https://inflection-quant.pages.dev/tags/brownian-motion/Recent content in Brownian Motion on Inflection Quant LabHugoen-usThu, 26 Mar 2026 00:00:00 +0000https://inflection-quant.pages.dev/articles/quant-foundations/understanding_brownian_motion/Thu, 26 Mar 2026 00:00:00 +0000https://inflection-quant.pages.dev/articles/quant-foundations/understanding_brownian_motion/<h2 id="why-this-matters">Why This Matters</h2> <p>Brownian motion, the mathematical model underlying everything from stock prices to heat diffusion, has one of its most elegant properties: the variance of its position at time $t$ grows linearly with time. Not $t^2$, not $\sqrt{t}$, but exactly $t$. This seemingly abstract fact has a concrete consequence in financial markets: under the idealised conditions of an at-the-money option with zero rates, it is precisely why option prices scale with $\sqrt{T}$ rather than $T$, a direct fingerprint of Brownian motion inside Black-Scholes. Understanding why requires looking at both <strong>physical observations</strong> and the <strong>mathematical construction</strong> of Brownian motion.</p>