En este trabajo se analiza la convexidad de una opción tipo call sobre la acción de NVIDIA Corporation, utilizando el modelo de Black-Scholes. Se simula el comportamiento del precio de la opción frente a diferentes escenarios del activo subyacente y a lo largo del tiempo, incorporando supuestos realistas como volatilidad variable y tasas de interés interpoladas. Adicionalmente, se estudian las sensibilidades de la opción mediante el cálculo de las griegas, con el fin de comprender su exposición al riesgo.
# Parámetros de la opción NVDA
K <- 200 # Precio de ejercicio
sigma <- 0.65 # Volatilidad implícita
r <- "tasa SOFR interpolada"
S0 <- 198.35 # Precio actual de NVIDIA CorporationBS_call <- function(S, K, r, sigma, T){
if(T <= 0) return(max(S-K,0))
d1 <- (log(S/K)+(r+sigma^2/2)*T)/(sigma*sqrt(T))
d2 <- d1 - sigma*sqrt(T)
S*pnorm(d1) - K*exp(-r*T)*pnorm(d2)
}library(dplyr)
library(knitr)
# Tabla inicio (mayor tiempo)
tabla_inicio <- results %>% filter(Tiempo == max(time_steps))
# Tabla intermedia (mes 6 aprox)
tabla_mitad <- results %>% filter(Tiempo == time_steps[6])
# Tabla final (cerca al vencimiento)
tabla_final <- results %>% filter(Tiempo == min(time_steps))
# Mostrar tablas (en RPubs se ven bonito)
kable(tabla_inicio, caption = "Tabla Convexidad - Inicio")| Precio_Activo | Precio_Opcion | Tiempo |
|---|---|---|
| 47.21948 | 1.768398 | 252 |
| 49.21948 | 2.020164 | 252 |
| 51.21948 | 2.291773 | 252 |
| 53.21948 | 2.583436 | 252 |
| 55.21948 | 2.895316 | 252 |
| 57.21948 | 3.227535 | 252 |
| 59.21948 | 3.580178 | 252 |
| 61.21948 | 3.953292 | 252 |
| 63.21948 | 4.346894 | 252 |
| 65.21948 | 4.760971 | 252 |
| 67.21948 | 5.195485 | 252 |
| 69.21948 | 5.650374 | 252 |
| 71.21948 | 6.125553 | 252 |
| 73.21948 | 6.620920 | 252 |
| 75.21948 | 7.136353 | 252 |
| 77.21948 | 7.671718 | 252 |
| 79.21948 | 8.226864 | 252 |
| 81.21948 | 8.801630 | 252 |
| 83.21948 | 9.395843 | 252 |
| 85.21948 | 10.009321 | 252 |
| 87.21948 | 10.641872 | 252 |
| 89.21948 | 11.293298 | 252 |
| 91.21948 | 11.963396 | 252 |
| 93.21948 | 12.651954 | 252 |
| 95.21948 | 13.358759 | 252 |
| 97.21948 | 14.083592 | 252 |
| 99.21948 | 14.826231 | 252 |
| 101.21948 | 15.586452 | 252 |
| 103.21948 | 16.364029 | 252 |
| 105.21948 | 17.158734 | 252 |
| 107.21948 | 17.970339 | 252 |
| 109.21948 | 18.798613 | 252 |
| 111.21948 | 19.643327 | 252 |
| 113.21948 | 20.504252 | 252 |
| 115.21948 | 21.381158 | 252 |
| 117.21948 | 22.273818 | 252 |
| 119.21948 | 23.182004 | 252 |
| 121.21948 | 24.105489 | 252 |
| 123.21948 | 25.044049 | 252 |
| 125.21948 | 25.997462 | 252 |
| 127.21948 | 26.965504 | 252 |
| 129.21948 | 27.947958 | 252 |
| 131.21948 | 28.944605 | 252 |
| 133.21948 | 29.955231 | 252 |
| 135.21948 | 30.979622 | 252 |
| 137.21948 | 32.017567 | 252 |
| 139.21948 | 33.068859 | 252 |
| 141.21948 | 34.133291 | 252 |
| 143.21948 | 35.210660 | 252 |
| 145.21948 | 36.300766 | 252 |
| 147.21948 | 37.403411 | 252 |
| 149.21948 | 38.518399 | 252 |
| 151.21948 | 39.645537 | 252 |
| 153.21948 | 40.784636 | 252 |
| 155.21948 | 41.935507 | 252 |
| 157.21948 | 43.097968 | 252 |
| 159.21948 | 44.271835 | 252 |
| 161.21948 | 45.456929 | 252 |
| 163.21948 | 46.653075 | 252 |
| 165.21948 | 47.860097 | 252 |
| 167.21948 | 49.077826 | 252 |
| 169.21948 | 50.306093 | 252 |
| 171.21948 | 51.544732 | 252 |
| 173.21948 | 52.793579 | 252 |
| 175.21948 | 54.052476 | 252 |
| 177.21948 | 55.321262 | 252 |
| 179.21948 | 56.599784 | 252 |
| 181.21948 | 57.887888 | 252 |
| 183.21948 | 59.185425 | 252 |
| 185.21948 | 60.492245 | 252 |
| 187.21948 | 61.808205 | 252 |
| 189.21948 | 63.133161 | 252 |
| 191.21948 | 64.466971 | 252 |
| 193.21948 | 65.809500 | 252 |
| 195.21948 | 67.160610 | 252 |
| 197.21948 | 68.520167 | 252 |
| 199.21948 | 69.888041 | 252 |
| 201.21948 | 71.264103 | 252 |
| 203.21948 | 72.648225 | 252 |
| 205.21948 | 74.040283 | 252 |
| 207.21948 | 75.440154 | 252 |
| 209.21948 | 76.847718 | 252 |
| 211.21948 | 78.262856 | 252 |
| 213.21948 | 79.685453 | 252 |
| 215.21948 | 81.115393 | 252 |
| 217.21948 | 82.552564 | 252 |
| 219.21948 | 83.996855 | 252 |
| 221.21948 | 85.448159 | 252 |
| 223.21948 | 86.906367 | 252 |
| 225.21948 | 88.371376 | 252 |
| 227.21948 | 89.843082 | 252 |
