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:

Gamma

Differentiating the price formula with respect to the forward rate twice, we get

$$ Caplet/Floorlet =A \; Notional \; (T_2-T_1) Black $$ $$ \frac{\partial Caplet/Floorlet}{\partial F}= Notional \; (T_2-T_1) \left( Black \frac{\partial A}{\partial F} + A \frac{\partial Black }{\partial F} \right) $$ $$ \frac{\partial^2 Caplet/Floorlet}{\partial F^2}= Notional \; (T_2-T_1) \left( Black \frac{\partial^2 A}{\partial F^2} + A \frac{\partial Black^2 }{\partial F^2} +2 \frac{\partial Black}{\partial F} \frac{\partial A}{\partial F} \right) $$

We know from Black's section that:

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

And we compute the second derivative of A below, using the first derivative result from the Delta section:

$$ \frac{\partial A}{\partial F} = - \frac{T_2-T_1}{(1+ (T_2-T_1) F)^2}$$ $$ \frac{\partial^2 A}{\partial F^2} = 2 \frac{(T_2-T_1)^2}{(1+ (T_2-T_1) F)^3} $$

Making the substitutions, we get

$$ \frac{\partial^2 Caplet/Floorlet}{\partial F^2}= Notional \; (T_2-T_1) \left( Black \frac{\partial^2 A}{\partial F^2} + A \frac{\partial Black^2 }{\partial F^2} +2 \frac{\partial Black}{\partial F} \frac{\partial A}{\partial F} \right) $$ $$ \quad \quad= Notional \; (T_2-T_1) \left( 2 \frac{(T_2-T_1)^2}{(1+ (T_2-T_1) F)^3} Black + A \frac {e^{-r T_1}} {F \sigma \sqrt{\tau}} n{\left (d_{1} \right)} - 2 e^{-r T_1} \phi N{\left( \phi d_{1} \right )} \frac{T_2-T_1}{(1+ (T_2-T_1) F)^2} \right) $$ $$ \quad \quad= Notional \; (T_2-T_1) \left( 2 \frac{(T_2-T_1)^2}{(1+ (T_2-T_1) F)^3} \left[ e^{-r\tau} \phi F N{\left( \phi d_{1} \right )}- e^{-r\tau} \phi K N{\left( \phi d_{2} \right )} \right] + A \frac {e^{-r T_1}} {F \sigma \sqrt{\tau}} n{\left (d_{1} \right)} - 2 e^{-r T_1} \phi N{\left( \phi d_{1} \right )} \frac{T_2-T_1}{(1+ (T_2-T_1) F)^2} \right) $$ $$ \quad \quad= Notional \; (T_2-T_1) \left( 2 \frac{(T_2-T_1)^2}{(1+ (T_2-T_1) F)^3} e^{-r\tau} \phi F N{\left( \phi d_{1} \right )}- 2 \frac{(T_2-T_1)^2}{(1+ (T_2-T_1) F)^3} e^{-r\tau} \phi K N{\left( \phi d_{2} \right )} + A \frac {e^{-r T_1}} {F \sigma \sqrt{\tau}} n{\left (d_{1} \right)} - 2 e^{-r T_1} \phi N{\left( \phi d_{1} \right )} \frac{T_2-T_1}{(1+ (T_2-T_1) F)^2} \right) $$ $$ \quad \quad= Notional \; (T_2-T_1) \left( 2 \frac{T_2-T_1}{(1+ (T_2-T_1) F)^2} e^{-r T_1} \phi N{\left( \phi d_{1} \right )} \left(\frac{F(T_2-T_1)}{1+ (T_2-T_1) F} - 1 \right) - 2 \frac{(T_2-T_1)^2}{(1+ (T_2-T_1) F)^3} e^{-r\tau} \phi K N{\left( \phi d_{2} \right )} + A \frac {e^{-r T_1}} {F \sigma \sqrt{\tau}} n{\left (d_{1} \right)} \right) $$ $$ \quad \quad= Notional \; (T_2-T_1) \left( -2 \frac{T_2-T_1}{(1+ (T_2-T_1) F)^3} e^{-r T_1} \phi N{\left( \phi d_{1} \right )} - 2 \frac{(T_2-T_1)^2}{(1+ (T_2-T_1) F)^3} e^{-r\tau} \phi K N{\left( \phi d_{2} \right )} + A \frac {e^{-r T_1}} {F \sigma \sqrt{\tau}} n{\left (d_{1} \right)} \right) $$