.883228 9.878746
#> [29] 9.874264 9.869783 9.865303 9.860824 9.856345 9.851867 9.847389
#> [36] 9.842913 9.838437 9.833961 9.829486 9.825012 9.820539 9.816066
#> [43] 9.811594 9.807123 9.802652 9.798182 9.793713 9.789244 9.784776
#> [50] 9.780308 9.775842 9.771376 9.766910 9.762446 9.757981 9.753518
#> [57] 9.749055 9.744593 9.740132 9.735671 9.731211 9.726752 9.722293
#> [64] 9.717835 9.713378 9.708921 9.704465 9.700010 9.695555 9.691101
#> [71] 9.686648 9.682195 9.677743 9.673292 9.668841 9.664391 9.659942
#> [78] 9.655493 9.651045 9.646598 9.642151 9.637705 9.633260 9.628815
#> [85] 9.624371 9.619928 9.615485 9.611043 9.606602 9.602162 9.597722
#> [92] 9.593282 9.588844 9.584406 9.579968 9.575532 9.571096 9.566660
#> [99] 9.562226 9.557792 9.553358 9.548926 9.544494 9.540062 9.535632
#> [106] 9.531202 9.526772 9.522344 9.517916 9.513489 9.509062 9.504636
#> [113] 9.500211 9.495786 9.491362 9.486939 9.482516 9.478094 9.473673
#> [120] 9.469253 9.464833 9.460413 9.455995 9.451577 9.447160 9.442743
#> [127] 9.438328 9.433912 9.429498 9.425084 9.420671 9.416259 9.411847
#> [134] 9.407436 9.403026 9.398616 9.394207 9.389799 9.385392 9.380985
#> [141] 9.376579 9.372173 9.367769 9.363365 9.358962 9.354559 9.350157
#> [148] 9.345756 9.341356 9.336956 9.332558 9.328159 9.323762 9.319365
#> [155] 9.314969 9.310574 9.306180 9.301786 9.297393 9.293001 9.288610
#> [162] 9.284219 9.279830 9.275441 9.271052 9.266665 9.262278 9.257892
#> [169] 9.253507 9.249123 9.244740 9.240357 9.235975 9.231594 9.227214
#> [176] 9.222835 9.218456 9.214079 9.209702 9.205326 9.200951 9.196577
#> [183] 9.192204 9.187831 9.183460 9.179089 9.174719 9.170350 9.165983
#> [190] 9.161616 9.157249 9.152884 9.148520 9.144157 9.139794 9.135433
#> [197] 9.131072 9.126713 9.122354 9.117997 9.113640 9.109285 9.104930
#> [204] 9.100577 9.096224 9.091873 9.087522 9.083173 9.078824 9.074477
#> [211] 9.070131 9.065785 9.061441 9.057098 9.052756 9.048415 9.044075
#> [218] 9.039737 9.035399 9.031062 9.026727 9.022393 9.018060 9.013728
#> [225] 9.009397 9.005067 9.000739 8.996412 8.992086 8.987761 8.983437
#> [232] 8.979115 8.974793 8.970473 8.966155 8.961837 8.957521 8.953206
#> [239] 8.948892 8.944580 8.940269 8.935959 8.931650 8.927343 8.923037
#> [246] 8.918732 8.914429 8.910127 8.905826 8.901527 8.897229 8.892933
#> [253] 8.888637 8.884344 8.880051 8.875760 8.871471 8.867183 8.862896
#> [260] 8.858611 8.854327 8.850044 8.845763 8.841484 8.837206 8.832930
#> [267] 8.828654 8.824381 8.820109 8.815838 8.811569 8.807302 8.803036
#> [274] 8.798772 8.794509 8.790247 8.785988 8.781730 8.777473 8.773218
#> [281] 8.768965 8.764713 8.760463 8.756214 8.751967 8.747722 8.743478
#> [288] 8.739236 8.734996 8.730757 8.726520 8.722285 8.718051 8.713819
#> [295] 8.709589 8.705360 8.701133 8.696908 8.692685 8.688463 8.684243
#> [302] 8.680025 8.675809 8.671594 8.667381 8.663170 8.658960 8.654753
#> [309] 8.650547 8.646343 8.642141 8.637941 8.633742 8.629545 8.625350
#> [316] 8.621157 8.616966 8.612777 8.608589 8.604404 8.600220 8.596038
#> [323] 8.591858 8.587680 8.583504 8.579329 8.575157 8.570986 8.566818
#> [330] 8.562651 8.558486 8.554324 8.550163 8.546004 8.541847 8.537692
#> [337] 8.533539 8.529388 8.525239 8.521092 8.516947 8.512803 8.508662
#> [344] 8.504523 8.500386 8.496251 8.492118 8.487987 8.483858 8.479731
#> [351] 8.475606 8.471483 8.467362 8.463243 8.459127 8.455012 8.450899
#> [358] 8.446789 8.442680 8.438574 8.434470 8.430368 8.426268 8.422170
#> [365] 8.418074 8.413980 8.409888 8.405799 8.401711 8.397626 8.393543
