Greeks formula implementation, from Black-Scholes-Merton model.
Usage
bsmgreeks(
greeks = c("delta", "gamma", "vega", "rho", "theta"),
type,
spot,
strike,
time,
rate,
yield,
sigma
)
bsmdelta(type, spot, strike, time, rate, yield, sigma)
bsmvega(type, spot, strike, time, rate, yield, sigma)
bsmgamma(type, spot, strike, time, rate, yield, sigma)
bsmtheta(type, spot, strike, time, rate, yield, sigma)
bsmrho(type, spot, strike, time, rate, yield, sigma)Arguments
- greeks
desired greek letter: delta, gamma, vega, rho or theta.
- 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.
bsmdelta
Delta greek letter measures the rate of change of the theoretical option value with respect to changes in the underlying asset's price.
bsmgamma
Gamma greek letter measures the rate of change in the delta with respect to changes in the underlying price.
bsmrho
Rho measures sensitivity to the interest rate: it is the derivative of the option value with respect to the risk free interest rate.
Examples
bsmgreeks("delta", "put", 20, 30, 0:2000 / 1000, 0.15, 0, 0.25)
#> [1] -1.0000000 -1.0000000 -1.0000000 -1.0000000 -1.0000000 -1.0000000
#> [7] -1.0000000 -1.0000000 -1.0000000 -1.0000000 -1.0000000 -1.0000000
#> [13] -1.0000000 -1.0000000 -1.0000000 -1.0000000 -1.0000000 -1.0000000
#> [19] -1.0000000 -1.0000000 -1.0000000 -1.0000000 -1.0000000 -1.0000000
#> [25] -1.0000000 -1.0000000 -1.0000000 -1.0000000 -1.0000000 -1.0000000
#> [31] -1.0000000 -1.0000000 -1.0000000 -1.0000000 -1.0000000 -1.0000000
#> [37] -1.0000000 -1.0000000 -1.0000000 -1.0000000 -1.0000000 -1.0000000
#> [43] -1.0000000 -1.0000000 -1.0000000 -1.0000000 -1.0000000 -1.0000000
#> [49] -1.0000000 -1.0000000 -1.0000000 -1.0000000 -1.0000000 -1.0000000
#> [55] -1.0000000 -1.0000000 -1.0000000 -1.0000000 -1.0000000 -1.0000000
#> [61] -1.0000000 -1.0000000 -1.0000000 -1.0000000 -1.0000000 -1.0000000
#> [67] -1.0000000 -1.0000000 -1.0000000 -1.0000000 -1.0000000 -1.0000000
#> [73] -1.0000000 -1.0000000 -1.0000000 -1.0000000 -1.0000000 -1.0000000
#> [79] -1.0000000 -1.0000000 -1.0000000 -1.0000000 -1.0000000 -1.0000000
#> [85] -1.0000000 -1.0000000 -0.9999999 -0.9999999 -0.9999999 -0.9999999
#> [91] -0.9999999 -0.9999999 -0.9999999 -0.9999998 -0.9999998 -0.9999998
#> [97] -0.9999997 -0.9999997 -0.9999996 -0.9999996 -0.9999995 -0.9999995
#> [103] -0.9999994 -0.9999993 -0.9999992 -0.9999991 -0.9999990 -0.9999988
#> [109] -0.9999987 -0.9999985 -0.9999983 -0.9999981 -0.9999979 -0.9999977
#> [115] -0.9999974 -0.9999971 -0.9999968 -0.9999965 -0.9999961 -0.9999957
#> [121] -0.9999953 -0.9999948 -0.9999943 -0.9999938 -0.9999932 -0.9999926
#> [127] -0.9999919 -0.9999912 -0.9999904 -0.9999896 -0.9999887 -0.9999877
#> [133] -0.9999867 -0.9999856 -0.9999844 -0.9999832 -0.9999819 -0.9999805
#> [139] -0.9999790 -0.9999775 -0.9999758 -0.9999741 -0.9999722 -0.9999702
#> [145] -0.9999682 -0.9999660 -0.9999637 -0.9999613 -0.9999587 -0.9999561
#> [151] -0.9999532 -0.9999503 -0.9999472 -0.9999439 -0.9999405 -0.9999370
#> [157] -0.9999333 -0.9999294 -0.9999253 -0.9999210 -0.9999166 -0.9999120
#> [163] -0.9999072 -0.9999021 -0.9998969 -0.9998915 -0.9998858 -0.9998799
#> [169] -0.9998738 -0.9998674 -0.9998608 -0.9998540 -0.9998469 -0.9998395
#> [175] -0.9998319 -0.9998240 -0.9998158 -0.9998074 -0.9997986 -0.9997896
#> [181] -0.9997803 -0.9997706 -0.9997607 -0.9997504 -0.9997398 -0.9997288
#> [187] -0.9997175 -0.9997059 -0.9996939 -0.9996816 -0.9996689 -0.9996559
#> [193] -0.9996424 -0.9996286 -0.9996144 -0.9995998 -0.9995848 -0.9995694
#> [199] -0.9995535 -0.9995373 -0.9995206 -0.9995035 -0.9994860 -0.9994680
#> [205] -0.9994496 -0.9994307 -0.9994114 -0.9993915 -0.9993713 -0.9993505
#> [211] -0.9993293 -0.9993075 -0.9992853 -0.9992625 -0.9992393 -0.9992155
#> [217] -0.9991913 -0.9991665 -0.9991411 -0.9991153 -0.9990889 -0.9990619
#> [223] -0.9990344 -0.9990063 -0.9989777 -0.9989485 -0.9989187 -0.9988884
#> [229] -0.9988574 -0.9988259 -0.9987938 -0.9987611 -0.9987277 -0.9986938
#> [235] -0.9986593 -0.9986241 -0.9985883 -0.9985519 -0.9985149 -0.9984772
#> [241] -0.9984388 -0.9983999 -0.9983602 -0.9983200 -0.9982790 -0.9982374
#> [247] -0.9981952 -0.9981522 -0.9981086 -0.9980643 -0.9980193 -0.9979736
#> [253] -0.9979273 -0.9978802 -0.9978325 -0.9977840 -0.9977348 -0.9976849
#> [259] -0.9976343 -0.9975830 -0.9975310 -0.9974782 -0.9974248 -0.9973705
#> [265] -0.9973156 -0.9972599 -0.9972035 -0.9971463 -0.9970884 -0.9970297
#> [271] -0.9969703 -0.9969101 -0.9968492 -0.9967875 -0.9967251 -0.9966618
#> [277] -0.9965979 -0.9965331 -0.9964676 -0.9964013 -0.9963342 -0.9962663
#> [283] -0.9961977 -0.9961283 -0.9960581 -0.9959871 -0.9959153 -0.9958427
#> [289] -0.9957693 -0.9956951 -0.9956202 -0.9955444 -0.9954678 -0.9953905
#> [295] -0.9953123 -0.9952333 -0.9951535 -0.9950730 -0.9949916 -0.9949093
#> [301] -0.9948263 -0.9947425 -0.9946578 -0.9945724 -0.9944861 -0.9943990
#> [307] -0.9943111 -0.9942223 -0.9941328 -0.9940424 -0.9939512 -0.9938592
#> [313] -0.9937663 -0.9936727 -0.9935782 -0.9934829 -0.9933867 -0.9932897
#> [319] -0.9931919 -0.9930933 -0.9929939 -0.9928936 -0.9927925 -0.9926906
#> [325] -0.9925878 -0.9924842 -0.9923798 -0.9922746 -0.9921685 -0.9920616
#> [331] -0.9919539 -0.9918453 -0.9917359 -0.9916257 -0.9915147 -0.9914028
#> [337] -0.9912901 -0.9911766 -0.9910622 -0.9909470 -0.9908310 -0.9907142
#> [343] -0.9905966 -0.9904781 -0.9903588 -0.9902387 -0.9901177 -0.9899960
#> [349] -0.9898734 -0.9897499 -0.9896257 -0.9895007 -0.9893748 -0.9892481
#> [355] -0.9891206 -0.9889923 -0.9888631 -0.9887332 -0.9886024 -0.9884709
#> [361] -0.9883385 -0.9882053 -0.9880713 -0.9879364 -0.9878008 -0.9876644
#> [367] -0.9875272 -0.9873891 -0.9872503 -0.9871106 -0.9869702 -0.9868289
#> [373] -0.9866869 -0.9865441 -0.9864004 -0.9862560 -0.9861108 -0.9859648
#> [379] -0.9858180 -0.9856704 -0.9855220 -0.9853728 -0.9852229 -0.9850722
#> [385] -0.9849207 -0.9847684 -0.9846153 -0.9844615 -0.9843068 -0.9841514
#> [391] -0.9839953 -0.9838384 -0.9836806 -0.9835222 -0.9833629 -0.9832030
#> [397] -0.9830422 -0.9828807 -0.9827184 -0.9825554 -0.9823916 -0.9822270
#> [403] -0.9820617 -0.9818957 -0.9817289 -0.9815614 -0.9813931 -0.9812241
#> [409] -0.9810543 -0.9808838 -0.9807126 -0.9805406 -0.9803679 -0.9801945
#> [415] -0.9800203 -0.9798454 -0.9796698 -0.9794935 -0.9793164 -0.9791387
#> [421] -0.9789602 -0.9787810 -0.9786010 -0.9784204 -0.9782391 -0.9780570
#> [427] -0.9778743 -0.9776908 -0.9775066 -0.9773218 -0.9771362 -0.9769500
#> [433] -0.9767630 -0.9765754 -0.9763871 -0.9761981 -0.9760084 -0.9758180
#> [439] -0.9756270 -0.9754352 -0.9752428 -0.9750498 -0.9748560 -0.9746616
#> [445] -0.9744665 -0.9742707 -0.9740743 -0.9738773 -0.9736795 -0.9734811
#> [451] -0.9732821 -0.9730824 -0.9728820 -0.9726810 -0.9724794 -0.9722771
#> [457] -0.9720742 -0.9718706 -0.9716664 -0.9714616 -0.9712561 -0.9710500
#> [463] -0.9708433 -0.9706359 -0.9704280 -0.9702194 -0.9700102 -0.9698003
#> [469] -0.9695899 -0.9693788 -0.9691671 -0.9689549 -0.9687420 -0.9685285
#> [475] -0.9683144 -0.9680997 -0.9678844 -0.9676685 -0.9674521 -0.9672350
#> [481] -0.9670173 -0.9667991 -0.9665803 -0.9663609 -0.9661409 -0.9659203
#> [487] -0.9656992 -0.9654775 -0.9652552 -0.9650323 -0.9648089 -0.9645850
#> [493] -0.9643604 -0.9641353 -0.9639097 -0.9636835 -0.9634567 -0.9632294
#> [499] -0.9630015 -0.9627731 -0.9625442 -0.9623147 -0.9620846 -0.9618541
#> [505] -0.9616229 -0.9613913 -0.9611591 -0.9609264 -0.9606932 -0.9604595
#> [511] -0.9602252 -0.9599904 -0.9597551 -0.9595193 -0.9592829 -0.9590461
#> [517] -0.9588087 -0.9585709 -0.9583325 -0.9580936 -0.9578542 -0.9576144
#> [523] -0.9573740 -0.9571331 -0.9568918 -0.9566500 -0.9564076 -0.9561648
#> [529] -0.9559215 -0.9556777 -0.9554335 -0.9551887 -0.9549435 -0.9546978
#> [535] -0.9544517 -0.9542051 -0.9539580 -0.9537104 -0.9534624 -0.9532139
#> [541] -0.9529650 -0.9527156 -0.9524658 -0.9522155 -0.9519647 -0.9517135
#> [547] -0.9514619 -0.9512098 -0.9509573 -0.9507043 -0.9504509 -0.9501971
#> [553] -0.9499428 -0.9496881 -0.9494330 -0.9491775 -0.9489215 -0.9486651
#> [559] -0.9484082 -0.9481510 -0.9478933 -0.9476353 -0.9473768 -0.9471179
