Black-Scholes: Derivation, Greeks, Limitations

Setup

Market Assumptions

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a filtered probability space supporting a standard Brownian motion $(W_t)_{t \geq 0}$. The stock price $(S_t)_{t \geq 0}$ follows geometric Brownian motion (GBM) under the physical measure $\mathbb{P}$:

$dS_t = \mu S_t \, dt + \sigma S_t \, dW_t, \qquad S_0 > 0.$

The following assumptions are in force throughout. Each is a modelling choice, not a physical law — and each is violated in practice:

1. Constant volatility. $\sigma > 0$ is a deterministic constant.
2. Constant risk-free rate. The risk-free rate $r \geq 0$ is constant and continuously compounded.
3. No dividends. The stock pays no cash dividends.
4. Continuous trading. Portfolio rebalancing can occur at every instant at zero cost.
5. No transaction costs, no bid-ask spread. All trades execute at the mid-price.
6. No short-selling constraints. The stock can be sold short without restriction.
7. Markets are frictionless and complete. Every contingent claim is replicable.

Under these assumptions, the Black-Scholes framework is internally consistent. The question is not whether it is true — it is not — but whether it is useful as a baseline and how its failure modes manifest in practice.

Derivation via Delta Hedging

The original Black-Scholes (1973) derivation proceeds by constructing a locally riskless portfolio.

Let $C(t, S) = C(t, S_t)$ be the price at time $t$ of a European call option with strike $K$ and maturity $T$. Assume $C \in C^{1,2}([0,T) \times (0,\infty))$.

Step 1: Apply Itô's lemma. Since $S_t$ satisfies the GBM SDE:

$dC = \left(\frac{\partial C}{\partial t} + \mu S \frac{\partial C}{\partial S} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 C}{\partial S^2}\right) dt + \sigma S \frac{\partial C}{\partial S} \, dW_t.$

Step 2: Form the delta-hedged portfolio. Define the portfolio

$\Pi_t = C(t, S_t) - \Delta_t \cdot S_t, \qquad \Delta_t = \frac{\partial C}{\partial S}(t, S_t).$

Its differential is:

$d\Pi_t = dC - \Delta_t \, dS_t = \left(\frac{\partial C}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 C}{\partial S^2}\right) dt.$

The $dW_t$ terms cancel exactly because $\Delta_t$ is chosen as the option's partial derivative with respect to $S$. The portfolio is instantaneously riskless.

Step 3: Apply no-arbitrage. A riskless portfolio must earn the risk-free rate:

$d\Pi_t = r \Pi_t \, dt = r(C - \Delta_t S) \, dt.$

Step 4: Equate and collect. Substituting:

$\frac{\partial C}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 C}{\partial S^2} = r C - r S \frac{\partial C}{\partial S}.$

Rearranging gives the Black-Scholes PDE:

$\frac{\partial C}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 C}{\partial S^2} + r S \frac{\partial C}{\partial S} - r C = 0, \qquad (t, S) \in [0, T) \times (0, \infty),$

subject to the terminal condition $C(T, S) = (S - K)^+$ and boundary conditions $C(t, 0) = 0$, $C(t, S) \sim S$ as $S \to \infty$.

Remark: The Drift $\mu$ Does Not Appear

The physical drift $\mu$ cancels in the portfolio construction. This is not a coincidence: it reflects the risk-neutral pricing principle derivable via Girsanov's theorem. The no-arbitrage price of any replicable claim depends only on $\sigma$ and $r$, not on the investor's expected return.

Derivation via Feynman-Kac

A cleaner derivation uses the Feynman-Kac representation. Under the risk-neutral measure $\mathbb{Q}$, defined by Girsanov's theorem with market price of risk $\lambda = (\mu - r)/\sigma$, the stock follows:

$dS_t = r S_t \, dt + \sigma S_t \, d\widetilde{W}_t,$

where $\widetilde{W}_t = W_t + \lambda t$ is a $\mathbb{Q}$-Brownian motion. The no-arbitrage price is:

$C(t, S) = e^{-r(T-t)} \mathbb{E}^{\mathbb{Q}}\!\left[(S_T - K)^+ \,\Big|\, \mathcal{F}_t\right].$

Under $\mathbb{Q}$, the log-price is:

$\ln S_T = \ln S_t + \left(r - \frac{\sigma^2}{2}\right)\tau + \sigma\sqrt{\tau}\, Z, \qquad Z \sim \mathcal{N}(0,1), \quad \tau = T - t.$

Evaluating the expectation by splitting the integration region $\{S_T > K\}$ yields the Black-Scholes formula: