I've been trying to generate correlated random numbers for Pricing a Basket option (Monte Carlo) using MATLAB. Nonetheless, I am now a bit confused when I use Cholesky decomposition i.e. when I decompose correlation matrix, I get one price. But when I decompose covariance matrix, I get another (much lower) price (I believe this is because values of element in cov matrix are much lower which in turn yields lower epsilon). So my questions are:1) when generating random numbers and using Cholesky decomposition, can one use covariance and correlation matrix interchangeably? (i.e. are they both "All positive definite"?) (It seems to me that most people are always talking about correlation matrix.)2) let X is the correlated random variable vector with a dimension of (path * number of stocks), and my X isX = (C'*Z)', where C is Matrix obtained from Cholesky decomposition of E (where E is n*n square covariance matrix) and Z is a vector of normally distributed random variable with a dimension of (number of stock * path).Did I make any mistakes in this formula? Please help advice.Cheers

hi iamOZyou always use correlation matrix -- which is as you've mentioned symmetric positive definite matrix with 1s in diagonals (covariance does not satisfy later condition).correlation matrix, and not covariance, is used for representing multivariate gaussian distribution...

Okay to go from cormat to covmatcovmat= vol_volmat %*% cormat %*% vol_vectorso if you did chol on covmat you are all set and if you did it on the cormat then you will miss the scaling by the volsin your random numbers.HTH.

hi i'am also trying to price a basket optioni see that antonio has sent the VBA Code for Cholesky Decompositioni took it but unfortunatelly,i don't know how to use it in Excel,i mean how do i enter my matrix A on Excel and how to obtain the L matrix wich is the cholesky decomposition of A.Can you please send me the excel File where i can find the pricing of the basket option if you have already done itthank you very much for your helpyou can send me the files at the following address : mohamed.rhrmaritlemcani@epfl.ch

sorry but I would use a covariance matrix...

> 1) when generating random numbers and using Cholesky decomposition, can one use > covariance and correlation matrix interchangeably? (i.e. are they both "All positive definite"?) (It > seems to me that most people are always talking about correlation matrix.)Cholesky decomposition is a special case of LU decomposition and for CD the matrix must be SYMMETRIC. The covariance matrix will be positive definite because the corresponding PDE is elliptic and hence the e-values are real and positive.I cannot say about correlation matix because I have nothing to go on to prove pd. My own feeling is that cov is a 'nicer' matrix than corr because it includes the essential PDE properties in it.

I've never tried it with the cov matrix.If you set it up to be a simulation, are you able to generate time series with the correct standard deviation and covariance matrix?I mean, it seems like this is the acid test. If you set up the standard correlated gaussian time series generator using the correlation matrix, the vols and using the Cholesky decomp, you definitely get time series generated with the right properties.Are you saying you can use the covariance matrix to generate time series with the correct statistical properties (vols and cov matrix) and this gives you a different option price, as compared to a price generated using the standard method?That would be astonishing! Please clarify.

Isn't it just a scaling issue?you start by N(0,1) (Multivariate Normal)to get to N(0,Sigma) (Sigma being covariance matrix) you use choleski since N(0,Sigma) = Sigma^(1/2) N(0,1) in distributionSigma^(1/2) is your choleski decomposition of the covariance matrix. (any other decomposition will do as well, Cholescki being nicer because easy to implement and giving you a lower diagonal matrix)If you use correlation instead of Sigma you need to scale it back by multiplying each element of N(0, Correl) by its volatility.Chuchlain, covariance and correlation matrix will both have the same properties since to get from one to the other your pre and post multiply by a diagonal matrix with elements 1/sigma_i (or sigma_i) which will not change the rank of your matrix, or the sign. Or am I missing something?Regards,G

> Chuchlain, covariance and correlation matrix will both have the same properties since to get from one to the other your pre and post > multiply by a diagonal matrix with elements 1/sigma_i (or sigma_i) which will not change the rank of your matrix, or the sign. Or am I > missing something?Gamanti,I am working on a related problem (ADI/PDE thread) but the covariance matrix is positive positive def. because of the ellipticity of the BS PDE. The corr. matrix looks 'less' positve definite. My interest is from a numerical background.Comparing, the cov matrix has all the coefficients from the BS PDE.From a QF viewpoint, I cannot say anything because I have not worked in this. So, my remark is more maths. than any clever QF argument.Still, (speaking as a novice), why choose one matrix over the other?You and Mr. erstwhile can say.

Very clearly explained in Glasserman (Monte Carlo Methods In Finance) P72 for a simple case with constant (over the time step of the MC) vols and correlations. It is just a scaling issue (on each variate) in this case.

In other words, it shouldn't matter which matrix you use, as a correct implementation of either will result in the same result - correct?

QuoteOriginally posted by: erstwhileIn other words, it shouldn't matter which matrix you use, as a correct implementation of either will result in the same result - correct?Indeed. That's kind of the answer we wish to hear either in the positive or the negative (Plan C, the 'I don't know" answer which is, BTW acceptable as well).As mentioned, I am a rooky, so I am counting on you

Cuchulainn: You're counting on me? But I'm only an amateur quant! You can count on me to ask questions tho ...

a correlation matrix is simply a covariance matrix where the variance of each RV is 1. which one you do Cholesky is pretty irrelevant.I generally do it on the covariance matrix because you have fewer operations in total.

QuoteOriginally posted by: erstwhileCuchulainn: You're counting on me? But I'm only an amateur quant! You can count on me to ask questions tho ...Hi erstwhile,Au contraire, I think that your posts are very good and factual!What's the typical number or rows and columns in a Cov matrix?