I have an assignment that requires me to calculate the implied volatility of a series of options using their parameters and market price. I understand that the easy way to do this would be to use the compute.implied.volatility function within R, however this question requires me to solve this using the nlm function. I understand that in this case I am wanting to minimise the distance between the actual price and my calculated price such that the distance is zero. To do this I obviously want to change the volatility in the option such that it sets my calculated price equal to the market price. The trouble I am having with this question is getting the nlm function to work, as we have not been taught much about it in this course.
I understand that I am meant to feed in a loop to nlm that enables it to iteratively calculate until it finds the minimum value that produces the result. I believe I'm not feeding in a function that works with the nlm, as I am currently getting an error of "Invalid function value in nlm optimizer".
I have attached my code as well as the inputs to work with, please let me know if I've written it incorrectly or if I need to tinker with it a little bit more to get an answer out for the required volatility. Thanks for any and all help!
```{r}
# Load in the library's and clear workspace
{cat("\014")
rm(list=ls(all=TRUE))
options(digits=6)}
library(fBasics)
library(knitr)
library(zoo)
library(psych)
library(lubridate)
library(stats)
library(boot)
library(matrixStats)
# First setup the parameter vectors to use in calculating IV
S <- rep(1200, 12) # Price at time = 0
r <- rep(0.01, 12) # Current interest rate
T <- rep(44/365, 12) # Time till maturity of the options
X <- c(1100,1120,1140,1160,1180,1200,1220,1240,1260,1280,1300,1320) #
Strike prices of each option
type <- c(1,1,1,1,1,1,0,0,0,0,0,0) # 1 = Put and 0 = Call variable
mktprice <- c(10.5,13.8,18.2,23.9,31.2,40,31.8,23.9,17.5,12.5,9.0,6.3)
# Market price of each option
sigma <- rep(0.2, 12) # Initial guess for sigma
options.df <- data.frame(S, X, r, T, type, mktprice, sigma)
# 1. First specify the Black-Scholes Function
BS.function.call <- function(sigma, options.df){
d1 <- (log(S/X) + (r + sigma^2/2)*T) / (sigma*sqrt(T))
d2 <- d1 - sigma*sqrt(T)
st <- S * pnorm(d1) - X*exp(-r*T)*pnorm(d2)
distance <- abs(mktprice - st)
return(distance) # We want to set the distance between market price
and calculated price = 0 using nlm by changing sigma
}
BS.function.put <- function(sigma, options.df){
d1 <- (log(S/X) + (r + sigma^2/2)*T) / (sigma*sqrt(T))
d2 <- d1 - sigma*sqrt(T)
st <- -S * pnorm(-d1) + X*exp(-r*T)*pnorm(-d2)
distance <- abs(mktprice - st)
return(distance)
}
# 2. Create an initial guess for sigma
sigma.guess <- 0.2
# 3. Run the optimization function
for (i in 1:nrow(options.df)){
if(type == 0){
result[i] <- nlm(BS.function.call, sigma.guess, options.df)
}
else{
result[i] <- nlm(BS.function.put, sigma.guess, options.df)
}
}
