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.