| 229.21948 | 91.321384 | 252 |
| 231.21948 | 92.806181 | 252 |
| 233.21948 | 94.297378 | 252 |
| 235.21948 | 95.794876 | 252 |
| 237.21948 | 97.298581 | 252 |
| 239.21948 | 98.808402 | 252 |
| 241.21948 | 100.324245 | 252 |
| 243.21948 | 101.846021 | 252 |
| 245.21948 | 103.373643 | 252 |
| 247.21948 | 104.907023 | 252 |
| 249.21948 | 106.446076 | 252 |
| 251.21948 | 107.990718 | 252 |
| 253.21948 | 109.540868 | 252 |
| 255.21948 | 111.096443 | 252 |
| 257.21948 | 112.657364 | 252 |
| 259.21948 | 114.223553 | 252 |
| 261.21948 | 115.794933 | 252 |
| 263.21948 | 117.371429 | 252 |
| 265.21948 | 118.952965 | 252 |
| 267.21948 | 120.539470 | 252 |
| 269.21948 | 122.130870 | 252 |
| 271.21948 | 123.727096 | 252 |
| 273.21948 | 125.328077 | 252 |
| 275.21948 | 126.933747 | 252 |
| 277.21948 | 128.544036 | 252 |
| 279.21948 | 130.158880 | 252 |
| 281.21948 | 131.778213 | 252 |
| 283.21948 | 133.401972 | 252 |
| 285.21948 | 135.030093 | 252 |
| 287.21948 | 136.662516 | 252 |
| 289.21948 | 138.299178 | 252 |
| 291.21948 | 139.940021 | 252 |
| 293.21948 | 141.584986 | 252 |
| 295.21948 | 143.234014 | 252 |
| 297.21948 | 144.887050 | 252 |
| 299.21948 | 146.544036 | 252 |
| 301.21948 | 148.204919 | 252 |
| 303.21948 | 149.869644 | 252 |
| 305.21948 | 151.538158 | 252 |
| 307.21948 | 153.210408 | 252 |
| 309.21948 | 154.886343 | 252 |
| 311.21948 | 156.565912 | 252 |
| 313.21948 | 158.249066 | 252 |
| 315.21948 | 159.935756 | 252 |
| 317.21948 | 161.625932 | 252 |
| 319.21948 | 163.319549 | 252 |
| 321.21948 | 165.016558 | 252 |
| 323.21948 | 166.716914 | 252 |
| 325.21948 | 168.420572 | 252 |
| 327.21948 | 170.127488 | 252 |
| 329.21948 | 171.837617 | 252 |
| 331.21948 | 173.550916 | 252 |
| 333.21948 | 175.267343 | 252 |
| 335.21948 | 176.986856 | 252 |
| 337.21948 | 178.709415 | 252 |
| 339.21948 | 180.434978 | 252 |
| 341.21948 | 182.163505 | 252 |
| 343.21948 | 183.894959 | 252 |
| 345.21948 | 185.629299 | 252 |
| 347.21948 | 187.366489 | 252 |
| 349.21948 | 189.106490 | 252 |
| 351.21948 | 190.849266 | 252 |
| 353.21948 | 192.594780 | 252 |
| 355.21948 | 194.342997 | 252 |
| 357.21948 | 196.093882 | 252 |
| 359.21948 | 197.847401 | 252 |
| 361.21948 | 199.603518 | 252 |
| 363.21948 | 201.362201 | 252 |
| 365.21948 | 203.123416 | 252 |
| 367.21948 | 204.887132 | 252 |
| 369.21948 | 206.653315 | 252 |
| 371.21948 | 208.421935 | 252 |
| 373.21948 | 210.192960 | 252 |
| 375.21948 | 211.966360 | 252 |
| 377.21948 | 213.742105 | 252 |
| 379.21948 | 215.520164 | 252 |
| 381.21948 | 217.300510 | 252 |
| 383.21948 | 219.083112 | 252 |
| 385.21948 | 220.867942 | 252 |
| 387.21948 | 222.654974 | 252 |
| 389.21948 | 224.444178 | 252 |
| 391.21948 | 226.235528 | 252 |
| 393.21948 | 228.028997 | 252 |
| 395.21948 | 229.824559 | 252 |
| 397.21948 | 231.622188 | 252 |
| 399.21948 | 233.421859 | 252 |
| 401.21948 | 235.223545 | 252 |
| 403.21948 | 237.027223 | 252 |
| 405.21948 | 238.832868 | 252 |
| 407.21948 | 240.640456 | 252 |
| 409.21948 | 242.449962 | 252 |
| 411.21948 | 244.261364 | 252 |
| 413.21948 | 246.074639 | 252 |
| 415.21948 | 247.889764 | 252 |
| 417.21948 | 249.706716 | 252 |
| 419.21948 | 251.525473 | 252 |
| 421.21948 | 253.346014 | 252 |
| 423.21948 | 255.168317 | 252 |
| 425.21948 | 256.992361 | 252 |
| 427.21948 | 258.818125 | 252 |
| 429.21948 | 260.645588 | 252 |
| 431.21948 | 262.474731 | 252 |
| 433.21948 | 264.305534 | 252 |
| 435.21948 | 266.137975 | 252 |
| 437.21948 | 267.972037 | 252 |
| 439.21948 | 269.807700 | 252 |
| 441.21948 | 271.644946 | 252 |
| 443.21948 | 273.483755 | 252 |
| 445.21948 | 275.324109 | 252 |
| 447.21948 | 277.165990 | 252 |
| 449.21948 | 279.009380 | 252 |
| 451.21948 | 280.854263 | 252 |
| 453.21948 | 282.700619 | 252 |
| 455.21948 | 284.548433 | 252 |
| 457.21948 | 286.397687 | 252 |
| 459.21948 | 288.248365 | 252 |
| 461.21948 | 290.100450 | 252 |
| 463.21948 | 291.953926 | 252 |
| 465.21948 | 293.808777 | 252 |
| 467.21948 | 295.664988 | 252 |
| 469.21948 | 297.522542 | 252 |
| 471.21948 | 299.381425 | 252 |
| 473.21948 | 301.241622 | 252 |
| 475.21948 | 303.103116 | 252 |
| 477.21948 | 304.965895 | 252 |
| 479.21948 | 306.829942 | 252 |
| 481.21948 | 308.695245 | 252 |
| 483.21948 | 310.561788 | 252 |
| 485.21948 | 312.429558 | 252 |
