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The Black-Scholes Formula

The Black-Scholes formula, derived independently by Fischer Black, Myron Scholes (1973), and Robert Merton (1973), provides a closed-form solution for pricing European options. It remains one of the most influential results in financial economics, earning Scholes and Merton the 1997 Nobel Prize.

Before the symbols arrive, the mechanism is simple: a European call pays \((S_T - K)^+\) at maturity, so its fair price must be the discounted expected payoff taken under a measure that prices the underlying correctly. Once the stock is a geometric Brownian motion under that measure, the expectation reduces to a Gaussian integral and collapses into a two-term closed form. The remainder of this section records the formula, its assumptions, and the meaning of its two terms; the rest of the chapter develops the six perspectives below.

Roadmap: Six Perspectives on the Black-Scholes Formula

The same closed-form result \(C = S\mathcal{N}(d_1) - Ke^{-rT}\mathcal{N}(d_2)\) admits six different mathematical viewpoints. Each row below names a perspective, its central object, the equation that captures it, what it expresses about the option, and the sibling section in which it is developed. The remaining files in this section — Put-call parity, Asymptotic behavior, and Computational examples — exploit these perspectives to derive constraints, limiting cases, and numerical implementation respectively.

Perspective Central object Main equation Interpretation Where developed
Replication Self-financing portfolio hedge equation \(dV = \Delta\, dS + r(V - \Delta S)\, dt\) eliminate risk by dynamic trading Girsanov derivation
PDE price function \(V(S, t)\) Black-Scholes PDE local dynamics of \(V\) Girsanov derivation
Probability risk-neutral expectation \(\mathbb{E}^{\mathbb{Q}}[\,\cdot\,]\) Feynman-Kac formula discounted expected payoff Probabilistic interpretation
Measure change Radon-Nikodym derivative Girsanov's theorem drift transformation \(\mathbb{P} \to \mathbb{Q} \to \mathbb{Q}^S\) Girsanov derivation, Probabilistic interpretation
Geometry convex payoff \((S_T - K)^+\) \(\Gamma > 0\) optionality (Jensen's inequality) Properties and bounds
Strike-space strike second derivative \(\partial^2 C / \partial K^2\) Breeden-Litzenberger implied risk-neutral density Digital option pricing

These are six coordinate systems on a single underlying object. The recurrence of \((d_1, d_2)\) across rows is the algebraic shadow of that unity: the same pair of numbers controls the exercise probability under \(\mathbb{Q}\), the delta under \(\mathbb{Q}^S\), the asymptotic limits, the strike derivative, and the Greeks.

Model Setup and Assumptions

Recall (see § GBM as Asset Price Model and § Assumptions): under the risk-neutral measure \(\mathbb{Q}\), the underlying follows

\[ dS_t = rS_t \, dt + \sigma S_t \, dW_t, \qquad S_t = S_0\exp\!\left((r - \tfrac{1}{2}\sigma^2)t + \sigma W_t\right), \]

with the standard frictionless / constant-parameter / no-dividend / European-exercise assumptions. We retain only the notational conventions needed below.

Contract specifications. \(S\) (or \(S_0\)) is the current asset price, \(K\) the strike, and \(T\) the time to maturity (in years). We price at time \(0\) throughout, so \(T\) plays the dual role of maturity date and time remaining; when discussing time evolution we use \(T \to 0\) instead of \(t \to T\).

The Black-Scholes Formulas

Section goal: the call and put pricing formulas, stated as a theorem with the rigour caveat.

1. European Call Option

Theorem (Black-Scholes Formula)

Under assumptions 1–7 above — together with the standard admissibility conditions on trading strategies and a Brownian filtration — the market is complete, the equivalent martingale measure \(\mathbb{Q}\) is unique, and the unique arbitrage-free time-\(0\) prices of European call and put options with strike \(K\) and maturity \(T\) are:

\[\begin{array}{llrrr} C &=& S\mathcal{N}(d_1) &-& Ke^{-rT}\mathcal{N}(d_2)\\ P &=& Ke^{-rT}\mathcal{N}(-d_2) &-& S\mathcal{N}(-d_1) \end{array}\]

where

\[ d_1 = \frac{\ln(S/K) + (r + \frac{1}{2}\sigma^2)T}{\sigma\sqrt{T}}, \qquad d_2 = \frac{\ln(S/K) + (r - \frac{1}{2}\sigma^2)T}{\sigma\sqrt{T}} = d_1 - \sigma\sqrt{T} \]

A fully rigorous proof requires the stochastic-calculus machinery (Itô's formula, Girsanov's theorem, martingale representation, admissibility) developed in later chapters; here we record the result and analyze its structure.

