Brownian Motion

Why This Matters In my earlier article on Brownian motion, I worked through the forward view: a process starting at a known value, diffusing into an uncertain future. Sometimes we know more than just the starting point. We also know where the process ended up, and we want to characterise the path in between. The object that answers this is the Brownian bridge: a Brownian motion conditioned on its terminal value. ...

Why This Matters 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 physical observations and the mathematical construction of Brownian motion. ...