R help - Aug 2011 - Calibrating the risk free interest rate using nlminb

Dear R-users

I am trying to find a value for the risk free rate minimizing the difference
between a BS call value with impl. volatilities minus the market price of a
call (assuming this is just the average bid ask price)

Here is my data:
http://r.789695.n4.nabble.com/file/n3747509/S%26P_500_calls%2C_jan-jun_2010.csv
S%26P_500_calls%2C_jan-jun_2010.csv 

S0 <- 1136.03
q <- 0.02145608
S0 <- spot[10,6]
S0
strike <- marketdata[1:460,9]
T <- marketdata[1:460,17]/365		#Notice the T is measured in years now
implvol <- marketdata[1:460,12]
ask <- marketdata[1:460,10]
bid <- marketdata[1:460,11]
mid_bidask <- (bid + ask) /2
r <- 0.020

And my code:
BS_Call <- function(S0, K, T, r, sigma, q)                  # Which works
fine
{
	sig <- sigma * sqrt(T)
	d1 <- (log (S0/K) + (r - q + sigma^2/2) * T ) / sig
	d2 <- d1 - sig
	Presentvalue <- exp(-r*T)
	return (S0 * exp(-q*T) * pnorm(d1) - K*Presentvalue*pnorm(d2))
}


#I now define the function I later want to minimize with respect to r
difference <- function (S0, K, T, r, sigma, q, mid)     
{
return(BS_Call(S0, strike, T, r, sigma, q)- mid_bidask))
} 
# which gives 460 values as it should, as T = 460

f <- function(x) difference(S0, strike, T, x, implvol, q, mid_bidask)
# If I now run
nlminb(start=r, f)

$par
[1] -0.9800001

$objective
[1] -395.85

$convergence
[1] 0

$iterations
[1] 2

$evaluations
function gradient 
       3        4 

$message
[1] "relative convergence (4)"
# But the risk free interest rate should not be negative, and the objective
function gives an negative values when in should be minizimed to 0. 

# I then try adding the following to nlminb, as I understand it to set the
objective function to non-negative
nlminb(start=r, f, control = list(rel.tol=1e-20))

$par
[1] 0.02

$objective
[1] 0

$convergence
[1] 1

$iterations
[1] 0

$evaluations
function gradient 
       0        0 

$message
[1] "'control' component 'rel.tol' = 1e-20, is out of
range"

#which always gives me the initial value
I have tried to read other relevant R help posts, but nothing I do seems to
work. Is there anyone who can help me with what I am doing wrong?

I have also try summing the difference function, but the same errors occur. 
Thanks

Rikke

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Newbie wrote:
> 
> Dear R-users
> 
> I am trying to find a value for the risk free rate minimizing the
> difference between a BS call value with impl. volatilities minus the
> market price of a call (assuming this is just the average bid ask price)
> 
> Here is my data:
> 
>
http://r.789695.n4.nabble.com/file/n3747509/S%26P_500_calls%2C_jan-jun_2010.csv
> S%26P_500_calls%2C_jan-jun_2010.csv 
> 
> S0 <- 1136.03
> q <- 0.02145608
> S0 <- spot[10,6]
> S0
> strike <- marketdata[1:460,9]
> T <- marketdata[1:460,17]/365		#Notice the T is measured in years now
> implvol <- marketdata[1:460,12]
> ask <- marketdata[1:460,10]
> bid <- marketdata[1:460,11]
> mid_bidask <- (bid + ask) /2
> r <- 0.020
> 
> And my code:
> BS_Call <- function(S0, K, T, r, sigma, q)                  # Which
works
> fine
> {
> 	sig <- sigma * sqrt(T)
> 	d1 <- (log (S0/K) + (r - q + sigma^2/2) * T ) / sig
> 	d2 <- d1 - sig
> 	Presentvalue <- exp(-r*T)
> 	return (S0 * exp(-q*T) * pnorm(d1) - K*Presentvalue*pnorm(d2))
> }
> 
> 
> #I now define the function I later want to minimize with respect to r
> difference <- function (S0, K, T, r, sigma, q, mid)     
> {
> return(BS_Call(S0, strike, T, r, sigma, q)- mid_bidask))
> } 
> # which gives 460 values as it should, as T = 460
> 
> f <- function(x) difference(S0, strike, T, x, implvol, q, mid_bidask)
> # If I now run
> nlminb(start=r, f)
> 
> $par
> [1] -0.9800001
> 
> $objective
> [1] -395.85
> 
> $convergence
> [1] 0
> 
> $iterations
> [1] 2
> 
> $evaluations
> function gradient 
>        3        4 
> 
> $message
> [1] "relative convergence (4)"
> # But the risk free interest rate should not be negative, and the
> objective function gives an negative values when in should be minizimed to
> 0. 
> 
> # I then try adding the following to nlminb, as I understand it to set the
> objective function to non-negative
> nlminb(start=r, f, control = list(rel.tol=1e-20))
> 
> $par
> [1] 0.02
> 
> $objective
> [1] 0
> 
> $convergence
> [1] 1
> 
> $iterations
> [1] 0
> 
> $evaluations
> function gradient 
>        0        0 
> 
> $message
> [1] "'control' component 'rel.tol' = 1e-20, is out of
range"
> 
> #which always gives me the initial value
> I have tried to read other relevant R help posts, but nothing I do seems
> to work. Is there anyone who can help me with what I am doing wrong?
> 
> I have also try summing the difference function, but the same errors
> occur. 
> Thanks
> 
> Rikke
> 

How did you read the .csv?
Where does variable "spot" come from?

Berend


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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

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