Calibrating the risk free interest rate using nlminb

I used:
marketdata <- read.csv(file="S&P 500 calls, jan-jun 2010.csv",
header=TRUE, sep=";")
after changing my directory to where the file is saved. 
The data imported should be correct. 
The spot is equal to S0, I typed it double in the post, sorry for that. 
So S0 = 1136.03 is the spot

1. it seems that your function difference contains an error: Superfluous
closing )
2. difference contains another possible error: it is not using the last
argument mid
It's using the global variable mid_bidask.

It should read

difference <- function (S0, K, T, r, sigma, q, mid)
{
return(BS_Call(S0, strike, T, r, sigma, q)- mid)
}

Your function returns a vector of length 460 as you correctly said.
But in  the documentation of nlminb it clearly states that the objective
must return a scalar value.
(NB. It doesn't seem to check?)

So in function f you should at least reduce the vector to a scalar.
In this case I would minimize the sum of the differences squared to prevent
the optimization routine
from getting negative values for differences.
So define f as follows

f <- function(x) sum(difference(S0, strike, T, x, implvol, q, mid_bidask)^2)

and then 

nlminb(start=r, f)

gives

nlminb(start=r, f)

$par
[1] 0.01268534

$objective
[1] 3140.738

$convergence
[1] 0

$iterations
[1] 5

$evaluations
function gradient 
       8        5 

$message
[1] "relative convergence (4)"

If you try nlm and optim (with method="BFGS) you'll get similar results.

It's up to you to decide if the outcome is plausible/acceptable.

Berend



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