Black Scholes Greeks

We derive the formulae for the Price and Greeks (derivatives with respect to inputs) of the European options under the Black-Scholes assumptions.

Rho

We now derive the formula for the first derivative of the Black-Scholes price formula with respect to the discount rate. Differentiating both sides of the Black-Scholes Call option formula with respect to \( r_{d} \), we get:

$$\frac {\partial BS Call Price } {\partial r_d} =\frac {\partial } {\partial r_d} \left( S e^{- r_{f} \tau} N{\left (d_{1} \right )}- K e^{- r_{d} \tau} N{\left (d_{1} - \sigma \sqrt{\tau} \right )} \right) $$ $$=\frac {\partial } {\partial r_d} \left( S e^{- r_{f} \tau} N{\left (d_{1} \right )}\right) -\frac {\partial } {\partial r_d} \left( K e^{- r_{d} \tau} N{\left (d_{1} - \sigma \sqrt{\tau} \right )} \right) $$ $$= S e^{- r_{f} \tau} \frac {\partial } {\partial r_d} \left( N{\left (d_{1} \right )}\right) -K \frac {\partial } {\partial r_d} \left( e^{- r_{d} \tau} N{\left (d_{1} - \sigma \sqrt{\tau} \right )} \right) $$ $$= S e^{- r_{f} \tau} \frac {\partial } {\partial r_d} N{\left (d_{1} \right )}-K \left( { e^{- r_{d} \tau} } \frac{\partial } {\partial r_d} { N\left (d_{1} - \sigma \sqrt{\tau} \right ) } + { N\left (d_{1} - \sigma \sqrt{\tau} \right ) } \frac{\partial } {\partial r_d} { e^{- r_{d} \tau} } \right) $$ $$= S e^{- r_{f} \tau} \frac {\partial } {\partial r_d} N{\left (d_{1} \right )}-K{ e^{- r_{d} \tau} } \frac{\partial } {\partial r_d} { N\left (d_{1} - \sigma \sqrt{\tau} \right ) } -K { N\left (d_{1} - \sigma \sqrt{\tau} \right ) } \frac{\partial } {\partial r_d} { e^{- r_{d} \tau} } $$ $$= S e^{- r_{f} \tau} n\left (d_{1} \right) \frac{\partial d_{1} } {\partial r_d} - K{ e^{- r_{d} \tau} } n\left (d_{1} - \sigma \sqrt{\tau} \right) \frac{\partial d_{1} }{\partial r_d} + K { N\left (d_{1} - \sigma \sqrt{\tau} \right ) } { e^{- r_{d} \tau} }\tau $$

Now substitute the expression for \( n \left( d_{1} -\sigma \sqrt{\tau} \right) \) from the Delta Section:

$$n \left( d_{1}-\sigma \sqrt{\tau}\right)=n \left( d_{1}\right) \frac{S}{K} e^{ \left (r_{d} - r_{f}\right) \tau}$$

Thus,

$$\frac {\partial BS Call Price } {\partial r_d} $$ $$= S e^{- r_{f} \tau} n\left (d_{1} \right) \frac{\partial d_{1} } {\partial r_d} - K{ e^{- r_{d} \tau} } n \left( d_{1}\right) \frac{S}{K} e^{ \left (r_{d} - r_{f}\right) \tau} \frac{\partial d_{1} }{\partial r_d} + K { N\left (d_{1} - \sigma \sqrt{\tau} \right ) } { e^{- r_{d} \tau} }\tau $$ $$= S e^{- r_{f} \tau} n\left (d_{1} \right) \frac{\partial d_{1} } {\partial r_d} - S e^{ - r_{f}\tau} n \left( d_{1}\right) \frac{\partial d_{1} }{\partial r_d} + K { N\left (d_{1} - \sigma \sqrt{\tau} \right ) } { e^{- r_{d} \tau} }\tau $$ $$= K { e^{- r_{d} \tau} }\tau { N\left (d_{1} - \sigma \sqrt{\tau} \right ) } $$

