Lesson 3.1: Itô's Lemma (Simple Case, for f(Wt))

Welcome to Module 3! This is where all our hard work from Modules 0, 1, and 2 finally pays off.

In Module 0, we built our "master tool" from normal calculus: the Taylor Expansion.

In Module 2, we discovered the "master rule" of the random world: the "Weird Algebra" where $(dW_t)^2 = dt$ .

In this lesson, we will collide these two ideas. We will apply the "normal" Taylor expansion to our "random" path $W_t$ , and then use our "weird" algebra to simplify the result.

The formula we discover is called Itô's Lemma. It is the "chain rule" for stochastic calculus, and it is the single most important tool we will use to build the Black-Scholes equation.

Part 1: The Problem (Why We Need a New Chain Rule)

In "normal" calculus (Lesson 1.4), the chain rule is simple. If we have a function $f(x)$ and $x = g(t)$ , the chain rule is:

\frac{df}{dt} = f'(x) \cdot g'(t)

Or, in differential notation, $df = f'(x)dx$ .

But this fails for a random path. We proved in Lesson 2.1 that our random path $W_t$ has no derivative (its "slope" is infinite). We can't use the normal chain rule.

So, we need to find a new chain rule. Our goal is to find a formula for the "tiny change" $df$ when our function is $f(W_t)$ .

Part 2: Assembling Our Tools

We have everything we need. We'll just put our tools on the workbench.

Tool #1: The 1-Variable Taylor Expansion (from Lesson 1.6)

This is our "prediction formula." It tells us how $f$ changes when we take a small step $\Delta x$ .

\Delta f \approx f'(x)\Delta x + \frac{1}{2}f''(x)(\Delta x)^2

Tool #2: The "Weird Algebra" (from Lesson 2.3)

This is our "simplification" cheat sheet. As $\Delta \to 0$ (as our steps become infinitesimal $d$ ):

(dt)^2 = 0

dt \cdot dW_t = 0

(dW_t)^2 = dt

Part 3: The Step-by-Step Derivation

Step 1: State the Goal

We want to find the formula for the infinitesimal change $df$ for a function $f(W_t)$ .

Step 2: Start with the 'Master Tool' (Taylor Expansion)

We begin with our 2nd-order Taylor formula for $\Delta f$ :

\Delta f \approx f'(x)(\Delta x) + \frac{1}{2}f''(x)(\Delta x)^2

Step 3: Rename the Variables

This is a small but critical step. We need to "map" our random path $W_t$ onto this "normal" formula.

Our function $f(x)$ is now $f(W_t)$ .
Our "anchor point" $x$ is our current position, $W_t$ .
Our "step" $\Delta x$ is our tiny random step, $\Delta W_t$ .

Let's substitute these new "roles" into the Taylor formula:

\Delta f \approx f'(W_t)(\Delta W_t) + \frac{1}{2}f''(W_t)(\Delta W_t)^2

Step 4: Take the Limit (Apply the 'Weird Algebra')

Now, we take the limit as our tiny steps $\Delta$ become infinitesimal $d$ . We check every term.

The 1st-Order Term: $f'(W_t)(\Delta W_t)$ . This is a "first-order" term. In the limit, it becomes: $f'(W_t)dW_t$ . This term SURVIVES. This is the "normal" part of the chain rule.

The 2nd-Order Term: $\frac{1}{2}f''(W_t)(\Delta W_t)^2$ . In "normal" calculus (Lesson 0.8), this term would be zero. But here, we must apply our "Master Rule" from Lesson 2.3: $(dW_t)^2 = dt$ . As we take the limit, $(\Delta W_t)^2$ becomes $dt$ . So, this term SURVIVES and becomes: $\frac{1}{2}f''(W_t)dt$ .

Step 5: Collect the 'Survivors'

We just add our two surviving terms back together.

df = f'(W_t)dW_t + \frac{1}{2}f''(W_t)dt

Part 4: The 'So What?' (The Itô Correction Term)

This is it. This is Itô's Lemma for a simple function.

Let's compare our "new" rule to the "old" one.

