Brownian Motion and Random Walk Models › Brownian Motion and Random Walk Models

The previous chapter documented the stylized facts of financial returns: fat tails, volatility clustering, and the approximate unpredictability of price movements. Now we turn to building mathematical models that capture at least the essence of these phenomena. The starting point, perhaps surprisingly, comes from 19th-century botany.

In 1827, Scottish botanist Robert Brown observed pollen grains suspended in water through a microscope. The particles moved erratically, jittering in seemingly random directions. This "Brownian motion" puzzled scientists until Einstein's 1905 paper explained it as the cumulative effect of countless molecular collisions, each individually invisible but collectively producing observable random movement.

Finance adopted this framework because stock prices exhibit similar behavior. Each trade, news item, or shift in sentiment acts like a molecular collision, nudging prices in unpredictable directions. The accumulated effect of millions of such nudges produces the irregular price paths we observe in markets.

This chapter builds Brownian motion from the ground up. We start with the simplest possible random process, a coin-flip walk on integers, then show how scaling this discrete model produces continuous Brownian motion in the limit. We then introduce Geometric Brownian Motion (GBM). This model adapts standard Brownian motion for asset prices, which cannot be negative and exhibit percentage returns. GBM forms the foundation for the Black-Scholes option pricing framework we'll develop in later chapters.