Lesson 4.2: The "Magic" (Part 1: Eliminating Risk)

Welcome to Lesson 4.2. In our last lesson, we successfully built our 'Magic Portfolio' (Π) and found its complete, 'messy' rule of change (SDE).

Our Portfolio SDE (from Lesson 4.1)

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 portfolio is still "poisoned" by randomness ( $dW_t$ ) and subjectivity ( $\mu$ ). Our mission is to cure it. In this lesson, we will perform the first, and most important, "magic trick": we will eliminate all the randomness.

Part 1: The Goal

Our goal is to create a risk-free portfolio. A risk-free portfolio is one that is deterministic—its change $d\Pi$ has zero randomness. Looking at our SDE, the entire random part of our portfolio is contained in the "Random Bin":

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

To make our portfolio risk-free, we must make this entire term equal to zero.

Part 2: The "Magic" Choice

How can we make this term zero? We can't stop the $dW_t$ "jiggle"—that's the market. But we can control the number of shares we buy: the variable $\Delta$ . The "magic trick" is to choose a perfect value for $\Delta$ that makes the entire bracketed expression `[...]` equal to zero.

The Derivation: Solving for Δ

We need to solve this equation: $- \sigma S_t \frac{\partial V}{\partial S} + \Delta (\sigma S_t) = 0$

Step 1: Isolate the Δ term

Move the negative term to the other side:

\Delta (\sigma S_t) = \sigma S_t \frac{\partial V}{\partial S}

Step 2: Solve for Δ

Divide both sides by $(\sigma S_t)$ :

\Delta = \frac{\sigma S_t \frac{\partial V}{\partial S}}{\sigma S_t}

Step 3: The "Magic" Result

The $(\sigma S_t)$ terms on the top and bottom cancel out perfectly, leaving us with:

\Delta = \frac{\partial V}{\partial S}

Part 3: The "So What?" (This is Delta-Hedging!)

We have just had our "eureka" moment. This isn't just a math trick; we've just discovered the most important concept in derivatives trading.

Let's translate what our result $\Delta = \frac{\partial V}{\partial S}$ means.

The Left Side ( $\Delta$ ): This is the number of shares we must buy for our portfolio (e.g., 0.5 shares).
The Right Side ( $\frac{\partial V}{\partial S}$ ): This is the "Delta" ( $\Delta$ ) of our option (from Lesson 3.3). It's the option's sensitivity to the stock's price.

We have just proved that to create a risk-free portfolio, the number of shares we buy must be exactly equal to the option's Delta. This is called "Delta-Hedging."

A Quick Example (The "Wow" Moment)

Let's say our option has a Delta ( $\frac{\partial V}{\partial S}$ ) of 0.6. Our "magic" rule tells us we must set $\Delta = 0.6$ . Let's check if the random parts cancel out:

Our portfolio is $\Pi = -V + 0.6 S_t$ .
The random part is $d\Pi_{\text{random}} = -dV_{\text{random}} + 0.6 dS_{\text{random}}$ .

Plugging in the random parts of our "ingredients":

d\Pi_{\text{random}} = - \left( \sigma S_t \cdot 0.6 \right)dW_t + 0.6 \left( \sigma S_t dW_t \right)

d\Pi_{\text{random}} = -0.6 \sigma S_t dW_t + 0.6 \sigma S_t dW_t = 0

It works. The randomness is gone. By holding 0.6 shares of the stock (long) against our 1 sold option (short), their random wiggles perfectly offset each other, and our portfolio's value is no longer random.

What's Next? (The 'Hook')

We have successfully "killed" the first "poison" term: the $dW_t$ . Our portfolio's SDE has no random part left.

So, our portfolio's change $d\Pi$ is now only equal to the "Predictable Bin":

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

But this equation still has the other "poison" term: $\mu$ (the stock's subjective, unknown drift).

In our next lesson, we will perform the second magic trick. We will plug our new "magic" value $\Delta = \frac{\partial V}{\partial S}$ into this equation.

We will watch in real-time as the $\mu$ terms also cancel out perfectly, leaving us with a final, non-random, non-subjective equation that only contains measurable, known values.

Up Next: Lesson 4.3: Eliminating Drift