Definition#

The Geometric Brownian motion can be defined by the following Stochastic Differential Equation (SDE)

with initial condition $X\mathrm{\_}0=x\mathrm{\_}0>0$, and constant parameters $\mu \in \mathbb{R}$, $\sigma >0$. Here, $W\mathrm{\_}t$ denotes a standard Brownian motion.

In order to find its solution, let us set $Y\mathrm{\_}t=\ln (X\mathrm{\_}t)$. Then, Ito’s formula implies

or equivalently

From this expression, we obtain the solution as follows:

Thus, for each $t>0$, the variable $\ln (X\mathrm{\_}t)|X\mathrm{\_}0$ (which will be denoted simply by $\ln (X\mathrm{\_}t)$) follows a normal distribution –since it can be expressed as a scaled marginal of the Brownian motion plus a deterministic part. Using the properties of the standard Brownian Motion we can calculate its expectation and variance, and verify that

and consequently $X\mathrm{\_}t$ follows a Log-normal distribution.

🔔 Random Facts 🔔#

• In 1952, Paul Samuelson and Robert Merton adapted the Wiener process into what we now recognise as Geometric Brownian Motion. Their modifications addressed the (so-called) shortcoming of the Bachelier’s model by ensuring that the process remains positive.

• GBM serves as the foundation for the famous Black-Scholes-Merton model for option pricing.

• The asymptotic properties of geometric Brownian motion (GBM) are determined by the quantity $\mu -\frac{{\sigma }^2}{2}$:

• $\mu -\frac{{\sigma }^2}{2}>0$: As $t$ approaches infinity, $X\mathrm{\_}t$ approaches infinity with probability 1

• $\mu -\frac{{\sigma }^2}{2}<0$: As $t$ approaches infinity, $X\mathrm{\_}t$ approaches zero with probability 1

• $\mu -\frac{{\sigma }^2}{2}=0$: As $t$ approaches infinity, $X\mathrm{\_}t$ has no limit with probability 1

More to Read#

• L. Bachelier, “Theory of Speculation: The Origins of Modern Finance,” Princeton University Press, Princeton, 1900.

• Samuelson, Paul A.. “Rational Theory of Warrant Pricing.” (2015).

• Brief History of the Black-Scholes-Merton Model

• Geometric Brownian Motion Python Notebook (part of my series of notes exploring concepts with Python)

• Stojkoski, V.; Sandev, T.; Basnarkov, L.; Kocarev, L.; Metzler, R. Generalised Geometric Brownian Motion: Theory and Applications to Option Pricing. Entropy 2020, 22, 1432. https://doi.org/10.3390/e22121432

P.s. If you are curious about probability distributions visit the Advent Calendar 2023 ✨