Notation: \(\mathcal{N}(x)\) denotes the cumulative distribution function of the standard normal distribution:

\[ \mathcal{N}(x) = \mathbb{P}(Z \leq x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^x e^{-\frac{z^2}{2}} dz \quad \text{where } Z \sim \mathcal{N}(0,1) \]

Component Analysis

Section goal: decomposition and interpretation of the parameters \(d_1\) and \(d_2\).

The pair \((d_1, d_2)\) is the algebraic shadow of the two perspectives that produce the formula: \(d_2\) records the standardised log-moneyness under the risk-neutral measure \(\mathbb{Q}\), and \(d_1\) does the same under the stock-numéraire measure \(\mathbb{Q}^S\) that arises from change of numéraire. The fixed gap \(d_1 - d_2 = \sigma\sqrt{T}\) is the Girsanov drift between these two measures; everything that follows in this section unpacks consequences of that single shift.

1. The d_1 Parameter

\[ d_1 = \frac{\ln(S/K) + (r + \frac{1}{2}\sigma^2)T}{\sigma\sqrt{T}} \]

Structure:

  • Numerator: Log-moneyness \(\ln(S/K)\) plus drift-adjusted growth \((r + \frac{1}{2}\sigma^2)T\)
  • Denominator: Volatility-adjusted time \(\sigma\sqrt{T}\)

Probabilistic interpretation (summary): \(\mathcal{N}(d_2) = \mathbb{Q}(S_T > K)\) is the risk-neutral exercise probability, and \(\mathcal{N}(d_1) = \mathbb{Q}^S(S_T > K)\) is the same exercise probability under the stock-numéraire measure. The relation \(d_1 = d_2 + \sigma\sqrt{T}\) encodes the Girsanov drift shift between the two measures. See probabilistic interpretation for the full derivation, including the conditional-expectation decomposition of the two-term formula and the connection to delta hedging.

2. The d_2 Parameter

\[ d_2 = d_1 - \sigma\sqrt{T} \]

The gap \(d_1 - d_2 = \sigma\sqrt{T}\) widens with volatility and time.

3. Key Identity

The normal density values at \(d_1\) and \(d_2\) satisfy a fundamental relation used throughout the Greeks derivations:

\[ S\,\phi(d_1) = Ke^{-rT}\phi(d_2) \]

where \(\phi(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2}\) is the standard normal density. This follows from \(d_1^2 - d_2^2 = 2\left[\ln(S/K) + rT\right]\).

4. The Normal CDF N(·)

Properties:

  • \(\mathcal{N}(0) = 0.5\) (median)
  • \(\mathcal{N}(x) \to 1\) as \(x \to \infty\)
  • \(\mathcal{N}(x) \to 0\) as \(x \to -\infty\)
  • \(\mathcal{N}(-x) = 1 - \mathcal{N}(x)\) (symmetry)

Computational note: \(\mathcal{N}(x)\) has no closed form in elementary functions, but it is exactly expressible through the error function via \(\mathcal{N}(x) = \tfrac{1}{2}\!\left[1 + \mathrm{erf}\!\left(x/\sqrt{2}\right)\right]\), and is efficiently computed using polynomial approximations or built-in functions (e.g., scipy.stats.norm.cdf).

Formula Structure

Section goal: what the two-term form \(S\mathcal{N}(d_1) - Ke^{-rT}\mathcal{N}(d_2)\) encodes.

Both formulas decompose into a stock term and a strike term:

\[ C = \underbrace{S\mathcal{N}(d_1)}_{\text{stock term}} - \underbrace{Ke^{-rT}\mathcal{N}(d_2)}_{\text{strike term}}, \qquad P = \underbrace{Ke^{-rT}\mathcal{N}(-d_2)}_{\text{strike term}} - \underbrace{S\mathcal{N}(-d_1)}_{\text{stock term}} \]

The two terms are the discounted \(\mathbb{Q}\)-expected payments received and paid conditional on exercise; the put expression mirrors the call via \(\mathcal{N}(-x) = 1 - \mathcal{N}(x)\). Full pricing-by-conditional-expectation derivation is in probabilistic interpretation. The shared \(d_1, d_2\) structure of both formulas reflects the underlying put-call parity relation derived in put-call parity.

