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

Delta

We derive the formua for the Delta of a European Swaption. Differentiating the price formula with respect to S, we get

$$ Swaption = A \; Black $$ $$ \frac{\partial Swaption}{\partial S}= \frac{\partial}{ \partial S} \left( A \; Black \right)$$ $$ \quad \quad= Black \frac{\partial A}{\partial S} + A \frac{\partial Black }{\partial S} \tag{1}$$

We know from Black's section that:

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

And we compute the derivative of A below:

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

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

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