Simulate an asset price with geometric Brownian motion; sample paths alongside the theoretical expected value.
Quantile fan (5–25–50–75–95%). Drag the horizontal level on the path cloud (or focus it and use the arrow keys). Select bins in the histogram (click or drag) to highlight the paths that ended there. Esc clears. Press Play to watch the paths unfold; the histogram shows the values at time t, so its spread grows like √t.
The log-normal density of S_T. Drag the σ track under the curve: the mean stays put while the median and mode slide left. The triangles mark the sample mean and median of the 400 simulated paths.
The average terminal price is S₀·e^(μT) = 108.33. Where does the typical path end, i.e. what is the median of S_T?
Geometric Brownian motion is the standard model for stock prices and the foundation of the Black–Scholes model. Prices change by random percentages rather than random amounts, so they stay positive and are lognormally distributed. This simulator shows sample paths for a given drift and volatility together with the theoretical expected value S₀e^{μt}.
It keeps prices positive and makes returns, not price changes, the random quantity, matching the idea that a stock is as likely to rise 10% whether it trades at 10 or 100.
The lognormal distribution is skewed: a few very high outcomes pull the mean up, while the median path grows only at μ − ½σ².