Moneyness Classification

Section goal: naming conventions for ITM, ATM, and OTM and their probabilistic meanings.

1. Definitions

For a call option with strike \(K\):

  • In-the-money (ITM): \(S > K\)

  • Intrinsic value = \(S - K > 0\)

  • Would have positive payoff if exercised immediately

  • At-the-money (ATM): \(S \approx K\)

  • Intrinsic value \(\approx 0\)

  • Most sensitive to volatility changes

  • Out-of-the-money (OTM): \(S < K\)

  • Intrinsic value = \(0\)

  • Pure time value

For a put: ITM when \(S < K\), OTM when \(S > K\).

2. Relationship to d_1 and d_2

Moneyness \(\ln(S/K)\) \(d_1, d_2\) \(\mathcal{N}(d_1), \mathcal{N}(d_2)\)
Deep OTM \(\ll 0\) Large negative Close to 0
OTM \(< 0\) Negative \(< 0.5\)
ATM \(\approx 0\) Small \(\approx 0.5\)
ITM \(> 0\) Positive \(> 0.5\)
Deep ITM \(\gg 0\) Large positive Close to 1

Special Cases

Section goal: limiting behaviour the formula reduces to in degenerate parameter regimes.

Three special configurations are referenced repeatedly throughout the chapter. Each is derived rigorously in asymptotic behavior; we record only the formulas here.

Case Condition Call price
ATM forward (ATMF) \(S = Ke^{-rT}\) \(d_1 = \tfrac{\sigma\sqrt{T}}{2}\), \(\;d_2 = -\tfrac{\sigma\sqrt{T}}{2}\); symmetric around \(\mathcal{N}(0) = 0.5\)
At maturity \(T = 0\) \(C \to (S-K)^+\), \(\;P \to (K-S)^+\) (recovers the terminal payoff)
Zero volatility \(\sigma \to 0\) \(C \to (S - Ke^{-rT})^+\) (forward value, no uncertainty premium)

The ATMF case is the natural reference point for delta hedging and the \(0.4\,S\sigma\sqrt{T}\) approximation; the at-maturity case verifies the formula's terminal boundary condition; the zero-volatility case isolates the deterministic ("forward") component of the price.

Comparison with Binomial Model

Section goal: how the discrete binomial price converges to Black-Scholes as \(\Delta t \to 0\).

The Black-Scholes formula is the continuous-time limit of the binomial model:

Binomial Model Black-Scholes Model
Discrete time steps Continuous time
Two possible outcomes per step Infinitesimal changes
Risk-neutral probability \(q\) Risk-neutral measure \(\mathbb{Q}\)
Backward induction PDE or expectation
\(\mathcal{N}(d_2) \approx\) binomial probability \(\mathcal{N}(d_2) = \lim_{n\to\infty}\) binomial

As the number of time steps \(n \to \infty\) in the binomial model with appropriate parameter scaling:

\[ \text{Binomial price} \to \text{Black-Scholes price} \]

The Black-Scholes PDE

Recall (see § Delta Hedging and § Discounting and Killing Term): the price function \(V(S, t)\) must satisfy

\[ \boxed{\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0}, \qquad V(S, T) = (S-K)^+ \text{ (call) or } (K-S)^+ \text{ (put)}. \]

The Black-Scholes call and put formulas are the unique solutions with these terminal conditions; the derivation routes (heat-equation transform, Feynman-Kac, characteristic functions, similarity solutions) are developed in § BS PDE Analytic Solution. Both trivial solutions \(V = S\) and \(V = e^{rt}\) are verified as Exercise 6 of § Delta Hedging.

Why This Formula?

Recall (see § One Equation, Five Perspectives): the formula admits four mutually consistent derivations — PDE, risk-neutral expectation, self-financing replication, and Girsanov / martingale change of measure. All three are unified by the no-arbitrage principle, and the unification is the subject of § One Equation, Five Perspectives.

