Caplet and Floorlet's Price and Greeks

We derive the formulae for the Greeks/derivatives of the Black equation for a European Caplet and Floorlet:

Formulae Summary

We list the Caplets/Floorlets' price and greeks formulae under the Black's model assumptions below:

$$ Price = A \; Notional \; (T_2-T_1) Black $$ $$ Delta (\Delta) = - Price \; A (T_2-T_1) + A \;Notional \; (T_2-T_1) e^{-r T_1} \phi N{\left( \phi d_{1} \right )} $$ $$ Gamma (\Gamma)= Notional \; (T_2-T_1) \left( -2 \frac{T_2-T_1}{(1+ (T_2-T_1) F)^3} e^{-r T_1} \phi N{\left( \phi d_{1} \right )} - 2 \frac{(T_2-T_1)^2}{(1+ (T_2-T_1) F)^3} e^{-r\tau} \phi K N{\left( \phi d_{2} \right )} + A \frac {e^{-r T_1}} {F \sigma \sqrt{\tau}} n{\left (d_{1} \right)} \right) $$ $$ Vega= A \; Notional \; (T_2-T_1) e^{-r T_1} F \sqrt{\tau} n \left( d_{1}\right)$$ $$ Theta (\Theta)= A \; Notional \; (T_2-T_1) \left( -\frac{e^{-r T_1} F \sigma}{2 \sqrt{\tau}} n \left( d_{1}\right) + r \; Black \right)$$

Where

$$ A=\frac{1}{1+ (T_2-T_1) F(0, T_1, T_2)} $$