Parameter estimation of multiple Geometric BM with Maximum Likelihood

beata · July 19th, 2004, 4:54 pm

say we have N firms and their firm value processes are given by correlated Geometric Brownian motionsd V_i_t = V_i_t * (/miu_i) * dt + V_i_t * sum_ j=1:M [/sigma_ij*dW_ j_t ], for i=1:N.I am looking for some REFERENSE which explains how to calculate the parameters of the correlated Geometric Brownian motions given historical data for each firm value V_i_t, and using the maximum likelihood method.thnx

obeelde · July 19th, 2004, 6:40 pm

Beata,Take a look at the following paper by Andrew Lo for the definitive Theory:Maximum Likelihood Estimation of Generalized Ito Processes with Discretely-Sampled Data, Econometric Theory 4(1988), 231-247. In this particular case, the estimation theory is very easy because we know that when we take the natural log transformation we get arithmetic brownian motion which has a normal distribution. In the multivariate case, we end up with a multivariate normal distribution after taking the natural log transformation.RegardsOwen

beata · July 19th, 2004, 7:07 pm

Owen,this paper is not available for free on the Internet.I guess I will have to look for something else.But thanks anyway

obeelde · July 19th, 2004, 8:56 pm

Beata,Look at equation (30) of "Conditional estimation of diffusion processes" by Minqiang Li, Neil D. Pearson,yand Allen M. PoteshmanFor an example of the density for the geometric brownian motion process. Simply generalize to the multivariateOwen