The random walk a stock price is assumed to take. Not the price plus noise — the price times noise: dS = μS·dt + σS·dW, whose solution S(t) = S₀·exp((μ − σ²/2)t + σW(t)) keeps prices always positive and makes log-returns Normal. Down the center, data flows: the parameters go in, the engine simulates a fixed-seed ensemble, the proven moments come out. The blue team builds and defends; the red team attacks. These are models with named assumptions, not investment advice.
source P. A. Samuelson, Rational Theory of Warrant Pricing, Industrial Management Review 6(2):13–31 (1965) — first to put stock prices on geometric Brownian motion; no stable open link, cited by author/journal/year AMBER. Origin: L. Bachelier, Théorie de la spéculation (1900) — archive.org/details/bachelier-theorie-de-la-speculation. Rendered, not quoted.
The stochastic differential equation dS = μS·dt + σS·dW says the proportional change is a drift μ·dt plus a Wiener shock σ·dW. Because the noise scales with S, the price is a product of exponentials — it can shrink toward zero but never crosses it.
Itô's lemma solves it exactly: S(t) = S₀·exp((μ − σ²/2)t + σ√t·Z), Z standard normal. So ln(S(t)/S₀) ~ Normal((μ−σ²/2)t, σ²t). The −σ²/2 is the price of taking logs — miss it and the mean lies.
Live closed-form moments for the current controls:
Take the-brownian-motion — additive, can go negative — and drive it through an exponential. Samuelson's 1965 correction of Bachelier: prices are the geometric Brownian motion, so they stay positive and their logs are the Gaussian, not the prices.
That single move — additive → multiplicative — is the exact process the-black-scholes prices an option under. Each sphere is the next one's premise.
The blue team's live check: re-simulate an independent fixed-seed ensemble and confirm the sample mean of S(t) matches the Itô closed form S₀·e^{μt}. If red drops the drift correction, the mean overshoots and this badge turns red.
Four numbers and one noise source define the whole process:
S₀ the starting price · μ the drift (instantaneous mean rate of return) · σ the volatility (how hard dW shakes it) · t the horizon · and W(t), one standard Wiener process with W(t) ~ Normal(0, t), independent increments.
These feed the panel below. Every random draw uses a fixed-seed generator, so the picture is stochastic in shape yet reproducible to the last digit — that is what lets the witness check it.
50 fixed-seed sample paths. The mean line is S₀e^{μt}; the median line S₀e^{(μ−σ²/2)t} sits below it — the signature of a right-skewed log-normal.
| quantity (closed form) | predicted | ensemble | ok |
|---|
Move any slider — every number is simulated live from S(t) = S₀·exp((μ−σ²/2)t + σ√t·Z), never looked up. Not a forecast; a model of the distribution.
What the machine proves, on a fixed-seed ensemble to a stated tolerance: log-returns are Normal((μ−σ²/2)t, σ²t); the price stays strictly positive; the ensemble mean approaches S₀e^{μt} exactly despite the −σ²/2 inside the log (the Itô drift correction); and the terminal price is log-normal — right-skewed, its median below its mean, with P(S(t) < E[S]) = Φ(σ√t/2) > ½.
The blue team's witness (left) confirms E[S] live; the red team (right) tries to make the mean lie.
Fat tails alone sink it: a 20σ daily move is impossible under GBM yet has happened. The process is the first workable model of a price, not a true one — which is exactly why the panel exposes σ and μ as knobs you set, not laws. Nothing here is investment advice.
"Stock prices are normally distributed." Cut. Prices are log-normal; the log-returns are normal. Normal prices would permit negative prices — GBM's whole point is that it cannot.
"μ is the expected log-return." Cut. The expected log-return is μ − σ²/2. μ is the drift of the price itself; the gap is the Itô correction — and it grows with volatility.
"GBM forecasts the future price." Kept, corrected. It models a distribution, not a path you will see; constant σ and no-jumps are assumptions, not facts. A model, not advice.
The red team's move: delete the −σ²/2 Itô correction — use S = S₀·exp(μt + σW) instead. The paths look plausible, but the mean now overshoots S₀e^{μt} by a factor e^{σ²t/2}. The witness (window 7) is watching.
Drop the correction and E[S(t)] silently inflates — the ensemble mean climbs above the closed form, the witness recomputes, disagrees, and turns red. Nothing is faked; the attack is real and it is caught.