For example, if I have a lattice that looks like this:
133.1
/
121
/ \
110 108.9
/ \ /
100 99
\ / \
90 89.1
\ /
81
\
72.9
Where the lattice starts at 100 and either goes up with a factor 1.1 and goes down with a factor 0.9. This lattice has 3 periods in which it goes up or down. It is clear that this matrix could be filled in for more periods.
The lattice in matrix form looks like this:
[,1] [,2] [,3] [,4]
[1,] 100 110 121 133.1
[2,] NA 90 99 108.9
[3,] NA NA 81 89.1
[4,] NA NA NA 72.9
I'm working in R. The code to generate the lattice matrix is as follows:
#Parameters
S0 <- 100 #price at t0
u <- 1.1 #up factor
d <- 0.9 #down factor
n <- 3 #number of periods
#Matrix for the prices
prices <- matrix(data=NA, nrow=(n+1), ncol=(n+1))
prices[1,1] <- S0
#Fill the matrix
for(column in 2:(n+1)){
for(row in 1:(column-1)){
prices[row,column] <- u*prices[row,column-1];
}
prices[column,column] <- d*prices[column-1,column-1];
}
I would like to create a code that generates a matrix with all possible paths through the lattice. For this example, it would look like this:
[,1] [,2] [,3] [,4]
[1,] 100 110 121 133.1
[2,] 100 110 121 108.9
[3,] 100 110 99 108.9
[4,] 100 110 99 89.1
[5,] 100 90 99 108.9
[6,] 100 90 99 89.1
[7,] 100 90 81 89.1
[8,] 100 90 81 72.9
I've been struggling with this piece of code for hours now, so any help would be much appreciated! Thanks in advance! :)
Each path of length n corresponds
to a sequence of up and down movements:
you just have to enumerate all those sequences.
If you already have the sequences of length n-1,
as a matrix u,
the sequences of length n can be obtained as
rbind(
cbind( u, .9 ),
cbind( u, 1.1 )
)
You can put it in a function, and call it n times.
n <- 4
up <- 1.1
down <- .9
m <- Reduce(
function(u,v) rbind( cbind( u, up ), cbind( u, down ) ),
rep(NA,n),
100
)
t(apply(m, 1, cumprod))
# [1,] 100 110 121 133.1 146.41
# [2,] 100 90 99 108.9 119.79
# [3,] 100 110 99 108.9 119.79
# [4,] 100 90 81 89.1 98.01
# [5,] 100 110 121 108.9 119.79
# [6,] 100 90 99 89.1 98.01
# [7,] 100 110 99 89.1 98.01
# [8,] 100 90 81 72.9 80.19
# [9,] 100 110 121 133.1 119.79
# [10,] 100 90 99 108.9 98.01
# [11,] 100 110 99 108.9 98.01
# [12,] 100 90 81 89.1 80.19
# [13,] 100 110 121 108.9 98.01
# [14,] 100 90 99 89.1 80.19
# [15,] 100 110 99 89.1 80.19
# [16,] 100 90 81 72.9 65.61