QuantPrep | AI-Powered Learning for Quantitative Finance

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 $V(t, S_t)$ , that tool is this big, complex-looking formula for the change, $\Delta V$ :

\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,

\Delta S

, is not random. It's just a simple, predictable drift (like an average return of 8% per year).

The Step in Time:

\Delta t

The Step in the Stock:

\Delta S \approx \mu S_t \Delta t

(Just a simple, predictable change. No

dW_t

term!)

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

Our goal is to see what happens to our terms as our time step,

\Delta t

, gets "infinitesimally small" (we say "approaches 0"). Let's use a concrete number. Let's say our small step

\Delta t

is 0.01 seconds.

Rule 1: (Δt)² vs. Δt

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

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

The $(\Delta t)^2$ term is "infinitely smaller" than $\Delta t$ . 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 $\Delta S \approx \mu S_t \Delta t$ .

So, $\Delta t \cdot \Delta S \approx \Delta t \cdot (\mu S_t \Delta t) = \mu S_t (\Delta t)^2$

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

Rule 3: (ΔS)² vs. Δt

We know $\Delta S \approx \mu S_t \Delta t$ .

So, $(\Delta S)^2 \approx (\mu S_t \Delta t)^2 = (\mu S_t)^2 (\Delta t)^2$

This term also has a $(\Delta t)^2$ 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"?

\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:

\Delta V \approx \frac{\partial V}{\partial t}\Delta t + \frac{\partial V}{\partial S}\Delta S

In infinitesimal (

d

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

dV = \frac{\partial V}{\partial t}dt + \frac{\partial V}{\partial S}dS

Lesson 1.8: The Rules of "Normal" Infinitesimals

The "Normal World" Assumption

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

The Simplification: What "Survives"?

The Result: The "Normal" Chain Rule