Thread: Applying Cholesky decomposition to Monte-Carlo simulation

Can anyone help me to understand how to apply Cholesky decomposition to Monte-Carlo simulation? I just about understand Monte-Carlo simulation and can model it in Excel but as I understand it catering for correlation is acheived by somehow constraining the random numbers that are used.

Originally Posted by JustAQuestion

Can anyone help me to understand how to apply Cholesky decomposition to Monte-Carlo simulation? I just about understand Monte-Carlo simulation and can model it in Excel but as I understand it catering for correlation is acheived by somehow constraining the random numbers that are used.

Be more specific. Cholesky decomposition is a efficient and convenient LU decomposition that is appropriate for symmetric positive definite matrices (like the covariance matrix for a vector of random variables).

CB

I am simulating growth by generating random numbers using e^(u.Delta T + E.sigma.sqrt(Delta T))
where
e = e
u = an interest rate %
Delta T = time period
E = a draw from the normal distribution curve
sigma = volatility

I am going to add the above to the starting number to get my new number. I am doing this to many different numbers, all of whose changes in value will obey given correlations with the other numbers.

I understand that Cholesky decomposition is a technique used to realise those correlations into the above (which I have noted don't appear to use matrices as things stand).

All I kn ow is that I need to 'manipulate the random selection process so that correlation relationships are preserved.'

The best thing I can think of is that you only generate one random number/growth estimate then assign a new random number to all the remaing numbers according to their correlations with the first (/each other?). I don't have anything to check my work against.

I know that if I'm doing this for just two numbers and the correlatio is one I overwrite the random variable of the second with the firsts'. If the correlation is -1 I overwrite the seconds' random variable with the firsts * -1.
If the correlation is somewhere in between, e.g., 0.9, I think I will need to do something using both numbers and 0.9. If there are more than two numbers to generate for I will need a matrix of correlations (which will have a 1 along the diagonal from top left to bottom right) and it is here I can see where I will need matrix maths.

The Cholesky part is in turn referred to as 'Cholesky Decomposition', 'Cholesky Adjustment' and 'Cholesky Factorisation', I can only assume these are synonymous.

An alternative method to Monte-Carlo simulation is depicted in the attached JPEG. It uses matrices wheras my Monte-carlo simulation so far hadn't but I guess I will need to somehow.