# how to extract moments of GB from moment generating function?

I'm searching for the moments of geometric brownian motion using the gmm optimization program. the aim is to make the process y(t) of returns follows a normal distribution Are there any packages in MATLAB that are capable of this?

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I found these nice lecture note by Karl Sigman on the web. On page three you see if $X\sim N(\mu,\sigma)$ then the moment generating function (mgf) of $X$ is given by $$M_X(s) = E(exp(sX)) = \exp( \mu s + \sigma^2 s^2 /2)$$ Thus for Brownian motion with drift $X_t$ you get $$M_{X_t}(s) = E(exp(s X_t)) = \exp( \mu t s + \sigma^2 s^2 t /2).$$ Finally for $S_t = S_0 \exp(X_t)$, i.e. the geometric Brownian motion you get $$E[S_t^n] = S_0^n E[\exp(X_t)^n] = S_0^n E[\exp(n X_t)] = S_0^n M_{X_t}(n),$$ which can be calculated by the mgf of $X_t$. Then you get all moments by a simple formula.