European Swaption's Price and Greeks

We derive the formulae for the Greeks/derivatives of the Black equation for a European Receiver and Payer Swaptions

Gamma

We now derive the formula for the Gamma of a European Swaption. Differentiating the price formula with respect to S twice, we get

$$\frac{\partial Swaption}{\partial S}= Black \frac{\partial A}{\partial S} + A \frac{\partial Black }{\partial S}$$ $$\frac{\partial^2 Swaption}{\partial S^2}= \frac{\partial}{\partial S} \left( Black \frac{\partial A}{\partial S} + A \frac{\partial Black }{\partial S} \right)$$ $$\quad \quad= Black \frac{\partial^2 A}{\partial S^2} + A \frac{\partial Black^2 }{\partial S^2} +2 \frac{\partial Black}{\partial S} \frac{\partial A}{\partial S}$$

We know from Black's section that:

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

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

$$\frac{\partial A}{\partial S} = -\frac{1}{S} \left[ A - \frac{ M }{ {\left( 1+ \frac{S}{f} \right)}^{f M + 1}} \right]$$ $$\frac{\partial^2 A}{\partial S^2} = -\frac{1}{S} \frac{\partial}{ \partial S} \left[ A - \frac{ M }{ {\left( 1+ \frac{S}{f} \right)}^{f M + 1}} \right] - \left[ A - \frac{ M }{ {\left( 1+ \frac{S}{f} \right)}^{f M + 1}} \right] \frac{\partial}{\partial S} \left( \frac{1}{S} \right)$$ $$\quad \quad = -\frac{1}{S} \left[ \frac{\partial A}{ \partial S} + \frac{ M \left( f M+1\right)}{ {\left( 1+ \frac{S}{f} \right)}^{f M + 2}} \frac{1}{f} \right] + \frac{1}{S^2} \left[ A - \frac{ M }{ {\left( 1+ \frac{S}{f} \right)}^{f M + 1}} \right]$$ $$\quad \quad = -\frac{1}{S} \left[ \frac{\partial A}{ \partial S} + \frac{ M \left( f M+1\right)}{ {\left( 1+ \frac{S}{f} \right)}^{f M + 2}} \frac{1}{f} \right] - \frac{1}{S} \frac{\partial A}{\partial S}$$ $$\quad \quad = -\frac{1}{S} \left[ 2\frac{\partial A}{ \partial S} + \frac{ M \left( M+ \frac{1}{f} \right)}{ {\left( 1+ \frac{S}{f} \right)}^{f M + 2}} \right]$$ $$\quad \quad = -\frac{1}{S} \left[ - \frac{2}{S} \left[ A - \frac{ M }{ {\left( 1+ \frac{S}{f} \right)}^{f M + 1}} \right] + \frac{ M \left( M+ \frac{1}{f} \right)}{ {\left( 1+ \frac{S}{f} \right)}^{f M + 2}} \right]$$

Making the substitutions, we get

$$\frac{\partial^2 Swaption}{\partial S^2} = A \frac{\partial Black^2 }{\partial S^2}+Black \frac{\partial^2 A}{\partial S^2} + 2 \frac{\partial Black}{\partial S} \frac{\partial A}{\partial S}$$ $$\quad = A \frac {e^{-r\tau}} {S \sigma \sqrt{\tau}} n{\left (d_{1} \right)} - e^{-r\tau} \left[ \phi S N{\left( \phi d_{1} \right )}- \phi K N{\left( \phi d_{2}\right )} \right] \frac{1}{S} \left[- \frac{2}{S} \left[ A - \frac{ M }{ {\left( 1+ \frac{S}{f} \right)}^{f M + 1}} \right] + \frac{ M \left( M+ \frac{1}{f} \right)}{ {\left( 1+ \frac{S}{f} \right)}^{f M + 2}} \right] + -2 e^{-r\tau} \phi N{\left( \phi d_{1} \right )} \frac{1}{S} \left[ A - \frac{ M }{ {\left( 1+ \frac{S}{f} \right)}^{f M + 1}} \right]$$ $$\quad =A \frac {e^{-r\tau}} {S \sigma \sqrt{\tau}} n{\left (d_{1} \right)} -e^{-r\tau} \phi N{\left( \phi d_{1} \right )} \frac{ M \left( M+ \frac{1}{f} \right)}{ {\left( 1+ \frac{S}{f} \right)}^{f M + 2}} + e^{-r\tau} \phi \frac{K}{S} N{\left( \phi d_{2}\right )} \left[- \frac{2}{S} \left( A - \frac{ M }{ {\left( 1+ \frac{S}{f} \right)}^{f M + 1}} \right) + \frac{ M \left( M+ \frac{1}{f} \right)}{ {\left( 1+ \frac{S}{f} \right)}^{f M + 2}} \right]$$ $$\quad = A \frac {e^{-r\tau}} {S \sigma \sqrt{\tau}} n{\left (d_{1} \right)} - \frac{ M \left( M+ \frac{1}{f} \right)}{ {\left( 1+ \frac{S}{f} \right)}^{f M + 2}} \frac{1}{S} \left( e^{-r\tau} \phi S N{\left( \phi d_{1} \right )} - e^{-r\tau} \phi K N{\left( \phi d_{2}\right )} \right) - e^{-r\tau} \phi \frac{K}{S} N{\left( \phi d_{2}\right )} \frac{2}{S} \left( A - \frac{ M }{ {\left( 1+ \frac{S}{f} \right)}^{f M + 1}} \right)$$ $$\quad =A \frac {e^{-r\tau}} {S \sigma \sqrt{\tau}} n{\left (d_{1} \right)} - \frac{ M \left( M+ \frac{1}{f} \right)}{ {\left( 1+ \frac{S}{f} \right)}^{f M + 2}} \frac{Black}{S} - e^{-r\tau} \phi K N{\left( \phi d_{2}\right )} \frac{2}{S^2} \left( A - \frac{ M }{ {\left( 1+ \frac{S}{f} \right)}^{f M + 1}} \right)$$