#> [565] -0.9468586 -0.9465988 -0.9463387 -0.9460782 -0.9458173 -0.9455559
#> [571] -0.9452942 -0.9450321 -0.9447696 -0.9445067 -0.9442434 -0.9439797
#> [577] -0.9437156 -0.9434512 -0.9431864 -0.9429211 -0.9426555 -0.9423896
#> [583] -0.9421232 -0.9418565 -0.9415894 -0.9413220 -0.9410542 -0.9407860
#> [589] -0.9405174 -0.9402485 -0.9399792 -0.9397096 -0.9394396 -0.9391693
#> [595] -0.9388986 -0.9386276 -0.9383562 -0.9380844 -0.9378124 -0.9375399
#> [601] -0.9372672 -0.9369941 -0.9367206 -0.9364469 -0.9361728 -0.9358983
#> [607] -0.9356235 -0.9353484 -0.9350730 -0.9347973 -0.9345212 -0.9342448
#> [613] -0.9339681 -0.9336910 -0.9334137 -0.9331360 -0.9328581 -0.9325798
#> [619] -0.9323012 -0.9320223 -0.9317430 -0.9314635 -0.9311837 -0.9309036
#> [625] -0.9306232 -0.9303425 -0.9300614 -0.9297801 -0.9294985 -0.9292166
#> [631] -0.9289345 -0.9286520 -0.9283692 -0.9280862 -0.9278029 -0.9275193
#> [637] -0.9272354 -0.9269512 -0.9266668 -0.9263821 -0.9260971 -0.9258119
#> [643] -0.9255263 -0.9252405 -0.9249545 -0.9246682 -0.9243816 -0.9240947
#> [649] -0.9238076 -0.9235202 -0.9232326 -0.9229447 -0.9226566 -0.9223682
#> [655] -0.9220796 -0.9217907 -0.9215015 -0.9212122 -0.9209225 -0.9206327
#> [661] -0.9203425 -0.9200522 -0.9197616 -0.9194708 -0.9191797 -0.9188884
#> [667] -0.9185969 -0.9183051 -0.9180131 -0.9177209 -0.9174284 -0.9171357
#> [673] -0.9168428 -0.9165497 -0.9162563 -0.9159628 -0.9156690 -0.9153750
#> [679] -0.9150808 -0.9147863 -0.9144917 -0.9141968 -0.9139017 -0.9136065
#> [685] -0.9133110 -0.9130153 -0.9127194 -0.9124233 -0.9121270 -0.9118305
#> [691] -0.9115337 -0.9112368 -0.9109397 -0.9106424 -0.9103450 -0.9100473
#> [697] -0.9097494 -0.9094513 -0.9091531 -0.9088546 -0.9085560 -0.9082572
#> [703] -0.9079582 -0.9076590 -0.9073597 -0.9070601 -0.9067604 -0.9064605
#> [709] -0.9061605 -0.9058602 -0.9055598 -0.9052592 -0.9049584 -0.9046575
#> [715] -0.9043564 -0.9040551 -0.9037537 -0.9034521 -0.9031504 -0.9028484
#> [721] -0.9025464 -0.9022441 -0.9019417 -0.9016391 -0.9013364 -0.9010336
#> [727] -0.9007305 -0.9004274 -0.9001240 -0.8998205 -0.8995169 -0.8992131
#> [733] -0.8989092 -0.8986051 -0.8983009 -0.8979966 -0.8976921 -0.8973874
#> [739] -0.8970827 -0.8967777 -0.8964727 -0.8961675 -0.8958622 -0.8955567
#> [745] -0.8952511 -0.8949454 -0.8946395 -0.8943335 -0.8940274 -0.8937211
#> [751] -0.8934148 -0.8931083 -0.8928016 -0.8924949 -0.8921880 -0.8918810
#> [757] -0.8915739 -0.8912667 -0.8909593 -0.8906519 -0.8903443 -0.8900366
#> [763] -0.8897288 -0.8894209 -0.8891128 -0.8888047 -0.8884964 -0.8881880
#> [769] -0.8878796 -0.8875710 -0.8872623 -0.8869535 -0.8866446 -0.8863356
#> [775] -0.8860265 -0.8857173 -0.8854080 -0.8850986 -0.8847891 -0.8844795
#> [781] -0.8841698 -0.8838600 -0.8835501 -0.8832401 -0.8829301 -0.8826199
#> [787] -0.8823097 -0.8819993 -0.8816889 -0.8813784 -0.8810678 -0.8807571
#> [793] -0.8804463 -0.8801355 -0.8798245 -0.8795135 -0.8792024 -0.8788912
#> [799] -0.8785799 -0.8782686 -0.8779572 -0.8776457 -0.8773341 -0.8770225
#> [805] -0.8767107 -0.8763989 -0.8760871 -0.8757751 -0.8754631 -0.8751511
#> [811] -0.8748389 -0.8745267 -0.8742144 -0.8739021 -0.8735896 -0.8732772
#> [817] -0.8729646 -0.8726520 -0.8723393 -0.8720266 -0.8717138 -0.8714010
#> [823] -0.8710880 -0.8707751 -0.8704621 -0.8701490 -0.8698358 -0.8695226
#> [829] -0.8692094 -0.8688961 -0.8685827 -0.8682693 -0.8679559 -0.8676424
#> [835] -0.8673288 -0.8670152 -0.8667016 -0.8663879 -0.8660741 -0.8657603
#> [841] -0.8654465 -0.8651326 -0.8648187 -0.8645047 -0.8641907 -0.8638767
#> [847] -0.8635626 -0.8632484 -0.8629343 -0.8626201 -0.8623058 -0.8619916
#> [853] -0.8616772 -0.8613629 -0.8610485 -0.8607341 -0.8604196 -0.8601051
#> [859] -0.8597906 -0.8594761 -0.8591615 -0.8588469 -0.8585323 -0.8582176
#> [865] -0.8579029 -0.8575882 -0.8572734 -0.8569587 -0.8566439 -0.8563290
#> [871] -0.8560142 -0.8556993 -0.8553844 -0.8550695 -0.8547546 -0.8544397
#> [877] -0.8541247 -0.8538097 -0.8534947 -0.8531797 -0.8528646 -0.8525496
#> [883] -0.8522345 -0.8519194 -0.8516043 -0.8512892 -0.8509740 -0.8506589
#> [889] -0.8503437 -0.8500286 -0.8497134 -0.8493982 -0.8490830 -0.8487678
#> [895] -0.8484526 -0.8481374 -0.8478222 -0.8475069 -0.8471917 -0.8468764
#> [901] -0.8465612 -0.8462459 -0.8459307 -0.8456154 -0.8453001 -0.8449849
#> [907] -0.8446696 -0.8443543 -0.8440391 -0.8437238 -0.8434085 -0.8430933
#> [913] -0.8427780 -0.8424628 -0.8421475 -0.8418322 -0.8415170 -0.8412018
#> [919] -0.8408865 -0.8405713 -0.8402561 -0.8399408 -0.8396256 -0.8393104
#> [925] -0.8389952 -0.8386801 -0.8383649 -0.8380497 -0.8377346 -0.8374194
#> [931] -0.8371043 -0.8367892 -0.8364741 -0.8361590 -0.8358439 -0.8355288
#> [937] -0.8352138 -0.8348988 -0.8345837 -0.8342687 -0.8339538 -0.8336388
#> [943] -0.8333238 -0.8330089 -0.8326940 -0.8323791 -0.8320642 -0.8317494
#> [949] -0.8314346 -0.8311198 -0.8308050 -0.8304902 -0.8301755 -0.8298607
#> [955] -0.8295460 -0.8292314 -0.8289167 -0.8286021 -0.8282875 -0.8279729
#> [961] -0.8276584 -0.8273439 -0.8270294 -0.8267149 -0.8264005 -0.8260861
#> [967] -0.8257717 -0.8254574 -0.8251431 -0.8248288 -0.8245145 -0.8242003
#> [973] -0.8238861 -0.8235719 -0.8232578 -0.8229437 -0.8226297 -0.8223156
#> [979] -0.8220016 -0.8216877 -0.8213738 -0.8210599 -0.8207460 -0.8204322
#> [985] -0.8201184 -0.8198047 -0.8194910 -0.8191773 -0.8188637 -0.8185501
#> [991] -0.8182366 -0.8179230 -0.8176096 -0.8172961 -0.8169828 -0.8166694
#> [997] -0.8163561 -0.8160428 -0.8157296 -0.8154164 -0.8151033 -0.8147902
#> [1003] -0.8144772 -0.8141642 -0.8138512 -0.8135383 -0.8132254 -0.8129126
#> [1009] -0.8125998 -0.8122871 -0.8119744 -0.8116618 -0.8113492 -0.8110366
#> [1015] -0.8107241 -0.8104117 -0.8100993 -0.8097869 -0.8094746 -0.8091624
#> [1021] -0.8088502 -0.8085380 -0.8082259 -0.8079139 -0.8076019 -0.8072900
#> [1027] -0.8069781 -0.8066662 -0.8063545 -0.8060427 -0.8057311 -0.8054194
#> [1033] -0.8051079 -0.8047963 -0.8044849 -0.8041735 -0.8038621 -0.8035508
#> [1039] -0.8032396 -0.8029284 -0.8026173 -0.8023063 -0.8019952 -0.8016843
#> [1045] -0.8013734 -0.8010626 -0.8007518 -0.8004411 -0.8001304 -0.7998198
#> [1051] -0.7995093 -0.7991988 -0.7988884 -0.7985781 -0.7982678 -0.7979575
#> [1057] -0.7976474 -0.7973373 -0.7970272 -0.7967172 -0.7964073 -0.7960975
#> [1063] -0.7957877 -0.7954779 -0.7951683 -0.7948587 -0.7945491 -0.7942397
#> [1069] -0.7939303 -0.7936209 -0.7933117 -0.7930024 -0.7926933 -0.7923842
#> [1075] -0.7920752 -0.7917663 -0.7914574 -0.7911486 -0.7908399 -0.7905312
#> [1081] -0.7902226 -0.7899141 -0.7896056 -0.7892972 -0.7889889 -0.7886806
#> [1087] -0.7883724 -0.7880643 -0.7877563 -0.7874483 -0.7871404 -0.7868326
#> [1093] -0.7865248 -0.7862171 -0.7859095 -0.7856020 -0.7852945 -0.7849871
#> [1099] -0.7846798 -0.7843725 -0.7840653 -0.7837582 -0.7834512 -0.7831442
#> [1105] -0.7828374 -0.7825305 -0.7822238 -0.7819171 -0.7816106 -0.7813041
#> [1111] -0.7809976 -0.7806913 -0.7803850 -0.7800788 -0.7797727 -0.7794666
#> [1117] -0.7791606 -0.7788547 -0.7785489 -0.7782432 -0.7779375 -0.7776319
#> [1123] -0.7773264 -0.7770210 -0.7767156 -0.7764104 -0.7761052 -0.7758001
#> [1129] -0.7754950 -0.7751901 -0.7748852 -0.7745804 -0.7742757 -0.7739711
#> [1135] -0.7736665 -0.7733621 -0.7730577 -0.7727534 -0.7724492 -0.7721450
#> [1141] -0.7718410 -0.7715370 -0.7712331 -0.7709293 -0.7706255 -0.7703219
#> [1147] -0.7700183 -0.7697148 -0.7694114 -0.7691081 -0.7688049 -0.7685018
#> [1153] -0.7681987 -0.7678957 -0.7675928 -0.7672900 -0.7669873 -0.7666847
#> [1159] -0.7663821 -0.7660797 -0.7657773 -0.7654750 -0.7651728 -0.7648706
#> [1165] -0.7645686 -0.7642667 -0.7639648 -0.7636630 -0.7633613 -0.7630597
#> [1171] -0.7627582 -0.7624568 -0.7621555 -0.7618542 -0.7615530 -0.7612520
#> [1177] -0.7609510 -0.7606501 -0.7603493 -0.7600486 -0.7597479 -0.7594474
#> [1183] -0.7591469 -0.7588466 -0.7585463 -0.7582461 -0.7579460 -0.7576460
#> [1189] -0.7573461 -0.7570463 -0.7567466 -0.7564469 -0.7561474 -0.7558479
#> [1195] -0.7555486 -0.7552493 -0.7549501 -0.7546510 -0.7543520 -0.7540531
#> [1201] -0.7537543 -0.7534555 -0.7531569 -0.7528584 -0.7525599 -0.7522616
