Create asset paths in R using GBM and Monte Carlo

Question

I would like to create asset paths using Geometric BM and Monte carlo simulation for a Basket option. I have the correlation matrix, the covariance matrix. I used Cholesky on correlation matrix and then I multiplied it with a vector of random numbers to create correlated random numbers. My issue is that when I try to put them into my equation, my paths are only decreasing. Could you please help me?

$i$ for number of simulations

$j$ for number of assets

$r_n$ is a matrix with random numbers

correl is the cholesky (from correlation matrix)

trial_par$sigma[1,1] is the covariance matrix

dt=1/365 are the steps

asset1[1,1]=price of the asset now

for (i in 1:5){ for (a in 2:(n+1)){ for (b in 1:3){ rn[a,b]=rnorm(1,0,sqrt(dt))}} crn=rn%*%correl for (j in 2:366){ asset1[i,j]=asset1[i,j-1]*exp((r-0.5*trial_par$sigma[1,1])*dt+\sqrt{trial_par$sigma[1,1])*crn[j,1]*sqrt(dt)} asset2[i,j]=asset2[i,j-1]*exp((r-0.5*trial_par$sigma[2,2])*dt+$\sqrt{trial_par sigma[2,2]}$*crn[j,2]*sqrt(dt)) asset3[i,j]=asset3[i,j-1]*exp((r-0.5*trial_par$sigma[3,3])*dt+sqrt(trial_par$sigma[3,3])*crn[j,3]*sqrt(dt)) }}

Answer 1

Hello and thank you for replying.

I would like to create asset paths using Geometric BM and Monte carlo simulation for a Basket option. I have the correlation matrix, the covariance matrix. I used Cholesky on correlation matrix and then I multiplied it with a vector of random numbers to create correlated random numbers. My issue is that when I try to put them into my equation, my paths are only decreasing. Could you please help me?

$i$ for number of simulations

$j$ for steps(j=2 is the first price after today)

$r_n$ is a matrix with random numbers

correl is the cholesky (from correlation matrix)

trial_par$sigma[1,1] is the covariance matrix

dt=1/365 are the steps

asset1[1,1]=price of the asset now

crn the matrix with correlated random numbers

edit:I changed the j-it was a mistake

My intention is the following: I would like to use cholesky(not sure if I have to use cholesky on covariance matrix or correlation matrix), and multiply it with uncorrelated numbers to produce correlated random numbers(it will happen i times, each for every simulation).

The code is the following:

for (i in 1:5){

for (a in 2:(n+1)){

for (b in 1:3){

  rn[a,b]=rnorm(1,0,sqrt(dt))}}

crn=rn%*%correl

for (j in 2:366){

asset1[i,j]=asset1[i,j-1]*exp((r-0.5*trial_par$sigma[1,1])*dt+sqrt(trial_par$sigma[1,1])*crn[j,1]*sqrt(dt))


asset2[i,j]=asset2[i,j-1]*exp((r-0.5*trial_par$sigma[2,2])*dt+sqrt(trial_par$sigma[2,2])*crn[j,2]*sqrt(dt))


asset3[i,j]=asset3[i,j-1]*exp((r-0.5*trial_par$sigma[3,3])*dt+sqrt(trial_par$sigma[3,3])*crn[j,3]*sqrt(dt))

}}