Black's Price and Greeks

We derive the formulae for the price and Greeks of a call and a put options under the Black's model assumptions:

Vega

In this section, we derive formula for Vega, which is the first derivative of the option price wrt \( \sigma \). Differentiating

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

Where we subsituted

$$n \left( d_{1}-\sigma \sqrt{\tau}\right)=n \left( d_{1}\right) \frac{S}{K} $$