| 487.21948 | 314.298542 | 252 |
| 489.21948 | 316.168724 | 252 |
| 491.21948 | 318.040093 | 252 |
| 493.21948 | 319.912635 | 252 |
| 495.21948 | 321.786336 | 252 |
| 497.21948 | 323.661185 | 252 |
| 499.21948 | 325.537168 | 252 |
| 501.21948 | 327.414272 | 252 |
| 503.21948 | 329.292486 | 252 |
| 505.21948 | 331.171797 | 252 |
| 507.21948 | 333.052193 | 252 |
| 509.21948 | 334.933662 | 252 |
| 511.21948 | 336.816193 | 252 |
| 513.21948 | 338.699773 | 252 |
| 515.21948 | 340.584392 | 252 |
| 517.21948 | 342.470039 | 252 |
| 519.21948 | 344.356701 | 252 |
| 521.21948 | 346.244368 | 252 |
| 523.21948 | 348.133030 | 252 |
| 525.21948 | 350.022675 | 252 |
| 527.21948 | 351.913293 | 252 |
| 529.21948 | 353.804873 | 252 |
| 531.21948 | 355.697405 | 252 |
| 533.21948 | 357.590879 | 252 |
| 535.21948 | 359.485286 | 252 |
| 537.21948 | 361.380614 | 252 |
| 539.21948 | 363.276854 | 252 |
| 541.21948 | 365.173997 | 252 |
| 543.21948 | 367.072032 | 252 |
| 545.21948 | 368.970952 | 252 |
| 547.21948 | 370.870745 | 252 |
| 549.21948 | 372.771404 | 252 |
| 551.21948 | 374.672918 | 252 |
| 553.21948 | 376.575279 | 252 |
| 555.21948 | 378.478478 | 252 |
| 557.21948 | 380.382507 | 252 |
| 559.21948 | 382.287356 | 252 |
| 561.21948 | 384.193017 | 252 |
| 563.21948 | 386.099482 | 252 |
| 565.21948 | 388.006743 | 252 |
| 567.21948 | 389.914790 | 252 |
| 569.21948 | 391.823617 | 252 |
| 571.21948 | 393.733214 | 252 |
| 573.21948 | 395.643575 | 252 |
| 575.21948 | 397.554691 | 252 |
| 577.21948 | 399.466554 | 252 |
| 579.21948 | 401.379158 | 252 |
| 581.21948 | 403.292493 | 252 |
| 583.21948 | 405.206553 | 252 |
| 585.21948 | 407.121331 | 252 |
| 587.21948 | 409.036820 | 252 |
| 589.21948 | 410.953011 | 252 |
| 591.21948 | 412.869898 | 252 |
| 593.21948 | 414.787474 | 252 |
| 595.21948 | 416.705732 | 252 |
| 597.21948 | 418.624665 | 252 |
| 599.21948 | 420.544266 | 252 |
| 601.21948 | 422.464529 | 252 |
| 603.21948 | 424.385447 | 252 |
kable(tabla_mitad, caption = "Tabla Convexidad - Mitad")| Precio_Activo | Precio_Opcion | Tiempo |
|---|---|---|
| 47.21948 | 0.1241300 | 142 |
| 49.21948 | 0.1575294 | 142 |
| 51.21948 | 0.1972178 | 142 |
| 53.21948 | 0.2438695 | 142 |
| 55.21948 | 0.2981673 | 142 |
| 57.21948 | 0.3607970 | 142 |
| 59.21948 | 0.4324439 | 142 |
| 61.21948 | 0.5137879 | 142 |
| 63.21948 | 0.6055004 | 142 |
| 65.21948 | 0.7082408 | 142 |
| 67.21948 | 0.8226534 | 142 |
| 69.21948 | 0.9493649 | 142 |
| 71.21948 | 1.0889820 | 142 |
| 73.21948 | 1.2420896 | 142 |
| 75.21948 | 1.4092491 | 142 |
| 77.21948 | 1.5909968 | 142 |
| 79.21948 | 1.7878434 | 142 |
| 81.21948 | 2.0002724 | 142 |
| 83.21948 | 2.2287403 | 142 |
| 85.21948 | 2.4736755 | 142 |
| 87.21948 | 2.7354788 | 142 |
| 89.21948 | 3.0145225 | 142 |
| 91.21948 | 3.3111515 | 142 |
| 93.21948 | 3.6256828 | 142 |
| 95.21948 | 3.9584063 | 142 |
| 97.21948 | 4.3095849 | 142 |
| 99.21948 | 4.6794556 | 142 |
| 101.21948 | 5.0682293 | 142 |
| 103.21948 | 5.4760926 | 142 |
| 105.21948 | 5.9032076 | 142 |
| 107.21948 | 6.3497129 | 142 |
| 109.21948 | 6.8157249 | 142 |
| 111.21948 | 7.3013381 | 142 |
| 113.21948 | 7.8066262 | 142 |
| 115.21948 | 8.3316428 | 142 |
| 117.21948 | 8.8764225 | 142 |
| 119.21948 | 9.4409818 | 142 |
| 121.21948 | 10.0253195 | 142 |
| 123.21948 | 10.6294182 | 142 |
| 125.21948 | 11.2532447 | 142 |
| 127.21948 | 11.8967510 | 142 |
| 129.21948 | 12.5598752 | 142 |
| 131.21948 | 13.2425422 | 142 |
| 133.21948 | 13.9446643 | 142 |
| 135.21948 | 14.6661422 | 142 |
| 137.21948 | 15.4068660 | 142 |
| 139.21948 | 16.1667151 | 142 |
| 141.21948 | 16.9455598 | 142 |
| 143.21948 | 17.7432612 | 142 |
| 145.21948 | 18.5596725 | 142 |
| 147.21948 | 19.3946391 | 142 |
| 149.21948 | 20.2479994 | 142 |
| 151.21948 | 21.1195855 | 142 |
| 153.21948 | 22.0092234 | 142 |
| 155.21948 | 22.9167337 | 142 |
| 157.21948 | 23.8419325 | 142 |
| 159.21948 | 24.7846309 | 142 |
| 161.21948 | 25.7446364 | 142 |
| 163.21948 | 26.7217529 | 142 |
| 165.21948 | 27.7157811 | 142 |
| 167.21948 | 28.7265189 | 142 |
| 169.21948 | 29.7537619 | 142 |
| 171.21948 | 30.7973036 | 142 |
| 173.21948 | 31.8569358 | 142 |
| 175.21948 | 32.9324488 | 142 |
| 177.21948 | 34.0236319 | 142 |
| 179.21948 | 35.1302734 | 142 |
