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:

Gamma

We now proceed to derive formula for the Gamma, which is the second partial derivative of the price wrt S. We have, from the Delta secttion, the formula for the Delta of an option:

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

Differentiating both sides wrt S

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