[R-SIG-Finance] sufficient n for a binomial option pricing model

[J Toll]

Hi,

I have a question regarding the selection of n, the number of time
steps, in a binomial option pricing model.  I suppose my question is
not strictly related to R.  As larger values should be more accurate,
what I've read on the subject simply suggests that you use a
sufficiently large value for your purposes.  So I've been trying to
evaluate what is a sufficiently large value of n for my purposes.  Is
there any rule of thumb regarding the value of n?

When using the fOptions package CRRBinomialTreeOption function, with
varying n, the price oscillates back and forth converging on a price.
This can be clearly seen through plotting.

require(fOptions)

x <- function(n) {

  CRRBinomialTreeOption(TypeFlag = "ca",
                        S = 50,
                        X = 50,
                        Time = 1/12,
                        r = 0.02,
                        b = 0.02,
                        sigma = 0.18,
                        n = n)@price
}

y <- sapply(1:100, x)               # mean(y) == 1.079693
plot(y)

Given this oscillation, my question is whether it would be "better" to
compute two prices using two smaller, consecutive values of n rather
than one large value?  Or is there some other better way?

For example, using n =1000 or 1001, the option prices are within 5
hundredths of a cent, but the calculation is extremely slow for
either.

x(1000)                                    # 1.077408
x(1001)                                    # 1.077926

mean(sapply(1000:1001, x))      # 1.077667

Comparatively, taking the mean of n= 40 and 41 yields a value very
close to the middle of the range, yet is much faster.

mean(sapply(40:41, x))             # 1.0776

It seems like averaging two smaller, consecutive values of n is
basically as accurate and far faster than using large values of n.  I
was hoping someone might have some insight into why this might or
might not be a valid approach.  Thanks.


James