Brownian motion

Brownian motion describes the motion of a particle in a fluid or gas. Such a particle bounces around “randomly” within the fluid surrounding it. The path it creates appears quite noisy and random. It can essentially be considered a limiting form of the random walk where both the time between steps and the step length approach zero (with some caveats: namely, if the time step is

, the step length must be

- this produces a process whose variance scales exactly with time). Consider flipping a coin to determine whether to take a step forward or backward. Now let the time between steps approach zero.

Monte-Carlo simulation of terminal values is a relatively simple simulation, and one can probably be completed analytically. The true power of Monte-Carlo simulation is unlocked when analysing scenarios that are more difficult to solve analytically, if not impossible. For example, if we wanted to...