Black–Scholes & Option Greeks Cheat Sheet

Black–Scholes prices, put–call parity and the Greeks, with intuition for each.

Pricing

C = S N (d_{1}) - K e^{- r T} N (d_{2})

P = K e^{- r T} N (- d_{2}) - S N (- d_{1})

d_{1} = \frac{ln ( S / K ) + ( r + σ ^{2} /2 ) T}{σ T}, d_{2} = d_{1} - σ T

C - P = S - K e^{- r T}

Δ_{C} = N (d_{1}), Δ_{P} = N (d_{1}) - 1

Sensitivity to the spot price.

Γ = \frac{N ^{'} ( d _{1} )}{S σ T}

Largest at the money, close to expiry.

ν = S N^{'} (d_{1}) T

Θ_{C} = - \frac{S N ^{'} ( d _{1} ) σ}{2 T} - r K e^{- r T} N (d_{2})

C \approx 0.4 S σ T

Useful for mental estimates when $r \approx 0$ .