📂Probability Theory

Geometric Brownian Motion

Definition ¹

Let’s say the following Stochastic Differential Equation (SDE) is given by $\mu \in \mathbb{R}$ and $\sigma^{2} > 0$. $$ d X_{t} = X_{t} \left( \mu dt + \sigma d B_{t} \right) $$ The solution of this SDE is found as a Stochastic Process for the initial value $X_{0}$, which is referred to as Geometric Brownian Motion. $$ X_{t} = X_{0} \exp \left[ \left( \mu - {{ \sigma^{2} } \over { 2 }} \right) t + \sigma B_{t} \right] $$

Explanation

Geometric Brownian Motion (GBM) is famously known as a basic model describing the trend of indices in the fields of finance and economics. The term geometric seems to derive from the Geometric Series, which is the sum of an exponentially growing sequence, and has no relation to Geometry.

Log-normal Distribution

If we consider only the mathematical properties, excluding applications, the most notable characteristic of GBM is that it follows a normal distribution when logarithmically transformed, i.e., it follows a Log-normal Distribution. This is evident since $\exp$ contains a Brownian Motion that follows a Normal Distribution.

Dynamics

From the perspective of [Population Dynamics](../../categories/Population Dynamics), the system represented by GBM is merely the Malthusian Growth Model $N ' = r N$ with an added noise term $X_{t} \sigma d B_{t}$. Although it’s not strictly necessary to define GBM as a solution to a Stochastic Differential Equation, knowing it as such provides great help in understanding due to its clear deterministic Ordinary Differential Equation properties.

Financial Engineering

A prime application of GBM is to explain the fluctuation in prices of underlying assets such as stock prices. Just as the change in population is proportional to the total population, the change in the price of an asset is also proportional to the price of the asset, and assuming the asset does not get delisted, it cannot become negative, among other favorable assumptions.

Let’s assume the price of a certain stock $p_{t}$ follows GBM. Taking the log of the division of the closing price on $t$ days by the closing price on $t-1$ days, $$ r_{t} = \nabla \log p_{t} = \log {{ p_{t} } \over { p_{t-1} }} $$ is referred to as the return—Return, which matches our intuition by being positive if the price has risen and negative if it has fallen, regardless of the size of the stock price. As explained in the Log-normal Distribution section, this return follows a normal distribution, and it’s interesting because it focuses on the essence of growth and degrowth of stock prices rather than simple ups and downs.