#> [1207] -0.7519633 -0.7516651 -0.7513671 -0.7510691 -0.7507712 -0.7504734
#> [1213] -0.7501757 -0.7498781 -0.7495806 -0.7492832 -0.7489858 -0.7486886
#> [1219] -0.7483915 -0.7480944 -0.7477975 -0.7475006 -0.7472039 -0.7469072
#> [1225] -0.7466106 -0.7463142 -0.7460178 -0.7457215 -0.7454253 -0.7451292
#> [1231] -0.7448332 -0.7445373 -0.7442415 -0.7439458 -0.7436502 -0.7433547
#> [1237] -0.7430593 -0.7427640 -0.7424687 -0.7421736 -0.7418786 -0.7415837
#> [1243] -0.7412888 -0.7409941 -0.7406994 -0.7404049 -0.7401105 -0.7398161
#> [1249] -0.7395219 -0.7392277 -0.7389337 -0.7386397 -0.7383459 -0.7380521
#> [1255] -0.7377584 -0.7374649 -0.7371714 -0.7368781 -0.7365848 -0.7362916
#> [1261] -0.7359986 -0.7357056 -0.7354127 -0.7351200 -0.7348273 -0.7345348
#> [1267] -0.7342423 -0.7339499 -0.7336577 -0.7333655 -0.7330734 -0.7327815
#> [1273] -0.7324896 -0.7321978 -0.7319062 -0.7316146 -0.7313231 -0.7310318
#> [1279] -0.7307405 -0.7304493 -0.7301583 -0.7298673 -0.7295765 -0.7292857
#> [1285] -0.7289951 -0.7287045 -0.7284141 -0.7281237 -0.7278335 -0.7275433
#> [1291] -0.7272533 -0.7269633 -0.7266735 -0.7263837 -0.7260941 -0.7258046
#> [1297] -0.7255151 -0.7252258 -0.7249366 -0.7246475 -0.7243584 -0.7240695
#> [1303] -0.7237807 -0.7234920 -0.7232034 -0.7229149 -0.7226265 -0.7223382
#> [1309] -0.7220500 -0.7217619 -0.7214739 -0.7211860 -0.7208982 -0.7206106
#> [1315] -0.7203230 -0.7200355 -0.7197481 -0.7194609 -0.7191737 -0.7188867
#> [1321] -0.7185997 -0.7183129 -0.7180261 -0.7177395 -0.7174530 -0.7171665
#> [1327] -0.7168802 -0.7165940 -0.7163079 -0.7160219 -0.7157359 -0.7154501
#> [1333] -0.7151644 -0.7148789 -0.7145934 -0.7143080 -0.7140227 -0.7137375
#> [1339] -0.7134525 -0.7131675 -0.7128827 -0.7125979 -0.7123133 -0.7120287
#> [1345] -0.7117443 -0.7114600 -0.7111757 -0.7108916 -0.7106076 -0.7103237
#> [1351] -0.7100399 -0.7097562 -0.7094726 -0.7091892 -0.7089058 -0.7086225
#> [1357] -0.7083394 -0.7080563 -0.7077733 -0.7074905 -0.7072078 -0.7069251
#> [1363] -0.7066426 -0.7063602 -0.7060779 -0.7057957 -0.7055136 -0.7052316
#> [1369] -0.7049497 -0.7046679 -0.7043863 -0.7041047 -0.7038233 -0.7035419
#> [1375] -0.7032607 -0.7029795 -0.7026985 -0.7024176 -0.7021368 -0.7018561
#> [1381] -0.7015755 -0.7012950 -0.7010146 -0.7007343 -0.7004541 -0.7001741
#> [1387] -0.6998941 -0.6996143 -0.6993346 -0.6990549 -0.6987754 -0.6984960
#> [1393] -0.6982167 -0.6979375 -0.6976584 -0.6973794 -0.6971005 -0.6968218
#> [1399] -0.6965431 -0.6962645 -0.6959861 -0.6957078 -0.6954295 -0.6951514
#> [1405] -0.6948734 -0.6945955 -0.6943177 -0.6940400 -0.6937624 -0.6934850
#> [1411] -0.6932076 -0.6929304 -0.6926532 -0.6923762 -0.6920993 -0.6918224
#> [1417] -0.6915457 -0.6912691 -0.6909926 -0.6907162 -0.6904400 -0.6901638
#> [1423] -0.6898877 -0.6896118 -0.6893360 -0.6890602 -0.6887846 -0.6885091
#> [1429] -0.6882337 -0.6879584 -0.6876832 -0.6874081 -0.6871331 -0.6868583
#> [1435] -0.6865835 -0.6863089 -0.6860343 -0.6857599 -0.6854856 -0.6852114
#> [1441] -0.6849373 -0.6846633 -0.6843894 -0.6841157 -0.6838420 -0.6835684
#> [1447] -0.6832950 -0.6830217 -0.6827484 -0.6824753 -0.6822023 -0.6819294
#> [1453] -0.6816566 -0.6813840 -0.6811114 -0.6808389 -0.6805666 -0.6802944
#> [1459] -0.6800222 -0.6797502 -0.6794783 -0.6792065 -0.6789348 -0.6786632
#> [1465] -0.6783917 -0.6781204 -0.6778491 -0.6775780 -0.6773069 -0.6770360
#> [1471] -0.6767652 -0.6764945 -0.6762239 -0.6759534 -0.6756830 -0.6754128
#> [1477] -0.6751426 -0.6748725 -0.6746026 -0.6743328 -0.6740631 -0.6737935
#> [1483] -0.6735240 -0.6732546 -0.6729853 -0.6727161 -0.6724470 -0.6721781
#> [1489] -0.6719093 -0.6716405 -0.6713719 -0.6711034 -0.6708350 -0.6705667
#> [1495] -0.6702985 -0.6700304 -0.6697625 -0.6694946 -0.6692269 -0.6689592
#> [1501] -0.6686917 -0.6684243 -0.6681570 -0.6678898 -0.6676227 -0.6673557
#> [1507] -0.6670889 -0.6668221 -0.6665555 -0.6662889 -0.6660225 -0.6657562
#> [1513] -0.6654900 -0.6652239 -0.6649579 -0.6646920 -0.6644263 -0.6641606
#> [1519] -0.6638951 -0.6636296 -0.6633643 -0.6630991 -0.6628340 -0.6625690
#> [1525] -0.6623041 -0.6620393 -0.6617747 -0.6615101 -0.6612456 -0.6609813
#> [1531] -0.6607171 -0.6604530 -0.6601890 -0.6599251 -0.6596613 -0.6593976
#> [1537] -0.6591340 -0.6588706 -0.6586072 -0.6583440 -0.6580808 -0.6578178
#> [1543] -0.6575549 -0.6572921 -0.6570294 -0.6567668 -0.6565044 -0.6562420
#> [1549] -0.6559798 -0.6557176 -0.6554556 -0.6551937 -0.6549319 -0.6546702
#> [1555] -0.6544086 -0.6541471 -0.6538857 -0.6536245 -0.6533633 -0.6531023
#> [1561] -0.6528413 -0.6525805 -0.6523198 -0.6520592 -0.6517987 -0.6515383
#> [1567] -0.6512780 -0.6510179 -0.6507578 -0.6504979 -0.6502380 -0.6499783
#> [1573] -0.6497187 -0.6494592 -0.6491998 -0.6489405 -0.6486813 -0.6484222
#> [1579] -0.6481633 -0.6479044 -0.6476457 -0.6473870 -0.6471285 -0.6468701
#> [1585] -0.6466118 -0.6463536 -0.6460955 -0.6458376 -0.6455797 -0.6453219
#> [1591] -0.6450643 -0.6448067 -0.6445493 -0.6442920 -0.6440348 -0.6437777
#> [1597] -0.6435207 -0.6432638 -0.6430071 -0.6427504 -0.6424938 -0.6422374
#> [1603] -0.6419811 -0.6417248 -0.6414687 -0.6412127 -0.6409568 -0.6407010
#> [1609] -0.6404453 -0.6401898 -0.6399343 -0.6396790 -0.6394237 -0.6391686
#> [1615] -0.6389136 -0.6386586 -0.6384038 -0.6381491 -0.6378945 -0.6376401
#> [1621] -0.6373857 -0.6371314 -0.6368773 -0.6366232 -0.6363693 -0.6361155
#> [1627] -0.6358618 -0.6356081 -0.6353546 -0.6351013 -0.6348480 -0.6345948
#> [1633] -0.6343417 -0.6340888 -0.6338359 -0.6335832 -0.6333306 -0.6330780
#> [1639] -0.6328256 -0.6325733 -0.6323211 -0.6320690 -0.6318171 -0.6315652
#> [1645] -0.6313134 -0.6310618 -0.6308102 -0.6305588 -0.6303075 -0.6300562
#> [1651] -0.6298051 -0.6295541 -0.6293032 -0.6290524 -0.6288018 -0.6285512
#> [1657] -0.6283007 -0.6280504 -0.6278001 -0.6275500 -0.6273000 -0.6270500
#> [1663] -0.6268002 -0.6265505 -0.6263009 -0.6260514 -0.6258021 -0.6255528
#> [1669] -0.6253036 -0.6250546 -0.6248056 -0.6245568 -0.6243080 -0.6240594
#> [1675] -0.6238109 -0.6235625 -0.6233142 -0.6230660 -0.6228179 -0.6225699
#> [1681] -0.6223221 -0.6220743 -0.6218266 -0.6215791 -0.6213317 -0.6210843
#> [1687] -0.6208371 -0.6205900 -0.6203430 -0.6200961 -0.6198493 -0.6196026
#> [1693] -0.6193560 -0.6191095 -0.6188632 -0.6186169 -0.6183708 -0.6181247
#> [1699] -0.6178788 -0.6176329 -0.6173872 -0.6171416 -0.6168961 -0.6166507
#> [1705] -0.6164054 -0.6161602 -0.6159151 -0.6156702 -0.6154253 -0.6151805
#> [1711] -0.6149359 -0.6146913 -0.6144469 -0.6142026 -0.6139583 -0.6137142
#> [1717] -0.6134702 -0.6132263 -0.6129825 -0.6127388 -0.6124952 -0.6122517
#> [1723] -0.6120084 -0.6117651 -0.6115219 -0.6112789 -0.6110359 -0.6107931
#> [1729] -0.6105504 -0.6103077 -0.6100652 -0.6098228 -0.6095805 -0.6093383
#> [1735] -0.6090962 -0.6088542 -0.6086123 -0.6083706 -0.6081289 -0.6078873
#> [1741] -0.6076459 -0.6074045 -0.6071633 -0.6069221 -0.6066811 -0.6064401
#> [1747] -0.6061993 -0.6059586 -0.6057180 -0.6054775 -0.6052371 -0.6049968
#> [1753] -0.6047566 -0.6045165 -0.6042765 -0.6040367 -0.6037969 -0.6035572
#> [1759] -0.6033177 -0.6030782 -0.6028389 -0.6025996 -0.6023605 -0.6021215
#> [1765] -0.6018826 -0.6016437 -0.6014050 -0.6011664 -0.6009279 -0.6006895
#> [1771] -0.6004512 -0.6002130 -0.5999749 -0.5997370 -0.5994991 -0.5992613
#> [1777] -0.5990237 -0.5987861 -0.5985486 -0.5983113 -0.5980741 -0.5978369
#> [1783] -0.5975999 -0.5973629 -0.5971261 -0.5968894 -0.5966528 -0.5964163
#> [1789] -0.5961799 -0.5959436 -0.5957074 -0.5954713 -0.5952353 -0.5949994
#> [1795] -0.5947636 -0.5945279 -0.5942924 -0.5940569 -0.5938215 -0.5935863
#> [1801] -0.5933511 -0.5931161 -0.5928811 -0.5926463 -0.5924115 -0.5921769
#> [1807] -0.5919423 -0.5917079 -0.5914736 -0.5912394 -0.5910053 -0.5907712
#> [1813] -0.5905373 -0.5903035 -0.5900698 -0.5898362 -0.5896027 -0.5893693
#> [1819] -0.5891360 -0.5889029 -0.5886698 -0.5884368 -0.5882039 -0.5879711
#> [1825] -0.5877385 -0.5875059 -0.5872734 -0.5870411 -0.5868088 -0.5865767
#> [1831] -0.5863446 -0.5861127 -0.5858808 -0.5856491 -0.5854174 -0.5851859
#> [1837] -0.5849545 -0.5847231 -0.5844919 -0.5842608 -0.5840298 -0.5837989
