The Black-Scholes Model

Setup

Market Context

The Black-Scholes model (Black and Scholes 1973, Merton 1973) is the foundation of modern derivatives pricing. It is not used naively in production — every desk knows its failures — but it remains the universal language of the options market. Implied volatility is defined as the Black-Scholes vol that matches a market price. Greeks are reported in Black-Scholes units. Hedging is delta-hedging, its daily P&L explained by the Black-Scholes Taylor expansion.

Understanding Black-Scholes deeply — its derivation, its assumptions, and exactly where and why it fails — is the prerequisite for every model in this course.

Assumptions

The following assumptions are required for the Black-Scholes derivation. Any violation is a potential source of model error in practice.

1. The underlying $S_t$ follows geometric Brownian motion: $dS_t = \mu S_t \, dt + \sigma S_t \, dW_t$ where $\mu \in \mathbb{R}$ is the constant drift and $\sigma > 0$ is the constant volatility. $W_t$ is a standard Brownian motion on a filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \geq 0}, \mathbb{P})$.

2. The risk-free rate $r \geq 0$ is constant and continuously compounded.

3. The dividend yield $q \geq 0$ is constant and paid continuously.

4. No transaction costs, no taxes, and continuous trading is possible.

5. Short selling is permitted with full use of proceeds.

6. No arbitrage: there is no self-financing portfolio that starts at zero cost and produces a non-negative payoff with positive probability.

Theory

1. From GBM to the Log-Normal Distribution

Applying Itô's lemma to $f(S_t) = \ln S_t$:

$df = \frac{\partial f}{\partial S} \, dS + \frac{1}{2} \frac{\partial^2 f}{\partial S^2} (dS)^2 = \frac{1}{S} dS - \frac{1}{2S^2} \sigma^2 S^2 \, dt$

Substituting $dS_t = \mu S_t \, dt + \sigma S_t \, dW_t$:

$d(\ln S_t) = \left(\mu - \tfrac{1}{2}\sigma^2\right)dt + \sigma \, dW_t$

Integrating from $0$ to $T$:

$\ln S_T = \ln S_0 + \left(\mu - \tfrac{1}{2}\sigma^2\right)T + \sigma W_T$

Since $W_T \sim N(0, T)$:

$\boxed{S_T = S_0 \exp\!\left(\left(\mu - \tfrac{1}{2}\sigma^2\right)T + \sigma\sqrt{T}\,Z\right)}, \quad Z \sim N(0,1)$

The term $-\tfrac{1}{2}\sigma^2$ is the Itô correction. It is essential: without it, taking expectations would give $\mathbb{E}[S_T] \neq S_0 e^{\mu T}$. The correction arises because Itô integration is a stochastic integral — the chain rule differs from the classical chain rule by the quadratic variation term $\frac{1}{2}\sigma^2 S^2 \, dt$.

Verification: $\mathbb{E}[S_T] = S_0 \exp\!\left[\left(\mu - \tfrac{1}{2}\sigma^2\right)T + \tfrac{1}{2}\sigma^2 T\right] = S_0 e^{\mu T}$ ✓

2. The Black-Scholes PDE via Delta-Hedging

Let $V(S, t)$ denote the price at time $t$ of any derivative on $S$ maturing at $T$. Construct the delta-hedged portfolio:

$\Pi = V - \Delta \cdot S \quad \text{(long option, short $\Delta$ shares)}$

Over an infinitesimal interval $dt$, the portfolio change is:

$d\Pi = dV - \Delta \, dS$

Applying Itô's lemma to $V(S_t, t)$:

$dV = \frac{\partial V}{\partial t}dt + \frac{\partial V}{\partial S}dS + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2}dt$

Substituting and choosing $\Delta = \frac{\partial V}{\partial S}$ eliminates the stochastic $dS$ term:

$d\Pi = \left(\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2}\right)dt$

This portfolio is now instantaneously riskless: it has no $dW_t$ exposure. By no-arbitrage, it must earn the risk-free rate:

$d\Pi = r\Pi \, dt = r\!\left(V - \frac{\partial V}{\partial S} \cdot S\right)dt$

Equating the two expressions for $d\Pi$:

$\boxed{\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0}$

This is the Black-Scholes PDE. It holds for any derivative $V$ whose payoff at $T$ depends only on $S_T$ — calls, puts, binary options, barriers.