Summary

The Black-Scholes formula for European options:

Call: \(C = S\mathcal{N}(d_1) - Ke^{-rT}\mathcal{N}(d_2)\)

Put: \(P = Ke^{-rT}\mathcal{N}(-d_2) - S\mathcal{N}(-d_1)\)

where

\[ d_1 = \frac{\ln(S/K) + (r + \frac{1}{2}\sigma^2)T}{\sigma\sqrt{T}}, \quad d_2 = d_1 - \sigma\sqrt{T} \]

Key features:

  • Closed-form solution (no numerical iteration needed)
  • Depends on five observable inputs: \(S\), \(K\), \(T\), \(r\), \(\sigma\)
  • \(\mathcal{N}(d_1)\) and \(\mathcal{N}(d_2)\) have probabilistic interpretations
  • Decomposes into "stock term" and "strike term"
  • Satisfies put-call parity, boundary conditions, and no-arbitrage constraints

The formula's elegance and practical utility have made it ubiquitous in financial markets, despite its simplifying assumptions.

Exercises

Exercise 1. A European call option has the following parameters: \(S_0 = 80\), \(K = 85\), \(r = 3\%\), \(\sigma = 25\%\), and \(T = 0.5\) years. Compute \(d_1\), \(d_2\), and the Black-Scholes call price \(C_0\). Classify the option as ITM, ATM, or OTM.

Solution to Exercise 1

Parameters: \(S_0 = 80\), \(K = 85\), \(r = 0.03\), \(\sigma = 0.25\), \(T = 0.5\).

Step 1: Compute \(d_1\)

\[ d_1 = \frac{\ln(80/85) + (0.03 + 0.5 \times 0.0625) \times 0.5}{0.25\sqrt{0.5}} \]

Numerator: \(\ln(0.9412) + (0.03 + 0.03125) \times 0.5 = -0.06062 + 0.030625 = -0.02999\).

Denominator: \(0.25 \times 0.7071 = 0.17678\).

\[ d_1 = \frac{-0.02999}{0.17678} = -0.1697 \]

Step 2: Compute \(d_2\)

\[ d_2 = -0.1697 - 0.17678 = -0.3465 \]

Step 3: Evaluate cumulative normal values

\[ \mathcal{N}(d_1) = \mathcal{N}(-0.1697) \approx 0.4326 \]
\[ \mathcal{N}(d_2) = \mathcal{N}(-0.3465) \approx 0.3645 \]

Step 4: Compute call price

\[ C_0 = 80 \times 0.4326 - 85 \times e^{-0.03 \times 0.5} \times 0.3645 \]
\[ = 34.61 - 85 \times 0.9851 \times 0.3645 = 34.61 - 30.52 = 4.09 \]

Classification: Since \(S_0 = 80 < K = 85\), the call is out-of-the-money (OTM). The entire value of $4.09 is time value.

Exercise 2. Verify that the Black-Scholes call formula satisfies the lower bound \(C \geq \max(S - Ke^{-rT}, 0)\) for the parameters \(S_0 = 100\), \(K = 90\), \(r = 5\%\), \(\sigma = 30\%\), \(T = 1\). Compute both sides explicitly.

Solution to Exercise 2

Parameters: \(S_0 = 100\), \(K = 90\), \(r = 0.05\), \(\sigma = 0.30\), \(T = 1\).

Lower bound: \(\max(S_0 - Ke^{-rT}, 0) = \max(100 - 90e^{-0.05}, 0) = \max(100 - 85.61, 0) = 14.39\).

Compute Black-Scholes price:

\[ d_1 = \frac{\ln(100/90) + (0.05 + 0.045) \times 1}{0.30} = \frac{0.10536 + 0.095}{0.30} = \frac{0.20036}{0.30} = 0.6679 \]
\[ d_2 = 0.6679 - 0.30 = 0.3679 \]
\[ \mathcal{N}(0.6679) \approx 0.7479, \quad \mathcal{N}(0.3679) \approx 0.6436 \]
\[ C_0 = 100 \times 0.7479 - 90 \times e^{-0.05} \times 0.6436 = 74.79 - 85.61 \times 0.6436 = 74.79 - 55.10 = 19.69 \]

Verification:

  • Upper bound: \(C_0 = 19.69 \leq 100 = S_0\) ✓
  • Lower bound: \(C_0 = 19.69 \geq 14.39 = S_0 - Ke^{-rT}\) ✓

Both bounds are satisfied.

Exercise 3. Starting from the Black-Scholes call formula \(C = S\mathcal{N}(d_1) - Ke^{-rT}\mathcal{N}(d_2)\), derive the put formula

\[ P = Ke^{-rT}\mathcal{N}(-d_2) - S\mathcal{N}(-d_1) \]

using put-call parity \(C - P = S - Ke^{-rT}\) and the identity \(\mathcal{N}(-x) = 1 - \mathcal{N}(x)\).

