QuantPrep | AI-Powered Learning for Quantitative Finance

The price of an option, $V$ , doesn't just depend on one thing. It depends on at least two: the stock price, $S_t$ , and the time, $t$ . So, our function is $V(t, S_t)$ . We need to upgrade our tool to handle functions of multiple variables.

The 'Why': The Mountain Map Problem

Imagine you are standing on the side of a mountain. Your position (x, y) (e.g., latitude and longitude) determines your altitude, $f(x, y)$ .

You are at an "anchor point" (a, b). You know:

Your current altitude: $f(a, b)$
Your slope in the North/South direction: $\frac{\partial f}{\partial y}$
Your slope in the East/West direction: $\frac{\partial f}{\partial x}$

The Problem: Can you use only the information at your "anchor point" $(a, b)$ to build a "simple map" (a flat plane) that predicts your altitude at a new point (x, y) nearby?

The Answer: Yes! This is the 1st-Order Taylor Expansion for two variables.

Building Our 'Map' Step-by-Step

Step 1: The 'Good' Guess (1st-Order: The Tangent Plane)

In the single-variable case, our "map" was a tangent line. In the two-variable case, our "map" is a tangent plane. The logic is the same: (New Value) ≈ (Old Value) + (Change from moving in the x-direction) + (Change from moving in the y-direction).

Goal: Predict $f(x, y)$ using our anchor point $(a, b)$ .

f(x, y) \approx f(a, b) + \frac{\partial f}{\partial x}(x-a) + \frac{\partial f}{\partial y}(y-b)

This is a great approximation, but it's flat. Our "mountain" (our option price function) is curved. We need to account for that.

Step 2: The 'Great' Guess (2nd-Order: The Parabolic Bowl)

This is the master tool we need for Itô's Lemma. We add curvature terms: "Pure x" curvature ( $\frac{\partial^2 f}{\partial x^2}$ ), "Pure y" curvature ( $\frac{\partial^2 f}{\partial y^2}$ ), and "Mixed" curvature ( $\frac{\partial^2 f}{\partial x \partial y}$ ).

f(x, y) \approx f(a, b) + \frac{\partial f}{\partial x}(x-a) + \frac{\partial f}{\partial y}(y-b) + \frac{1}{2}\left( \frac{\partial^2 f}{\partial x^2}(x-a)^2 + 2\frac{\partial^2 f}{\partial x \partial y}(x-a)(y-b) + \frac{\partial^2 f}{\partial y^2}(y-b)^2 \right)

This looks terrifying, but it's just: Prediction ≈ (Old Value) + (Slope Terms) + (Curvature Terms).

The Most Important Formula (The 'Change' Formula)

For our class, we need to adapt this formula for our Option Price, $V(t, S_t)$ . We rename our variables:

Function $f$ becomes $V$ (Option Price).
Variable "x" becomes $t$ (Time).
Variable "y" becomes $S_t$ (Stock Price).
Step $x-a$ becomes $\Delta t$ .
Step $y-b$ becomes $\Delta S$ .

We want the "change in V," or $\Delta V$ . We just move the $f(a,b)$ term to the left and substitute our new names. This gives us the MASTER TOOL for Itô's Lemma:

Master Tool for Itô's Lemma

\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}}