#> [1843] -0.5835680 -0.5833373 -0.5831067 -0.5828762 -0.5826458 -0.5824155
#> [1849] -0.5821853 -0.5819552 -0.5817252 -0.5814953 -0.5812655 -0.5810358
#> [1855] -0.5808062 -0.5805767 -0.5803473 -0.5801181 -0.5798889 -0.5796598
#> [1861] -0.5794308 -0.5792019 -0.5789732 -0.5787445 -0.5785159 -0.5782875
#> [1867] -0.5780591 -0.5778308 -0.5776027 -0.5773746 -0.5771466 -0.5769188
#> [1873] -0.5766910 -0.5764634 -0.5762358 -0.5760084 -0.5757810 -0.5755538
#> [1879] -0.5753266 -0.5750996 -0.5748726 -0.5746458 -0.5744190 -0.5741924
#> [1885] -0.5739658 -0.5737394 -0.5735131 -0.5732868 -0.5730607 -0.5728346
#> [1891] -0.5726087 -0.5723829 -0.5721571 -0.5719315 -0.5717060 -0.5714805
#> [1897] -0.5712552 -0.5710300 -0.5708048 -0.5705798 -0.5703549 -0.5701300
#> [1903] -0.5699053 -0.5696807 -0.5694561 -0.5692317 -0.5690074 -0.5687831
#> [1909] -0.5685590 -0.5683350 -0.5681111 -0.5678872 -0.5676635 -0.5674399
#> [1915] -0.5672163 -0.5669929 -0.5667696 -0.5665463 -0.5663232 -0.5661002
#> [1921] -0.5658772 -0.5656544 -0.5654317 -0.5652090 -0.5649865 -0.5647641
#> [1927] -0.5645417 -0.5643195 -0.5640974 -0.5638753 -0.5636534 -0.5634316
#> [1933] -0.5632098 -0.5629882 -0.5627666 -0.5625452 -0.5623238 -0.5621026
#> [1939] -0.5618815 -0.5616604 -0.5614395 -0.5612186 -0.5609979 -0.5607772
#> [1945] -0.5605567 -0.5603362 -0.5601158 -0.5598956 -0.5596754 -0.5594554
#> [1951] -0.5592354 -0.5590155 -0.5587958 -0.5585761 -0.5583565 -0.5581371
#> [1957] -0.5579177 -0.5576984 -0.5574792 -0.5572602 -0.5570412 -0.5568223
#> [1963] -0.5566035 -0.5563848 -0.5561662 -0.5559478 -0.5557294 -0.5555111
#> [1969] -0.5552929 -0.5550748 -0.5548568 -0.5546389 -0.5544210 -0.5542033
#> [1975] -0.5539857 -0.5537682 -0.5535508 -0.5533335 -0.5531163 -0.5528991
#> [1981] -0.5526821 -0.5524652 -0.5522483 -0.5520316 -0.5518150 -0.5515984
#> [1987] -0.5513820 -0.5511656 -0.5509494 -0.5507332 -0.5505171 -0.5503012
#> [1993] -0.5500853 -0.5498695 -0.5496539 -0.5494383 -0.5492228 -0.5490074
#> [1999] -0.5487922 -0.5485770 -0.5483619
bsmdelta(c("call", "put"), c(50, 49, 48, 47), 45, 0.25, 0.13, 0.01, 0.2)
#> [1] 0.9174852 -0.1144770
bsmdelta("call", 50, 45, 0.25, 0.13, 0.01, 1:1000 / 1000)
#> [1] 0.9975031
bsmdelta("put", 20, 30, 0:2000 / 1000, 0.15, 0, 0.25)
#> [1] -1
bsmvega(c("call", "put"), c(50, 49, 48, 47), 45, 0.25, 0.13, 0.01, 0.2)
#> [1] 3.715018 4.736695 5.819530 6.872763
bsmvega("call", 50, 45, 0.25, 0.13, 0.01, 1:1000 / 1000)
#> [1] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> [6] 0.000000e+00 0.000000e+00 2.001097e-248 3.094994e-196 6.628640e-159
#> [11] 2.766740e-131 2.816159e-110 6.288958e-94 5.901296e-81 1.723425e-70
#> [16] 6.332747e-62 7.947308e-55 7.062187e-49 7.644338e-44 1.519251e-39
#> [21] 7.596656e-36 1.221137e-32 7.653768e-30 2.181149e-27 3.197801e-25
#> [26] 2.666333e-23 1.372475e-21 4.666650e-20 1.108722e-18 1.929146e-17
#> [31] 2.557419e-16 2.670935e-15 2.261059e-14 1.589713e-13 9.479102e-13
#> [36] 4.880864e-12 2.204458e-11 8.853092e-11 3.199194e-10 1.051152e-09
#> [41] 3.169212e-09 8.839171e-09 2.296973e-08 5.596865e-08 1.285983e-07
#> [46] 2.800401e-07 5.805799e-07 1.150589e-06 2.187649e-06 4.003681e-06
#> [51] 7.073801e-06 1.209825e-05 2.007819e-05 3.240540e-05 5.096524e-05
#> [56] 7.825098e-05 1.174873e-04 1.727592e-04 2.491435e-04 3.528368e-04
#> [61] 4.912773e-04 6.732556e-04 9.090089e-04 1.210297e-03 1.590458e-03
#> [66] 2.064434e-03 2.648776e-03 3.361625e-03 4.222655e-03 5.253005e-03
#> [71] 6.475177e-03 7.912917e-03 9.591072e-03 1.153544e-02 1.377260e-02
#> [76] 1.632972e-02 1.923439e-02 2.251444e-02 2.619771e-02 3.031191e-02
#> [81] 3.488441e-02 3.994208e-02 4.551110e-02 5.161685e-02 5.828372e-02
#> [86] 6.553499e-02 7.339276e-02 8.187778e-02 9.100940e-02 1.008055e-01
#> [91] 1.112824e-01 1.224548e-01 1.343359e-01 1.469371e-01 1.602683e-01
#> [96] 1.743377e-01 1.891517e-01 2.047154e-01 2.210321e-01 2.381036e-01
#> [101] 2.559302e-01 2.745107e-01 2.938425e-01 3.139218e-01 3.347433e-01
#> [106] 3.563006e-01 3.785862e-01 4.015913e-01 4.253064e-01 4.497208e-01
#> [111] 4.748229e-01 5.006004e-01 5.270403e-01 5.541288e-01 5.818515e-01
#> [116] 6.101934e-01 6.391391e-01 6.686727e-01 6.987780e-01 7.294384e-01
#> [121] 7.606370e-01 7.923566e-01 8.245801e-01 8.572900e-01 8.904687e-01
#> [126] 9.240988e-01 9.581627e-01 9.926427e-01 1.027521e+00 1.062781e+00
#> [131] 1.098405e+00 1.134376e+00 1.170676e+00 1.207289e+00 1.244198e+00
#> [136] 1.281387e+00 1.318840e+00 1.356540e+00 1.394471e+00 1.432620e+00
#> [141] 1.470970e+00 1.509506e+00 1.548214e+00 1.587080e+00 1.626089e+00
#> [146] 1.665229e+00 1.704486e+00 1.743846e+00 1.783298e+00 1.822830e+00
#> [151] 1.862428e+00 1.902081e+00 1.941779e+00 1.981510e+00 2.021263e+00
#> [156] 2.061029e+00 2.100796e+00 2.140556e+00 2.180299e+00 2.220015e+00
#> [161] 2.259697e+00 2.299335e+00 2.338922e+00 2.378449e+00 2.417909e+00
#> [166] 2.457294e+00 2.496597e+00 2.535813e+00 2.574933e+00 2.613951e+00
#> [171] 2.652862e+00 2.691661e+00 2.730340e+00 2.768895e+00 2.807321e+00
#> [176] 2.845613e+00 2.883766e+00 2.921775e+00 2.959637e+00 2.997348e+00
#> [181] 3.034903e+00 3.072299e+00 3.109533e+00 3.146600e+00 3.183499e+00
#> [186] 3.220225e+00 3.256776e+00 3.293150e+00 3.329344e+00 3.365356e+00
#> [191] 3.401183e+00 3.436823e+00 3.472275e+00 3.507536e+00 3.542605e+00
#> [196] 3.577480e+00 3.612161e+00 3.646645e+00 3.680931e+00 3.715018e+00
#> [201] 3.748906e+00 3.782593e+00 3.816078e+00 3.849361e+00 3.882442e+00
#> [206] 3.915319e+00 3.947991e+00 3.980460e+00 4.012724e+00 4.044783e+00
#> [211] 4.076637e+00 4.108286e+00 4.139730e+00 4.170968e+00 4.202002e+00
#> [216] 4.232830e+00 4.263454e+00 4.293873e+00 4.324088e+00 4.354099e+00
#> [221] 4.383907e+00 4.413511e+00 4.442913e+00 4.472113e+00 4.501112e+00
#> [226] 4.529909e+00 4.558506e+00 4.586904e+00 4.615103e+00 4.643103e+00
#> [231] 4.670906e+00 4.698513e+00 4.725924e+00 4.753139e+00 4.780161e+00
#> [236] 4.806989e+00 4.833625e+00 4.860070e+00 4.886324e+00 4.912388e+00
#> [241] 4.938264e+00 4.963952e+00 4.989454e+00 5.014770e+00 5.039902e+00
#> [246] 5.064850e+00 5.089616e+00 5.114201e+00 5.138605e+00 5.162830e+00
#> [251] 5.186877e+00 5.210747e+00 5.234441e+00 5.257961e+00 5.281306e+00
#> [256] 5.304479e+00 5.327481e+00 5.350313e+00 5.372975e+00 5.395470e+00
#> [261] 5.417797e+00 5.439959e+00 5.461956e+00 5.483790e+00 5.505462e+00
#> [266] 5.526972e+00 5.548323e+00 5.569515e+00 5.590548e+00 5.611426e+00
#> [271] 5.632148e+00 5.652716e+00 5.673130e+00 5.693393e+00 5.713505e+00
#> [276] 5.733466e+00 5.753280e+00 5.772946e+00 5.792465e+00 5.811839e+00
#> [281] 5.831069e+00 5.850156e+00 5.869101e+00 5.887906e+00 5.906570e+00
#> [286] 5.925096e+00 5.943484e+00 5.961736e+00 5.979852e+00 5.997834e+00
#> [291] 6.015682e+00 6.033399e+00 6.050984e+00 6.068439e+00 6.085764e+00
#> [296] 6.102962e+00 6.120032e+00 6.136977e+00 6.153796e+00 6.170492e+00
#> [301] 6.187064e+00 6.203515e+00 6.219844e+00 6.236053e+00 6.252143e+00
#> [306] 6.268115e+00 6.283970e+00 6.299708e+00 6.315332e+00 6.330841e+00
#> [311] 6.346236e+00 6.361519e+00 6.376691e+00 6.391751e+00 6.406702e+00
#> [316] 6.421545e+00 6.436279e+00 6.450906e+00 6.465427e+00 6.479843e+00
#> [321] 6.494155e+00 6.508362e+00 6.522467e+00 6.536470e+00 6.550373e+00
#> [326] 6.564174e+00 6.577877e+00 6.591481e+00 6.604987e+00 6.618396e+00
#> [331] 6.631709e+00 6.644927e+00 6.658050e+00 6.671080e+00 6.684016e+00
#> [336] 6.696860e+00 6.709613e+00 6.722275e+00 6.734847e+00 6.747329e+00
#> [341] 6.759724e+00 6.772030e+00 6.784249e+00 6.796382e+00 6.808430e+00
#> [346] 6.820392e+00 6.832270e+00 6.844065e+00 6.855777e+00 6.867407e+00
#> [351] 6.878955e+00 6.890422e+00 6.901809e+00 6.913117e+00 6.924346e+00
#> [356] 6.935497e+00 6.946570e+00 6.957566e+00 6.968486e+00 6.979330e+00
#> [361] 6.990099e+00 7.000793e+00 7.011414e+00 7.021962e+00 7.032436e+00
#> [366] 7.042839e+00 7.053171e+00 7.063431e+00 7.073621e+00 7.083741e+00
#> [371] 7.093793e+00 7.103775e+00 7.113690e+00 7.123537e+00 7.133317e+00
#> [376] 7.143030e+00 7.152678e+00 7.162260e+00 7.171777e+00 7.181230e+00
