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.

Gamma

We now proceed to derive formula for Gamma, which is the second partial derivative of the price with respect to S. We have, from the Delta section, the formula for the Delta of a Call option:

$$\frac {\partial BS Call Price} {\partial S}= e^{- r_{f} \tau} N{\left (d_{1} \right )}$$

Differentiating both sides with respect to S,

$$\frac{\partial^{2} BS Call Price}{\partial S^{2}} =\frac {\partial } {\partial S} \left( { e^{- r_{f} \tau} N{\left (d_{1} \right )} } \right)$$ $$= e^{- r_{f} \tau} \frac {\partial } {\partial S} \left( N{\left (d_{1} \right )} \right) $$ $$= e^{- r_{f} \tau} n{\left (d_{1} \right)} \frac {\partial } {\partial S} \left( d_{1} \right) $$ $$= \frac{ e^{- r_{f} \tau} }{S \sigma \sqrt{\tau}} n{\left (d_{1} \right)} $$

Similarly, for a Put option, we have

$$ \frac {\partial BS Put Price } {\partial S} =-e^{- r_{f} \tau} N{\left (-d_{1} \right )}$$

Differentiating both sides with respect to S,

$$\frac{\partial^{2} BS Put Price}{\partial S^{2}} =\frac {\partial } {\partial S} \left( { -e^{- r_{f} \tau} N{\left (-d_{1} \right )} } \right) $$ $$=- e^{- r_{f} \tau} \frac {\partial } {\partial S} \left( N{\left (-d_{1} \right )} \right) $$ $$=- e^{- r_{f} \tau} n{\left (-d_{1} \right)} \frac {\partial } {\partial S} \left(- d_{1} \right) $$ $$= \frac{ e^{- r_{f} \tau} }{S \sigma \sqrt{\tau}} n{\left (-d_{1} \right)} $$ $$= \frac{ e^{- r_{f} \tau} }{S \sigma \sqrt{\tau}} n{\left (d_{1} \right)} $$