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:

Theta

And in the final section, we derive formula for Theta, which is the first derivative of the option price with respect to t. Note that in our representation so far \( \tau=(T-t) \) so the dependence on t comes through \( \tau \).

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

Where we subsituted

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