Brownian Motion and Random Walk Models › Summary

This chapter built the mathematical foundation for modeling random price movements in continuous time:

Random walks provide an intuitive starting point. A simple random walk adds independent $\pm 1$ steps, producing positions with zero mean and variance proportional to time. The Central Limit Theorem ensures that the distribution approaches normal as the number of steps grows.

Brownian motion emerges as the continuous-time limit of appropriately scaled random walks. It is characterized by four properties: starting at zero, continuous paths, normally distributed increments with variance proportional to time, and independence of increments over non-overlapping intervals. The quadratic variation of Brownian motion equals elapsed time rather than zero, a property that underlies all of stochastic calculus.

Geometric Brownian Motion adapts Brownian motion for asset prices by modeling log-prices rather than prices. This ensures prices remain positive and captures the multiplicative nature of returns. Under GBM, prices follow a lognormal distribution and log-returns are normally distributed with constant volatility.

These models provide the mathematical foundation for option pricing theory. In the next chapter, we'll develop Itô's lemma, which tells us how functions of Brownian motion evolve and allows us to derive the Black-Scholes equation for option prices.