Brownian motion simulator
The random walk behind Black-Scholes, drawn live. Sixty share-price paths grow from the same starting point; slide the drift and volatility and watch the fan tighten, widen and skew.
Weekly steps over 3 years · simulated median finish 113.1 vs theoretical 116.2
Each step multiplies the price by exp((μ − σ²/2)Δt + σ√Δt·Z) with Z standard normal — the exact GBM increment, not an approximation. Push σ up and the fan widens while the median finish drifts DOWN even with μ fixed: that's the −σ²/2 volatility drag the exams love. The same assumption underpins the Black-Scholes price in the options tools.
What to notice
The spread of finishing prices is lognormal: paths can multiply without limit above but can only fall to zero below, so the fan is skewed upward even though each step is symmetric in log terms. And watch the median as you raise volatility with drift fixed — it falls. The average is dragged by the lucky tail while the typical path suffers the volatility drag of −σ²/2, which is exactly the correction term in the Black-Scholes derivation and in the exams' lognormal questions.
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For education only, not financial advice.