Equations used to derive greeks for European options.
Delta: $$ e^{-r_ft} . N(d1) $$
Gamma: $$ \frac{e^{-r_ft}}{S_tσ_s\sqrt{t}}.n(d1) $$
Vega: $$ S_te^{-r_ft}.\sqrt{t}.n(d1) $$
Rho: $$ Ke^{-r_dt}t.N(d1 - σ_s\sqrt{t}) $$
Theta: $$ S_te^{r_ft}r_f.N(d1) - Ke^{-r_dt}r_d.N(d1 - σ_s\sqrt{t}) -S_te^{-r_ft}\frac{σ_s}{2\sqrt{t}}.n(d1) $$
Delta: $$ -e^{-r_ft} . N(-d1) $$
Gamma: $$ \frac{e^{-r_ft}}{S_tσ_s\sqrt{t}}.n(d1) $$
Vega: $$ S_te^{-r_ft}.\sqrt{t}.n(d1) $$
Rho: $$ -Ke^{-r_dt}t.N(-d1 + σ_s\sqrt{t}) $$
Theta: $$ -S_te^{r_ft}r_f.N(-d1) + Ke^{-r_dt}r_d.N(-d1 + σ_s\sqrt{t}) -S_te^{-r_ft}\frac{σ_s}{2\sqrt{t}}.n(d1) $$