The Black-Scholes formula and the Greeks

Solving the Black-Scholes PDE for a European call payoff gives the most famous formula in derivatives. The closed form has remarkable simplicity and is the basis on which option-Greek risk management is built.

European call price

C(S, t) = S Φ(d_1) - K e^{-r(T-t)} Φ(d_2)
d_1 = (ln(S/K) + (r + σ²/2)(T-t)) / (σ √(T-t))
d_2 = d_1 - σ √(T-t)

Φ is the standard normal CDF; K is the strike; T - t is time to expiry. The put price follows by put-call parity:

P(S, t) = K e^{-r(T-t)} Φ(-d_2) - S Φ(-d_1)
P - C = K e^{-r(T-t)} - S    (put-call parity)

Derivation outline

Under Q, log S(T) ~ N(log S(t) + (r - σ²/2)(T-t), σ²(T-t)). The call payoff is (S(T) - K)⁺. Discounted Q-expectation:

C = e^{-r(T-t)} E^Q[(S(T) - K)⁺]
  = e^{-r(T-t)} ∫_{K}^∞ (S - K) f_Q(S) dS

Two integrals: ∫ S f_Q(S) dS and ∫ K f_Q(S) dS. Both have closed forms in terms of Φ, yielding the BS formula. Detailed derivation in any derivatives text (Hull, Shreve, Wilmott).

The Greeks

Sensitivities of the option price to its inputs. Critical for hedging and risk management.

Delta (Δ)

∂C/∂S = Φ(d_1) for a call. For a put: Φ(d_1) - 1. The hedge ratio: the number of shares to short per long call to delta-hedge. Bounded in [0, 1] for calls, [-1, 0] for puts.

Gamma (Γ)

∂²C/∂S² = φ(d_1) / (S σ √(T-t)). Φ-density divided by stock-units-of-vol. Same for calls and puts. Peaks near the money. Captures the curvature; the rate at which delta changes.

Vega

∂C/∂σ = S φ(d_1) √(T-t). Same for calls and puts. Always positive (long-option positions are long vol). Important for vol traders.

Theta (Θ)

∂C/∂t. Typically negative — options decay in time, all else equal. Accelerates near expiry for near-the-money options.

Rho (ρ)

∂C/∂r = K(T-t) e^{-r(T-t)} Φ(d_2). Sensitivity to interest rates. Long calls have positive rho; long puts have negative rho.

Greek identity (BS PDE in Greek form)

Θ + rS Δ + (σ²S²/2) Γ - rV = 0

The BS PDE, rewritten with Greek notation. Time decay (Θ) and interest accrual (rSΔ) balance the gamma-times-variance term, which is the 'hedging error' a delta-hedged portfolio accumulates when the stock moves. This identity is the operational way option-traders think about positions.

Implied volatility

Given a market price for an option, invert the BS formula to find the σ that reproduces it. This is the implied volatility — the market's forecast of future volatility embedded in option prices. In practice, IV varies systematically by strike (skew) and maturity (term structure), giving the implied-volatility surface that vol traders monitor obsessively.

Limitations

• Constant volatility: addressed by Heston, SABR, local vol.
• Constant rates: addressed by stochastic-rate models (Hull-White).
• Continuous trading: real markets have discrete trading; Leland (1985) corrected for transaction costs.
• No jumps: addressed by Merton jump-diffusion.
• Lognormal distributional assumption: addressed by alternative distributions and stochastic-vol models.