Lesson 1.8: The Rules of "Normal" Infinitesimals

Why We Ignore Curvature

Welcome! In our last two lessons, we built our "master tool"—the 2nd-Order Taylor Expansion.

For a function of two variables, like our option price V(t,St)V(t, S_t), that tool is this big, complex-looking formula for the change, ΔV\Delta V:

ΔV≈∂V∂tΔt+∂V∂SΔS⏟1st-Order ’Slope’ Terms+12(∂2V∂t2(Δt)2+2∂2V∂t∂S(ΔtΔS)+∂2V∂S2(ΔS)2)⏟2nd-Order ’Curvature’ Terms\Delta V \approx \underbrace{\frac{\partial V}{\partial t}\Delta t + \frac{\partial V}{\partial S}\Delta S}_{\text{1st-Order 'Slope' Terms}} + \underbrace{\frac{1}{2}\left( \frac{\partial^2 V}{\partial t^2}(\Delta t)^2 + 2\frac{\partial^2 V}{\partial t \partial S}(\Delta t \Delta S) + \frac{\partial^2 V}{\partial S^2}(\Delta S)^2 \right)}_{\text{2nd-Order 'Curvature' Terms}}

This formula is 100% correct, but in the world of normal (non-random) calculus, it's massive overkill. Why? Because in the "normal" world, all those 2nd-order "curvature" terms are effectively zero.

This lesson is crucial. We're going to prove why they are zero in normal calculus. This will establish a "baseline" so you can see exactly what rule stochastic calculus breaks to change all of finance.

The "Normal World" Assumption

In normal calculus, all our functions are smooth and predictable. There is no randomness. This means the change in our stock price, ΔS\Delta S, is not random. It's just a simple, predictable drift (like an average return of 8% per year).

  • The Step in Time: Δt\Delta t
  • The Step in the Stock: ΔS≈μStΔt\Delta S \approx \mu S_t \Delta t (Just a simple, predictable change. No dWtdW_t term!)

The Rules of "Normal" Algebra (Orders of Magnitude)

Our goal is to see what happens to our terms as our time step, Δt\Delta t, gets "infinitesimally small" (we say "approaches 0"). Let's use a concrete number. Let's say our small step Δt\Delta t is 0.01 seconds.

Rule 1: (Δt)² vs. Δt

Δt=0.01\Delta t = 0.01 (a small number)

(Δt)2=(0.01)2=0.0001(\Delta t)^2 = (0.01)^2 = \mathbf{0.0001} (a much, much smaller number)

The (Δt)2(\Delta t)^2 term is "infinitely smaller" than Δt\Delta t. It's so small that we can treat it as zero in our approximation. Rule #1: (Δt)² ≈ 0

Rule 2: (Δt ⋅ ΔS) vs. Δt

We know ΔS≈μStΔt\Delta S \approx \mu S_t \Delta t.

So, Δt⋅ΔS≈Δt⋅(μStΔt)=μSt(Δt)2\Delta t \cdot \Delta S \approx \Delta t \cdot (\mu S_t \Delta t) = \mu S_t (\Delta t)^2

This term also has a (Δt)2(\Delta t)^2 in it! Rule #2: (Δt ⋅ ΔS) ≈ 0

Rule 3: (ΔS)² vs. Δt

We know ΔS≈μStΔt\Delta S \approx \mu S_t \Delta t.

So, (ΔS)2≈(μStΔt)2=(μSt)2(Δt)2(\Delta S)^2 \approx (\mu S_t \Delta t)^2 = (\mu S_t)^2 (\Delta t)^2

This term also has a (Δt)2(\Delta t)^2 in it! Rule #3: (ΔS)² ≈ 0

The Simplification: What "Survives"?

Let's go back to our big "master tool" and apply these rules. Which terms "die" (go to 0) and which "survive"?

ΔV≈∂V∂tΔt⏟Has Δt→SURVIVES+∂V∂SΔS⏟Has ΔS≈Δt→SURVIVES+12(∂2V∂t2(Δt)2⏟Rule 1→DIES+2∂2V∂t∂S(ΔtΔS)⏟Rule 2→DIES+∂2V∂S2(ΔS)2⏟Rule 3→DIES)\Delta V \approx \underbrace{\frac{\partial V}{\partial t}\Delta t}_{\text{Has }\Delta t \to \textbf{SURVIVES}} + \underbrace{\frac{\partial V}{\partial S}\Delta S}_{\text{Has }\Delta S \approx \Delta t \to \textbf{SURVIVES}} + \frac{1}{2}\left( \underbrace{\frac{\partial^2 V}{\partial t^2}(\Delta t)^2}_{\text{Rule 1} \to \textbf{DIES}} + \underbrace{2\frac{\partial^2 V}{\partial t \partial S}(\Delta t \Delta S)}_{\text{Rule 2} \to \textbf{DIES}} + \underbrace{\frac{\partial^2 V}{\partial S^2}(\Delta S)^2}_{\text{Rule 3} \to \textbf{DIES}} \right)

As you can see, in normal calculus, the entire 2nd-order "curvature" bracket disappears!

The Result: The "Normal" Chain Rule

After all the 2nd-order terms go to zero, we're left with just the simple, 1st-order "slope" terms:

ΔV≈∂V∂tΔt+∂V∂SΔS\Delta V \approx \frac{\partial V}{\partial t}\Delta t + \frac{\partial V}{\partial S}\Delta S

In infinitesimal (dd) notation, this is the Multivariable Chain Rule you learn in a standard calculus class:

dV=∂V∂tdt+∂V∂SdSdV = \frac{\partial V}{\partial t}dt + \frac{\partial V}{\partial S}dS
What's Next? (The 'Hook')
  • You've now proven why normal calculus only cares about the 1st-order "slope" terms (Delta).
  • But this is where our entire course changes.
  • In Module 2, we will introduce our random path WtW_t.
  • In Module 3, we will discover a new, "weird rule" of algebra: (dWt)² → dt
  • This means that when our step ΔS\Delta S is random, the (ΔS)2(\Delta S)^2 term does NOT go to zero. It "survives" and becomes a Δt\Delta t term.
  • This single, tiny change is what forces us to keep the 2nd-order "curvature" term (∂2V∂S2\frac{\partial^2 V}{\partial S^2}), which in finance is called Gamma. This one surviving term is the "Itô correction term," and it is the key to all of modern finance.

Up Next: Module 2: The Foundations of Randomness