| 181.21948 | 36.2521612 | 142 |
| 183.21948 | 37.3890826 | 142 |
| 185.21948 | 38.5408249 | 142 |
| 187.21948 | 39.7071756 | 142 |
| 189.21948 | 40.8879224 | 142 |
| 191.21948 | 42.0828533 | 142 |
| 193.21948 | 43.2917571 | 142 |
| 195.21948 | 44.5144235 | 142 |
| 197.21948 | 45.7506431 | 142 |
| 199.21948 | 47.0002074 | 142 |
| 201.21948 | 48.2629094 | 142 |
| 203.21948 | 49.5385433 | 142 |
| 205.21948 | 50.8269049 | 142 |
| 207.21948 | 52.1277915 | 142 |
| 209.21948 | 53.4410019 | 142 |
| 211.21948 | 54.7663368 | 142 |
| 213.21948 | 56.1035988 | 142 |
| 215.21948 | 57.4525923 | 142 |
| 217.21948 | 58.8131237 | 142 |
| 219.21948 | 60.1850013 | 142 |
| 221.21948 | 61.5680357 | 142 |
| 223.21948 | 62.9620394 | 142 |
| 225.21948 | 64.3668272 | 142 |
| 227.21948 | 65.7822160 | 142 |
| 229.21948 | 67.2080252 | 142 |
| 231.21948 | 68.6440760 | 142 |
| 233.21948 | 70.0901923 | 142 |
| 235.21948 | 71.5462002 | 142 |
| 237.21948 | 73.0119279 | 142 |
| 239.21948 | 74.4872063 | 142 |
| 241.21948 | 75.9718684 | 142 |
| 243.21948 | 77.4657495 | 142 |
| 245.21948 | 78.9686875 | 142 |
| 247.21948 | 80.4805225 | 142 |
| 249.21948 | 82.0010969 | 142 |
| 251.21948 | 83.5302555 | 142 |
| 253.21948 | 85.0678456 | 142 |
| 255.21948 | 86.6137166 | 142 |
| 257.21948 | 88.1677203 | 142 |
| 259.21948 | 89.7297109 | 142 |
| 261.21948 | 91.2995449 | 142 |
| 263.21948 | 92.8770809 | 142 |
| 265.21948 | 94.4621800 | 142 |
| 267.21948 | 96.0547054 | 142 |
| 269.21948 | 97.6545226 | 142 |
| 271.21948 | 99.2614993 | 142 |
| 273.21948 | 100.8755054 | 142 |
| 275.21948 | 102.4964130 | 142 |
| 277.21948 | 104.1240963 | 142 |
| 279.21948 | 105.7584317 | 142 |
| 281.21948 | 107.3992977 | 142 |
| 283.21948 | 109.0465747 | 142 |
| 285.21948 | 110.7001454 | 142 |
| 287.21948 | 112.3598945 | 142 |
| 289.21948 | 114.0257085 | 142 |
| 291.21948 | 115.6974762 | 142 |
| 293.21948 | 117.3750881 | 142 |
| 295.21948 | 119.0584368 | 142 |
| 297.21948 | 120.7474166 | 142 |
| 299.21948 | 122.4419239 | 142 |
| 301.21948 | 124.1418568 | 142 |
| 303.21948 | 125.8471153 | 142 |
| 305.21948 | 127.5576013 | 142 |
| 307.21948 | 129.2732182 | 142 |
| 309.21948 | 130.9938714 | 142 |
| 311.21948 | 132.7194679 | 142 |
| 313.21948 | 134.4499165 | 142 |
| 315.21948 | 136.1851274 | 142 |
| 317.21948 | 137.9250128 | 142 |
| 319.21948 | 139.6694863 | 142 |
| 321.21948 | 141.4184631 | 142 |
| 323.21948 | 143.1718600 | 142 |
| 325.21948 | 144.9295953 | 142 |
| 327.21948 | 146.6915890 | 142 |
| 329.21948 | 148.4577622 | 142 |
| 331.21948 | 150.2280379 | 142 |
| 333.21948 | 152.0023402 | 142 |
| 335.21948 | 153.7805948 | 142 |
| 337.21948 | 155.5627288 | 142 |
| 339.21948 | 157.3486705 | 142 |
| 341.21948 | 159.1383496 | 142 |
| 343.21948 | 160.9316972 | 142 |
| 345.21948 | 162.7286458 | 142 |
| 347.21948 | 164.5291288 | 142 |
| 349.21948 | 166.3330812 | 142 |
| 351.21948 | 168.1404390 | 142 |
| 353.21948 | 169.9511397 | 142 |
| 355.21948 | 171.7651215 | 142 |
| 357.21948 | 173.5823243 | 142 |
| 359.21948 | 175.4026888 | 142 |
| 361.21948 | 177.2261568 | 142 |
| 363.21948 | 179.0526715 | 142 |
| 365.21948 | 180.8821768 | 142 |
| 367.21948 | 182.7146180 | 142 |
| 369.21948 | 184.5499412 | 142 |
| 371.21948 | 186.3880936 | 142 |
| 373.21948 | 188.2290234 | 142 |
| 375.21948 | 190.0726799 | 142 |
| 377.21948 | 191.9190131 | 142 |
| 379.21948 | 193.7679742 | 142 |
| 381.21948 | 195.6195152 | 142 |
| 383.21948 | 197.4735890 | 142 |
| 385.21948 | 199.3301494 | 142 |
| 387.21948 | 201.1891513 | 142 |
| 389.21948 | 203.0505501 | 142 |
| 391.21948 | 204.9143022 | 142 |
| 393.21948 | 206.7803649 | 142 |
| 395.21948 | 208.6486963 | 142 |
| 397.21948 | 210.5192552 | 142 |
| 399.21948 | 212.3920011 | 142 |
| 401.21948 | 214.2668947 | 142 |
| 403.21948 | 216.1438968 | 142 |
| 405.21948 | 218.0229696 | 142 |
| 407.21948 | 219.9040755 | 142 |
| 409.21948 | 221.7871779 | 142 |
| 411.21948 | 223.6722407 | 142 |
| 413.21948 | 225.5592288 | 142 |
| 415.21948 | 227.4481074 | 142 |
| 417.21948 | 229.3388426 | 142 |
| 419.21948 | 231.2314011 | 142 |
| 421.21948 | 233.1257500 | 142 |
| 423.21948 | 235.0218574 | 142 |
| 425.21948 | 236.9196918 | 142 |
| 427.21948 | 238.8192223 | 142 |
| 429.21948 | 240.7204185 | 142 |
