Random walks

Each gray line represents a simulated one-year equity market trajectory generated by a geometric Brownian motion. This model, introduced in early 20th-century probability theory and later formalized in financial economics as the foundation of modern asset-pricing theory, describes prices as continuous-time processes driven by a deterministic drift and a random shock. In comparison, the red line shows the realized US market path in 2023. The dispersion across simulated paths illustrates that even when average return and volatility are known, short-term price realizations remain dominated by randomness.

Data: Geometric Brownian motion simulations (100 paths) following dS/S = μdt + σdW, where S is portfolio value, μ and σ are mean and volatility estimated from daily US total market log returns (2023), and dW is a Wiener process. Calibrated using data from the Kenneth R. French Data Library.