Solution to Exercise 3

Starting from put-call parity:

\[ C - P = S - Ke^{-rT} \]

Solving for \(P\):

\[ P = C - S + Ke^{-rT} \]

Substituting the Black-Scholes call formula:

\[ P = S\mathcal{N}(d_1) - Ke^{-rT}\mathcal{N}(d_2) - S + Ke^{-rT} \]

Rearranging:

\[ P = -S[1 - \mathcal{N}(d_1)] + Ke^{-rT}[1 - \mathcal{N}(d_2)] \]

Using the identity \(\mathcal{N}(-x) = 1 - \mathcal{N}(x)\):

\[ 1 - \mathcal{N}(d_1) = \mathcal{N}(-d_1), \quad 1 - \mathcal{N}(d_2) = \mathcal{N}(-d_2) \]

Therefore:

\[ P = Ke^{-rT}\mathcal{N}(-d_2) - S\mathcal{N}(-d_1) \]

This is exactly the Black-Scholes put formula.

Exercise 4. Consider the at-the-money forward case where \(S = Ke^{-rT}\). Show that in this case \(d_1 = \frac{\sigma\sqrt{T}}{2}\) and \(d_2 = -\frac{\sigma\sqrt{T}}{2}\). Derive an approximate formula for the ATM forward call price when \(\sigma\sqrt{T}\) is small, using the Taylor expansion \(\mathcal{N}(x) \approx \frac{1}{2} + \frac{x}{\sqrt{2\pi}}\) for small \(x\).

Solution to Exercise 4

When \(S = Ke^{-rT}\), we have \(\ln(S/K) = -rT\).

\[ d_1 = \frac{-rT + (r + \frac{1}{2}\sigma^2)T}{\sigma\sqrt{T}} = \frac{\frac{1}{2}\sigma^2T}{\sigma\sqrt{T}} = \frac{\sigma\sqrt{T}}{2} \]
\[ d_2 = d_1 - \sigma\sqrt{T} = \frac{\sigma\sqrt{T}}{2} - \sigma\sqrt{T} = -\frac{\sigma\sqrt{T}}{2} \]

Let \(\epsilon = \frac{\sigma\sqrt{T}}{2}\), which is small when \(\sigma\sqrt{T}\) is small. Using \(\mathcal{N}(x) \approx \frac{1}{2} + \frac{x}{\sqrt{2\pi}}\):

\[ C = S\mathcal{N}(\epsilon) - Ke^{-rT}\mathcal{N}(-\epsilon) \]

Since \(S = Ke^{-rT}\), denote this common value by \(F\):

\[ C = F\left(\frac{1}{2} + \frac{\epsilon}{\sqrt{2\pi}}\right) - F\left(\frac{1}{2} - \frac{\epsilon}{\sqrt{2\pi}}\right) = \frac{2F\epsilon}{\sqrt{2\pi}} \]

Substituting back \(\epsilon = \frac{\sigma\sqrt{T}}{2}\) and \(F = Ke^{-rT}\):

\[ C_{\text{ATMF}} \approx \frac{Ke^{-rT} \sigma\sqrt{T}}{\sqrt{2\pi}} \approx 0.3989 \cdot Ke^{-rT} \cdot \sigma\sqrt{T} \]

Exercise 5. Show that the Black-Scholes formula recovers the correct terminal payoff. That is, prove that as \(T \to 0\) (time-to-maturity vanishing):

\[ C \to (S - K)^+ \quad \text{and} \quad P \to (K - S)^+ \]

by analyzing the limits of \(d_1\) and \(d_2\) separately for the cases \(S > K\), \(S < K\), and \(S = K\).

Solution to Exercise 5

As \(T \to 0\):

Case 1: \(S > K\)

\[ d_1 = \frac{\ln(S/K) + (r + \frac{1}{2}\sigma^2)T}{\sigma\sqrt{T}} \]

Since \(\ln(S/K) > 0\) is fixed and \(\sigma\sqrt{T} \to 0\), we get \(d_1 \to +\infty\). Similarly \(d_2 \to +\infty\).

\[ C \to S \cdot 1 - K \cdot e^0 \cdot 1 = S - K = (S-K)^+ \]
\[ P \to K \cdot 0 - S \cdot 0 = 0 = (K-S)^+ \]

Case 2: \(S < K\)

