QuantPrep | AI-Powered Learning for Quantitative Finance

Our "normal" calculus tool for measuring length, $\sum |\Delta W|$ , failed completely. This is the mathematical proof that a random path is "infinitely wiggly" and has no derivative.

This is the most important failure in modern finance. It forces us to ask the single most important question of this course:

"If summing the first power fails, what happens if we try summing the second power?"

This new type of measurement is called Quadratic Variation. "Quadratic" just means "squared." This lesson is the "Aha!" moment that makes all of finance possible.

Part 1: Defining Our New 'Measurement'

Our goal is to find the Quadratic Variation (QV) of our Brownian Motion $W_t$ from time 0 to $T$ .

This is the mathematical "recipe" for it:

QV_T = \lim_{n \to \infty} \sum_{i=1}^n (\Delta W_i)^2

Let's break this down in plain English. We are going to:

Slice our total time $T$ into $n$ tiny steps.
Find the random change $\Delta W_i$ for each tiny step.
Square that tiny random change: $(\Delta W_i)^2$ .
Sum all those squared changes together.
Take the Limit as our steps become infinitely small ( $n \to \infty$ ).

In our last lesson, the sum $\sum |\Delta W|$ exploded. What will happen to $\sum (\Delta W)^2$ ? Let's find out.

Part 2: The Step-by-Step Derivation (The "Aha!" Moment)

Derivation Steps

Just like in Lesson 2.1, we can't get an exact value for this sum because

\Delta W_i

is a random number. But we can find its "typical size" or "expected value."

Step 1: What is the 'typical size' of one squared step, (ΔWᵢ)²?

We can solve this using our "weird scaling" rule from Lesson 1.3. We proved that the "typical size" of a single random step $\Delta W_i$ is $\sqrt{\Delta t}$ .

Therefore, the "typical size" of a squared step $(\Delta W_i)^2$ is just:

(\text{'Typical' } \Delta W_i)^2 \approx (\sqrt{\Delta t})^2 = \Delta t

This is the single most important discovery in our course. In "normal" calculus, a squared step $(\Delta t)^2$ is tiny and we treat it as zero. In "stochastic" calculus, a squared random step $(\Delta W)^2$ is not zero. It "behaves like" a first-order step, $\Delta t$ .

Step 2: Replace the random term with its "typical size."

Let's replace every $(\Delta W_i)^2$ in our sum with its "typical size," $\Delta t$ . We are now trying to calculate:

QV_T \approx \sum_{i=1}^n \Delta t

Step 3: Solve the "normal" sum.

What is $\sum_{i=1}^n \Delta t$ ? This is just "summing up all the tiny time steps." If we sum up all the tiny time steps from our start (time 0) to our finish (time $T$ ), the total time that has passed is just $T$ .

Step 4: The Final Result (The "Aha!" Moment).

We've just shown that as $n \to \infty$ , our sum $\sum (\Delta W_i)^2$ doesn't go to infinity, and it doesn't go to zero. It converges to a finite, non-random, and beautiful number: the total time $T$ .

\lim_{n \to \infty} \sum_{i=1}^n (\Delta W_i)^2 = T

This is the Quadratic Variation of a Brownian Motion.

Part 3: The "So What?" (Comparing Two Worlds)

This discovery reveals the "weird algebra" of the random world. Let's compare what we found for a "smooth" path (like $f(t) = t^2$ ) versus our "random" path ( $W_t$ ).

Measurement	"Normal" Smooth Path ( $f(t)$ )	"Random" Wiggly Path ( $W_t$ )
Sum of 1st Power $\sum \|\Delta f\|$ vs. $\sum \|\Delta W\|$	→ Finite Path Length (This works.)	→ Infinity ( $\infty$ ) (This fails.)
Sum of 2nd Power $\sum (\Delta f)^2$ vs. $\sum (\Delta W)^2$	→ Zero (0) (This is useless.)	→ Time ( $T$ ) (This works!)

This table is the essence of stochastic calculus. The "normal" tools are completely backward in the random world. In the random world:

Measuring "length" (1st power) is impossible.
Measuring "quadratic variation" (2nd power) is the only way to get a useful, finite measurement of the path's total "wiggliness."

Lesson 2.2: Quadratic Variation (The "Aha!" Moment)