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:

Delta

Differentiating the interest rate Caplet/Floorlet price formula with respect to the forward rate, we get

$$ Caplet/Floorlet = A \; Notional \; (T_2-T_1) Black $$ $$ \frac{\partial Caplet/Floorlet}{\partial F}= Notional \; (T_2-T_1) \frac{\partial}{ \partial F} \left( A \; Black\right)$$ $$ \quad \quad= Notional \; (T_2-T_1) \left( Black \frac{\partial A}{\partial F} + A \frac{\partial Black }{\partial F} \right) \tag{1}$$

We know from Black's section (replacing S with F and \( \tau\) with \( T_1\) ) that:

$$ \frac{\partial Black }{\partial F} = e^{-r T_1} \phi N{\left( \phi d_{1} \right )} $$

And we compute the derivative of A below:

$$ A = \frac{1}{1+ (T_2-T_1) F} $$ $$ \frac{\partial A}{\partial F} = - \frac{1}{(1+ (T_2-T_1) F)^2} \frac{\partial}{\partial F} \left(1+ (T_2-T_1) F \right)$$ $$ \quad = - \frac{1}{(1+ (T_2-T_1) F)^2} (T_2-T_1) $$ $$ \quad = - \frac{T_2-T_1}{(1+ (T_2-T_1) F)^2} $$

Substituting the Black formula, and the derivaitves of A and Black computed above into equation 1, we get

$$ \frac{\partial Caplet/Floorlet}{\partial F}= Notional \; (T_2-T_1) \left( Black \frac{\partial A}{\partial F} + A \frac{\partial Black }{\partial F} \right) $$ $$ \quad \quad= Notional \; (T_2-T_1) \left( - Black \frac{T_2-T_1}{(1+ (T_2-T_1) F)^2} + A e^{-r T_1} \phi N{\left( \phi d_{1} \right )} \right) $$ $$ \quad \quad= Notional \; (T_2-T_1) \left( - Black \; A^2 (T_2-T_1) + A e^{-r T_1} \phi N{\left( \phi d_{1} \right )} \right) $$ $$ \quad \quad= - Notional \; (T_2-T_1) Black \; A^2 (T_2-T_1) + A \;Notional \; (T_2-T_1) e^{-r T_1} \phi N{\left( \phi d_{1} \right )} $$ $$ \quad \quad= - Caplet/Floorlet \;Price \; A (T_2-T_1) + A \;Notional \; (T_2-T_1) e^{-r T_1} \phi N{\left( \phi d_{1} \right )} $$