We repeat the above procedure to compute the Rho for a Put option:

$$\frac {\partial BS Put Price } {\partial r_d} $$ $$=\frac {\partial } {\partial r_d} \left(-S e^{- r_{f} \tau} N{\left (-d_{1} \right )}+ K e^{- r_{d} \tau} N{\left (-d_{1} + \sigma \sqrt{\tau} \right )} \right) $$ $$=\frac {\partial } {\partial r_d} \left( -S e^{- r_{f} \tau} N{\left (-d_{1} \right )}\right) + \frac {\partial } {\partial r_d} \left( K e^{- r_{d} \tau} N{\left (-d_{1} + \sigma \sqrt{\tau} \right )} \right) $$ $$=- S e^{- r_{f} \tau} \frac {\partial } {\partial r_d} \left( N{\left (-d_{1} \right )}\right) + K \frac {\partial } {\partial r_d} \left( e^{- r_{d} \tau} N{\left (-d_{1} + \sigma \sqrt{\tau} \right )} \right) $$ $$= -S e^{- r_{f} \tau} \frac {\partial } {\partial r_d} N{\left (-d_{1} \right )} + K \left( { e^{- r_{d} \tau} } \frac{\partial } {\partial r_d} { N\left (-d_{1} + \sigma \sqrt{\tau} \right ) } + { N\left (-d_{1} + \sigma \sqrt{\tau} \right ) } \frac{\partial } {\partial r_d} { e^{- r_{d} \tau} } \right) $$ $$=- S e^{- r_{f} \tau} \frac {\partial } {\partial r_d} N{\left(-d_{1} \right )}+ K{ e^{- r_{d} \tau} } \frac{\partial } {\partial r_d} { N\left(-d_{1} + \sigma \sqrt{\tau} \right ) } + K { N\left(-d_{1} + \sigma \sqrt{\tau} \right ) } \frac{\partial } {\partial r_d} { e^{- r_{d} \tau} } $$ $$= S e^{- r_{f} \tau} n\left(-d_{1} \right) \frac{\partial d_{1} } {\partial r_d} - K{ e^{- r_{d} \tau} } n\left(-d_{1} + \sigma \sqrt{\tau} \right) \frac{\partial d_{1} }{\partial r_d} - K { N\left(-d_{1} + \sigma \sqrt{\tau} \right ) } { e^{- r_{d} \tau} }\tau $$ $$= S e^{- r_{f} \tau} n\left(d_{1} \right) \frac{\partial d_{1} } {\partial r_d} - K{ e^{- r_{d} \tau} } n\left(d_{1} - \sigma \sqrt{\tau} \right) \frac{\partial d_{1} }{\partial r_d} - K { N\left(-d_{1} + \sigma \sqrt{\tau} \right ) } { e^{- r_{d} \tau} }\tau $$ $$= S e^{- r_{f} \tau} n\left(d_{1} \right) \frac{\partial d_{1} } {\partial r_d} - K{ e^{- r_{d} \tau} } n \left( d_{1}\right) \frac{S}{K} e^{ \left (r_{d} - r_{f}\right) \tau} \frac{\partial d_{1} }{\partial r_d} - K { N\left(-d_{1} + \sigma \sqrt{\tau} \right ) } { e^{- r_{d} \tau} }\tau $$ $$= S e^{- r_{f} \tau} n\left(d_{1} \right) \frac{\partial d_{1} } {\partial r_d} -S e^{- r_{f} \tau} n\left(d_{1} \right) \frac{\partial d_{1} } {\partial r_d} - K { N\left(-d_{1} + \sigma \sqrt{\tau} \right ) } { e^{- r_{d} \tau} }\tau $$ $$= - K { N\left(-d_{1} + \sigma \sqrt{\tau} \right ) } { e^{- r_{d} \tau} }\tau $$ $$= - K { e^{- r_{d} \tau} }\tau { N\left(-d_{1} + \sigma \sqrt{\tau} \right ) } $$