#> [381] 7.190620e+00 7.199946e+00 7.209209e+00 7.218410e+00 7.227549e+00
#> [386] 7.236626e+00 7.245643e+00 7.254600e+00 7.263497e+00 7.272334e+00
#> [391] 7.281113e+00 7.289833e+00 7.298495e+00 7.307099e+00 7.315647e+00
#> [396] 7.324138e+00 7.332572e+00 7.340951e+00 7.349275e+00 7.357544e+00
#> [401] 7.365758e+00 7.373919e+00 7.382026e+00 7.390079e+00 7.398080e+00
#> [406] 7.406029e+00 7.413926e+00 7.421771e+00 7.429565e+00 7.437309e+00
#> [411] 7.445002e+00 7.452645e+00 7.460238e+00 7.467783e+00 7.475278e+00
#> [416] 7.482726e+00 7.490125e+00 7.497476e+00 7.504781e+00 7.512038e+00
#> [421] 7.519249e+00 7.526413e+00 7.533532e+00 7.540605e+00 7.547633e+00
#> [426] 7.554616e+00 7.561554e+00 7.568449e+00 7.575299e+00 7.582106e+00
#> [431] 7.588870e+00 7.595591e+00 7.602270e+00 7.608906e+00 7.615501e+00
#> [436] 7.622054e+00 7.628565e+00 7.635036e+00 7.641466e+00 7.647856e+00
#> [441] 7.654206e+00 7.660516e+00 7.666786e+00 7.673017e+00 7.679210e+00
#> [446] 7.685364e+00 7.691480e+00 7.697557e+00 7.703597e+00 7.709600e+00
#> [451] 7.715565e+00 7.721493e+00 7.727385e+00 7.733240e+00 7.739060e+00
#> [456] 7.744843e+00 7.750591e+00 7.756304e+00 7.761981e+00 7.767624e+00
#> [461] 7.773232e+00 7.778806e+00 7.784346e+00 7.789851e+00 7.795324e+00
#> [466] 7.800763e+00 7.806169e+00 7.811542e+00 7.816882e+00 7.822191e+00
#> [471] 7.827467e+00 7.832711e+00 7.837923e+00 7.843104e+00 7.848253e+00
#> [476] 7.853372e+00 7.858460e+00 7.863517e+00 7.868544e+00 7.873540e+00
#> [481] 7.878507e+00 7.883444e+00 7.888351e+00 7.893230e+00 7.898078e+00
#> [486] 7.902899e+00 7.907690e+00 7.912453e+00 7.917187e+00 7.921894e+00
#> [491] 7.926572e+00 7.931223e+00 7.935846e+00 7.940442e+00 7.945010e+00
#> [496] 7.949552e+00 7.954067e+00 7.958555e+00 7.963017e+00 7.967452e+00
#> [501] 7.971862e+00 7.976245e+00 7.980603e+00 7.984936e+00 7.989242e+00
#> [506] 7.993524e+00 7.997781e+00 8.002013e+00 8.006220e+00 8.010403e+00
#> [511] 8.014561e+00 8.018695e+00 8.022805e+00 8.026891e+00 8.030954e+00
#> [516] 8.034992e+00 8.039008e+00 8.043000e+00 8.046969e+00 8.050916e+00
#> [521] 8.054839e+00 8.058740e+00 8.062618e+00 8.066474e+00 8.070308e+00
#> [526] 8.074119e+00 8.077909e+00 8.081677e+00 8.085423e+00 8.089148e+00
#> [531] 8.092852e+00 8.096534e+00 8.100196e+00 8.103836e+00 8.107455e+00
#> [536] 8.111054e+00 8.114633e+00 8.118191e+00 8.121729e+00 8.125246e+00
#> [541] 8.128744e+00 8.132221e+00 8.135679e+00 8.139118e+00 8.142537e+00
#> [546] 8.145936e+00 8.149316e+00 8.152677e+00 8.156020e+00 8.159343e+00
#> [551] 8.162647e+00 8.165933e+00 8.169200e+00 8.172449e+00 8.175680e+00
#> [556] 8.178892e+00 8.182086e+00 8.185262e+00 8.188421e+00 8.191562e+00
#> [561] 8.194685e+00 8.197790e+00 8.200878e+00 8.203949e+00 8.207003e+00
#> [566] 8.210039e+00 8.213059e+00 8.216062e+00 8.219048e+00 8.222017e+00
#> [571] 8.224970e+00 8.227906e+00 8.230826e+00 8.233729e+00 8.236617e+00
#> [576] 8.239488e+00 8.242343e+00 8.245183e+00 8.248006e+00 8.250814e+00
#> [581] 8.253607e+00 8.256384e+00 8.259145e+00 8.261891e+00 8.264622e+00
#> [586] 8.267338e+00 8.270039e+00 8.272724e+00 8.275395e+00 8.278051e+00
#> [591] 8.280693e+00 8.283320e+00 8.285932e+00 8.288530e+00 8.291114e+00
#> [596] 8.293683e+00 8.296238e+00 8.298779e+00 8.301306e+00 8.303819e+00
#> [601] 8.306318e+00 8.308804e+00 8.311276e+00 8.313734e+00 8.316178e+00
#> [606] 8.318609e+00 8.321027e+00 8.323432e+00 8.325823e+00 8.328201e+00
#> [611] 8.330566e+00 8.332918e+00 8.335257e+00 8.337584e+00 8.339897e+00
#> [616] 8.342198e+00 8.344486e+00 8.346762e+00 8.349025e+00 8.351276e+00
#> [621] 8.353514e+00 8.355740e+00 8.357954e+00 8.360156e+00 8.362346e+00
#> [626] 8.364523e+00 8.366689e+00 8.368843e+00 8.370985e+00 8.373115e+00
#> [631] 8.375234e+00 8.377341e+00 8.379437e+00 8.381521e+00 8.383593e+00
#> [636] 8.385655e+00 8.387705e+00 8.389743e+00 8.391771e+00 8.393788e+00
#> [641] 8.395793e+00 8.397787e+00 8.399771e+00 8.401744e+00 8.403706e+00
#> [646] 8.405657e+00 8.407597e+00 8.409527e+00 8.411446e+00 8.413355e+00
#> [651] 8.415253e+00 8.417141e+00 8.419019e+00 8.420886e+00 8.422743e+00
#> [656] 8.424590e+00 8.426426e+00 8.428253e+00 8.430070e+00 8.431876e+00
#> [661] 8.433673e+00 8.435460e+00 8.437237e+00 8.439004e+00 8.440762e+00
#> [666] 8.442510e+00 8.444248e+00 8.445977e+00 8.447696e+00 8.449406e+00
#> [671] 8.451107e+00 8.452798e+00 8.454479e+00 8.456152e+00 8.457815e+00
#> [676] 8.459469e+00 8.461115e+00 8.462750e+00 8.464377e+00 8.465995e+00
#> [681] 8.467604e+00 8.469205e+00 8.470796e+00 8.472378e+00 8.473952e+00
#> [686] 8.475517e+00 8.477074e+00 8.478622e+00 8.480161e+00 8.481692e+00
#> [691] 8.483214e+00 8.484728e+00 8.486233e+00 8.487730e+00 8.489219e+00
#> [696] 8.490699e+00 8.492172e+00 8.493636e+00 8.495092e+00 8.496539e+00
#> [701] 8.497979e+00 8.499411e+00 8.500835e+00 8.502250e+00 8.503658e+00
#> [706] 8.505058e+00 8.506450e+00 8.507835e+00 8.509211e+00 8.510580e+00
#> [711] 8.511941e+00 8.513295e+00 8.514641e+00 8.515979e+00 8.517310e+00
#> [716] 8.518634e+00 8.519950e+00 8.521258e+00 8.522559e+00 8.523853e+00
#> [721] 8.525140e+00 8.526419e+00 8.527691e+00 8.528956e+00 8.530213e+00
#> [726] 8.531464e+00 8.532707e+00 8.533943e+00 8.535173e+00 8.536395e+00
#> [731] 8.537610e+00 8.538819e+00 8.540020e+00 8.541215e+00 8.542402e+00
#> [736] 8.543583e+00 8.544757e+00 8.545925e+00 8.547086e+00 8.548240e+00
#> [741] 8.549387e+00 8.550528e+00 8.551662e+00 8.552790e+00 8.553911e+00
#> [746] 8.555025e+00 8.556133e+00 8.557235e+00 8.558330e+00 8.559419e+00
#> [751] 8.560502e+00 8.561578e+00 8.562648e+00 8.563712e+00 8.564769e+00
#> [756] 8.565821e+00 8.566866e+00 8.567905e+00 8.568938e+00 8.569964e+00
#> [761] 8.570985e+00 8.572000e+00 8.573008e+00 8.574011e+00 8.575008e+00
#> [766] 8.575999e+00 8.576984e+00 8.577963e+00 8.578936e+00 8.579903e+00
#> [771] 8.580865e+00 8.581821e+00 8.582771e+00 8.583715e+00 8.584654e+00
#> [776] 8.585587e+00 8.586514e+00 8.587436e+00 8.588352e+00 8.589263e+00
#> [781] 8.590168e+00 8.591067e+00 8.591961e+00 8.592850e+00 8.593733e+00
#> [786] 8.594610e+00 8.595483e+00 8.596350e+00 8.597211e+00 8.598067e+00
#> [791] 8.598918e+00 8.599764e+00 8.600604e+00 8.601439e+00 8.602269e+00
#> [796] 8.603094e+00 8.603914e+00 8.604728e+00 8.605537e+00 8.606342e+00
#> [801] 8.607141e+00 8.607935e+00 8.608724e+00 8.609508e+00 8.610287e+00
#> [806] 8.611061e+00 8.611830e+00 8.612594e+00 8.613353e+00 8.614107e+00
#> [811] 8.614857e+00 8.615601e+00 8.616341e+00 8.617076e+00 8.617806e+00
#> [816] 8.618531e+00 8.619252e+00 8.619968e+00 8.620679e+00 8.621386e+00
#> [821] 8.622087e+00 8.622785e+00 8.623477e+00 8.624165e+00 8.624848e+00
#> [826] 8.625527e+00 8.626201e+00 8.626871e+00 8.627536e+00 8.628196e+00
#> [831] 8.628853e+00 8.629504e+00 8.630151e+00 8.630794e+00 8.631433e+00
#> [836] 8.632066e+00 8.632696e+00 8.633321e+00 8.633942e+00 8.634559e+00
#> [841] 8.635171e+00 8.635779e+00 8.636382e+00 8.636982e+00 8.637577e+00
#> [846] 8.638168e+00 8.638755e+00 8.639337e+00 8.639916e+00 8.640490e+00
#> [851] 8.641060e+00 8.641626e+00 8.642188e+00 8.642745e+00 8.643299e+00
#> [856] 8.643849e+00 8.644394e+00 8.644936e+00 8.645473e+00 8.646007e+00
#> [861] 8.646536e+00 8.647062e+00 8.647583e+00 8.648101e+00 8.648615e+00
#> [866] 8.649125e+00 8.649631e+00 8.650133e+00 8.650631e+00 8.651125e+00
#> [871] 8.651616e+00 8.652103e+00 8.652585e+00 8.653065e+00 8.653540e+00
#> [876] 8.654011e+00 8.654479e+00 8.654943e+00 8.655404e+00 8.655861e+00
#> [881] 8.656314e+00 8.656763e+00 8.657208e+00 8.657650e+00 8.658089e+00
#> [886] 8.658524e+00 8.658955e+00 8.659382e+00 8.659806e+00 8.660227e+00
#> [891] 8.660643e+00 8.661057e+00 8.661466e+00 8.661873e+00 8.662275e+00
#> [896] 8.662675e+00 8.663070e+00 8.663463e+00 8.663852e+00 8.664237e+00
#> [901] 8.664619e+00 8.664998e+00 8.665373e+00 8.665745e+00 8.666113e+00
#> [906] 8.666478e+00 8.666840e+00 8.667198e+00 8.667553e+00 8.667905e+00
#> [911] 8.668253e+00 8.668598e+00 8.668940e+00 8.669278e+00 8.669614e+00
#> [916] 8.669946e+00 8.670274e+00 8.670600e+00 8.670922e+00 8.671241e+00
#> [921] 8.671557e+00 8.671870e+00 8.672180e+00 8.672486e+00 8.672790e+00