| 431.21948 | 242.6232508 | 142 |
| 433.21948 | 244.5276900 | 142 |
| 435.21948 | 246.4337074 | 142 |
| 437.21948 | 248.3412750 | 142 |
| 439.21948 | 250.2503653 | 142 |
| 441.21948 | 252.1609511 | 142 |
| 443.21948 | 254.0730060 | 142 |
| 445.21948 | 255.9865040 | 142 |
| 447.21948 | 257.9014194 | 142 |
| 449.21948 | 259.8177274 | 142 |
| 451.21948 | 261.7354032 | 142 |
| 453.21948 | 263.6544227 | 142 |
| 455.21948 | 265.5747624 | 142 |
| 457.21948 | 267.4963988 | 142 |
| 459.21948 | 269.4193093 | 142 |
| 461.21948 | 271.3434715 | 142 |
| 463.21948 | 273.2688634 | 142 |
| 465.21948 | 275.1954635 | 142 |
| 467.21948 | 277.1232506 | 142 |
| 469.21948 | 279.0522040 | 142 |
| 471.21948 | 280.9823034 | 142 |
| 473.21948 | 282.9135287 | 142 |
| 475.21948 | 284.8458603 | 142 |
| 477.21948 | 286.7792790 | 142 |
| 479.21948 | 288.7137660 | 142 |
| 481.21948 | 290.6493026 | 142 |
| 483.21948 | 292.5858706 | 142 |
| 485.21948 | 294.5234523 | 142 |
| 487.21948 | 296.4620300 | 142 |
| 489.21948 | 298.4015866 | 142 |
| 491.21948 | 300.3421052 | 142 |
| 493.21948 | 302.2835692 | 142 |
| 495.21948 | 304.2259623 | 142 |
| 497.21948 | 306.1692686 | 142 |
| 499.21948 | 308.1134724 | 142 |
| 501.21948 | 310.0585583 | 142 |
| 503.21948 | 312.0045112 | 142 |
| 505.21948 | 313.9513162 | 142 |
| 507.21948 | 315.8989588 | 142 |
| 509.21948 | 317.8474247 | 142 |
| 511.21948 | 319.7966998 | 142 |
| 513.21948 | 321.7467705 | 142 |
| 515.21948 | 323.6976231 | 142 |
| 517.21948 | 325.6492444 | 142 |
| 519.21948 | 327.6016214 | 142 |
| 521.21948 | 329.5547412 | 142 |
| 523.21948 | 331.5085913 | 142 |
| 525.21948 | 333.4631593 | 142 |
| 527.21948 | 335.4184331 | 142 |
| 529.21948 | 337.3744009 | 142 |
| 531.21948 | 339.3310509 | 142 |
| 533.21948 | 341.2883717 | 142 |
| 535.21948 | 343.2463520 | 142 |
| 537.21948 | 345.2049808 | 142 |
| 539.21948 | 347.1642471 | 142 |
| 541.21948 | 349.1241403 | 142 |
| 543.21948 | 351.0846500 | 142 |
| 545.21948 | 353.0457659 | 142 |
| 547.21948 | 355.0074778 | 142 |
| 549.21948 | 356.9697759 | 142 |
| 551.21948 | 358.9326503 | 142 |
| 553.21948 | 360.8960916 | 142 |
| 555.21948 | 362.8600902 | 142 |
| 557.21948 | 364.8246371 | 142 |
| 559.21948 | 366.7897231 | 142 |
| 561.21948 | 368.7553393 | 142 |
| 563.21948 | 370.7214769 | 142 |
| 565.21948 | 372.6881275 | 142 |
| 567.21948 | 374.6552824 | 142 |
| 569.21948 | 376.6229336 | 142 |
| 571.21948 | 378.5910727 | 142 |
| 573.21948 | 380.5596918 | 142 |
| 575.21948 | 382.5287830 | 142 |
| 577.21948 | 384.4983387 | 142 |
| 579.21948 | 386.4683512 | 142 |
| 581.21948 | 388.4388131 | 142 |
| 583.21948 | 390.4097170 | 142 |
| 585.21948 | 392.3810558 | 142 |
| 587.21948 | 394.3528224 | 142 |
| 589.21948 | 396.3250099 | 142 |
| 591.21948 | 398.2976114 | 142 |
| 593.21948 | 400.2706202 | 142 |
| 595.21948 | 402.2440298 | 142 |
| 597.21948 | 404.2178337 | 142 |
| 599.21948 | 406.1920254 | 142 |
| 601.21948 | 408.1665989 | 142 |
| 603.21948 | 410.1415479 | 142 |
kable(tabla_final, caption = "Tabla Convexidad - Vencimiento")| Precio_Activo | Precio_Opcion | Tiempo |
|---|---|---|
| 47.21948 | 0.0000000 | 10 |
| 49.21948 | 0.0000000 | 10 |
| 51.21948 | 0.0000000 | 10 |
| 53.21948 | 0.0000000 | 10 |
| 55.21948 | 0.0000000 | 10 |
| 57.21948 | 0.0000000 | 10 |
| 59.21948 | 0.0000000 | 10 |
| 61.21948 | 0.0000000 | 10 |
| 63.21948 | 0.0000000 | 10 |
| 65.21948 | 0.0000000 | 10 |
| 67.21948 | 0.0000000 | 10 |
| 69.21948 | 0.0000000 | 10 |
| 71.21948 | 0.0000000 | 10 |
| 73.21948 | 0.0000000 | 10 |
| 75.21948 | 0.0000000 | 10 |
| 77.21948 | 0.0000000 | 10 |
| 79.21948 | 0.0000000 | 10 |
| 81.21948 | 0.0000000 | 10 |
| 83.21948 | 0.0000000 | 10 |
| 85.21948 | 0.0000000 | 10 |
| 87.21948 | 0.0000000 | 10 |
| 89.21948 | 0.0000000 | 10 |
| 91.21948 | 0.0000000 | 10 |
| 93.21948 | 0.0000000 | 10 |
| 95.21948 | 0.0000000 | 10 |
| 97.21948 | 0.0000000 | 10 |
| 99.21948 | 0.0000001 | 10 |
| 101.21948 | 0.0000003 | 10 |
| 103.21948 | 0.0000006 | 10 |
| 105.21948 | 0.0000013 | 10 |
| 107.21948 | 0.0000029 | 10 |
| 109.21948 | 0.0000060 | 10 |
| 111.21948 | 0.0000122 | 10 |
| 113.21948 | 0.0000240 | 10 |
| 115.21948 | 0.0000458 | 10 |
| 117.21948 | 0.0000851 | 10 |
| 119.21948 | 0.0001541 | 10 |
| 121.21948 | 0.0002719 | 10 |
