There are detailed explanations of the math and theory elsewhere. I've just written some code to see it in action. This generates 100 simulations for the S&P 500 ETF with the ticker SPY, modeling the past year:
What's especially interesting today is that the current value of SPY is on the extreme lower end of what we see simulated. Even increasing 100 to 1000 or more, we are still on a rare path.
There are some known problems with the model that may help explain this. First, volatility isn't really constant in financial markets. Second, the randomness in GBM is normally distributed but we know that stock returns are not. They have fatter tails, or higher kurtosis. Also stock prices react to specific geopolitical events that definitely are not random, even opening a day at a different level than the previous day's closing.
Out of curiosity, and because the second issue is the easiest to tweak, I replaced the normal distribution with a Laplace distribution and see a slightly wider dispersion of results (in both directions). In reality, though, the 52 week range is 218.26 - 339.08 so we still aren't capturing the extremes witnessed.
Any ideas on what's going on here? It must have something to do with the massive increase in volatility at the end of what was otherwise a calm year. Please comment.