Writing
Pricing an option requires modeling how uncertainty evolves over an entire time horizon rather than at a single future date, something classical probability was not designed to do. This note explains how that challenge led to the development of Brownian motion.
This technical note examines why financial price movements are better modeled as relative rather than absolute changes. While arithmetic Brownian motion captures continuous fluctuations, its absolute, additive structure fails to reflect the proportional scaling of speculative returns and permits negative asset prices.
In our previous note, The Geometry of Fluctuation, we established the geometric framework of multiplicative asset dynamics, leading to the formulation of Geometric Brownian Motion (GBM) and the identification of the −½σ²dt volatility correction. However, that derivation relied on the stochastic multiplication table and second-order Taylor expansions as algebraic axioms.