R Programming Implement Newton Raphson Algorithm

Question

I'm a student and I want to solve a textbook exercise which is implementing Newton-Raphson Algorithm in R Programming.

This is the exercise:

#Inputs:
  s0 <- 2.36
  E <- 2.36 
  r <- 0.01 
  t <- 1 
  c <- 0.1875

#Initial values
  sigma <-0.10 
  sig <- rep(0,10)
  sig[1] <- sigma

#Newton-Raphson method:
for(i in 2:100){
  d1 <- (log(s0/E)+(r+sigma^2/2)*t)/(sigma*sqrt(t)) 
  d2 <- d1-sigma*sqrt(t) 
  f <- s0*pnorm(d1)-E*exp(r*t)*pnorm(d2)-c 

#Derivative of d1 w.r.t. sigma:
      d11 <- (sigma^2*t*sqrt(t)-(log(s0/E)+(r+sigma^2/2)*t)*sqrt(t))/(sigma^2*t) 

#Derivative of d2 w.r.t. sigma: 
      d22 <- d11-sqrt(t) 

#Derivative of f(sigma):
      f1 <- s0*dnorm(d1)*d11-E*exp(-r*t)*dnorm(d2)*d22 

#Update sigma: 
    sigma <- sigma - f/f1 
    sig[i] <- sigma
    if(abs(sig[i]-sig[i-1]) < 0.00000001){sig <- sig[1:i]; break}}sig

The result for sig is [1] 0.1000000 0.2140636 0.2117527 0.2117864 0.2117859 0.2117859

But the answer in my textbook is: [1] 0.1000000 0.1877024 0.1876218 0.1876218

Can some one tell me where I've gone wrong?

Answer 1

The problem is that the expressions for f and f1 aren't matching. The term exp(r*t) in the expression for f should read exp(-r*t) if the expression for f1 is correct. Define f like this:

f <- s0*pnorm(d1)-E*exp(-r*t)*pnorm(d2)-c

When you change this, you'll get an output:

[1] 0.1000000 0.1877024 0.1876218 0.1876218