Lesson 1.2: Brownian Motion (The 'Perfect' Model of Randomness)

Welcome to Lesson 1.2! In our last lesson, we discovered that financial data is 'infinitely wiggly' (a property called self-similarity) and that this property breaks normal calculus because we can't find a 'slope' (or derivative) at any point.

Our problem was that we had a concept of randomness, but no mathematical model for it.

In this lesson, we will define that model. We are going to build, from the ground up, the "perfect" or "ideal" model of a random, wiggly path. This model is the cornerstone of all of quantitative finance.

It is called Brownian Motion or, more formally, a Wiener Process. We will give it the symbol WtW_t.

What is Wt?

First, what do the letters mean?

  • tt: This is just time (e.g., 1 second, 2.5 seconds, 10 minutes).
  • WtW_t: This is the position (or "value") of our random path at time tt.

Analogy: Think of a single grain of pollen floating on a drop of water. It's being constantly bombarded by millions of tiny, invisible water molecules.

  • tt is the time on your stopwatch.
  • WtW_t is the exact x-y coordinate of that pollen grain at time tt.

To be a "perfect" mathematical model of this random motion, our WtW_t path must follow four simple, iron-clad rules.

The 4 Rules of Brownian Motion

These four rules are our axioms. We accept them as the definition of WtW_t.

Rule 1: It Starts at Zero
W0=0W_0 = 0

The Intuition: This is just our "anchor point." We have to start somewhere, so we all agree to start our model at the origin.

Rule 2: It is a Continuous Path

The Intuition: The path is "infinitely wiggly," but it never teleports. It can't "jump" from a position of 5 to 10 in an instant. To get from 5 to 10, it must pass through every single number in between. This is the property that allows us to use calculus on it at all.

Rule 3: It Has Independent Increments

The Intuition: The path has no memory. The random change that happens between 10:00 AM and 10:01 AM has absolutely no connection to what happened between 9:00 AM and 9:01 AM.

Rule 4: The Increments are Normally Distributed (The "Wow" Rule)
This is the most important rule. It's the "engine" of our new calculus. It states that the change in the path's position (an "increment") between any two points in time, ss and tt, is a random number drawn from a Normal Distribution.
Wt−Ws∼N(0,t−s)W_t - W_s \sim N(0, t-s)
  • Wt−WsW_t - W_s: This is the "increment," or the total change in position.
  • ∼\sim: "is distributed as".
  • N(...)N(...): Our Normal Distribution "recipe".
  • 00 (The Mean): The center of the bell curve is 0. The path is "unbiased."
  • t−st-s (The Variance): The "spread" or "messiness" of the possible outcomes is exactly equal to the time that has passed.

The Physical Meaning of Rule #4

This rule gives "randomness" a unit. It says Variance = Time.

  • If you wait 1 second (t−s=1t-s = 1), the change in the path, W1−W0W_1 - W_0, is a random number from N(0,1)N(0, 1).
  • If you wait 10 seconds (t−s=10t-s = 10), the change, W10−W0W_{10} - W_0, is a random number from N(0,10)N(0, 10).

The bell curve for the 10-second change is much wider and flatter than for the 1-second change. This makes perfect physical sense: More time = More wiggling = More uncertainty about the final position.

What's Next? (The 'Hook')

    This "Variance = Time" rule has a huge and non-obvious consequence. We know Standard Deviation=Variance\text{Standard Deviation} = \sqrt{\text{Variance}}. So if the variance of our random step ΔW\Delta W is Δt\Delta t, then the "typical size" (standard deviation) of that step must be Δt\sqrt{\Delta t}.

    This will lead us directly to the "weird scaling" property, which is the foundation for the "new algebra" we'll build in Module 2.

Up Next: The "Weird" Scaling Property