3. Risk-Neutral Pricing and the Black-Scholes Formula

Under the risk-neutral measure $\mathbb{Q}$, the stock drift is replaced by $r - q$ (the cost of carry):

$dS_t = (r - q) S_t \, dt + \sigma S_t \, dW_t^{\mathbb{Q}}$

so that:

$S_T = S_0 \exp\!\left(\left(r - q - \tfrac{1}{2}\sigma^2\right)T + \sigma\sqrt{T}\,Z\right), \quad Z \sim N(0,1) \text{ under } \mathbb{Q}$

The no-arbitrage price of any derivative is:

$V(0) = e^{-rT}\,\mathbb{E}^{\mathbb{Q}}\!\left[\text{Payoff}(S_T)\right]$

For a European call with payoff $\max(S_T - K, 0)$, this expectation evaluates to:

$\boxed{C = S_0 e^{-qT} N(d_1) - K e^{-rT} N(d_2)}$

where $N(\cdot)$ is the standard normal CDF and:

$d_1 = \frac{\ln(S_0/K) + (r - q + \tfrac{1}{2}\sigma^2)T}{\sigma\sqrt{T}}, \qquad d_2 = d_1 - \sigma\sqrt{T}$

For a European put with payoff $\max(K - S_T, 0)$:

$\boxed{P = K e^{-rT} N(-d_2) - S_0 e^{-qT} N(-d_1)}$

Interpretation of the terms:

• $N(d_2)$ is the risk-neutral probability $\mathbb{Q}(S_T > K)$ — the probability of exercise under $\mathbb{Q}$.
• $N(d_1)$ is the delta of the call: $e^{-qT} N(d_1) = \partial C / \partial S_0$.
• $S_0 e^{-qT} N(d_1)$ is the present value of receiving $S_T$ conditional on exercise.
• $K e^{-rT} N(d_2)$ is the present value of paying $K$ conditional on exercise.

Put-call parity (model-independent):

$C - P = S_0 e^{-qT} - K e^{-rT}$

This follows from the no-arbitrage argument: a long call and a short put with the same strike and maturity replicates a long forward. It holds for any model, not just Black-Scholes.

4. The Five Standard Greeks

The Greeks measure the sensitivity of the option price to each model input. For a European call:

where $N'(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2/2}$ is the standard normal PDF.

Interpretations:

• Delta is the hedge ratio: to delta-hedge a long call, short $\Delta$ shares. $\Delta \in (0, e^{-qT})$ for calls.
• Gamma measures the rate of change of delta. It is always positive for vanilla options (calls and puts are both convex in $S$). Peak gamma occurs at-the-money.
• Vega is always positive for vanilla options: longer options or higher vol → higher price.
• Theta is the time decay of the option value. It is typically negative for long options — you pay for the optionality daily.
• Rho is small for short-dated options but matters for multi-year maturities.

5. The Gamma-Theta P&L Relationship

The P&L of a delta-hedged long call over a small interval $dt$ is:

$\text{P\&L} \approx \theta \cdot dt + \frac{1}{2}\Gamma (\Delta S)^2$

where $\Delta S = S_{t+dt} - S_t$ is the actual spot move. This is the gamma-theta trade-off:

• $\theta \cdot dt < 0$: you pay time decay for holding the position.
• $\frac{1}{2}\Gamma (\Delta S)^2 > 0$: you earn positive convexity P&L from every spot move (regardless of direction).

The breakeven condition is:

$\frac{1}{2}\Gamma (\Delta S)^2 = -\theta \cdot dt \implies \frac{(\Delta S)^2}{dt} = \sigma_{\text{impl}}^2 S^2$

A delta-hedged long call earns positive P&L if and only if the realised variance $(\Delta S)^2 / dt$ exceeds the implied variance $\sigma_{\text{impl}}^2 S^2$ priced into the option.

Implementation

This module's interactive notebook demonstrates the following computations in Python:

1. Log-normal distribution of $S_T$: how $\sigma$ controls the spread of the terminal distribution, and how the Itô correction separates the mean from the median.
2. Black-Scholes formula: call and put prices across strikes, with put-call parity verified to machine precision.
3. Greeks as functions of spot: Delta, Gamma, Vega, and Theta visualised simultaneously, showing the characteristic shapes (peak gamma at-the-money, theta most negative at-the-money).
4. Gamma-theta P&L decomposition: Monte Carlo simulation of a delta-hedged call's daily P&L under various realised volatilities, confirming the breakeven at $\sigma_{\text{realised}} = \sigma_{\text{implied}}$.
5. Implied volatility inversion: Newton-Raphson solver recovering $\sigma$ from a market price, demonstrating the round-trip to machine precision.
6. Homogeneity and put-call parity surface: verification of degree-1 homogeneity in $(S, K)$ and the put-call parity error surface across all strikes and maturities.

