Wiener Process / Brownian Motion: The Continuous Random Walk

What happens when we take a simple stochastic coin-flip process, but compress time so tightly that a new random step occurs every single millisecond?

When randomness becomes continuous, it is called a Wiener Process, more commonly known in physics and finance as Brownian Motion.

A Wiener Process (W_t) is the foundational building block of all quantitative finance. It has two defining characteristics:

• Continuous Time: The movement never stops; the path is infinitely jagged and fractal in nature.
• Normal Distribution: The size of the steps follows a strict Gaussian distribution (Bell Curve). Microscopic movements happen constantly, while massive, violent swings are statistically rare.

When quantitative developers build models to price options (like the famous Black-Scholes model), they use the Wiener Process to inject raw, mathematical randomness directly into their differential equations.