Lesson 3.2: Itô's Lemma (The Full Version, for f(t, X_t))

Welcome to Lesson 3.2. This is the most important and powerful derivation in our entire course.

In Lesson 3.1, we derived Itô's Lemma for a simple function f(Wt)f(W_t). We proved that df=f′(Wt)dWt+12f′′(Wt)dtdf = f'(W_t)dW_t + \frac{1}{2}f''(W_t)dt. This was our first look at the "Itô Correction Term."

Why We Need an Upgrade

That simple formula is not powerful enough to price a real stock option. An option's price (which we'll call ff) isn't just a simple function of WtW_t. It's a function of two variables that change simultaneously:

  1. Time (tt): As time passes, the option's value "melts" (this is Theta).
  2. The Stock Price (XtX_t): And the stock XtX_t is *not* a simple WtW_t; it's a complex SDE (our model from Lesson 1.4).

Our goal is to find the "chain rule" for this much more complex function, f(t,Xt)f(t, X_t). This is the master tool that unlocks the Black-Scholes equation. We will derive it from first principles, assuming nothing.

Part 1: Assembling Our Three 'Master Tools'

Tool #1: The 2-Variable Taylor Expansion (from Lesson 1.7)

This is our "prediction formula" for the change in ff, or Δf\Delta f. We use xx as a placeholder for XtX_t. The total change Δf\Delta f is the sum of all the partial changes:

Δf≈∂f∂tΔt+∂f∂xΔX+12(∂2f∂t2(Δt)2+2∂2f∂t∂x(ΔtΔX)+∂2f∂x2(ΔX)2)\Delta f \approx \frac{\partial f}{\partial t}\Delta t + \frac{\partial f}{\partial x}\Delta X + \frac{1}{2}\left( \frac{\partial^2 f}{\partial t^2}(\Delta t)^2 + 2\frac{\partial^2 f}{\partial t \partial x}(\Delta t \Delta X) + \frac{\partial^2 f}{\partial x^2}(\Delta X)^2 \right)
Tool #2: The General SDE (Our "Ingredient" from Lesson 1.4)

Our function ff depends on XtX_t. So we need the "rule" for the step ΔX\Delta X. We'll use the *general* form of an SDE, using aa for drift and bb for diffusion.

ΔX≈aΔt+bΔWt\Delta X \approx a \Delta t + b \Delta W_t
Tool #3: The "Weird Algebra" (Our "Simplifier" from Lesson 2.3)

This is our "cheat sheet" for what happens when we take the limit as Δt→0\Delta t \to 0:

  1. (Δt)2→0(\Delta t)^2 \to 0
  2. ΔtΔWt→0\Delta t \Delta W_t \to 0
  3. (ΔWt)2→Δt(\Delta W_t)^2 \to \Delta t (The Master Rule!)

Part 2: The Step-by-Step Derivation (The 'Collision')

We are now going to plug Tool #2 into Tool #1, and then use Tool #3 to simplify the resulting mess. We will go through the 2-Variable Taylor expansion, term by term, and see which ones "survive" and which ones "die" (go to 0).

Step 1: Term 1: Time Decay Term

∂f∂tΔt\frac{\partial f}{\partial t}\Delta t

This is 1st-order in Δt\Delta t. It's not squared. Result: SURVIVES.

Step 2: Term 2: Delta Term

∂f∂xΔX\frac{\partial f}{\partial x}\Delta X

This is the "normal" part of the chain rule. We plug in our SDE for ΔX\Delta X:

∂f∂x(aΔt+bΔWt)\frac{\partial f}{\partial x}\left( a \Delta t + b \Delta W_t \right)

This gives two sub-terms: a∂f∂xΔta \frac{\partial f}{\partial x} \Delta t and b∂f∂xΔWtb \frac{\partial f}{\partial x} \Delta W_t. Both are 1st-order (in Δt\Delta t or ΔWt∼Δt\Delta W_t \sim \sqrt{\Delta t}). Result: BOTH SURVIVE.

Step 3: Term 3: Time Curvature Term

12∂2f∂t2(Δt)2\frac{1}{2}\frac{\partial^2 f}{\partial t^2}(\Delta t)^2

This term has a (Δt)2(\Delta t)^2. By Rule #1, (Δt)2→0(\Delta t)^2 \to 0. Result: DIES (goes to 0).

Step 4: Term 4: Cross Term

∂2f∂t∂x(ΔtΔX)\frac{\partial^2 f}{\partial t \partial x}(\Delta t \Delta X)

This term contains factors like (Δt)2(\Delta t)^2 and (ΔtΔWt)(\Delta t \Delta W_t). By Rule #1 and Rule #2, both go to zero. Result: DIES (goes to 0).

Step 5: Term 5: Gamma / Convexity Term

12∂2f∂x2(ΔX)2\frac{1}{2}\frac{\partial^2 f}{\partial x^2}(\Delta X)^2

We plug in our SDE for ΔX\Delta X: 12∂2f∂x2⋅(aΔt+bΔWt)2\frac{1}{2}\frac{\partial^2 f}{\partial x^2} \cdot \left( a \Delta t + b \Delta W_t \right)^2

When we "FOIL" the squared part, all the resulting sub-terms (like a2(Δt)2a^2(\Delta t)^2 and 2ab(ΔtΔWt)2ab(\Delta t \Delta W_t)) die, except for the (ΔWt)2(\Delta W_t)^2 term.

The surviving sub-term is: (bΔWt)2=b2(ΔWt)2(b \Delta W_t)^2 = b^2(\Delta W_t)^2. By Rule #3, The Master Rule, this becomes b2(Δt)b^2(\Delta t).

Result: The entire 5th term SURVIVES and becomes 12b2∂2f∂x2Δt\frac{1}{2}b^2 \frac{\partial^2 f}{\partial x^2} \Delta t.

Part 3: Collect the 'Survivors' and Write the Formula

We add up all the pieces that didn't go to zero and group them by the differential (dtdt or dWtdW_t).

dt bin (The "Drift"):

[∂f∂t+a∂f∂x+12b2∂2f∂x2]Δt\left[ \frac{\partial f}{\partial t} + a \frac{\partial f}{\partial x} + \frac{1}{2}b^2\frac{\partial^2 f}{\partial x^2} \right] \Delta t

dWt bin (The "Diffusion"):

[b∂f∂x]ΔWt\left[ b \frac{\partial f}{\partial x} \right] \Delta W_t

Writing this in the infinitesimal dd notation gives us the Full Itô's Lemma:

The Full Itô's Lemma (Master Formula)
df=(∂f∂t+a∂f∂x+12b2∂2f∂x2)dt+(b∂f∂x)dWtdf = \left( \frac{\partial f}{\partial t} + a \frac{\partial f}{\partial x} + \frac{1}{2}b^2\frac{\partial^2 f}{\partial x^2} \right)dt + \left( b \frac{\partial f}{\partial x} \right)dW_t

Part 4: What's Next? (The 'Hook')

This formula is the engine. In Module 4, we will:

  1. Plug in our specific stock model (GBM, where a=μSta = \mu S_t and b=σStb = \sigma S_t).
  2. Perform the "magic trick" of delta-hedging to cancel out the dWtdW_t term.
  3. This will lead us directly to the Black-Scholes-Merton PDE.