Day 5: Arithmetic Brownian Motion

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)

\[dX\_t = \mu dt + \sigma dW\_t, \quad t >0,\]

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

\[X\_t = \int\_0^t \mu ds + \int\_0^t \sigma dW\_t = \mu t + \sigma W\_t,\]

which gives us the solution.

A drifted dāˆ’dimensional Brownian motion is a vector-valued stochastic process defined as

\[X(t) = (X\_1(t) ,X\_2(t), \cdots, X\_d(t)), \qquad \geq 0,\]

whose components \(X\_i\) are independent, one-dimensional Arithmetic Brownian motions.

šŸ”” Random Facts šŸ””#

  • The Arithmetic Brownian model was postulated byĀ Louis BachelierĀ on his PhD thesis ā€œTheory of Speculationā€ (1900) as a mathematical model for stock prices movements –known 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 Black–Scholes modelĀ to the Bachelier model.

More to Read#

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