#> [926] 8.673090e+00 8.673387e+00 8.673681e+00 8.673971e+00 8.674259e+00
#> [931] 8.674544e+00 8.674825e+00 8.675104e+00 8.675379e+00 8.675652e+00
#> [936] 8.675921e+00 8.676187e+00 8.676451e+00 8.676711e+00 8.676969e+00
#> [941] 8.677223e+00 8.677474e+00 8.677723e+00 8.677968e+00 8.678211e+00
#> [946] 8.678451e+00 8.678687e+00 8.678921e+00 8.679152e+00 8.679380e+00
#> [951] 8.679605e+00 8.679827e+00 8.680047e+00 8.680263e+00 8.680477e+00
#> [956] 8.680688e+00 8.680896e+00 8.681101e+00 8.681304e+00 8.681504e+00
#> [961] 8.681700e+00 8.681895e+00 8.682086e+00 8.682274e+00 8.682460e+00
#> [966] 8.682643e+00 8.682824e+00 8.683001e+00 8.683176e+00 8.683348e+00
#> [971] 8.683518e+00 8.683685e+00 8.683849e+00 8.684010e+00 8.684169e+00
#> [976] 8.684325e+00 8.684479e+00 8.684630e+00 8.684778e+00 8.684924e+00
#> [981] 8.685067e+00 8.685207e+00 8.685345e+00 8.685480e+00 8.685613e+00
#> [986] 8.685743e+00 8.685871e+00 8.685996e+00 8.686118e+00 8.686238e+00
#> [991] 8.686355e+00 8.686470e+00 8.686583e+00 8.686692e+00 8.686800e+00
#> [996] 8.686905e+00 8.687007e+00 8.687107e+00 8.687204e+00 8.687299e+00
bsmvega("put", 20, 30, 0:2000 / 1000, 0.15, 0, 0.25)
#> [1] 0.000000e+00 0.000000e+00 2.933904e-286 5.673607e-191 2.602887e-143
#> [6] 1.055215e-114 1.266233e-95 5.440716e-82 9.212451e-72 8.376144e-64
#> [11] 1.960533e-57 3.203324e-52 7.105338e-48 3.391031e-44 4.838619e-41
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#> [36] 2.295642e-16 6.610485e-16 1.798419e-15 4.643228e-15 1.142261e-14
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#> [1781] 1.032180e+01 1.032627e+01 1.033073e+01 1.033519e+01 1.033965e+01
#> [1786] 1.034410e+01 1.034854e+01 1.035299e+01 1.035742e+01 1.036186e+01
#> [1791] 1.036628e+01 1.037071e+01 1.037513e+01 1.037954e+01 1.038395e+01
#> [1796] 1.038836e+01 1.039276e+01 1.039716e+01 1.040155e+01 1.040594e+01
#> [1801] 1.041032e+01 1.041470e+01 1.041908e+01 1.042345e+01 1.042782e+01
#> [1806] 1.043218e+01 1.043653e+01 1.044089e+01 1.044524e+01 1.044958e+01
#> [1811] 1.045392e+01 1.045825e+01 1.046258e+01 1.046691e+01 1.047123e+01
#> [1816] 1.047555e+01 1.047986e+01 1.048417e+01 1.048847e+01 1.049277e+01
#> [1821] 1.049707e+01 1.050136e+01 1.050565e+01 1.050993e+01 1.051421e+01
#> [1826] 1.051848e+01 1.052275e+01 1.052701e+01 1.053127e+01 1.053553e+01
#> [1831] 1.053978e+01 1.054403e+01 1.054827e+01 1.055251e+01 1.055674e+01
#> [1836] 1.056097e+01 1.056519e+01 1.056942e+01 1.057363e+01 1.057784e+01
#> [1841] 1.058205e+01 1.058625e+01 1.059045e+01 1.059465e+01 1.059884e+01
#> [1846] 1.060302e+01 1.060720e+01 1.061138e+01 1.061555e+01 1.061972e+01
#> [1851] 1.062389e+01 1.062805e+01 1.063220e+01 1.063635e+01 1.064050e+01
#> [1856] 1.064464e+01 1.064878e+01 1.065291e+01 1.065704e+01 1.066117e+01
#> [1861] 1.066529e+01 1.066940e+01 1.067352e+01 1.067762e+01 1.068173e+01
#> [1866] 1.068583e+01 1.068992e+01 1.069401e+01 1.069810e+01 1.070218e+01
#> [1871] 1.070626e+01 1.071033e+01 1.071440e+01 1.071847e+01 1.072253e+01
#> [1876] 1.072658e+01 1.073064e+01 1.073468e+01 1.073873e+01 1.074277e+01
#> [1881] 1.074680e+01 1.075083e+01 1.075486e+01 1.075888e+01 1.076290e+01
#> [1886] 1.076691e+01 1.077092e+01 1.077493e+01 1.077893e+01 1.078292e+01
#> [1891] 1.078692e+01 1.079091e+01 1.079489e+01 1.079887e+01 1.080285e+01
#> [1896] 1.080682e+01 1.081078e+01 1.081475e+01 1.081871e+01 1.082266e+01
#> [1901] 1.082661e+01 1.083056e+01 1.083450e+01 1.083844e+01 1.084237e+01
#> [1906] 1.084630e+01 1.085022e+01 1.085414e+01 1.085806e+01 1.086197e+01
#> [1911] 1.086588e+01 1.086979e+01 1.087369e+01 1.087758e+01 1.088147e+01
#> [1916] 1.088536e+01 1.088924e+01 1.089312e+01 1.089700e+01 1.090087e+01
#> [1921] 1.090473e+01 1.090860e+01 1.091245e+01 1.091631e+01 1.092016e+01
#> [1926] 1.092400e+01 1.092785e+01 1.093168e+01 1.093552e+01 1.093935e+01
#> [1931] 1.094317e+01 1.094699e+01 1.095081e+01 1.095462e+01 1.095843e+01
#> [1936] 1.096223e+01 1.096603e+01 1.096983e+01 1.097362e+01 1.097741e+01
#> [1941] 1.098119e+01 1.098497e+01 1.098875e+01 1.099252e+01 1.099629e+01
#> [1946] 1.100005e+01 1.100381e+01 1.100756e+01 1.101131e+01 1.101506e+01
#> [1951] 1.101880e+01 1.102254e+01 1.102628e+01 1.103001e+01 1.103373e+01
#> [1956] 1.103746e+01 1.104117e+01 1.104489e+01 1.104860e+01 1.105231e+01
#> [1961] 1.105601e+01 1.105970e+01 1.106340e+01 1.106709e+01 1.107077e+01
#> [1966] 1.107446e+01 1.107813e+01 1.108181e+01 1.108548e+01 1.108914e+01
#> [1971] 1.109280e+01 1.109646e+01 1.110011e+01 1.110376e+01 1.110741e+01
#> [1976] 1.111105e+01 1.111469e+01 1.111832e+01 1.112195e+01 1.112557e+01
#> [1981] 1.112920e+01 1.113281e+01 1.113643e+01 1.114004e+01 1.114364e+01
#> [1986] 1.114724e+01 1.115084e+01 1.115443e+01 1.115802e+01 1.116161e+01
#> [1991] 1.116519e+01 1.116876e+01 1.117234e+01 1.117591e+01 1.117947e+01
#> [1996] 1.118303e+01 1.118659e+01 1.119014e+01 1.119369e+01 1.119724e+01
#> [2001] 1.120078e+01
bsmgamma(c("call", "put"), c(50, 49, 48, 47), 45, 0.25, 0.13, 0.01, 0.2)
#> [1] 0.02972015 0.03945602 0.05051675 0.06222511
bsmgamma("call", 50, 45, 0.25, 0.13, 0.01, 1:1000 / 1000)
#> [1] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
#> [6] 0.000000e+00 0.000000e+00 4.002194e-249 5.502211e-197 1.060582e-159
#> [11] 4.024350e-132 3.754879e-111 7.740257e-95 6.744339e-82 1.838320e-71
#> [16] 6.332747e-63 7.479820e-56 6.277499e-50 6.437337e-45 1.215400e-40
#> [21] 5.787929e-37 8.880995e-34 5.324360e-31 1.454099e-28 2.046593e-26
#> [26] 1.640821e-24 8.133188e-23 2.666657e-21 6.117090e-20 1.028878e-18
#> [31] 1.319958e-17 1.335468e-16 1.096271e-15 7.481002e-15 4.333304e-14
#> [36] 2.169273e-13 9.532791e-13 3.727618e-12 1.312490e-11 4.204607e-11
#> [41] 1.236766e-10 3.367303e-10 8.546878e-10 2.035224e-09 4.572385e-09
#> [46] 9.740526e-09 1.976442e-08 3.835298e-08 7.143344e-08 1.281178e-07
#> [51] 2.219232e-07 3.722539e-07 6.061339e-07 9.601601e-07 1.482625e-06
#> [56] 2.235742e-06 3.297888e-06 4.765772e-06 6.756435e-06 9.408980e-06
#> [61] 1.288596e-05 1.737434e-05 2.308594e-05 3.025744e-05 3.914974e-05
#> [66] 5.004687e-05 6.325436e-05 7.909705e-05 9.791664e-05 1.200687e-04
#> [71] 1.459195e-04 1.758426e-04 2.102153e-04 2.494149e-04 2.938154e-04
#> [76] 3.437835e-04 3.996757e-04 4.618346e-04 5.305865e-04 6.062382e-04
#> [81] 6.890748e-04 7.793576e-04 8.773224e-04 9.831781e-04 1.097105e-03
#> [86] 1.219256e-03 1.349752e-03 1.488687e-03 1.636124e-03 1.792097e-03
#> [91] 1.956613e-03 2.129648e-03 2.311155e-03 2.501057e-03 2.699256e-03
#> [96] 2.905628e-03 3.120029e-03 3.342293e-03 3.572237e-03 3.809658e-03
#> [101] 4.054340e-03 4.306050e-03 4.564544e-03 4.829565e-03 5.100850e-03
#> [106] 5.378122e-03 5.661102e-03 5.949501e-03 6.243030e-03 6.541393e-03
#> [111] 6.844294e-03 7.151435e-03 7.462518e-03 7.777246e-03 8.095325e-03
#> [116] 8.416461e-03 8.740364e-03 9.066749e-03 9.395335e-03 9.725845e-03
#> [121] 1.005801e-02 1.039156e-02 1.072624e-02 1.106181e-02 1.139800e-02
#> [126] 1.173459e-02 1.207134e-02 1.240803e-02 1.274445e-02 1.308039e-02
#> [131] 1.341564e-02 1.375001e-02 1.408332e-02 1.441539e-02 1.474605e-02
#> [136] 1.507514e-02 1.540251e-02 1.572800e-02 1.605147e-02 1.637280e-02
#> [141] 1.669185e-02 1.700851e-02 1.732267e-02 1.763422e-02 1.794305e-02
#> [146] 1.824908e-02 1.855223e-02 1.885239e-02 1.914951e-02 1.944351e-02
#> [151] 1.973433e-02 2.002191e-02 2.030619e-02 2.058712e-02 2.086465e-02
#> [156] 2.113875e-02 2.140939e-02 2.167652e-02 2.194011e-02 2.220015e-02
#> [161] 2.245662e-02 2.270948e-02 2.295874e-02 2.320438e-02 2.344639e-02
#> [166] 2.368476e-02 2.391950e-02 2.415060e-02 2.437806e-02 2.460189e-02
#> [171] 2.482211e-02 2.503870e-02 2.525170e-02 2.546110e-02 2.566693e-02
#> [176] 2.586921e-02 2.606794e-02 2.626315e-02 2.645486e-02 2.664309e-02
#> [181] 2.682787e-02 2.700922e-02 2.718717e-02 2.736174e-02 2.753296e-02
#> [186] 2.770086e-02 2.786547e-02 2.802681e-02 2.818492e-02 2.833984e-02
#> [191] 2.849158e-02 2.864019e-02 2.878570e-02 2.892813e-02 2.906753e-02
#> [196] 2.920392e-02 2.933735e-02 2.946783e-02 2.959542e-02 2.972015e-02
#> [201] 2.984204e-02 2.996113e-02 3.007746e-02 3.019107e-02 3.030198e-02
