A Gentle Introduction to Geometric Brownian Motion in Finance
In this article I will give a primer on how to think about finance in terms of stochastic processes, how to solve the Geometric Brownian Motion and the intuition on how we arrive to such a model.
Ordinary Calculus Vs. Stochastic Calculus in Financial Markets:
One of the main points to grasp here is why we use stochastic calculus in financial markets in the first place. This comes from the fact that financial markets contain an element of randomness and therefore cannot be solved using ordinary calculus alone.
a. Ordinary Calculus:
When we look at any differentiable function graphically, if we zoom in close enough we will always encounter a point in which the function is flat, therefore is differentiable and its derivative is zero.
This gives us a function that is continuous and differentiable everywhere. If we were to take its area, i.e. its integral, we would use standard calculus and it would look like this:
