No. 171 · Stochastic Processes
A thousand random walks leave the same point. Each obeys one rule — a little drift, a little noise, compounded step by step — and each becomes a different future. No single path is a forecast. Their spread is.
Read the method below
Double the horizon and the band widens by √2, not two. That sub-linear creep is the fingerprint of diffusion — the same reason a drop of ink, or a rumour, or an error bar, flares into a cone rather than a wedge.
The shaded envelope holds the middle eighty percent of futures. Ten percent stray above the upper edge, ten below the lower. At the horizon the band opens to ±0.90 in log-return — the ninetieth percentile a full step clear of the tenth.
Every thread runs the identical geometric Brownian motion; only its coin-flips differ. Pile enough of them and the chaos resolves into a smooth, computable shape — the log-normal cone that underwrites diffusion physics and the Black–Scholes price alike.
No instrument, no asset, no advice. Every number on this page is synthetic — a bare geometric Brownian motion chosen to show the mathematics clearly. It describes no market and predicts nothing. The subject is the shape of chance itself, not any future you could trade.