#> [206] 3.041024e-02 3.051588e-02 3.061892e-02 3.071942e-02 3.081739e-02
#> [211] 3.091289e-02 3.100593e-02 3.109656e-02 3.118481e-02 3.127071e-02
#> [216] 3.135430e-02 3.143560e-02 3.151466e-02 3.159151e-02 3.166617e-02
#> [221] 3.173869e-02 3.180909e-02 3.187740e-02 3.194366e-02 3.200790e-02
#> [226] 3.207015e-02 3.213044e-02 3.218880e-02 3.224526e-02 3.229985e-02
#> [231] 3.235260e-02 3.240354e-02 3.245269e-02 3.250010e-02 3.254578e-02
#> [236] 3.258976e-02 3.263207e-02 3.267274e-02 3.271179e-02 3.274925e-02
#> [241] 3.278515e-02 3.281952e-02 3.285237e-02 3.288374e-02 3.291365e-02
#> [246] 3.294212e-02 3.296917e-02 3.299484e-02 3.301915e-02 3.304211e-02
#> [251] 3.306376e-02 3.308411e-02 3.310319e-02 3.312101e-02 3.313761e-02
#> [256] 3.315300e-02 3.316720e-02 3.318023e-02 3.319212e-02 3.320289e-02
#> [261] 3.321255e-02 3.322112e-02 3.322863e-02 3.323509e-02 3.324052e-02
#> [266] 3.324495e-02 3.324838e-02 3.325083e-02 3.325233e-02 3.325289e-02
#> [271] 3.325253e-02 3.325127e-02 3.324911e-02 3.324609e-02 3.324221e-02
#> [276] 3.323749e-02 3.323194e-02 3.322559e-02 3.321844e-02 3.321051e-02
#> [281] 3.320182e-02 3.319238e-02 3.318220e-02 3.317130e-02 3.315969e-02
#> [286] 3.314739e-02 3.313441e-02 3.312075e-02 3.310645e-02 3.309150e-02
#> [291] 3.307592e-02 3.305972e-02 3.304291e-02 3.302552e-02 3.300753e-02
#> [296] 3.298898e-02 3.296987e-02 3.295021e-02 3.293001e-02 3.290929e-02
#> [301] 3.288805e-02 3.286630e-02 3.284406e-02 3.282133e-02 3.279813e-02
#> [306] 3.277446e-02 3.275033e-02 3.272576e-02 3.270075e-02 3.267531e-02
#> [311] 3.264945e-02 3.262317e-02 3.259650e-02 3.256943e-02 3.254198e-02
#> [316] 3.251415e-02 3.248595e-02 3.245739e-02 3.242848e-02 3.239922e-02
#> [321] 3.236962e-02 3.233969e-02 3.230944e-02 3.227887e-02 3.224799e-02
#> [326] 3.221681e-02 3.218533e-02 3.215357e-02 3.212152e-02 3.208919e-02
#> [331] 3.205660e-02 3.202375e-02 3.199063e-02 3.195727e-02 3.192366e-02
#> [336] 3.188981e-02 3.185573e-02 3.182142e-02 3.178689e-02 3.175214e-02
#> [341] 3.171718e-02 3.168201e-02 3.164664e-02 3.161108e-02 3.157533e-02
#> [346] 3.153939e-02 3.150326e-02 3.146697e-02 3.143050e-02 3.139386e-02
#> [351] 3.135706e-02 3.132010e-02 3.128299e-02 3.124573e-02 3.120832e-02
#> [356] 3.117077e-02 3.113309e-02 3.109527e-02 3.105732e-02 3.101924e-02
#> [361] 3.098105e-02 3.094273e-02 3.090430e-02 3.086577e-02 3.082712e-02
#> [366] 3.078837e-02 3.074952e-02 3.071057e-02 3.067153e-02 3.063240e-02
#> [371] 3.059318e-02 3.055387e-02 3.051449e-02 3.047502e-02 3.043548e-02
#> [376] 3.039587e-02 3.035619e-02 3.031644e-02 3.027663e-02 3.023676e-02
#> [381] 3.019683e-02 3.015684e-02 3.011680e-02 3.007671e-02 3.003657e-02
#> [386] 2.999638e-02 2.995615e-02 2.991588e-02 2.987556e-02 2.983522e-02
#> [391] 2.979483e-02 2.975442e-02 2.971397e-02 2.967350e-02 2.963300e-02
#> [396] 2.959248e-02 2.955193e-02 2.951136e-02 2.947078e-02 2.943018e-02
#> [401] 2.938956e-02 2.934893e-02 2.930829e-02 2.926764e-02 2.922698e-02
#> [406] 2.918632e-02 2.914565e-02 2.910498e-02 2.906431e-02 2.902364e-02
#> [411] 2.898297e-02 2.894231e-02 2.890165e-02 2.886100e-02 2.882035e-02
#> [416] 2.877971e-02 2.873909e-02 2.869847e-02 2.865787e-02 2.861729e-02
#> [421] 2.857672e-02 2.853616e-02 2.849563e-02 2.845511e-02 2.841462e-02
#> [426] 2.837414e-02 2.833369e-02 2.829327e-02 2.825286e-02 2.821249e-02
#> [431] 2.817214e-02 2.813182e-02 2.809153e-02 2.805127e-02 2.801104e-02
#> [436] 2.797084e-02 2.793067e-02 2.789054e-02 2.785045e-02 2.781038e-02
#> [441] 2.777036e-02 2.773037e-02 2.769042e-02 2.765051e-02 2.761064e-02
#> [446] 2.757081e-02 2.753102e-02 2.749128e-02 2.745157e-02 2.741191e-02
#> [451] 2.737229e-02 2.733272e-02 2.729319e-02 2.725371e-02 2.721428e-02
#> [456] 2.717489e-02 2.713555e-02 2.709626e-02 2.705701e-02 2.701782e-02
#> [461] 2.697868e-02 2.693959e-02 2.690055e-02 2.686156e-02 2.682262e-02
#> [466] 2.678374e-02 2.674490e-02 2.670613e-02 2.666740e-02 2.662873e-02
#> [471] 2.659012e-02 2.655156e-02 2.651306e-02 2.647461e-02 2.643622e-02
#> [476] 2.639789e-02 2.635961e-02 2.632140e-02 2.628324e-02 2.624513e-02
#> [481] 2.620709e-02 2.616911e-02 2.613118e-02 2.609332e-02 2.605552e-02
#> [486] 2.601777e-02 2.598009e-02 2.594247e-02 2.590491e-02 2.586741e-02
#> [491] 2.582997e-02 2.579259e-02 2.575528e-02 2.571803e-02 2.568084e-02
#> [496] 2.564372e-02 2.560665e-02 2.556965e-02 2.553272e-02 2.549585e-02
#> [501] 2.545904e-02 2.542230e-02 2.538562e-02 2.534900e-02 2.531245e-02
#> [506] 2.527597e-02 2.523955e-02 2.520319e-02 2.516690e-02 2.513067e-02
#> [511] 2.509452e-02 2.505842e-02 2.502239e-02 2.498643e-02 2.495054e-02
#> [516] 2.491471e-02 2.487894e-02 2.484324e-02 2.480761e-02 2.477205e-02
#> [521] 2.473655e-02 2.470112e-02 2.466575e-02 2.463045e-02 2.459522e-02
#> [526] 2.456006e-02 2.452496e-02 2.448993e-02 2.445497e-02 2.442007e-02
#> [531] 2.438524e-02 2.435048e-02 2.431578e-02 2.428116e-02 2.424660e-02
#> [536] 2.421210e-02 2.417768e-02 2.414332e-02 2.410903e-02 2.407480e-02
#> [541] 2.404065e-02 2.400656e-02 2.397254e-02 2.393858e-02 2.390469e-02
#> [546] 2.387088e-02 2.383712e-02 2.380344e-02 2.376982e-02 2.373627e-02
#> [551] 2.370279e-02 2.366937e-02 2.363602e-02 2.360274e-02 2.356953e-02
#> [556] 2.353638e-02 2.350330e-02 2.347029e-02 2.343734e-02 2.340446e-02
#> [561] 2.337165e-02 2.333890e-02 2.330623e-02 2.327361e-02 2.324107e-02
#> [566] 2.320859e-02 2.317618e-02 2.314384e-02 2.311156e-02 2.307935e-02
#> [571] 2.304720e-02 2.301512e-02 2.298311e-02 2.295116e-02 2.291928e-02
#> [576] 2.288747e-02 2.285572e-02 2.282404e-02 2.279242e-02 2.276087e-02
#> [581] 2.272938e-02 2.269796e-02 2.266661e-02 2.263532e-02 2.260409e-02
#> [586] 2.257294e-02 2.254184e-02 2.251081e-02 2.247985e-02 2.244895e-02
#> [591] 2.241812e-02 2.238735e-02 2.235665e-02 2.232601e-02 2.229543e-02
#> [596] 2.226492e-02 2.223447e-02 2.220409e-02 2.217377e-02 2.214352e-02
#> [601] 2.211333e-02 2.208320e-02 2.205314e-02 2.202314e-02 2.199320e-02
#> [606] 2.196333e-02 2.193351e-02 2.190377e-02 2.187408e-02 2.184446e-02
#> [611] 2.181490e-02 2.178541e-02 2.175597e-02 2.172660e-02 2.169729e-02
#> [616] 2.166805e-02 2.163886e-02 2.160974e-02 2.158068e-02 2.155168e-02
#> [621] 2.152274e-02 2.149387e-02 2.146505e-02 2.143630e-02 2.140760e-02
#> [626] 2.137897e-02 2.135040e-02 2.132189e-02 2.129344e-02 2.126505e-02
#> [631] 2.123673e-02 2.120846e-02 2.118025e-02 2.115210e-02 2.112401e-02
#> [636] 2.109599e-02 2.106802e-02 2.104011e-02 2.101226e-02 2.098447e-02
#> [641] 2.095674e-02 2.092907e-02 2.090145e-02 2.087390e-02 2.084640e-02
#> [646] 2.081896e-02 2.079159e-02 2.076426e-02 2.073700e-02 2.070980e-02
#> [651] 2.068265e-02 2.065556e-02 2.062853e-02 2.060156e-02 2.057464e-02
#> [656] 2.054778e-02 2.052098e-02 2.049423e-02 2.046754e-02 2.044091e-02
#> [661] 2.041434e-02 2.038782e-02 2.036136e-02 2.033495e-02 2.030860e-02
#> [666] 2.028231e-02 2.025607e-02 2.022989e-02 2.020376e-02 2.017769e-02
#> [671] 2.015167e-02 2.012571e-02 2.009980e-02 2.007395e-02 2.004815e-02
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#> [571] 2.934400e-02 2.942927e-02 2.951442e-02 2.959945e-02 2.968434e-02
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#> [1436] 5.920362e-02 5.920523e-02 5.920680e-02 5.920832e-02 5.920982e-02
#> [1441] 5.921127e-02 5.921269e-02 5.921406e-02 5.921540e-02 5.921671e-02
#> [1446] 5.921797e-02 5.921920e-02 5.922039e-02 5.922155e-02 5.922266e-02
#> [1451] 5.922374e-02 5.922478e-02 5.922579e-02 5.922676e-02 5.922769e-02
#> [1456] 5.922858e-02 5.922944e-02 5.923026e-02 5.923105e-02 5.923180e-02
#> [1461] 5.923251e-02 5.923319e-02 5.923383e-02 5.923443e-02 5.923500e-02
#> [1466] 5.923553e-02 5.923603e-02 5.923649e-02 5.923692e-02 5.923731e-02
#> [1471] 5.923766e-02 5.923798e-02 5.923827e-02 5.923852e-02 5.923873e-02
#> [1476] 5.923891e-02 5.923905e-02 5.923916e-02 5.923924e-02 5.923928e-02
#> [1481] 5.923928e-02 5.923925e-02 5.923919e-02 5.923909e-02 5.923896e-02
#> [1486] 5.923879e-02 5.923859e-02 5.923836e-02 5.923809e-02 5.923778e-02
#> [1491] 5.923745e-02 5.923708e-02 5.923667e-02 5.923624e-02 5.923576e-02
