Chapter 7 — Black–Scholes from First Principles

Guiding question

Given a stock following $dS = \mu S\,dt + \sigma S\,dW$, what is the fair price of an option on $S$? The answer should depend only on observable quantities, not on $\mu$ — the investor's subjective view of expected return. How can this be, and how do we derive the formula rigorously?

Black–Scholes is the synthesis of everything built in Chapters 1–6. It requires:

• GBM as the stock model (Chapter 3, 5)
• Itô's formula to compute option price dynamics (Chapter 3)
• Self-financing portfolio and replication (new concept here)
• The PDE or risk-neutral pricing formula (Chapters 6, 10)

---

7.1 The model setup

Stock: $dS_t = \mu S_t\,dt + \sigma S_t\,dW_t^{\mathbb P}$, with $S_0 > 0$, $\sigma > 0$.

Bond: $dB_t = r B_t\,dt$ (risk-free savings account), so $B_t = B_0 e^{rt}$.

Option: a European call with payoff $g(S_T) = (S_T - K)^+$ at time $T$.

Objective: find the replicating portfolio and its cost at time $0$.

---

7.2 The self-financing condition

A portfolio $(\Delta_t, \psi_t)$ consists of $\Delta_t$ shares of stock and $\psi_t$ units of bond. Its value is:

$$V_t = \Delta_t S_t + \psi_t B_t.$$

The portfolio is self-financing if all changes in value come from capital gains — no external cash flows:

$$dV_t = \Delta_t\,dS_t + \psi_t\,dB_t.$$

In words: the change in wealth equals the gain from the stock position plus the gain from the bond position, with no injections or withdrawals.

Using $dS_t = \mu S_t\,dt + \sigma S_t\,dW_t$ and $dB_t = rB_t\,dt$:

$$dV_t = \Delta_t(\mu S_t\,dt + \sigma S_t\,dW_t) + \psi_t rB_t\,dt = (\mu\Delta_t S_t + r\psi_t B_t)\,dt + \sigma\Delta_t S_t\,dW_t.$$

---

7.3 The replication argument

Assume the option price has the form $V_t = f(t,S_t)$ for some smooth function $f$. Apply Itô's formula:

$$dV_t = df(t,S_t) = \left[\partial_t f + \mu S_t\partial_S f + \tfrac12\sigma^2 S_t^2\partial_{SS}f\right]dt + \sigma S_t\partial_S f\,dW_t.$$

Matching the $dW_t$ term with the self-financing condition:

$$\sigma S_t\partial_S f = \sigma\Delta_t S_t \implies \boxed{\Delta_t = \partial_S f(t,S_t).}$$

The hedge ratio is $\Delta = \partial_S f$, the delta of the option.

Matching the $dt$ terms:

$$\partial_t f + \mu S\partial_S f + \tfrac12\sigma^2 S^2\partial_{SS}f = \mu\Delta S + r\psi B.$$

Substituting $\Delta = \partial_S f$ and $V = f = \Delta S + \psi B$, so $\psi B = f - \Delta S = f - S\partial_S f$:

$$\partial_t f + \mu S\partial_S f + \tfrac12\sigma^2 S^2\partial_{SS}f = \mu S\partial_S f + r(f - S\partial_S f).$$

The $\mu$ terms cancel:

$$\boxed{\partial_t f + rS\,\partial_S f + \tfrac12\sigma^2 S^2\,\partial_{SS}f - rf = 0.}$$

This is the Black–Scholes PDE.

Why $\mu$ disappears

The hedge $\Delta = \partial_S f$ exactly cancels the stochastic term. The remaining PDE has no $\mu$ — the investor's subjective expected return is irrelevant. This is the mathematical statement of no-arbitrage: if two investors with different $\mu$ beliefs both delta-hedge, they hold the same portfolio and must agree on the option price.

Economically, delta-hedging eliminates the risk of the position. The hedged portfolio earns the risk-free rate $r$, regardless of $\mu$.

