Every blue line is one sample path of the process behind every continuous-time pricing model, from Black-Scholes to the risk engines running at large funds.
Brownian motion W(t) is the limit of a random walk whose steps get smaller and more frequent. Three properties define it. It starts at zero, its increments are independent, and each increment is normal with variance equal to the elapsed time, so W(t) ~ N(0, t).
The dashed gold band marks ±2√t. Notice that the paths spread with the square root of time rather than with time itself. That is exactly why annual volatility is the daily figure times √252 ≈ 15.9, the square-root-of-time scaling every desk uses.
Hit "new simulation" a few times. Each set of paths is different, and the statistical envelope is always the same. Individual disorder obeying a collective order is the tension at the heart of stochastic calculus.
Both simulators on this site are open source, MIT licensed, together with the Python scripts that verify the numbers against closed-form solutions.
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