[R] How to remove double loop?

[Ray Brownrigg]

On Thu, 20 Mar 2008, Alberto Monteiro wrote:
> Jonas Malmros wrote:
> > But what if I had a matrix, where the last column was filled with
> > values first (again, a for loop), and the rest was filled by using a
> > double loop?
>
> Killing one of those loops is quite simple. The other may be
> harder.
>
> > OVal <- matrix(0, n+1, n+1)
> >
> > for(i in 0:n){
> >     OVal[i+1, n+1] <- max(Val[i+1, n+1]-K, 0)
> > }
>
> This is straightforward: notice that most R arithmetic
> functions that operate in scalars also operate in vectors,
> matrices and arrays. For example, sin(pi) is zero, but
> sin(c(0,pi/2,pi,3*pi/2)) is c(0, 1, 0, -1) (with some
> rounding errors).
>
> The loop in _i_ is just to take the max of a column? So:
>
> OVal[1:(n+1), n+1] <- # this is a vector
>   max(Val[1:(n+1), n+1] -  # this is another vector
>     - K, 0)   # vector - scalar = vector, max(vector) = vector
>
> or, in one line:
>
> OVal[1:(n+1), n+1] <- max(Val[1:(n+1), n+1] - K, 0)
>
> You can drop the indices, as you are taking all of them:
>
> OVal[,n+1] <- max(Val[,n+1] - K, 0)
>
> > for(i in seq(n,1, by=-1)){
> >     for(j in 0:(i-1)){
> >         OVal[j+1, i] <- a*((1-p)*OVal[j+1, i+1]+p*OVal[j+2, i+1])
> >     }
> > }
>
> The inner loop is simple, but somehow tricky.
>
> You are computing each j-th term as the same j-th term combined
> with the (j+1)-th term. So, you take a combination of js in
> the 1:i range and combine with js in the 2:(i+1) range... So:
>
>     OVal[1:i, i] <- a*((1-p)*OVal[1:i, i+1] + p*OVal[2:(i+1), i+1])
>
> The outer loop (in i) probably can't be optimized.
>
> Alberto Monteiro

Unfortunately, neither of these loop removals gives the same answers as the 
original code :-( [Hence the request for reproducible code].  Try:
TreeMy(K=0.5)

What does work though is the following:

TreeMy <- 
function(S=50,K=50,sigma=0.4,r=0.1,Time=1,n=3){
dtime <- Time/n
u <- exp(sigma*sqrt(dtime))
d <- 1/u
p <- (exp(r*dtime)-d)/(u-d)
a <- exp(-r*dtime)

OVal <- matrix(0, n+1, n+1)

Val <- outer(0:n, 0:n, function(i, j) ifelse(i <= j, u^i*d^(j-i), 0))

OVal[, n+1] <- pmax(Val[, n+1]-K, 0)

for (i in n:1) {
    OVal[1:i, i] <-  colSums(matrix(a*c((1 - p), p)*
    OVal[, i + 1][rep(1:2, i) + rep(0:(i - 1), each=2)], ncol=i))
}

list("Stock Price"=Val, "Option Price"=OVal)
}

Here the first loop is removed with a simple pmax() (the 'vector' form of 
max()).  The second loop uses the trick of expanding the column operations 
into a vector using recycling then restructuring them back into a matrix.  
This provides a significant speed improvement.

HTH
Ray Brownrigg