| 123.21948 | 0.0004686 | 10 |
| 125.21948 | 0.0007892 | 10 |
| 127.21948 | 0.0013003 | 10 |
| 129.21948 | 0.0020980 | 10 |
| 131.21948 | 0.0033179 | 10 |
| 133.21948 | 0.0051475 | 10 |
| 135.21948 | 0.0078406 | 10 |
| 137.21948 | 0.0117346 | 10 |
| 139.21948 | 0.0172690 | 10 |
| 141.21948 | 0.0250066 | 10 |
| 143.21948 | 0.0356554 | 10 |
| 145.21948 | 0.0500903 | 10 |
| 147.21948 | 0.0693757 | 10 |
| 149.21948 | 0.0947850 | 10 |
| 151.21948 | 0.1278182 | 10 |
| 153.21948 | 0.1702152 | 10 |
| 155.21948 | 0.2239643 | 10 |
| 157.21948 | 0.2913039 | 10 |
| 159.21948 | 0.3747180 | 10 |
| 161.21948 | 0.4769240 | 10 |
| 163.21948 | 0.6008519 | 10 |
| 165.21948 | 0.7496171 | 10 |
| 167.21948 | 0.9264847 | 10 |
| 169.21948 | 1.1348278 | 10 |
| 171.21948 | 1.3780801 | 10 |
| 173.21948 | 1.6596847 | 10 |
| 175.21948 | 1.9830398 | 10 |
| 177.21948 | 2.3514433 | 10 |
| 179.21948 | 2.7680386 | 10 |
| 181.21948 | 3.2357626 | 10 |
| 183.21948 | 3.7572969 | 10 |
| 185.21948 | 4.3350254 | 10 |
| 187.21948 | 4.9709971 | 10 |
| 189.21948 | 5.6668974 | 10 |
| 191.21948 | 6.4240260 | 10 |
| 193.21948 | 7.2432843 | 10 |
| 195.21948 | 8.1251695 | 10 |
| 197.21948 | 9.0697786 | 10 |
| 199.21948 | 10.0768174 | 10 |
| 201.21948 | 11.1456188 | 10 |
| 203.21948 | 12.2751649 | 10 |
| 205.21948 | 13.4641155 | 10 |
| 207.21948 | 14.7108399 | 10 |
| 209.21948 | 16.0134514 | 10 |
| 211.21948 | 17.3698442 | 10 |
| 213.21948 | 18.7777308 | 10 |
| 215.21948 | 20.2346796 | 10 |
| 217.21948 | 21.7381514 | 10 |
| 219.21948 | 23.2855343 | 10 |
| 221.21948 | 24.8741772 | 10 |
| 223.21948 | 26.5014196 | 10 |
| 225.21948 | 28.1646195 | 10 |
| 227.21948 | 29.8611774 | 10 |
| 229.21948 | 31.5885578 | 10 |
| 231.21948 | 33.3443068 | 10 |
| 233.21948 | 35.1260668 | 10 |
| 235.21948 | 36.9315884 | 10 |
| 237.21948 | 38.7587385 | 10 |
| 239.21948 | 40.6055070 | 10 |
| 241.21948 | 42.4700101 | 10 |
| 243.21948 | 44.3504920 | 10 |
| 245.21948 | 46.2453242 | 10 |
| 247.21948 | 48.1530040 | 10 |
| 249.21948 | 50.0721509 | 10 |
| 251.21948 | 52.0015025 | 10 |
| 253.21948 | 53.9399089 | 10 |
| 255.21948 | 55.8863274 | 10 |
| 257.21948 | 57.8398159 | 10 |
| 259.21948 | 59.7995263 | 10 |
| 261.21948 | 61.7646982 | 10 |
| 263.21948 | 63.7346518 | 10 |
| 265.21948 | 65.7087817 | 10 |
| 267.21948 | 67.6865504 | 10 |
| 269.21948 | 69.6674820 | 10 |
| 271.21948 | 71.6511568 | 10 |
| 273.21948 | 73.6372054 | 10 |
| 275.21948 | 75.6253036 | 10 |
| 277.21948 | 77.6151678 | 10 |
| 279.21948 | 79.6065506 | 10 |
| 281.21948 | 81.5992365 | 10 |
| 283.21948 | 83.5930385 | 10 |
| 285.21948 | 85.5877946 | 10 |
| 287.21948 | 87.5833647 | 10 |
| 289.21948 | 89.5796282 | 10 |
| 291.21948 | 91.5764811 | 10 |
| 293.21948 | 93.5738342 | 10 |
| 295.21948 | 95.5716112 | 10 |
| 297.21948 | 97.5697468 | 10 |
| 299.21948 | 99.5681853 | 10 |
| 301.21948 | 101.5668791 | 10 |
| 303.21948 | 103.5657879 | 10 |
| 305.21948 | 105.5648774 | 10 |
| 307.21948 | 107.5641187 | 10 |
| 309.21948 | 109.5634872 | 10 |
| 311.21948 | 111.5629622 | 10 |
| 313.21948 | 113.5625262 | 10 |
| 315.21948 | 115.5621646 | 10 |
| 317.21948 | 117.5618650 | 10 |
| 319.21948 | 119.5616170 | 10 |
| 321.21948 | 121.5614119 | 10 |
| 323.21948 | 123.5612425 | 10 |
| 325.21948 | 125.5611027 | 10 |
| 327.21948 | 127.5609875 | 10 |
| 329.21948 | 129.5608926 | 10 |
| 331.21948 | 131.5608145 | 10 |
| 333.21948 | 133.5607503 | 10 |
| 335.21948 | 135.5606975 | 10 |
| 337.21948 | 137.5606543 | 10 |
| 339.21948 | 139.5606188 | 10 |
| 341.21948 | 141.5605897 | 10 |
| 343.21948 | 143.5605659 | 10 |
| 345.21948 | 145.5605464 | 10 |
| 347.21948 | 147.5605305 | 10 |
| 349.21948 | 149.5605175 | 10 |
| 351.21948 | 151.5605069 | 10 |
| 353.21948 | 153.5604982 | 10 |
| 355.21948 | 155.5604912 | 10 |
| 357.21948 | 157.5604854 | 10 |
| 359.21948 | 159.5604808 | 10 |
| 361.21948 | 161.5604770 | 10 |
| 363.21948 | 163.5604739 | 10 |
| 365.21948 | 165.5604714 | 10 |
| 367.21948 | 167.5604693 | 10 |
| 369.21948 | 169.5604677 | 10 |
| 371.21948 | 171.5604663 | 10 |
| 373.21948 | 173.5604652 | 10 |
| 375.21948 | 175.5604643 | 10 |
| 377.21948 | 177.5604636 | 10 |
| 379.21948 | 179.5604630 | 10 |
| 381.21948 | 181.5604626 | 10 |
| 383.21948 | 183.5604622 | 10 |
| 385.21948 | 185.5604619 | 10 |
