Geometric Brownian Motion, Drift, and Volatility in Financial Markets

Summary

This introduction explains why financial prices are modeled as stochastic processes rather than deterministic functions. It reviews the calculus chain rule, distinguishes deterministic and random processes, and introduces Brownian motion through the analogy of particles moving unpredictably in air. It then presents geometric Brownian motion as a model for stock prices, separating the price change into a drift component and a volatility component driven by a Wiener process.

A Microsoft price chart and simulated Wiener paths provide visual illustrations. The article describes drift as the overall direction of the price path and volatility as its random variation, then outlines properties of Wiener increments, including their zero mean and time-dependent variance. It is an intuition-building first installment, not a trading system or empirical validation that real prices satisfy GBM assumptions. It acknowledges that stock returns are used as a proxy and points to a later article for Ito’s lemma and its trading applications.

Key ideas

• Classical calculus rules for deterministic functions do not directly describe random price paths.
• Geometric Brownian motion models price changes as a drift term plus a stochastic volatility term.
• The model represents randomness with a Wiener process whose increments have zero mean and variance proportional to elapsed time.
• Drift describes the directional component of a price path, while volatility describes its random variation.
• The article offers conceptual and visual intuition, while leaving Ito’s lemma and trading applications to a later installment.

Tags

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.