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.
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.