Deriving the Black-Scholes PDE

The Black-Scholes partial differential equation is the equation governing the value of any European derivative on a GBM underlying. It is derived by the most-celebrated argument in finance: dynamically hedge the option with the underlying to eliminate all randomness, requiring the resulting riskless portfolio to earn the risk-free rate.

Setup

Stock follows GBM: dS = μ S dt + σ S dW. Riskless asset earns r per unit time. Consider a European derivative V(S, t) maturing at T with payoff V(S, T) = f(S). The value V depends on S, t, and parameters (σ, r, T).

The Itô differential of V

Apply Itô's lemma to V(S, t) with dS as above:

dV = ∂V/∂t dt + ∂V/∂S dS + (1/2)(σ²S²)(∂²V/∂S²) dt
   = (∂V/∂t + μS ∂V/∂S + (σ²S²/2) ∂²V/∂S²) dt + σS ∂V/∂S · dW

The delta-hedging argument

Form a portfolio: long one option, short Δ shares of stock. Choose Δ to eliminate the dW term. Portfolio value: Π = V - Δ S. Differential: dΠ = dV - Δ dS.

dΠ = (∂V/∂t + μS ∂V/∂S + (σ²S²/2) ∂²V/∂S²) dt + σS ∂V/∂S dW - Δ μS dt - Δ σS dW
   = (∂V/∂t + (μS)(∂V/∂S - Δ) + (σ²S²/2) ∂²V/∂S²) dt + σS (∂V/∂S - Δ) dW

Choose Δ = ∂V/∂S: the dW term vanishes. Π is now riskless instantaneously. By no-arbitrage, it must grow at the risk-free rate r:

dΠ = r Π dt = r (V - S ∂V/∂S) dt

Equating to the drift expression above:

∂V/∂t + (σ²S²/2) ∂²V/∂S² = r (V - S ∂V/∂S)

Rearranging gives the Black-Scholes PDE:

∂V/∂t + r S ∂V/∂S + (σ²S²/2) ∂²V/∂S² - r V = 0

Boundary conditions

• Terminal: V(S, T) = f(S) — the option payoff.
• For calls: V(0, t) = 0; V(S, t) → S as S → ∞.
• For puts: V(0, t) = K e^{-r(T-t)}; V(S, t) → 0 as S → ∞.
• For exotic options: barrier conditions, smooth-pasting, etc.

What the PDE says

• Drift μ is gone! The real-world drift doesn't matter for option pricing — only the risk-free rate r appears. This is Girsanov in action.
• The option value depends on σ, not on μ. Two investors with different views on μ but the same σ agree on the option price.
• The PDE is parabolic; running backwards in time (from T to 0). Finite-difference and other PDE solvers apply.

Equivalence to risk-neutral expectation

Feynman-Kac says: V(S, t) = e^{-r(T-t)} E^Q[f(S(T)) | S(t) = S]. Under Q, S follows GBM with drift r. Solving the PDE is equivalent to computing this Q-expectation. Same answer, two computational routes.

Multi-asset extensions

For a derivative on multiple assets, V(S_1, ..., S_n, t), the PDE becomes:

∂V/∂t + Σ_i r S_i ∂V/∂S_i + (1/2) Σ_{i,j} ρ_{ij} σ_i σ_j S_i S_j ∂²V/∂S_i ∂S_j - rV = 0

Multi-dimensional PDE solving is expensive; Monte Carlo wins for n > 3 or so. Curse of dimensionality is fundamental to PDE methods.