Lesson 4.3: The "Magic" (Part 2: Eliminating Subjective Drift μ)

Welcome to Lesson 4.3. This is the second half of the 'magic trick' and one of the most profound 'Aha!' moments in finance.

In Lesson 4.2, we proved that we could make our portfolio Π\Pi perfectly risk-free by setting our number of shares Δ\Delta equal to the option's Delta, ∂V∂S\frac{\partial V}{\partial S}.

This successfully "killed" the entire "Random Bin" in our portfolio's SDE, dΠd\Pi:

dΠ=[−∂V∂t−μSt∂V∂S−12σ2St2∂2V∂S2+Δ(μSt)]dt⏟The ’Predictable Bin’+[… ]dWt⏟= 0d\Pi = \underbrace{\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}_{\text{The 'Predictable Bin'}} + \underbrace{\cancel{\left[ \dots \right] dW_t}}_{\text{= 0}}

Our portfolio's change dΠd\Pi is now 100% risk-free and deterministic.

The Problem: The "Subjectivity" Poison

But we still have a huge problem. Look at the "Predictable Bin" that's left over:

dΠ=[−∂V∂t−μSt∂V∂S−12σ2St2∂2V∂S2+Δ(μSt)]dtd\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

This equation is "poisoned" by the μ\mu (mu) term.

  • μ\mu is the stock's expected drift (e.g., 8% per year, 10% per year, etc.).
  • It's a subjective guess about the future. My μ\mu is different from your μ\mu.
  • If the option's price depends on μ\mu, then you and I will *never* agree on a single price.

This is where the second magic trick happens. We are now going to plug our "magic value" for Δ\Delta into this equation.

Part 1: The Derivation (The Second Cancellation)

Let's perform the substitution.

Tool #1: Our "Predictable Bin" (from Lesson 4.2)
dΠ=[−∂V∂t−μSt∂V∂S−12σ2St2∂2V∂S2+Δ(μSt)]dtd\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
Tool #2: Our "Magic Value" for Δ (from Lesson 4.2)
Δ=∂V∂S\Delta = \frac{\partial V}{\partial S}

Now, let's substitute the "Magic Value" into the "Predictable Bin":

dΠ=[−∂V∂t−μSt∂V∂S⏟Term 1−12σ2St2∂2V∂S2+(∂V∂S)(μSt)⏟Term 2]dtd\Pi = \left[ - \frac{\partial V}{\partial t} \underbrace{- \mu S_t \frac{\partial V}{\partial S}}_{\text{Term 1}} - \frac{1}{2}\sigma^2 S_t^2 \frac{\partial^2 V}{\partial S^2} + \underbrace{\left( \frac{\partial V}{\partial S} \right) (\mu S_t)}_{\text{Term 2}} \right] dt

Look closely at Term 1 and Term 2.

  • Term 1 is: −μSt∂V∂S- \mu S_t \frac{\partial V}{\partial S}
  • Term 2 is: +μSt∂V∂S+ \mu S_t \frac{\partial V}{\partial S}

They are identical, but with opposite signs. They perfectly cancel each other out.

dΠ=[−∂V∂t−μSt∂V∂S−12σ2St2∂2V∂S2+(∂V∂S)(μSt)]dtd\Pi = \left[ - \frac{\partial V}{\partial t} \cancel{- \mu S_t \frac{\partial V}{\partial S}} - \frac{1}{2}\sigma^2 S_t^2 \frac{\partial^2 V}{\partial S^2} + \cancel{\left( \frac{\partial V}{\partial S} \right) (\mu S_t)} \right] dt

Part 2: The "Cured" Portfolio

After this second magical cancellation, what are we left with? The change in our portfolio dΠd\Pi is now only equal to the two terms that survived:

The Cured Portfolio's Change
dΠ=(−∂V∂t−12σ2St2∂2V∂S2)dtd\Pi = \left( - \frac{\partial V}{\partial t} - \frac{1}{2}\sigma^2 S_t^2 \frac{\partial^2 V}{\partial S^2} \right) dt

Take a moment to appreciate this equation. This is the profound "Aha!" moment. Our portfolio's change dΠd\Pi is now:

  1. 100% Risk-Free: The dWtdW_t term is gone.
  2. 100% Objective: The subjective μ\mu term is *also* gone.

The change in our portfolio's value depends *only* on:

  • ∂V∂t\frac{\partial V}{\partial t} (Theta, or time decay)
  • σ2\sigma^2 (Volatility, which we can measure)
  • StS_t (The stock price, which we can see)
  • ∂2V∂S2\frac{\partial^2 V}{\partial S^2} (Gamma, or the option's curvature)

All of these are objective, measurable numbers. Everyone in the world can agree on them. This means everyone, regardless of their "bullish" or "bearish" view (μ\mu), *must* agree on the change in this portfolio's value.

This proves that the price of an option must be independent of the stock's expected return.

The Physical Meaning (The "Wow" Moment)

Why did μ\mu disappear? By delta-hedging, we created a portfolio Π=−V+ΔSt\Pi = -V + \Delta S_t. This portfolio is "long" the stock's drift (from the +ΔSt+\Delta S_t part) and "short" the stock's drift (from the −V-V part, via the a∂V∂Sa\frac{\partial V}{\partial S} term in Itô's Lemma). We proved that these two "drift bets" are perfectly equal and opposite, so they cancel out, leaving us with a portfolio that is "market-neutral" to the stock's expected return.

What's Next? (The 'Hook')

    We have successfully built a perfectly risk-free, objective portfolio. Its change is given by the deterministic equation we just derived.

    This is the "hard part." The final step, Lesson 4.4, is just a simple, elegant argument.

    We will ask: "In a fair market with no free-lunch opportunities (no 'arbitrage'), what rate of return *must* a perfectly risk-free asset earn?"

    It *must* earn the risk-free interest rate, rr (like a government bond).

    So, in our final lesson, we will create a second equation for dΠd\Pi: dΠ=r⋅Π⋅dtd\Pi = r \cdot \Pi \cdot dt.

    We will then set our two equations for dΠd\Pi equal to each other, which will give us the final Black-Scholes-Merton PDE.

Up Next: Lesson 4.4: The "No-Free-Lunch" Argument & The Final PDE