Originally Posted by avito009 We all know this. Even a fifth grader knows that brownian motion measures distance. Distance is change in position. 
Really? That's what they teach? I shudder to think what teachers have in mind when the say such things.
Brownian motion is
defined as the random motion of particles suspended in a fluid. See:
https://en.wikipedia.org/wiki/Brownian_motion
Originally Posted by avito009 Distance is change in position. 
W
Where did you get that idea? That's not what distance is. Distance is the measure of how far apart two points in space is.
Originally Posted by avito009 This is similar to geometric brownian motion which measures change in stock price. 
See:
https://en.wikipedia.org/wiki/Geometric_Brownian_motion
Geometric Brownian motion is a the continuoustime stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion.

Originally Posted by avito009 ... similar to variance concept in quantitative finance. 
Variance is a concept that comes from probability and statistics theory.