#> [372] 8.389462 8.385383 8.381306 8.377232 8.373159 8.369089 8.365021
#> [379] 8.360955 8.356891 8.352829 8.348770 8.344713 8.340658 8.336605
#> [386] 8.332554 8.328506 8.324459 8.320415 8.316373 8.312334 8.308296
#> [393] 8.304261 8.300228 8.296197 8.292169 8.288142 8.284118 8.280096
#> [400] 8.276077 8.272059 8.268044 8.264031 8.260021 8.256012 8.252006
#> [407] 8.248003 8.244001 8.240002 8.236005 8.232010 8.228017 8.224027
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#> [442] 8.109314 8.105393 8.101474 8.097558 8.093645 8.089733 8.085825
#> [449] 8.081918 8.078014 8.074112 8.070212 8.066315 8.062420 8.058528
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#> [848] 6.710038 6.707038 6.704039 6.701043 6.698049 6.695056 6.692066
#> [855] 6.689078 6.686091 6.683107 6.680124 6.677143 6.674165 6.671188
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#> [904] 6.545015 6.542123 6.539232 6.536344 6.533457 6.530572 6.527689
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#> [918] 6.504692 6.501825 6.498961 6.496098 6.493238 6.490379 6.487522
#> [925] 6.484666 6.481813 6.478962 6.476112 6.473264 6.470418 6.467574
#> [932] 6.464731 6.461891 6.459052 6.456215 6.453380 6.450547 6.447716
#> [939] 6.444886 6.442058 6.439232 6.436408 6.433586 6.430765 6.427947
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#> [953] 6.405462 6.402659 6.399859 6.397060 6.394263 6.391467 6.388674
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#> [1037] 6.176146 6.173488 6.170832 6.168178 6.165525 6.162874 6.160225
#> [1044] 6.157577 6.154931 6.152287 6.149644 6.147003 6.144364 6.141726
#> [1051] 6.139090 6.136455 6.133822 6.131191 6.128561 6.125933 6.123307
#> [1058] 6.120682 6.118059 6.115438 6.112818 6.110200 6.107583 6.104968
#> [1065] 6.102355 6.099743 6.097133 6.094524 6.091917 6.089312 6.086708
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#> [1079] 6.065936 6.063347 6.060759 6.058173 6.055589 6.053006 6.050424
#> [1086] 6.047845 6.045266 6.042690 6.040115 6.037541 6.034969 6.032399
#> [1093] 6.029831 6.027263 6.024698 6.022134 6.019571 6.017011 6.014451
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#> [1114] 5.976249 5.973715 5.971182 5.968651 5.966121 5.963593 5.961066
#> [1121] 5.958541 5.956017 5.953495 5.950975 5.948456 5.945938 5.943422
#> [1128] 5.940908 5.938395 5.935883 5.933373 5.930865 5.928358 5.925853
#> [1135] 5.923349 5.920847 5.918346 5.915847 5.913349 5.910853 5.908358
#> [1142] 5.905865 5.903373 5.900883 5.898394 5.895907 5.893421 5.890937
#> [1149] 5.888454 5.885973 5.883493 5.881015 5.878538 5.876063 5.873589
#> [1156] 5.871117 5.868646 5.866177 5.863709 5.861242 5.858777 5.856314
#> [1163] 5.853852 5.851392 5.848933 5.846475 5.844019 5.841565 5.839111
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#> [1184] 5.802490 5.800061 5.797632 5.795205 5.792780 5.790356 5.787933
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#> [1205] 5.751767 5.749367 5.746969 5.744572 5.742177 5.739783 5.737390
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#> [1233] 5.685108 5.682748 5.680389 5.678031 5.675675 5.673320 5.670967
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#> [1282] 5.571052 5.568758 5.566465 5.564174 5.561884 5.559595 5.557307
#> [1289] 5.555021 5.552736 5.550453 5.548171 5.545890 5.543610 5.541332
#> [1296] 5.539055 5.536779 5.534505 5.532232 5.529960 5.527690 5.525421
#> [1303] 5.523153 5.520886 5.518621 5.516357 5.514095 5.511833 5.509573
#> [1310] 5.507315 5.505057 5.502801 5.500546 5.498293 5.496040 5.493789
