Simulate arithmetic Brownian motion with drift and volatility; watch several sample paths evolve over time.
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.
Drag the barrier on the path cloud (or focus it and use the arrow keys). Paths that touch it turn coloured from the moment they first hit it.
The theory curve and share assume no drift (μ = 0) and a start at 0. Then, by the reflection principle, P(τ_b ≤ T) = 2·P(W_T ≥ b), and τ_b has density |b|/(σ√(2πt³))·e^(−b²/(2σ²t)). The paths are simulated on a grid of 200 steps, so a path can cross b between two grid points without being caught; the simulated share is therefore slightly below the continuous theory.
With μ ≠ 0 the formula above doesn't apply, so only the simulated values are shown.
What share of paths touch the barrier b = 0.50 at least once by T = 1.00 (in %)?
Brownian motion is the continuous-time random walk at the heart of financial models. This simulator draws arithmetic Brownian motion with a chosen drift and volatility, using reproducible seeded paths, so you can see how the drift sets the trend and the volatility sets the spread that grows over time.
Drift μ is the average change per unit time, the slope of the mean path. Volatility σ scales the random part; the spread around the mean grows like σ√t.
Yes. Its value is normally distributed, so it can take any real value. That is why stock prices are usually modelled with geometric Brownian motion instead.