Deriving Geometric Brownian Motion Moments from a Normal MGF

Summary

The document shows how to derive moments of geometric Brownian motion from the moment generating function of its underlying Brownian motion with drift. For a normally distributed variable, the moment generating function has an exponential form involving its mean and variance. Applying this to Brownian motion at time t gives an MGF determined by drift, volatility, and elapsed time.

For a geometric Brownian motion written as an initial value multiplied by the exponential of that process, the nth moment is the initial value raised to n times the Brownian MGF evaluated at n. This yields a direct expression for each moment and can support calculations used in estimation methods such as generalized method of moments. The response provides the derivation but does not address the question about MATLAB packages, nor does it discuss estimation design, data assumptions, or empirical validation.

Key ideas

• A Brownian motion with drift has a normal distribution at each fixed time, with mean and variance scaled by time.
• Its moment generating function can be evaluated at an integer power to obtain the corresponding asset-price moment.
• The nth moment of geometric Brownian motion is the initial price raised to n multiplied by the underlying Brownian MGF at n.
• The derivation gives a formula for moments but does not identify software packages or explain a complete estimation workflow.

Tags

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


# 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?

## Answer by Richi Wa (score 4)

https://quant.stackexchange.com/a/9089

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.

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