Now \(\ln(S/K) < 0\), so \(d_1 \to -\infty\) and \(d_2 \to -\infty\).

\[ C \to S \cdot 0 - K \cdot 0 = 0 = (S-K)^+ \]
\[ P \to K \cdot 1 - S \cdot 1 = K - S = (K-S)^+ \]

Case 3: \(S = K\)

\[ d_1 = \frac{(r + \frac{1}{2}\sigma^2)T}{\sigma\sqrt{T}} = \frac{(r + \frac{1}{2}\sigma^2)\sqrt{T}}{\sigma} \to 0 \]

Similarly \(d_2 \to 0\). So \(\mathcal{N}(d_1), \mathcal{N}(d_2) \to \frac{1}{2}\).

\[ C \to K \cdot \frac{1}{2} - K \cdot 1 \cdot \frac{1}{2} = 0 = (S-K)^+ \]
\[ P \to K \cdot \frac{1}{2} - K \cdot \frac{1}{2} = 0 = (K-S)^+ \]

In all three cases, \(C \to (S-K)^+\) and \(P \to (K-S)^+\) as \(T \to 0\).

Exercise 6. Verify that \(V(S, t) = S\) (holding the stock) and \(V(S, t) = e^{rt}\) (the risk-free bond) both satisfy the Black-Scholes PDE

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

by computing each partial derivative and substituting.

Solution to Exercise 6

For \(V(S,t) = S\):

\[ \frac{\partial V}{\partial t} = 0, \quad \frac{\partial V}{\partial S} = 1, \quad \frac{\partial^2 V}{\partial S^2} = 0 \]

Substituting into the PDE:

\[ 0 + \frac{1}{2}\sigma^2 S^2 \cdot 0 + rS \cdot 1 - rS = rS - rS = 0 \quad \checkmark \]

For \(V(S,t) = e^{rt}\):

\[ \frac{\partial V}{\partial t} = re^{rt}, \quad \frac{\partial V}{\partial S} = 0, \quad \frac{\partial^2 V}{\partial S^2} = 0 \]

Substituting into the PDE:

\[ re^{rt} + \frac{1}{2}\sigma^2 S^2 \cdot 0 + rS \cdot 0 - re^{rt} = re^{rt} - re^{rt} = 0 \quad \checkmark \]

Both trivial solutions satisfy the Black-Scholes PDE.

Exercise 7. In the zero-volatility limit \(\sigma \to 0\), explain why the call price reduces to \(C = \max(S - Ke^{-rT}, 0)\). What happens to \(d_1\) and \(d_2\) in this limit when \(S > Ke^{-rT}\)? When \(S < Ke^{-rT}\)? Relate your answer to the deterministic evolution of the stock price when \(\sigma = 0\).

Solution to Exercise 7

When \(\sigma = 0\), the stock evolves deterministically: \(S_T = Se^{rT}\). There is no randomness, so the option payoff is known with certainty.

When \(S > Ke^{-rT}\), equivalently \(Se^{rT} > K\):

\[ d_1 = \frac{\ln(S/K) + rT}{\sigma\sqrt{T}} + \frac{\sigma\sqrt{T}}{2} \]

The numerator \(\ln(S/K) + rT = \ln(Se^{rT}/K) > 0\) since \(Se^{rT} > K\). Dividing by \(\sigma\sqrt{T} \to 0^+\) gives \(d_1 \to +\infty\). Similarly \(d_2 \to +\infty\).

\[ C \to S \cdot 1 - Ke^{-rT} \cdot 1 = S - Ke^{-rT} \]

When \(S < Ke^{-rT}\), equivalently \(Se^{rT} < K\):

The numerator \(\ln(Se^{rT}/K) < 0\), so dividing by \(\sigma\sqrt{T} \to 0^+\) gives \(d_1 \to -\infty\) and \(d_2 \to -\infty\).

\[ C \to S \cdot 0 - Ke^{-rT} \cdot 0 = 0 \]

Combining: \(C \to \max(S - Ke^{-rT}, 0)\).

This makes sense because with \(\sigma = 0\), the stock grows deterministically at rate \(r\), reaching \(Se^{rT}\) at maturity. The call payoff is \((Se^{rT} - K)^+ = (S - Ke^{-rT})^+ \cdot e^{rT}\), and discounting back gives \(\max(S - Ke^{-rT}, 0)\). Without randomness, there is no option premium beyond the forward intrinsic value.