Greeks

Analytic derivatives in desk units, checked numerically.

Analytic derivatives, converted to desk units on return. See notation for the conversions.

ν = \frac{\partial V}{\partial σ} \times 0.01 = S e^{- q T} φ (d_{1}) \sqrt{T} \times 0.01

Checked against numerical derivatives

Each analytic Greek is verified against the numerical derivative of the function it claims to differentiate: a central difference for vega, a second difference for gamma. An analytic expression that disagrees with its own numerical derivative is wrong however it was derived.

Anchor case: S = K = 100, T = 1, r = 0.05, q = 0, σ = 0.20. Asserted to 1e−10.
Quantity	Value
delta	0.636830651175619
gamma	0.018762017345847
vega	0.375240346916938
theta	−0.017572678209420
rho	0.532324815453763

Structural invariants

Gamma and vega are identical for a call and its matching put.
Put delta equals call delta less the dividend discount e^−qT.
Put-call parity holds across strikes, tenors and a non-zero dividend yield.

Parity in particular catches almost every sign error that survives inspection, which is why it is asserted across a grid rather than at a single point.