Stochastic Processes
Brownian motion with drift and diffusion, exponentiated into geometric Brownian motion, is the first workable model of a stock price.
A stochastic process is a random quantity that evolves in time, the natural language for prices. Brownian motion supplies the building block: continuous paths, independent increments, and variance that grows linearly with time. Add a drift and a diffusion coefficient and you get an Itô process; exponentiate it and you get geometric Brownian motion, the first serious model of a stock price. It captures two essentials, prices that stay positive and volatility that scales with the square root of time, and its failures, jumps and volatility clustering, are exactly the stylized facts of the first node that later models must repair.