| 387.21948 | 187.5604617 | 10 |
| 389.21948 | 189.5604615 | 10 |
| 391.21948 | 191.5604613 | 10 |
| 393.21948 | 193.5604612 | 10 |
| 395.21948 | 195.5604611 | 10 |
| 397.21948 | 197.5604610 | 10 |
| 399.21948 | 199.5604609 | 10 |
| 401.21948 | 201.5604608 | 10 |
| 403.21948 | 203.5604608 | 10 |
| 405.21948 | 205.5604608 | 10 |
| 407.21948 | 207.5604607 | 10 |
| 409.21948 | 209.5604607 | 10 |
| 411.21948 | 211.5604607 | 10 |
| 413.21948 | 213.5604607 | 10 |
| 415.21948 | 215.5604607 | 10 |
| 417.21948 | 217.5604607 | 10 |
| 419.21948 | 219.5604607 | 10 |
| 421.21948 | 221.5604606 | 10 |
| 423.21948 | 223.5604606 | 10 |
| 425.21948 | 225.5604606 | 10 |
| 427.21948 | 227.5604606 | 10 |
| 429.21948 | 229.5604606 | 10 |
| 431.21948 | 231.5604606 | 10 |
| 433.21948 | 233.5604606 | 10 |
| 435.21948 | 235.5604606 | 10 |
| 437.21948 | 237.5604606 | 10 |
| 439.21948 | 239.5604606 | 10 |
| 441.21948 | 241.5604606 | 10 |
| 443.21948 | 243.5604606 | 10 |
| 445.21948 | 245.5604606 | 10 |
| 447.21948 | 247.5604606 | 10 |
| 449.21948 | 249.5604606 | 10 |
| 451.21948 | 251.5604606 | 10 |
| 453.21948 | 253.5604606 | 10 |
| 455.21948 | 255.5604606 | 10 |
| 457.21948 | 257.5604606 | 10 |
| 459.21948 | 259.5604606 | 10 |
| 461.21948 | 261.5604606 | 10 |
| 463.21948 | 263.5604606 | 10 |
| 465.21948 | 265.5604606 | 10 |
| 467.21948 | 267.5604606 | 10 |
| 469.21948 | 269.5604606 | 10 |
| 471.21948 | 271.5604606 | 10 |
| 473.21948 | 273.5604606 | 10 |
| 475.21948 | 275.5604606 | 10 |
| 477.21948 | 277.5604606 | 10 |
| 479.21948 | 279.5604606 | 10 |
| 481.21948 | 281.5604606 | 10 |
| 483.21948 | 283.5604606 | 10 |
| 485.21948 | 285.5604606 | 10 |
| 487.21948 | 287.5604606 | 10 |
| 489.21948 | 289.5604606 | 10 |
| 491.21948 | 291.5604606 | 10 |
| 493.21948 | 293.5604606 | 10 |
| 495.21948 | 295.5604606 | 10 |
| 497.21948 | 297.5604606 | 10 |
| 499.21948 | 299.5604606 | 10 |
| 501.21948 | 301.5604606 | 10 |
| 503.21948 | 303.5604606 | 10 |
| 505.21948 | 305.5604606 | 10 |
| 507.21948 | 307.5604606 | 10 |
| 509.21948 | 309.5604606 | 10 |
| 511.21948 | 311.5604606 | 10 |
| 513.21948 | 313.5604606 | 10 |
| 515.21948 | 315.5604606 | 10 |
| 517.21948 | 317.5604606 | 10 |
| 519.21948 | 319.5604606 | 10 |
| 521.21948 | 321.5604606 | 10 |
| 523.21948 | 323.5604606 | 10 |
| 525.21948 | 325.5604606 | 10 |
| 527.21948 | 327.5604606 | 10 |
| 529.21948 | 329.5604606 | 10 |
| 531.21948 | 331.5604606 | 10 |
| 533.21948 | 333.5604606 | 10 |
| 535.21948 | 335.5604606 | 10 |
| 537.21948 | 337.5604606 | 10 |
| 539.21948 | 339.5604606 | 10 |
| 541.21948 | 341.5604606 | 10 |
| 543.21948 | 343.5604606 | 10 |
| 545.21948 | 345.5604606 | 10 |
| 547.21948 | 347.5604606 | 10 |
| 549.21948 | 349.5604606 | 10 |
| 551.21948 | 351.5604606 | 10 |
| 553.21948 | 353.5604606 | 10 |
| 555.21948 | 355.5604606 | 10 |
| 557.21948 | 357.5604606 | 10 |
| 559.21948 | 359.5604606 | 10 |
| 561.21948 | 361.5604606 | 10 |
| 563.21948 | 363.5604606 | 10 |
| 565.21948 | 365.5604606 | 10 |
| 567.21948 | 367.5604606 | 10 |
| 569.21948 | 369.5604606 | 10 |
| 571.21948 | 371.5604606 | 10 |
| 573.21948 | 373.5604606 | 10 |
| 575.21948 | 375.5604606 | 10 |
| 577.21948 | 377.5604606 | 10 |
| 579.21948 | 379.5604606 | 10 |
| 581.21948 | 381.5604606 | 10 |
| 583.21948 | 383.5604606 | 10 |
| 585.21948 | 385.5604606 | 10 |
| 587.21948 | 387.5604606 | 10 |
| 589.21948 | 389.5604606 | 10 |
| 591.21948 | 391.5604606 | 10 |
| 593.21948 | 393.5604606 | 10 |
| 595.21948 | 395.5604606 | 10 |
| 597.21948 | 397.5604606 | 10 |
| 599.21948 | 399.5604606 | 10 |
| 601.21948 | 401.5604606 | 10 |
| 603.21948 | 403.5604606 | 10 |
Se construyó una tabla de precios del activo subyacente en un rango definido por los percentiles 2.5% y 97.5%, con incrementos de 2 dólares. A partir de esta, se calcularon los precios de la opción para distintos momentos en el tiempo, evidenciando un comportamiento no lineal característico de la convexidad. Las tablas muestran la evolución del precio de la opción para distintos niveles del activo subyacente en diferentes momentos del tiempo. Se observa que, a medida que se acerca el vencimiento, el comportamiento del precio de la opción se vuelve más sensible a cambios en el activo, evidenciando una mayor convexidad.
