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:

$\boxed{C(t, S) = S \, N(d_1) - K e^{-r\tau} N(d_2),}$

$d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)\,\tau}{\sigma\sqrt{\tau}}, \qquad d_2 = d_1 - \sigma\sqrt{\tau}.$

Here $N(\cdot)$ denotes the standard normal CDF and $n(\cdot)$ its density. By put-call parity,

$P(t, S) = K e^{-r\tau} N(-d_2) - S\, N(-d_1).$

Notation

Throughout: $\tau = T - t$ (time to expiry), $n(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2}$ (standard normal pdf), $N(x) = \int_{-\infty}^x n(u)\,du$ (standard normal CDF).

The Five Greeks

The Greeks measure sensitivity of the option price to each model input. All five are derived analytically from the closed-form formula. A useful identity, easily verified from the definition of $d_1$:

$S \, n(d_1) = K e^{-r\tau} n(d_2).$

This identity allows equivalent representations and simplifies many derivations.

Delta: $\Delta = \partial C / \partial S$

$\Delta_{\mathrm{call}} = N(d_1), \qquad \Delta_{\mathrm{put}} = N(d_1) - 1 = -N(-d_1).$

Interpretation. $\Delta$ is both the hedge ratio (shares needed to replicate) and, under $\mathbb{Q}$, the probability that the call expires in the money when the underlying is a forward: $\Delta_{\mathrm{call}} = N(d_1) \approx \mathbb{Q}(S_T > K)$ only if $r = 0$; otherwise $\mathbb{Q}(S_T > K) = N(d_2)$. The distinction matters at long maturities.

Gamma: $\Gamma = \partial^2 C / \partial S^2$

$\Gamma = \frac{n(d_1)}{S \sigma \sqrt{\tau}}.$

$\Gamma$ is identical for calls and puts (put-call parity). It measures convexity: the rate at which the hedge ratio must be adjusted. High $\Gamma$ → frequent rebalancing → high transaction costs.

Theta: $\Theta = \partial C / \partial t$

$\Theta_{\mathrm{call}} = -\frac{S\, n(d_1)\, \sigma}{2\sqrt{\tau}} - r K e^{-r\tau} N(d_2).$

$\Theta$ is almost always negative for long calls (the option loses time value). For deep in-the-money calls, $\Theta$ can be positive (early exercise premium in American options). The Black-Scholes PDE is often written as:

$\Theta + \frac{1}{2}\sigma^2 S^2 \Gamma + r S \Delta - r C = 0,$

which expresses the P&L decomposition: time decay offset by gamma income and financing of the delta position.

Vega: $\nu = \partial C / \partial \sigma$

$\nu = S \sqrt{\tau}\, n(d_1).$

Vega is positive for both calls and puts and is the same for both (put-call parity). It measures the option's sensitivity to volatility — the input that Black-Scholes treats as known but that must in practice be inferred from market prices (implied volatility).

Rho: $\varrho = \partial C / \partial r$

$\varrho_{\mathrm{call}} = K \tau e^{-r\tau} N(d_2), \qquad \varrho_{\mathrm{put}} = -K \tau e^{-r\tau} N(-d_2).$

Rho is small for short-dated equity options but material for long-dated ones and for interest rate derivatives.

Validation

Several internal consistency checks confirm the formula is correct:

Put-call parity. $C - P = S - Ke^{-r\tau}$. Verified directly from the formula.

Boundary behaviour. As $\tau \to 0$: $C \to (S - K)^+$. As $S \to 0$: $d_1, d_2 \to -\infty$, $N(d_1), N(d_2) \to 0$, so $C \to 0$. As $S \to \infty$: $N(d_1), N(d_2) \to 1$, so $C \to S - Ke^{-r\tau}$ (intrinsic value).

Delta bounds. $0 \leq N(d_1) \leq 1$ always; $N(d_1) \to 1$ as $S/K \to \infty$ (call becomes a forward), $N(d_1) \to 0$ as $S/K \to 0$ (call becomes worthless).

Limitations

Volatility Smile

The single most important empirical failure of Black-Scholes is the volatility smile: in the market, the implied volatility $\hat{\sigma}(K, T)$ — defined as the unique $\sigma$ such that $\text{BS}(S, K, r, T, \sigma) = C_{\text{market}}$ — is not constant. It varies with strike and maturity.

In equity markets, the smile is typically a skew: implied vol is higher for low strikes (out-of-the-money puts) than high strikes, reflecting fear of large downside moves. In FX markets, the smile is more symmetric. This violates the core assumption of constant $\sigma$.

Discrete Hedging Error

Black-Scholes assumes continuous rebalancing. In practice, hedging occurs at discrete intervals $\{t_0, t_1, \ldots, t_n\}$. The replication error per step is:

$\varepsilon_k = \frac{1}{2}\Gamma(t_k, S_{t_k})\left[(\Delta S)^2 - \sigma^2 S_{t_k}^2 \Delta t\right],$

which is non-zero when realised variance deviates from implied variance. Discrete hedging converts a continuous P&L of zero into a discrete P&L that fluctuates. The root-mean-square hedging error scales as $\sigma S \sqrt{\Gamma \cdot \Delta t}$.

Jump Risk

GBM has continuous paths. Real asset prices exhibit jumps (earnings announcements, central bank decisions, geopolitical events). A jump of size $J$ at time $\tau$ creates an instantaneous P&L of $\frac{1}{2}\Gamma J^2$ — unhedgeable in the Black-Scholes framework. Jump-diffusion models (Merton 1976, Kou 2002) incorporate this at the cost of model complexity and incomplete markets.

Lognormal Tails

GBM implies lognormal $S_T$, which has thin tails relative to empirical equity return distributions. Extreme losses are systematically underpriced. This is particularly dangerous for short volatility strategies and barrier option pricing near the barrier.

Constant Rate Assumption

For long-dated options (swaptions, caps, equity LTEPs), the constant-$r$ assumption introduces material pricing errors. Interest rate options require models where rates themselves are stochastic (see Hull-White, LMM).