What mathematical concepts are required before I can understand what exactly is a Geometric Brownian motion as applicable to stock prices? I mean which branches of probability, calculus, statistics etc. are needed to understand GBM?
By 'understand', I mean gain an intuitive understanding.
you need to know multivariable calculus and partial differential equations
If you know basic probability and basic programming you can write a MATLAB program less than 10 lines long to simulate (in discrete time) geometric brownian motion and thus gain a basic understanding of how GBM works. To understand what happens as the time step goes to zero, and to prove properties of the resulting continuous limit see the other answer above..
In order to really understand Geometric Brownian motion (GBM) you should study the basics of so called "stochastic analysis". You could start with the book Stochastic Differenctial Equations by Bernt Oksendal.
If you want to simulate it, either basic understanding of the above suffices, or you have a look at the numerics of SDEs Numerical Solution of Stochastic Differential Equations by Eckhard Platen.
Stochastic calculus is different from ordinary calculus. PDEs will help but you don't need them for GBM.
In my experience, in order to understand in depth what GBM is you need to know some single/multi variable calculus, knowledge of ordinary differential equations, some probability (even in several variables), lebesgue integration and some basics of functional analysis. These concepts are fundamental if you want to understand what an Itö process is (since GBM is a particular case of it) and how to use Itö's lemma.
No fancy theory is needed to understand why a GBM is applied to model stock prices. To get an intuitive understand, simple Macro-economics should suffice to understand why it is being applied:
it has a Brownian component
it has (exponential) drift - this makes the model able to deal with stock prices growing in line with GDP (actually faster than gdp since equity by definition already contains implied leverage)
it can`t become zero
it is relatively simple - the time-T distribution is lognormal and hence the derivation of asset prices at time T is easy to compute
in the long run, only a few realizations of the GBM will be above average, but these will then be signficantly above average (think: Google, Apple). The vast majority of realizations will be below average.
the reason why people use the GBM is because it also can be expressed with stochastic differential equations, but is still relatively simple to express (and solve) in mathematical terms. With a few modifications to the GBM SDE, you can obtain powerful models that allow you to model anything you observe in the markets (e.g. jumps, stochastic volatilites, etc.). If you want to understand what is going on mathematically, I highly recommend Oksendal.
