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:

Theta

Differentiating the price formula with respect to t, we get

$$ Caplet/Floorlet = A \; Notional \; (T_2-T_1) Black $$ $$ \frac{\partial Caplet/Floorlet}{\partial t}= A \; Notional \; (T_2-T_1) \frac{\partial}{ \partial t} \left( Black\right)$$

We know from Black's section that:

$$ \frac{\partial Black }{\partial t} = -\frac{e^{-r T_1} F \sigma}{2 \sqrt{\tau}} n \left( d_{1}\right) + r \; Black $$

Hence:

$$ \frac{\partial Caplet/Floorlet}{\partial t}= A \; Notional \; (T_2-T_1) \left( -\frac{e^{-r T_1} F \sigma}{2 \sqrt{\tau}} n \left( d_{1}\right) + r \; Black \right)$$