Normal Chain Rule: $df = f'(W_t) dW_t$

Itô's Chain Rule: $df = f'(W_t) dW_t + \mathbf{\frac{1}{2}f''(W_t)dt}$

That new term, $\frac{1}{2}f''(W_t)dt$ , is the famous "Itô correction term."

What is its physical meaning?

It's a predictable, non-random drift (it's a $dt$ term) that your function $f$ earns, simply because:

Your function is "curved" (the $f''$ term. If $f''=0$ , the term disappears!)
Your path is "random" (the $(dW_t)^2 = dt$ rule that created it)

The Skateboard Analogy (The "Wow" Moment):

Imagine $W_t$ as the side-to-side position of a skateboard in a half-pipe.

Imagine $f(W_t)$ as the height of the skateboard.

The half-pipe is "curved" (it has $f'' > 0$ , i.e., it's "convex").

If you just sit at the bottom ( $dW_t = 0$ ), your height is constant.

But what if you "jiggle" randomly side-to-side ( $dW_t$ )? Because the pipe is curved, every time you jiggle away from the center, your height goes up.

This "jiggling" ( $dW_t$ ) on a "curved" function ( $f''$ ) creates a net upward drift in your height.

This upward drift is the Itô correction term. It's a "free" bit of profit you get from volatility.

Part 5: Let's Use It! (Solving the 'Weird' Integral)

Solving the Integral from Lesson 2.4

We were stuck on

\int_0^T W_t dW_t

. Now we can solve it in 30 seconds.

Goal: Find $\int_0^T W_t dW_t$

Strategy: We need to find a function $f(W_t)$ where the output of Itô's Lemma contains the term $W_t dW_t$ . Our Itô formula is $df = f'(W_t)dW_t + \dots$ . We need $f'(W_t)$ to equal $W_t$ .

What's the anti-derivative of $W_t$ ? It's $f(W_t) = \frac{1}{2}W_t^2$ . Let's try that!

Step 1: Define f(W_t) and find its derivatives

\begin{gathered} f(W_t) = \frac{1}{2}W_t^2 \\ f'(W_t) = W_t \\ f''(W_t) = 1 \end{gathered}

Step 2: Plug into Itô's Lemma

df = f'(W_t)dW_t + \frac{1}{2}f''(W_t)dt

d(\frac{1}{2}W_t^2) = (W_t)dW_t + \frac{1}{2}(1)dt

We have our rule: $d(\frac{1}{2}W_t^2) = W_t dW_t + \frac{1}{2}dt$

Step 3: Rearrange and Integrate

We rearrange the formula to solve for the $W_t dW_t$ part we want:

W_t dW_t = d(\frac{1}{2}W_t^2) - \frac{1}{2}dt

To find the integral from 0 to T, we just integrate all the pieces:

\int_0^T W_t dW_t = \int_0^T d(\frac{1}{2}W_t^2) - \int_0^T \frac{1}{2}dt

The first term on the right is the sum of all "changes in f," which is just $f(T) - f(0)$ .

\int_0^T d(\frac{1}{2}W_t^2) = \frac{1}{2}W_T^2 - \frac{1}{2}W_0^2 = \frac{1}{2}W_T^2

The second term is a normal integral: $\int_0^T \frac{1}{2}dt = \frac{1}{2}T$

Step 4: The Final Answer

\int_0^T W_t dW_t = \frac{1}{2}W_T^2 - \frac{1}{2}T

We've done it! We've proved that the "weird" $-\frac{1}{2}T$ term from Lesson 2.4 is real, and it's the direct, physical consequence of the Itô correction term.

What's Next?

This is a huge achievement. We have our "chain rule" for a simple function $f(W_t)$ .

But this still isn't powerful enough for finance. Our stock model ( $S_t$ ) isn't just $W_t$ , and our option price ( $V$ ) isn't just a function of $S_t$ ; it's a function of $S_t$ and $t$ .

In our next lesson, we will "upgrade" our tool one last time. We will derive the full Itô's Lemma for $f(t, S_t)$ . We'll use the exact same logic we just learned, but with our 2-variable Taylor expansion (Lesson 1.7). This will be the "final boss" of our theory and the tool we use to derive the Black-Scholes equation.

Up Next: Itô's Lemma (The Full Version)