Lesson 2.3: The "Weird Algebra" of Infinitesimals (Our New Rules)

Welcome to Lesson 2.3. In our last two lessons, we made two critical discoveries.

Failure (Lesson 2.1): Summing the 1st power of random steps, $\sum |\Delta W|$ , goes to infinity.
Success! (Lesson 2.2): Summing the 2nd power of random steps, $\sum (\Delta W)^2$ , converges to a finite, non-random number: $T$ (the total time).

This is our "Aha!" Moment. We now have a "Rosetta Stone" that lets us translate between the random world and the normal world.

In this lesson, we will convert that "big-picture" sum (a calculus concept) into a simple, powerful set of "multiplication rules" (an algebra concept). These rules are the "cheat sheet" we will use to build Itô's Lemma in the next module.

Part 1: Deriving the Master Rule: (dWt)² = dt

This is the most important rule in all of finance. Let's see exactly where it comes from.

What We Found (Lesson 2.2):

We proved that the sum of all the squared random steps is equal to the sum of all the tiny time steps.

\lim_{n \to \infty} \sum_{i=1}^n (\Delta W_i)^2 = T

We also know, from Lesson 1.5, that:

\lim_{n \to \infty} \sum_{i=1}^n \Delta t = T

The Logical Leap:

If the total sum of all the $(\Delta W_i)^2$ pieces is equal to the total sum of all the $\Delta t$ pieces... then any one tiny piece from the first sum, $(\Delta W_i)^2$ , must be "equal in weight" to its corresponding tiny piece from the second sum, $\Delta t$ .

The "Piles of Sand" Analogy

Imagine you have a pile of a million tiny, random pebbles (the $\sum (\Delta W_i)^2$ pile) and a pile of a million tiny, identical grains of sand (the $\sum \Delta t$ pile). From Lesson 2.2, you know the total weight of both piles is exactly the same (they both equal $T$ ). Therefore, the "typical" weight of one random pebble, $(\Delta W_i)^2$ , must be equal to the weight of one grain of sand, $\Delta t$ .

The Final Step: Δ → d (Infinitesimal Notation)

In calculus, we move from "delta" ( $\Delta$ a tiny step) to "differential" ( $d$ an infinitely small step) to make our rules exact.

$\Delta t$ becomes $dt$ (an infinitely small step in time)

$\Delta W_t$ becomes $dW_t$ (an infinitely small random step)

By replacing the $\Delta$ with $d$ in our new rule, we get the Master Rule of Itô Calculus:

(dW_t)^2 = dt

This rule is the "weird algebra" in its final form. It states that the "squared" infinitesimal random step is not zero, but instead becomes a "first-order" infinitesimal time step.

Part 2: Deriving the "Normal" Rules

Now that we have our "master" rule, we need to formalize the other rules for multiplication. These are the "normal" rules we learned in Lesson 0.8.

Rule #2: What is (dt)²?

$\Delta t$ is a small number (e.g., 0.01). $(\Delta t)^2$ is a much smaller number (e.g., 0.0001). In the limit as $\Delta t \to 0$ , the $(\Delta t)^2$ term is "infinitely smaller" than $\Delta t$ and becomes zero. Rule #1: (Δt)² ≈ 0

(dt)^2 = 0

Rule #3: What is dt ⋅ dWt?

We find its "order of magnitude" using our "typical size" logic from Lesson 1.3: The size of $dt$ is of order 1. The "typical size" of $dW_t$ is $\sqrt{dt}$ , which is order 0.5. So, the "typical size" of their product is: $(dt) \cdot (\sqrt{dt}) = (dt)^{1.5}$ . If $dt = 0.01$ , then $(dt)^{1.5} \approx 0.001$ . Just like $(dt)^2$ , this term is "infinitely smaller" than $dt$ and goes to zero in the limit.

dt \cdot dW_t = 0

Part 3: The "So What?" (Our New "Multiplication Table")

We have just built the complete "cheat sheet" for Itô calculus. This is the new "algebra" that all of our derivations will be based on. Anytime we are expanding a formula (like a Taylor series) and we see these terms, we can now simplify them.

Itô "Weird Algebra" Rules

×	dt	dWt
dt	$\to 0$	$\to 0$
dWt	$\to 0$	$\to dt$

In simple terms: Any term "squared" with a $dt$ (like $(dt)^2$ ) is 0. Any "mixed" term ( $dt \cdot dW_t$ ) is 0. The only second-order term that survives is $(dW_t)^2$ , which magically becomes a first-order term: $dt$ .

What's Next? (The 'Hook')

You now have your "rules of engagement." This multiplication table is the only thing we need (in addition to our Taylor expansion from Module 0) to build the "master tool" of finance.

In our next module, Module 3: Itô's Lemma, we will finally build the "chain rule" for stochastic calculus.

We will start with our 2-variable Taylor expansion (from Lesson 1.7):

\Delta V \approx \dots + \frac{1}{2}\left( \dots + 2\frac{\partial^2 V}{\partial t \partial S}(\Delta t \Delta S) + \frac{\partial^2 V}{\partial S^2}(\Delta S)^2 \right)

We will plug in our stock SDE: $\Delta S = \mu S_t \Delta t + \sigma S_t \Delta W_t$ . When we do, we will get terms like $(\Delta t)^2$ , $\Delta t \Delta W_t$ , and $(\Delta W_t)^2$ . And now, thanks to this lesson, we have the simple rules to know exactly which terms "die" (go to 0) and which "survive."

The one term that survives, $(\Delta W_t)^2 \to \Delta t$ , will give us the famous "Itô correction term" that makes the Black-Scholes equation possible. You are now ready to derive.

Up Next: The Itô Integral