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:

Vega

Differentiating the price formula with respect to \(\sigma\), we get

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

Where we used the result from the Black's model section.