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title: “Analisis de Opciones sobre McDonalds (MCD)” author: “Tatiana Zuleta Hincapie” date: “2026-04-18” output: html_document: toc: true number_sections: true theme: cosmo highlight: tango —# 1. Resumen Ejecutivo

Análisis de convexidad y griegas para opciones europeas sobre McDonalds (MCD).

2. Parametros del modelo

S <- seq(250, 350, by=1)
K_call <- 310
K_put <- 295
r <- 0.05
sigma_call <- 0.20
sigma_put <- 0.22
T <- 1

3. Funciones Black-Scholes

d1 <- function(S,K,r,sigma,T){
  (log(S/K) + (r + 0.5*sigma^2)*T)/(sigma*sqrt(T))
}

d2 <- function(S,K,r,sigma,T){
  d1(S,K,r,sigma,T) - sigma*sqrt(T)
}

call_price <- function(S,K,r,sigma,T){
  S*pnorm(d1(S,K,r,sigma,T)) - K*exp(-r*T)*pnorm(d2(S,K,r,sigma,T))
}

put_price <- function(S,K,r,sigma,T){
  K*exp(-r*T)*pnorm(-d2(S,K,r,sigma,T)) - S*pnorm(-d1(S,K,r,sigma,T))
}

4. Convexidad

call_vals <- call_price(S, K_call, r, sigma_call, T)
put_vals  <- put_price(S, K_put, r, sigma_put, T)

plot(S, call_vals, type="l", col="blue", lwd=2,
     main="Convexidad Call vs Put", ylab="Precio")
lines(S, put_vals, col="red", lwd=2)
legend("topleft", legend=c("Call","Put"), col=c("blue","red"), lwd=2)

5. Delta

delta_call <- pnorm(d1(S,K_call,r,sigma_call,T))
delta_put  <- pnorm(d1(S,K_put,r,sigma_put,T)) - 1

plot(S, delta_call, type="l", col="blue", lwd=2,
     main="Delta", ylab="Valor")
lines(S, delta_put, col="red", lwd=2)
legend("topleft", legend=c("Call","Put"), col=c("blue","red"), lwd=2)

6. Gamma

gamma <- dnorm(d1(S,K_call,r,sigma_call,T)) / (S*sigma_call*sqrt(T))

plot(S, gamma, type="l", col="purple", lwd=2,
     main="Gamma", ylab="Valor")

7. Vega

vega <- S * dnorm(d1(S,K_call,r,sigma_call,T)) * sqrt(T)

plot(S, vega, type="l", col="darkgreen", lwd=2,
     main="Vega", ylab="Valor")

8. Theta

theta <- -(S*dnorm(d1(S,K_call,r,sigma_call,T))*sigma_call)/(2*sqrt(T)) -
          r*K_call*exp(-r*T)*pnorm(d2(S,K_call,r,sigma_call,T))

plot(S, theta, type="l", col="orange", lwd=2,
     main="Theta", ylab="Valor")

9. Conclusiones

Las opciones presentan convexidad y comportamiento no lineal.
La Gamma aumenta cerca del strike, y Theta refleja la perdida por tiempo.