#> [1317] 5.491540 5.489291 5.487044 5.484798 5.482554 5.480310 5.478068
#> [1324] 5.475828 5.473588 5.471350 5.469113 5.466877 5.464643 5.462410
#> [1331] 5.460178 5.457948 5.455718 5.453490 5.451264 5.449038 5.446814
#> [1338] 5.444591 5.442369 5.440149 5.437930 5.435712 5.433495 5.431280
#> [1345] 5.429065 5.426853 5.424641 5.422431 5.420221 5.418014 5.415807
#> [1352] 5.413601 5.411397 5.409194 5.406993 5.404792 5.402593 5.400395
#> [1359] 5.398199 5.396003 5.393809 5.391616 5.389424 5.387234 5.385044
#> [1366] 5.382856 5.380670 5.378484 5.376300 5.374116 5.371935 5.369754
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#> [1380] 5.352352 5.350183 5.348014 5.345847 5.343681 5.341516 5.339353
#> [1387] 5.337190 5.335029 5.332869 5.330710 5.328553 5.326396 5.324241
#> [1394] 5.322087 5.319934 5.317783 5.315632 5.313483 5.311335 5.309188
#> [1401] 5.307043 5.304898 5.302755 5.300613 5.298472 5.296333 5.294194
#> [1408] 5.292057 5.289921 5.287786 5.285653 5.283520 5.281389 5.279259
#> [1415] 5.277130 5.275002 5.272875 5.270750 5.268626 5.266503 5.264381
#> [1422] 5.262260 5.260141 5.258022 5.255905 5.253789 5.251674 5.249561
#> [1429] 5.247448 5.245337 5.243226 5.241117 5.239010 5.236903 5.234797
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#> [1443] 5.217995 5.215900 5.213806 5.211713 5.209621 5.207531 5.205441
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#> [1457] 5.188768 5.186689 5.184611 5.182534 5.180458 5.178384 5.176310
#> [1464] 5.174238 5.172167 5.170097 5.168028 5.165960 5.163894 5.161828
#> [1471] 5.159764 5.157701 5.155639 5.153578 5.151518 5.149459 5.147401
#> [1478] 5.145345 5.143290 5.141235 5.139182 5.137130 5.135079 5.133029
#> [1485] 5.130981 5.128933 5.126887 5.124841 5.122797 5.120754 5.118712
#> [1492] 5.116671 5.114631 5.112593 5.110555 5.108519 5.106483 5.104449
#> [1499] 5.102416 5.100384 5.098353 5.096323 5.094294 5.092266 5.090240
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#> [1513] 5.074066 5.072050 5.070034 5.068019 5.066006 5.063994 5.061982
#> [1520] 5.059972 5.057963 5.055955 5.053948 5.051942 5.049937 5.047933
#> [1527] 5.045930 5.043929 5.041928 5.039929 5.037930 5.035933 5.033937
#> [1534] 5.031941 5.029947 5.027954 5.025962 5.023971 5.021981 5.019993
#> [1541] 5.018005 5.016018 5.014033 5.012048 5.010065 5.008082 5.006101
#> [1548] 5.004120 5.002141 5.000163 4.998186 4.996210 4.994235 4.992261
#> [1555] 4.990288 4.988316 4.986345 4.984375 4.982407 4.980439 4.978472
#> [1562] 4.976507 4.974542 4.972579 4.970616 4.968655 4.966694 4.964735
#> [1569] 4.962777 4.960819 4.958863 4.956908 4.954954 4.953001 4.951049
#> [1576] 4.949098 4.947148 4.945199 4.943251 4.941304 4.939358 4.937413
#> [1583] 4.935469 4.933527 4.931585 4.929644 4.927704 4.925766 4.923828
#> [1590] 4.921891 4.919956 4.918021 4.916087 4.914155 4.912223 4.910293
#> [1597] 4.908363 4.906435 4.904507 4.902581 4.900655 4.898731 4.896807
#> [1604] 4.894885 4.892964 4.891043 4.889124 4.887206 4.885288 4.883372
#> [1611] 4.881456 4.879542 4.877629 4.875716 4.873805 4.871895 4.869985
#> [1618] 4.868077 4.866170 4.864263 4.862358 4.860454 4.858550 4.856648
#> [1625] 4.854746 4.852846 4.850947 4.849048 4.847151 4.845255 4.843359
#> [1632] 4.841465 4.839571 4.837679 4.835787 4.833897 4.832007 4.830119
#> [1639] 4.828231 4.826345 4.824459 4.822575 4.820691 4.818808 4.816927
#> [1646] 4.815046 4.813166 4.811288 4.809410 4.807533 4.805657 4.803783
#> [1653] 4.801909 4.800036 4.798164 4.796293 4.794423 4.792554 4.790686
#> [1660] 4.788819 4.786953 4.785088 4.783224 4.781361 4.779498 4.777637
#> [1667] 4.775777 4.773917 4.772059 4.770202 4.768345 4.766490 4.764635
#> [1674] 4.762782 4.760929 4.759077 4.757227 4.755377 4.753528 4.751680
#> [1681] 4.749833 4.747987 4.746142 4.744298 4.742455 4.740613 4.738772
#> [1688] 4.736932 4.735092 4.733254 4.731417 4.729580 4.727744 4.725910
#> [1695] 4.724076 4.722244 4.720412 4.718581 4.716751 4.714922 4.713094
#> [1702] 4.711267 4.709441 4.707616 4.705791 4.703968 4.702146 4.700324
#> [1709] 4.698504 4.696684 4.694865 4.693048 4.691231 4.689415 4.687600
#> [1716] 4.685786 4.683973 4.682161 4.680349 4.678539 4.676730 4.674921
#> [1723] 4.673113 4.671307 4.669501 4.667696 4.665892 4.664089 4.662287
#> [1730] 4.660486 4.658686 4.656887 4.655088 4.653291 4.651494 4.649699
#> [1737] 4.647904 4.646110 4.644317 4.642525 4.640734 4.638944 4.637154
#> [1744] 4.635366 4.633579 4.631792 4.630006 4.628222 4.626438 4.624655
#> [1751] 4.622873 4.621092 4.619311 4.617532 4.615754 4.613976 4.612199
#> [1758] 4.610424 4.608649 4.606875 4.605102 4.603330 4.601558 4.599788
#> [1765] 4.598018 4.596250 4.594482 4.592715 4.590949 4.589184 4.587420
#> [1772] 4.585657 4.583895 4.582133 4.580373 4.578613 4.576854 4.575096
#> [1779] 4.573339 4.571583 4.569828 4.568073 4.566320 4.564567 4.562815
#> [1786] 4.561064 4.559314 4.557565 4.555817 4.554069 4.552323 4.550577
#> [1793] 4.548833 4.547089 4.545346 4.543604 4.541862 4.540122 4.538382
#> [1800] 4.536644 4.534906 4.533169 4.531433 4.529698 4.527963 4.526230
#> [1807] 4.524497 4.522766 4.521035 4.519305 4.517576 4.515847 4.514120
#> [1814] 4.512394 4.510668 4.508943 4.507219 4.505496 4.503774 4.502052
#> [1821] 4.500332 4.498612 4.496893 4.495175 4.493458 4.491742 4.490027
#> [1828] 4.488312 4.486598 4.484885 4.483173 4.481462 4.479752 4.478042
#> [1835] 4.476334 4.474626 4.472919 4.471213 4.469508 4.467804 4.466100
#> [1842] 4.464397 4.462695 4.460994 4.459294 4.457595 4.455896 4.454199
#> [1849] 4.452502 4.450806 4.449111 4.447417 4.445723 4.444031 4.442339
#> [1856] 4.440648 4.438958 4.437268 4.435580 4.433892 4.432206 4.430520
#> [1863] 4.428834 4.427150 4.425467 4.423784 4.422102 4.420421 4.418741
#> [1870] 4.417062 4.415383 4.413706 4.412029 4.410353 4.408678 4.407003
#> [1877] 4.405330 4.403657 4.401985 4.400314 4.398644 4.396974 4.395305
#> [1884] 4.393638 4.391971 4.390304 4.388639 4.386975 4.385311 4.383648
#> [1891] 4.381986 4.380324 4.378664 4.377004 4.375345 4.373687 4.372030
#> [1898] 4.370374 4.368718 4.367063 4.365409 4.363756 4.362104 4.360452
#> [1905] 4.358801 4.357151 4.355502 4.353854 4.352206 4.350560 4.348914
#> [1912] 4.347268 4.345624 4.343981 4.342338 4.340696 4.339055 4.337414
#> [1919] 4.335775 4.334136 4.332498 4.330861 4.329225 4.327589 4.325954
#> [1926] 4.324320 4.322687 4.321055 4.319423 4.317792 4.316162 4.314533
#> [1933] 4.312904 4.311277 4.309650 4.308024 4.306399 4.304774 4.303150
#> [1940] 4.301527 4.299905 4.298284 4.296663 4.295044 4.293425 4.291806
#> [1947] 4.290189 4.288572 4.286956 4.285341 4.283727 4.282113 4.280501
#> [1954] 4.278889 4.277277 4.275667 4.274057 4.272448 4.270840 4.269233
#> [1961] 4.267626 4.266021 4.264416 4.262811 4.261208 4.259605 4.258003
#> [1968] 4.256402 4.254802 4.253202 4.251603 4.250005 4.248408 4.246812
#> [1975] 4.245216 4.243621 4.242027 4.240433 4.238840 4.237248 4.235657
#> [1982] 4.234067 4.232477 4.230888 4.229300 4.227713 4.226126 4.224540
#> [1989] 4.222955 4.221371 4.219787 4.218205 4.216623 4.215041 4.213461
#> [1996] 4.211881 4.210302 4.208724 4.207146 4.205569 4.203993