I would like to construct a portfolio with two stocks (say 50 of stock A and 100 of stock B) each assumed to follow geometric brownian motion with known drift and volatility. Is there a way to calculate the resulting drift and volatility of the combined portfolio? Considering both cases where the two stocks are correlated and totally uncorrelated.Thanks so much in advance!

the resulting portfolio isn't a geometric brownian motion unless the two stocks are perfectly correlated and have the same drift and volatilityin all other cases the concepts of drift and volatility of the portfolio process don't mean muchmaybe you mean the expectation and variance of the portfolio, in the sense of markowitz's portfolio theory?

Yes that's actually what I meant... sorry that I wasn't clear enough the first time

anyone please?I have been searching online, couldn't find any answer.any help would be much appreciated. thanks

you can combine as many geometric motions as you want if you have the covariance matrix of the processess. say you have n assets for time 1-t (normal random series )then you have a T,N matrix. You also have a N,N covariance matrix. Take a cholesky decomposition of the covaraince matrix which is also a N,N matrix. then the product of the first matirx (T,N) and the cholesky decomposition (T,N) will give you a (T,N) matrix which will have the same covariances the you described in the cov matrix. You can now combine thhe series with what ever weights you want. Hope this helps

Thanks! I know where to get started now.... Just 2 follow up questions1) After I found the correlated random parameters from the uncorrelated random parameters using the method you outlined, I would still need to do the simulations for the two assets (with the correlated random parameters) and add them up with weights and solve for the return and volatility of the combined portfolio right? In other words, there is no way I could figure out the return and volatility of the combined portfolio without actually having to do simulations right?2) if the two assets in my portofolio are uncorrelated, I won't even have to worry about finding the cov matrix at all right (since it's just an identity matrix)?Thank you so much again. This is really a great help! Appreciate it much

you start by generating 2 random simulations (uncorellated). then the cholesky adjustment makes those 2 processes corellated according to the covaraince matrix you chose

Hi minvosky;If you are simulating stocks or index brownian motion multi path case ,you have to use correlation matrix for uncorrelated yiled change.On the other hand,you can use cholesky matrix for single path simulation.If you have uncorrelated yield change ,covariance and correlation matrix,you can do easily.Best regards.

Hi,Thanks for your reply? How does the correlation matrix come into play? From what I heard and read, I thought I will generate two random normal distribution separately, and then use the cholesky decomposition for the covariance matrix to convert the two independent distributions to correlated distributions. epsilon_1 = Z_1epsilon_2 = rho Z_1 + sqrt(1-rho^2)Z_2where rho is the correlation coefficient, and Z_1 and Z_2 are the two uncorrelated random normal distributionsand finally I would find the price of the two assets using:S_(t+delta t) = S_t exp[(mu - sigma^2/2)delta t + sigma sqrt(delta t) epsilon]Is that correct?

YES

Hi,I was trying to model a two assets portfolio as well but I get complex numbers for the white noise of my second, correlated asset.Any idea where this could come from?I generate my white noise epsilon_1 from a normal dist.Then, I use Cholesky decomposition to come up with my second white noise epsilon_2 as in the message above but the result is complex numbers...which obviously gives me a complex number for my option price!Thanks for your help.

Indeed, it works now!Thanks for your help, asking imaginary dollars for my option would have probably not been in my best interest!Your website is very nice by the way.