Options & Derivatives

Geometric Brownian Motion

Also known as: GBM

The standard model for asset prices, in which log returns follow Brownian motion with drift.

dS = mu S dt + sigma S dW. Prices stay positive and are lognormally distributed.

The subtlety worth knowing. The drift of the log price is mu - sigma^2/2, not mu. That gap is volatility drag, and it is why the median outcome falls below the mean.

What it gets wrong: constant volatility (real volatility clusters), continuous paths (real prices jump), and thin tails. It remains the baseline because it is tractable, not because it is accurate.

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