---

7.4 Boundary conditions and the solution

The PDE is solved on $(t,S)\in[0,T)\times(0,\infty)$ with terminal condition

$$f(T,S) = (S-K)^+.$$

For $S\to 0$: $f\to 0$. For $S\to\infty$: $f\sim S - Ke^{-r(T-t)}$ (deep in-the-money).

The change of variables $x = \log S$ transforms the PDE to the heat equation. The solution is:

Black–Scholes formula

The price of a European call option is

$$C(t,S) = S\,N(d_1) - Ke^{-r(T-t)}\,N(d_2),$$

where

$$d_1 = \frac{\log(S/K) + (r+\sigma^2/2)(T-t)}{\sigma\sqrt{T-t}}, \qquad d_2 = d_1 - \sigma\sqrt{T-t},$$

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

Derivation via heat equation

Let $\tau = T-t$ (time to maturity) and $x = \log(S/K) + (r-\sigma^2/2)\tau$. The Black–Scholes PDE becomes the heat equation $\partial_\tau u = \frac{\sigma^2}{2}\partial_{xx}u$ with initial condition $u(0,x) = K(e^x - 1)^+$.

The solution is given by convolution with the Gaussian kernel:

$$u(\tau,x) = \frac{1}{\sigma\sqrt{2\pi\tau}}\int_{-\infty}^\infty K(e^y-1)^+ e^{-(x-y)^2/(2\sigma^2\tau)}\,dy.$$

Evaluating this integral (split at $y=0$) yields the Black–Scholes formula.

Black–Scholes call option: C = S·N(d₁) − Ke^(−rT)·N(d₂)

K = 100T = 0.50σ = 0.20r = 0.05

C(100) — Call price$8.216

Δ(100) — Shares to hold0.6255

Γ(100) — Convexity0.02736

Delta = ∂C/∂S = N(d₁) is the hedge ratio: hold Δ shares to replicate the option. Gamma = ∂²C/∂S² measures convexity — its interaction with (dS)² = σ²S²dt in Itô's formula is what creates the ½σ²S²Γ term in the Black–Scholes PDE.

---

7.5 The Greeks

The Greeks are sensitivities of the option price to model parameters:

$$\Delta = \partial_S C = N(d_1), \qquad \Gamma = \partial_{SS} C = \frac{N'(d_1)}{S\sigma\sqrt{\tau}},$$ $$\Theta = \partial_t C = -\frac{SN'(d_1)\sigma}{2\sqrt{\tau}} - rKe^{-r\tau}N(d_2), \qquad \mathcal V = \partial_\sigma C = SN'(d_1)\sqrt{\tau}.$$

The Black–Scholes PDE can be written compactly as:

$$\Theta + rS\Delta + \tfrac12\sigma^2 S^2\Gamma = rC.$$

This says: time decay plus drift income plus convexity income equals the risk-free return on the option value.

Gamma is the curvature cost

$\Gamma = \partial_{SS}C > 0$ means the option is a convex function of $S$. By Itô's formula, $(dS)^2 = \sigma^2 S^2\,dt$ generates a contribution $\frac12\Gamma\sigma^2 S^2\,dt$ to the instantaneous P&L of the delta-hedged portfolio. This convexity income is what the option holder "earns" by gamma scalping — rebalancing the delta hedge as $S$ moves. It is exactly balanced by time decay $\Theta$.

---

7.6 Put–call parity

By no-arbitrage, the put price $P(t,S) = Ke^{-r\tau}N(-d_2) - SN(-d_1)$ satisfies:

$$C - P = S - Ke^{-r\tau}.$$

This is put–call parity and requires no model assumptions — only the absence of arbitrage and the definitions of the payoffs.

---

7.7 The risk-neutral derivation

Alternatively, by Chapter 6, under $\mathbb Q$:

$$dS = rS\,dt + \sigma S\,d\widetilde W, \qquad S_T = S_t\exp\!\left[(r-\tfrac12\sigma^2)(T-t) + \sigma(\widetilde W_T - \widetilde W_t)\right].$$

The call price is:

$$C(t,S) = e^{-r\tau}\mathbb E^{\mathbb Q}\!\left[(S_T-K)^+\mid S_t = S\right].$$

Since $\log(S_T/S)\sim N((r-\sigma^2/2)\tau, \sigma^2\tau)$ under $\mathbb Q$, the expectation evaluates to the Black–Scholes formula. This derivation makes explicit that $C$ is an expectation — it can be computed by Monte Carlo simulation.

