Lesson 4.1: The Black-Scholes-Merton Derivation (Part 1: The "Magic Portfolio")

Welcome to Module 4. This is the 'summit' of our course. All the abstract theory we've built—Brownian Motion, the 'weird algebra' (dWt² = dt), and Itô's Lemma—was created to solve this one problem.

Our goal is to find a fair, non-random price for a European option, which we'll call $V(S_t, t)$ .

The Problem We Must Solve

In Lesson 3.2, we found the SDE (the "rule of change") for our option $V$ . It was a problem, not an answer:

dV = \left( \frac{\partial V}{\partial t} + \mu S_t \frac{\partial V}{\partial S_t} + \frac{1}{2}\sigma^2 S_t^2 \frac{\partial^2 V}{\partial S_t^2} \right)dt + \left( \sigma S_t \frac{\partial V}{\partial S_t} \right)dW_t

This equation is unusable for finding a single price. Why? It has two "poison" terms:

It's random: It has a $dW_t$ term.
It's subjective: It depends on $\mu$ (the stock's expected return), which is just a "guess." My guess for $\mu$ (8%) might be different from yours (10%), and we would get two different prices.

This lesson is about a "magic trick" so brilliant it won a Nobel Prize. We will show that we can combine the option ( $V$ ) and the stock ( $S_t$ ) into a single portfolio ( $\Pi$ ) that is perfectly non-random.

This "delta-hedging" trick will make both "poison" terms—the $dW_t$ and the $\mu$ —vanish completely.

Part 1: List Our "Ingredients" (The SDEs)

Let's lay out our two "ingredients" and their "rules of change."

Ingredient 1: The Stock (

S_t

)

This is our Geometric Brownian Motion (GBM) model from Lesson 1.4.

dS_t = \mu S_t dt + \sigma S_t dW_t

Ingredient 2: The Option (

V

)

This is the Full Itô's Lemma we derived in Lesson 3.2. We will use the shorthand $\frac{\partial V}{\partial S}$ for $\frac{\partial V}{\partial S_t}$ , etc.

dV = \left( \frac{\partial V}{\partial t} + \mu S_t \frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2 S_t^2 \frac{\partial^2 V}{\partial S^2} \right)dt + \left( \sigma S_t \frac{\partial V}{\partial S} \right)dW_t

Crucial Insight: Both $dS_t$ and $dV$ are driven by the same underlying source of randomness, $dW_t$ . This is what allows us to cancel them.

Part 2: Build the "Magic Portfolio" (Π)

Now, we build our portfolio. We'll call it $\Pi$ (the Greek letter "Pi"). We need to hold both assets, one "long" (we own it) and one "short" (we sold it).

Let's construct a portfolio where we:

Sell 1 Option (Our position value is $-V$ )
Buy $\Delta$ shares of Stock (Our position value is $+\Delta S_t$ )

The total value of our portfolio at any time is:

Portfolio Value (

\Pi

)

\Pi = -V + \Delta S_t

The Key: $\Delta$ (Delta) is just a number (like 0.5, or 60 shares). It is the *one thing we get to choose*. We're going to find a "magic" value for $\Delta$ that makes all our problems go away.

Part 3: Find the Change in the Portfolio (dΠ)

Now we need to find the "rule of change" for our portfolio. The tiny, infinitesimal change $d\Pi$ is just the sum of the changes in the parts we hold:

d\Pi = d(-V) + d(\Delta S_t)

Assuming we hold the number of shares $\Delta$ constant for this one, tiny infinitesimal step (we re-balance at the *next* step), the change is:

d\Pi = -dV + \Delta dS_t

Part 4: Substitute and Group (The "Messy" Part)

This is the big step. We are going to plug our two "Ingredient" SDEs from Part 1 into our portfolio equation from Part 3. We will write out *every single term*.

d\Pi = - \left[ \left( \frac{\partial V}{\partial t} + \mu S_t \frac{partial V}{\partial S} + \frac{1}{2}\sigma^2 S_t^2 \frac{\partial^2 V}{\partial S^2} \right)dt + \left( \sigma S_t \frac{\partial V}{\partial S} \right)dW_t \right] + \Delta \left[ (\mu S_t) dt + (\sigma S_t) dW_t \right]

This looks like a total mess. But let's be systematic. Our $d\Pi$ equation has two types of terms: predictable $dt$ terms and random $dW_t$ terms. Let's group them into two "bins."

The "Random Bin" (all the $dW_t$ terms):

Let's find all the pieces multiplied by $dW_t$ :

From the $-dV$ part: $- \left( \sigma S_t \frac{\partial V}{\partial S} \right) dW_t$
From the $+\Delta dS_t$ part: $+ \Delta (\sigma S_t) dW_t$

Putting them together, the total random part of our portfolio is:

\text{Random Bin} = \left[ - \sigma S_t \frac{\partial V}{\partial S} + \Delta (\sigma S_t) \right] dW_t

The "Predictable Bin" (all the $dt$ terms):

This is all the "leftovers." We will write out the full expression, not `(...)`:

From the $-dV$ part: $- \left( \frac{\partial V}{\partial t} + \mu S_t \frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2 S_t^2 \frac{\partial^2 V}{\partial S^2} \right) dt$
From the $+\Delta dS_t$ part: $+ \Delta (\mu S_t) dt$

Putting them together, the total predictable part is:

\text{Predictable Bin} = \left[ - \frac{\partial V}{\partial t} - \mu S_t \frac{\partial V}{\partial S} - \frac{1}{2}\sigma^2 S_t^2 \frac{\partial^2 V}{\partial S^2} + \Delta (\mu S_t) \right] dt

Part 5: Our New SDE for the Portfolio

We have successfully found the "rule of change" for our portfolio. It is:

d\Pi = \left[ - \frac{\partial V}{\partial t} - \mu S_t \frac{\partial V}{\partial S} - \frac{1}{2}\sigma^2 S_t^2 \frac{\partial^2 V}{\partial S^2} + \Delta (\mu S_t) \right] dt + \left[ - \sigma S_t \frac{\partial V}{\partial S} + \Delta (\sigma S_t) \right] dW_t

This is the end of Lesson 4.1. We have successfully set up the entire problem, writing out every single term in full detail.

What's Next? (The 'Hook')

We have set the stage for the "magic trick." Our portfolio's change $d\Pi$ is still random (it has a $dW_t$ term) and subjective (it has a $\mu$ term).

But look at the "Random Bin." It has one variable we can control: $\Delta$ .

\text{Random Bin} = \left[ - \sigma S_t \frac{\partial V}{\partial S} + \Delta (\sigma S_t) \right] dW_t

In our next lesson, Lesson 4.2, we will ask the most important question in finance:

"What 'magic' value can we choose for $\Delta$ that will make this entire 'Random Bin' equal to zero?"

We will solve for this $\Delta$ , which will "kill" all the randomness in our portfolio. This is the "delta-hedging" discovery, and it's the first major step to solving for the option's price.

Up Next: Lesson 4.2: Eliminating Risk