Valuation models

Black-Scholes option pricer

European call and put prices with a continuous dividend yield, plus delta, gamma, vega, theta and rho.

Inputs

SpotStrikeTime to expiryyDividend yield%Volatility%CurrencyRisk-free curveRisk-free curve: flat rate% cc

Price and Greeks

Methodology

Conventions

European exercise, lognormal underlying, constant volatility and rates.
Rate r and dividend yield q are continuously compounded.
Vega per 1 volatility point, rho per 1% change in rate, theta per calendar day (÷365).

References

Black and Scholes (1973), Merton (1973)
Hull, Options, Futures and Other Derivatives, ch. 15 and 19

Formulas

d₁, d₂: d₁ = [ln(S/K) + (r − q + σ²/2)T] / (σ√T), d₂ = d₁ − σ√T
Call: C = S e^(−qT) N(d₁) − K e^(−rT) N(d₂)
Put: P = K e^(−rT) N(−d₂) − S e^(−qT) N(−d₁)
Delta: Δc = e^(−qT) N(d₁), Δp = −e^(−qT) N(−d₁)
Gamma: Γ = e^(−qT) φ(d₁) / (S σ √T)
Vega: ν = S e^(−qT) φ(d₁) √T
Put-call parity: C − P = S e^(−qT) − K e^(−rT)