How did Black, Scholes, and Merton actually derive the model? Can someone outline the math at a high level?

Question

AcadiFi · Accepted Answer

You do not need to reproduce the math for Level II, but understanding the chain of reasoning helps cement the formula. Here is the conceptual flow in 5 steps.

**Step 1 — Assume stock follows geometric Brownian motion (GBM):**

$$dS = \mu \cdot S \cdot dt + \sigma \cdot S \cdot dW$$

Where $dW$ is a Wiener-process increment. GBM has two parameters: drift $\mu$ and volatility $\sigma$. Both are constants.

**Step 2 — Apply Itô's lemma to the option price $f(S, t)$:**

Because $S$ is stochastic and $f$ is a function of $S$, Itô's lemma (the calculus rule for stochastic processes) gives:

$$df = \left(\frac{\partial f}{\partial t} + \mu \cdot S \cdot \frac{\partial f}{\partial S} + 	frac{1}{2} \cdot \sigma^2 \cdot S^2 \cdot \frac{\partial^2 f}{\partial S^2}ight) \cdot dt + \sigma \cdot S \cdot \frac{\partial f}{\partial S} \cdot dW$$

Notice the $dW$ term — the option price has random fluctuations driven by the same shock as the stock.

**Step 3 — Build a delta-neutral portfolio:**

Hold one option short and $\Delta$ shares long. The portfolio value is $\Pi = -f + \Delta \cdot S$. Choose $\Delta = \partial f / \partial S$ so the $dW$ terms cancel:

$$d\Pi = \left(-\frac{\partial f}{\partial t} - 	frac{1}{2} \cdot \sigma^2 \cdot S^2 \cdot \frac{\partial^2 f}{\partial S^2}ight) \cdot dt$$

The portfolio is locally risk-free (no $dW$ term) over an instant.

**Step 4 — Set return = risk-free rate:**

Since the portfolio is risk-free, no-arbitrage forces its return to equal $r$. So:

$$d\Pi = r \cdot \Pi \cdot dt = r \cdot (-f + \Delta \cdot S) \cdot dt$$

Equating with Step 3 yields the **Black-Scholes PDE**:

$$\frac{\partial f}{\partial t} + r \cdot S \cdot \frac{\partial f}{\partial S} + 	frac{1}{2} \cdot \sigma^2 \cdot S^2 \cdot \frac{\partial^2 f}{\partial S^2} = r \cdot f$$

**Step 5 — Apply terminal boundary condition and solve:**

For a European call, $f(S, T) = \max(S - K, 0)$. Solving the PDE with that boundary condition gives the closed-form BSM formula.

```mermaid
flowchart TD
    A[GBM assumption: dS=μS dt + σS dW] --> B[Apply Ito lemma to f]
    B --> C[Form delta-neutral portfolio]
    C --> D[Cancel dW terms, choose Δ = ∂f/∂S]
    D --> E[Set portfolio return = r]
    E --> F[Black-Scholes PDE]
    F --> G[Apply boundary condition f(S,T) = max(S−K,0)]
    G --> H[Closed-form BSM call price]
    style H fill:#c9a84c,color:#0a0a0f
```

**Why the drift $\mu$ drops out at Step 4:** when you set $\Delta = \partial f / \partial S$, the only $dt$ term that survives does not contain $\mu$. That is the mathematical incarnation of the replication insight from Lesson 2.

Bottom line: you do not need to derive this on the exam, but knowing the chain means BSM is no longer a magic formula — it is the consequence of three ideas (GBM, Itô, no-arbitrage).

How did Black, Scholes, and Merton actually derive the model? Can someone outline the math at a high level?

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