Day 5: Arithmetic Brownian Motion#
The Arithmetic Brownian motion, or Brownian motion with drift and scale, is a stochastic process describing the dynamics of a value that changes over time with two key components: a steady, deterministic trend plus a random part driven by a Brownian motion.
Definition#
The Arithmetic Brownian motion can be defined by the following Stochastic Differential Equation (SDE)
with initial condition \(X\_0 =x\_0\), and constant parameters \(\mu\in \mathbb{R}\), \(\sigma>0\). Without loss of generality we are going to assume that \(x\_0=0\). Here, \(W\_t\) denotes a standard Brownian motion.
This equation is equivalent to
which gives us the solution.
A drifted dādimensional Brownian motion is a vector-valued stochastic process defined as
whose components \(X\_i\) are independent, one-dimensional Arithmetic Brownian motions.
š Random Facts š#
The Arithmetic Brownian model was postulated byĀ Louis BachelierĀ on his PhD thesis āTheory of Speculationā (1900) as a mathematical model for stock prices movements āknown today as the Bachelier model.
One early criticism of the Bachelier model is that the probability distribution which he chose to use to describe stock prices allowed for negative prices. His doctoral dissertation was graded down because of that feature. Today, such feature has became an advantage of the model!
On April 8, 2020, theĀ CME GroupĀ posted the noteĀ CME Clearing Plan to Address the Potential of a Negative Underlying in Certain Energy Options Contracts,[1]Ā saying that after a threshold on price, it would change its standard energy options model from one based onĀ Geometric Brownian MotionĀ and theĀ BlackāScholes modelĀ to the Bachelier model.
More to Read#
Brownian Motion Notebook (part of my series of notes exploring concepts with Python)
P.s. If you are curious about probability distributions visit the Advent Calendar 2023 āØ