La representación gráfica de la convexidad en la opción de NVDA evidencia la naturaleza no lineal del precio de los derivados financieros frente al activo subyacente. Se observa que, a medida que el precio de la acción aumenta, el valor de la opción crece de manera acelerada, confirmando la asimetría inherente a estos instrumentos. Asimismo, las curvas muestran que el tiempo hasta el vencimiento amplifica la sensibilidad del precio de la opción, lo cual resalta la importancia de las funciones griegas —especialmente la Gamma— en la gestión del riesgo. En consecuencia, el análisis de convexidad permite concluir que las opciones no solo ofrecen un perfil de ganancias potencialmente ilimitadas, sino que también requieren un control riguroso de su exposición, dado que pequeñas variaciones en el subyacente pueden generar cambios significativos en su valoración (Hull, 2018).
Figura 1 Convexidad de la opción de NVDA en distintos horizontes temporales. Fuente: elaboración propia en RStudio.
FUNCIONES GRIEGAS
#GRÁFICAS DE GRIEGAS
La evolución de la Delta en distintos horizontes temporales muestra que la sensibilidad de la opción frente al activo subyacente se intensifica a medida que se aproxima el vencimiento, mientras que en periodos intermedios la respuesta es más gradual. Este comportamiento confirma la naturaleza dinámica de las griegas y su relevancia para la gestión del riesgo en derivados (Hull, 2018).
la Gamma alcanza su máximo cuando el precio del activo se aproxima al precio de ejercicio, especialmente en vencimientos cortos. A medida que el tiempo hasta vencimiento aumenta, la Gamma se reduce y se distribuye de manera más amplia, lo que implica una menor variabilidad en la Delta y, por tanto, un riesgo más estable en la cobertura de opciones (Hull, 2018)
la Theta es más negativa en vencimientos cortos, reflejando una pérdida acelerada del valor temporal de la opción. En horizontes intermedios y largos, la sensibilidad al paso del tiempo se reduce, lo que confirma que el time decay es más relevante en opciones cercanas al vencimiento (Hull, 2018).”
la Vega es más alta en vencimientos largos, lo que indica que el valor de la opción es más sensible a variaciones en la volatilidad del activo subyacente. En vencimientos cortos, la sensibilidad es menor, confirmando que la volatilidad impacta de manera más significativa en opciones de largo plazo (Hull, 2018).
La Rho es prácticamente nula en vencimientos cortos, pero aumenta significativamente en horizontes largos. Esto evidencia que las tasas de interés tienen un impacto más relevante en la valoración de opciones de largo plazo, lo cual debe ser considerado en escenarios de cambios en la política monetaria (Hull, 2018).
#CONCLUSIONES: El análisis realizado permitió evaluar la convexidad y las sensibilidades de una opción tipo call sobre la acción de NVIDIA mediante el modelo de Black-Scholes, incorporando supuestos más realistas como volatilidad dinámica y tasas de interés interpoladas.
A partir de la simulación del precio del activo en un rango probabilístico definido por los percentiles 2.5% y 97.5%, se evidenció que el valor de la opción presenta un comportamiento no lineal, confirmando la convexidad característica de los instrumentos derivados. Esta propiedad implica que el valor de la opción responde de manera asimétrica a cambios en el precio del subyacente, especialmente a medida que se aproxima la fecha de vencimiento.
El análisis de las griegas (Delta, Gamma, Theta, Vega y Rho) permitió comprender de forma integral la sensibilidad del precio de la opción frente a distintos factores de riesgo. Se observó que la Delta tiende a comportarse de manera más abrupta cerca del vencimiento, mientras que la Gamma alcanza sus valores máximos en niveles cercanos al precio de ejercicio, reflejando una mayor inestabilidad en la cobertura. Por su parte, la Theta evidencia la pérdida acelerada de valor temporal en horizontes cortos, la Vega destaca la relevancia de la volatilidad en opciones de largo plazo y la Rho muestra la influencia de las tasas de interés principalmente en vencimientos más largos.
En conjunto, los resultados confirman que la valoración de opciones depende de múltiples factores interrelacionados, y que su comportamiento es altamente dinámico y no lineal. Esto resalta la importancia de comprender tanto la convexidad como las sensibilidades para una adecuada gestión del riesgo y una toma de decisiones informada en los mercados financieros.
#REFERENCIAS
Hull, J. C. (2018). Options, Futures, and Other Derivatives. Pearson. Yahoo Finance. (2026). Datos históricos y opciones de NVIDIA (NVDA). CME Group. (2026). Term SOFR Rates. Plotly Technologies Inc. (2026). Plotly for R. R Core Team. (2026). R: A Language and Environment for Statistical Computing.