---

7.8 Model assumptions and their violations

Assumption	Violation	Effect on pricing
Constant $\sigma$	Implied vol varies by strike	Volatility smile/skew
Continuous trading	Jumps in price	Jump premium
No transaction costs	Friction	Bid-ask spread in delta
Normal log-returns	Fat tails	Out-of-the-money option mispricing
Constant $r$	Stochastic rates	For long-dated options

---

7.9 Problems

Problem 1: PDE verification

Verify directly that $C(t,S) = SN(d_1) - Ke^{-r\tau}N(d_2)$ satisfies the Black–Scholes PDE. (Compute $\partial_\tau C$, $\partial_S C$, $\partial_{SS}C$ and substitute.)

Problem 2: Deep in-the-money limit

Show that $\lim_{S\to\infty} C(t,S) - (S - Ke^{-r\tau}) = 0$. Interpret geometrically.

Problem 3: At-the-money approximation

For $S = K$ and small $\tau$, show that $C \approx S\sigma\sqrt{\tau/2\pi}$. This is the leading-order at-the-money option price.

Problem 4: Implied volatility

Given an observed call price $C^{\text{obs}}$, define the implied volatility $\sigma_{\text{imp}}$ as the value of $\sigma$ in the Black–Scholes formula that matches $C^{\text{obs}}$. Show that $C$ is strictly increasing in $\sigma$, so $\sigma_{\text{imp}}$ is uniquely defined.

---

7.10 Selected solutions

Solution 3

At $S = K$, $d_1 = (r+\sigma^2/2)\sqrt{\tau}/\sigma$ and $d_2 = (r-\sigma^2/2)\sqrt{\tau}/\sigma$. For small $\tau$, $d_1\approx d_2\approx 0$ and $N(0) = 1/2$. Using the approximation $N(d)-1/2\approx d\cdot N'(0) = d/\sqrt{2\pi}$:

$$C \approx S(N(d_1)-N(d_2)) = S\cdot\frac{d_1-d_2}{\sqrt{2\pi}} = S\cdot\frac{\sigma\sqrt{\tau}}{\sqrt{2\pi}}.$$

Simplifying: $C \approx S\sigma\sqrt{\tau/(2\pi)}$. The leading-order ATM call price is linear in $\sigma$ and grows as $\sqrt{\tau}$ — a direct consequence of Brownian scaling.

Solution 4

The vega $\mathcal V = \partial_\sigma C = SN'(d_1)\sqrt{\tau} > 0$ for all $S,K,\tau,r,\sigma > 0$. Since $\mathcal V > 0$, $C$ is strictly increasing in $\sigma$. Therefore for any $C^{\text{obs}}$ in $(C(\sigma=0^+), C(\sigma=\infty))$, there is exactly one $\sigma_{\text{imp}}$ with $C(\sigma_{\text{imp}}) = C^{\text{obs}}$.

---

7.11 What this chapter gives you

Result	Formula
Black–Scholes PDE	$\partial_t f + rS\partial_S f + \frac12\sigma^2 S^2\partial_{SS}f = rf$
Hedge ratio (delta)	$\Delta = \partial_S C = N(d_1)$
Call price	$C = SN(d_1) - Ke^{-r\tau}N(d_2)$
Put–call parity	$C - P = S - Ke^{-r\tau}$
Risk-neutral pricing	$C = e^{-r\tau}\mathbb E^{\mathbb Q}[(S_T-K)^+]$
PDE $\leftrightarrow$ expectation	Black–Scholes formula = Feynman–Kac solution

The next chapters extend these ideas: Chapter 8 handles multiple assets, Chapter 10 establishes Feynman–Kac in full generality.