Brownian Motion

Brownian motion is the simple continuous stochastic process used to explain a random process. An example of Brownian motion in finance is the fluctuation in an asset’s price. Other real life examples are the random movement of a molecule of the gas, light beam entering a dark room.
 

In finance, we will use the geometric Brownian motion (GBM), which is technically a Markov process. It states that stock prices follow a random walk and is consistent with the weak efficient market hypothesis which states that an asset's prices fully reflect all past available information.

The formula for GBM is:

where,

S = stock price

σ  =  standard deviation of the returns

μ  = expected returns

t = time

ε = random variable
 

The first term on the right side of GBM formula is a "drift" and the second term is a "shock". For each time period, our model assumes the price will "drift" up by the expected return. But the drift will be added or subtracted by a random shock.