#> [1496] 5.923526e-02 5.923472e-02 5.923415e-02 5.923354e-02 5.923291e-02
#> [1501] 5.923224e-02 5.923153e-02 5.923080e-02 5.923003e-02 5.922922e-02
#> [1506] 5.922839e-02 5.922752e-02 5.922662e-02 5.922569e-02 5.922472e-02
#> [1511] 5.922372e-02 5.922270e-02 5.922163e-02 5.922054e-02 5.921941e-02
#> [1516] 5.921826e-02 5.921707e-02 5.921584e-02 5.921459e-02 5.921331e-02
#> [1521] 5.921199e-02 5.921064e-02 5.920926e-02 5.920785e-02 5.920641e-02
#> [1526] 5.920493e-02 5.920343e-02 5.920189e-02 5.920033e-02 5.919873e-02
#> [1531] 5.919710e-02 5.919544e-02 5.919375e-02 5.919203e-02 5.919028e-02
#> [1536] 5.918850e-02 5.918669e-02 5.918485e-02 5.918297e-02 5.918107e-02
#> [1541] 5.917914e-02 5.917717e-02 5.917518e-02 5.917316e-02 5.917110e-02
#> [1546] 5.916902e-02 5.916691e-02 5.916477e-02 5.916260e-02 5.916039e-02
#> [1551] 5.915816e-02 5.915590e-02 5.915361e-02 5.915130e-02 5.914895e-02
#> [1556] 5.914657e-02 5.914416e-02 5.914173e-02 5.913927e-02 5.913677e-02
#> [1561] 5.913425e-02 5.913170e-02 5.912912e-02 5.912651e-02 5.912388e-02
#> [1566] 5.912121e-02 5.911852e-02 5.911580e-02 5.911305e-02 5.911027e-02
#> [1571] 5.910747e-02 5.910463e-02 5.910177e-02 5.909888e-02 5.909596e-02
#> [1576] 5.909302e-02 5.909004e-02 5.908704e-02 5.908402e-02 5.908096e-02
#> [1581] 5.907788e-02 5.907477e-02 5.907163e-02 5.906846e-02 5.906527e-02
#> [1586] 5.906205e-02 5.905880e-02 5.905553e-02 5.905223e-02 5.904890e-02
#> [1591] 5.904555e-02 5.904217e-02 5.903876e-02 5.903532e-02 5.903186e-02
#> [1596] 5.902837e-02 5.902486e-02 5.902132e-02 5.901775e-02 5.901416e-02
#> [1601] 5.901054e-02 5.900689e-02 5.900322e-02 5.899952e-02 5.899580e-02
#> [1606] 5.899205e-02 5.898828e-02 5.898448e-02 5.898065e-02 5.897680e-02
#> [1611] 5.897292e-02 5.896901e-02 5.896508e-02 5.896113e-02 5.895715e-02
#> [1616] 5.895315e-02 5.894911e-02 5.894506e-02 5.894098e-02 5.893687e-02
#> [1621] 5.893274e-02 5.892859e-02 5.892441e-02 5.892020e-02 5.891597e-02
#> [1626] 5.891171e-02 5.890744e-02 5.890313e-02 5.889880e-02 5.889445e-02
#> [1631] 5.889007e-02 5.888567e-02 5.888124e-02 5.887679e-02 5.887232e-02
#> [1636] 5.886782e-02 5.886330e-02 5.885875e-02 5.885418e-02 5.884958e-02
#> [1641] 5.884497e-02 5.884032e-02 5.883566e-02 5.883097e-02 5.882625e-02
#> [1646] 5.882152e-02 5.881676e-02 5.881197e-02 5.880716e-02 5.880233e-02
#> [1651] 5.879748e-02 5.879260e-02 5.878770e-02 5.878278e-02 5.877783e-02
#> [1656] 5.877286e-02 5.876787e-02 5.876285e-02 5.875782e-02 5.875276e-02
#> [1661] 5.874767e-02 5.874257e-02 5.873744e-02 5.873228e-02 5.872711e-02
#> [1666] 5.872191e-02 5.871669e-02 5.871145e-02 5.870619e-02 5.870090e-02
#> [1671] 5.869560e-02 5.869027e-02 5.868491e-02 5.867954e-02 5.867414e-02
#> [1676] 5.866872e-02 5.866328e-02 5.865782e-02 5.865234e-02 5.864683e-02
#> [1681] 5.864130e-02 5.863576e-02 5.863019e-02 5.862459e-02 5.861898e-02
#> [1686] 5.861335e-02 5.860769e-02 5.860201e-02 5.859631e-02 5.859059e-02
#> [1691] 5.858485e-02 5.857909e-02 5.857331e-02 5.856750e-02 5.856168e-02
#> [1696] 5.855583e-02 5.854996e-02 5.854408e-02 5.853817e-02 5.853224e-02
#> [1701] 5.852629e-02 5.852032e-02 5.851433e-02 5.850831e-02 5.850228e-02
#> [1706] 5.849623e-02 5.849016e-02 5.848406e-02 5.847795e-02 5.847182e-02
#> [1711] 5.846566e-02 5.845949e-02 5.845330e-02 5.844708e-02 5.844085e-02
#> [1716] 5.843459e-02 5.842832e-02 5.842203e-02 5.841571e-02 5.840938e-02
#> [1721] 5.840303e-02 5.839666e-02 5.839026e-02 5.838385e-02 5.837742e-02
#> [1726] 5.837097e-02 5.836450e-02 5.835801e-02 5.835151e-02 5.834498e-02
#> [1731] 5.833843e-02 5.833186e-02 5.832528e-02 5.831868e-02 5.831205e-02
#> [1736] 5.830541e-02 5.829875e-02 5.829207e-02 5.828537e-02 5.827865e-02
#> [1741] 5.827192e-02 5.826516e-02 5.825839e-02 5.825160e-02 5.824479e-02
#> [1746] 5.823796e-02 5.823111e-02 5.822424e-02 5.821736e-02 5.821046e-02
#> [1751] 5.820353e-02 5.819660e-02 5.818964e-02 5.818266e-02 5.817567e-02
#> [1756] 5.816866e-02 5.816163e-02 5.815458e-02 5.814751e-02 5.814043e-02
#> [1761] 5.813333e-02 5.812621e-02 5.811907e-02 5.811192e-02 5.810474e-02
#> [1766] 5.809755e-02 5.809035e-02 5.808312e-02 5.807588e-02 5.806862e-02
#> [1771] 5.806134e-02 5.805405e-02 5.804673e-02 5.803941e-02 5.803206e-02
#> [1776] 5.802469e-02 5.801731e-02 5.800992e-02 5.800250e-02 5.799507e-02
#> [1781] 5.798762e-02 5.798015e-02 5.797267e-02 5.796517e-02 5.795765e-02
#> [1786] 5.795012e-02 5.794257e-02 5.793500e-02 5.792742e-02 5.791982e-02
#> [1791] 5.791220e-02 5.790457e-02 5.789692e-02 5.788925e-02 5.788157e-02
#> [1796] 5.787387e-02 5.786616e-02 5.785843e-02 5.785068e-02 5.784292e-02
#> [1801] 5.783514e-02 5.782734e-02 5.781953e-02 5.781170e-02 5.780385e-02
#> [1806] 5.779599e-02 5.778812e-02 5.778023e-02 5.777232e-02 5.776439e-02
#> [1811] 5.775646e-02 5.774850e-02 5.774053e-02 5.773254e-02 5.772454e-02
#> [1816] 5.771652e-02 5.770849e-02 5.770044e-02 5.769238e-02 5.768430e-02
#> [1821] 5.767620e-02 5.766809e-02 5.765997e-02 5.765183e-02 5.764367e-02
#> [1826] 5.763550e-02 5.762732e-02 5.761912e-02 5.761090e-02 5.760267e-02
#> [1831] 5.759442e-02 5.758616e-02 5.757788e-02 5.756959e-02 5.756129e-02
#> [1836] 5.755297e-02 5.754463e-02 5.753628e-02 5.752792e-02 5.751954e-02
#> [1841] 5.751115e-02 5.750274e-02 5.749431e-02 5.748588e-02 5.747742e-02
#> [1846] 5.746896e-02 5.746048e-02 5.745198e-02 5.744347e-02 5.743495e-02
#> [1851] 5.742641e-02 5.741786e-02 5.740929e-02 5.740071e-02 5.739212e-02
#> [1856] 5.738351e-02 5.737489e-02 5.736625e-02 5.735760e-02 5.734894e-02
#> [1861] 5.734026e-02 5.733157e-02 5.732286e-02 5.731414e-02 5.730541e-02
#> [1866] 5.729666e-02 5.728790e-02 5.727913e-02 5.727034e-02 5.726154e-02
#> [1871] 5.725272e-02 5.724389e-02 5.723505e-02 5.722620e-02 5.721733e-02
#> [1876] 5.720845e-02 5.719955e-02 5.719064e-02 5.718172e-02 5.717279e-02
#> [1881] 5.716384e-02 5.715488e-02 5.714590e-02 5.713691e-02 5.712791e-02
#> [1886] 5.711890e-02 5.710987e-02 5.710083e-02 5.709178e-02 5.708272e-02
#> [1891] 5.707364e-02 5.706455e-02 5.705544e-02 5.704633e-02 5.703720e-02
#> [1896] 5.702806e-02 5.701890e-02 5.700973e-02 5.700055e-02 5.699136e-02
#> [1901] 5.698216e-02 5.697294e-02 5.696371e-02 5.695447e-02 5.694522e-02
#> [1906] 5.693595e-02 5.692667e-02 5.691738e-02 5.690807e-02 5.689876e-02
#> [1911] 5.688943e-02 5.688009e-02 5.687074e-02 5.686137e-02 5.685200e-02
#> [1916] 5.684261e-02 5.683321e-02 5.682380e-02 5.681437e-02 5.680494e-02
#> [1921] 5.679549e-02 5.678603e-02 5.677656e-02 5.676707e-02 5.675758e-02
#> [1926] 5.674807e-02 5.673855e-02 5.672902e-02 5.671948e-02 5.670993e-02
#> [1931] 5.670036e-02 5.669079e-02 5.668120e-02 5.667160e-02 5.666199e-02
#> [1936] 5.665237e-02 5.664273e-02 5.663309e-02 5.662343e-02 5.661377e-02
#> [1941] 5.660409e-02 5.659440e-02 5.658470e-02 5.657498e-02 5.656526e-02
#> [1946] 5.655553e-02 5.654578e-02 5.653603e-02 5.652626e-02 5.651648e-02
#> [1951] 5.650669e-02 5.649689e-02 5.648708e-02 5.647726e-02 5.646742e-02
#> [1956] 5.645758e-02 5.644772e-02 5.643786e-02 5.642798e-02 5.641810e-02
#> [1961] 5.640820e-02 5.639829e-02 5.638837e-02 5.637844e-02 5.636850e-02
#> [1966] 5.635855e-02 5.634859e-02 5.633862e-02 5.632864e-02 5.631865e-02
#> [1971] 5.630865e-02 5.629863e-02 5.628861e-02 5.627858e-02 5.626853e-02
#> [1976] 5.625848e-02 5.624842e-02 5.623834e-02 5.622826e-02 5.621816e-02
#> [1981] 5.620806e-02 5.619795e-02 5.618782e-02 5.617769e-02 5.616754e-02
#> [1986] 5.615739e-02 5.614722e-02 5.613705e-02 5.612686e-02 5.611667e-02
#> [1991] 5.610647e-02 5.609625e-02 5.608603e-02 5.607580e-02 5.606555e-02
#> [1996] 5.605530e-02 5.604504e-02 5.603477e-02 5.602449e-02 5.601420e-02
#> [2001] 5.600389e-02
bsmtheta(c("call", "put"), c(50, 49, 48, 47), 45, 0.25, 0.13, 0.01, 0.2)
#> [1] -6.145512 -1.184449
bsmtheta("call", 50, 45, 0.25, 0.13, 0.01, 1:1000 / 1000)
#> [1] -5.16418
bsmtheta("put", 20, 30, 0:2000 / 1000, 0.15, 0, 0.25)
#> [1] NaN
bsmrho(c("call", "put"), c(50, 49, 48, 47), 45, 0.25, 0.13, 0.01, 0.2)
#> [1] 9.842783 -1.047469
bsmrho("call", 50, 45, 0.25, 0.13, 0.01, 1:1000 / 1000)
#> [1] 10.89025
bsmrho("put", 20, 30, 0:2000 / 1000, 0.15, 0, 0.25)
#> [1] 0