The notebook runs entirely in-browser via Pyodide. No installation required.

Validation

Any correct implementation of the Black-Scholes formula must satisfy the following checks:

Put-call parity: $C - P = S_0 e^{-qT} - K e^{-rT}$ to machine precision (errors should be $O(10^{-15})$).

Boundary limits:

• Deep ITM call ($S \gg K$): $C \to S e^{-qT} - K e^{-rT}$ (forward value); $\Delta \to e^{-qT}$.
• Deep OTM call ($S \ll K$): $C \to 0$; $\Delta \to 0$.
• At expiry ($T \to 0$): $C \to \max(S - K, 0)$ (intrinsic value).

Homogeneity: For any $\lambda > 0$: $C(\lambda S, \lambda K, r, q, \sigma, T) = \lambda \, C(S, K, r, q, \sigma, T)$ The call price is degree-1 homogeneous in $(S, K)$. This follows because $d_1$ and $d_2$ depend on $S/K$, which is invariant under the scaling.

Greeks consistency:

• $\Delta \in (0, e^{-qT})$ for calls; $\Delta \in (-e^{-qT}, 0)$ for puts.
• $\Gamma > 0$ for all vanilla options.
• As $\sigma \to 0$: $C \to e^{-rT} \max(S e^{(r-q)T} - K, 0)$ (the option becomes a forward).

Limitations

The Black-Scholes model fails in the following structurally important ways:

Constant volatility — most critical failure. Real options markets exhibit a volatility smile: OTM puts trade at higher implied vol than ATM, and OTM calls trade at lower implied vol (equity skew). There is also a term structure: vol is not constant across maturities. Black-Scholes with flat vol misprices any position with exposure to the vol surface shape — which is nearly every real options position.

Log-normal returns. Empirically, equity returns have fat tails and exhibit crash risk (jump risk). The log-normal distribution underprices OTM puts (the market's demand for crash protection) and overprices OTM calls relative to what the market charges. This is the root cause of the equity skew.

Continuous trading. Real hedging is discrete — at best daily, often less frequent. The variance of the P&L of a delta-hedged portfolio scales as $O(1/\sqrt{n})$ in the number of rebalancing steps $n$, not zero. Continuous hedging is a mathematical idealisation.

No transaction costs. In practice, every delta-rebalancing trade incurs bid-offer spread and market impact. The optimal hedging frequency is a trade-off between hedging error and transaction cost — not the continuous limit.

Constant rates. For options with maturities beyond 1–2 years, interest rate uncertainty affects the option price. Stochastic rate corrections (e.g., Rabinovitch 1989) are required for equity/rate hybrid products.

No jumps. Equity prices can gap overnight, on earnings announcements, or during market crises. Jump-diffusion models (Merton 1976, Kou 2002) extend Black-Scholes by adding a Poisson jump component. Standard Black-Scholes cannot price this risk correctly.

Interview Angle

Junior (L1) — Implementation and intuition:

• Write the Black-Scholes call formula. Define $d_1$ and $d_2$.
• What is the interpretation of $N(d_2)$? What is $N(d_1)$?
• Why doesn't the stock drift $\mu$ appear in the formula?
• State put-call parity. Is it model-dependent?
• What is the sign of gamma for a long call? For a long put?

Senior (L2) — Derivation and edge cases:

• Derive the Black-Scholes PDE via delta-hedging. Where does the Itô correction appear, and why?
• Write down the gamma-theta P&L relationship. When does a delta-hedged long call make money?
• The Black-Scholes formula is degree-1 homogeneous in $(S, K)$. Prove it and give an economic interpretation.
• What happens to the delta of a call as $T \to 0$ and the option is at-the-money? (Answer: it goes to 0.5 for $q = 0$, reflecting a 50-50 chance of expiring ITM.)
• Describe Newton-Raphson implied vol inversion. Where can it fail?

Researcher (L3) — Model critique and extensions:

• The Black-Scholes model implies a flat implied vol surface. In practice, equities have a pronounced skew. Name three structural features of equity markets that Black-Scholes cannot capture, and identify which model class addresses each one.
• The Dupire local vol model is Black-Scholes with $\sigma = \sigma(S, t)$. It can exactly fit any observed vol surface. Does this mean it is a better model? What does it get wrong about vol dynamics?
• In Black-Scholes, the vega-weighted P&L from a delta-hedged position converges to zero as hedging frequency increases. Is this still true under stochastic volatility? What is the correct statement?