GBM models asset prices as following a random walk with drift. It's the foundation of options pricing and portfolio simulation, capturing both expected growth and random fluctuations.
Formula
S(t+dt) = S(t) × exp[(μ - σ²/2) × dt + σ × √dt × Z]Methodology
Geometric Brownian Motion (GBM) is the standard model for simulating asset price movements. It has two components:
1. Drift term (μ × dt): The expected directional movement 2. Diffusion term (σ × √dt × Z): Random fluctuations
The critical detail is the Ito correction (-σ²/2). Under GBM, the expected value of future wealth is exp(μ×t), but the median is exp((μ - σ²/2)×t). The Ito correction ensures our simulation produces the correct distribution.
Without this correction, simulations would systematically overestimate future wealth, especially for high-volatility portfolios over long time horizons.
GBM assumes: - Returns are log-normally distributed - Volatility is constant over time - Markets are efficient (no predictable patterns)
These assumptions are imperfect but provide a reasonable baseline for long-term projections.
Data Source
Uses portfolio expected return (μ) and volatility (σ) from the risk metrics calculations. Z is a random draw from a standard normal distribution.
Reference
Hull, J.C. (2022). Options, Futures, and Other Derivatives. Pearson, 11th Edition, Chapter 14
Limitations
Assumes constant volatility and log-normal returns. Does not capture fat tails, volatility clustering, or regime changes.
Related Metrics
Project Your Portfolio
Run Monte Carlo simulations to see potential future outcomes.
For Educational Purposes Only
This analysis is not investment advice. Results are based on simplified models using historical data. Past performance does not guarantee future results. All investments carry risk of loss. Consult a qualified financial advisor before making investment decisions.