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.

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 \( \tau \). We begin with the Theta of a Call Option:

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

Due to the prevalence of t in the above expression, we deviate from our presentation style in that we carry out the two sub differentiations separately, and then combine the results. Firstly,

$$ S \frac {\partial } {\partial t} \left( e^{- r_{f} \tau} N{\left (d_{1} \right )} \right) $$ $$=S\left( { e^{- r_{f} \tau} } \frac{\partial } {\partial t} { N\left (d_{1} \right ) } + { N\left (d_{1} \right ) } \frac{\partial } {\partial t} { e^{- r_{f} \tau} } \right) $$ $$=S { e^{- r_{f} \tau} } n\left (d_{1} \right) \frac{\partial d_{1} }{\partial t}+ S{ N\left (d_{1} \right ) } { e^{- r_{f} \tau} } r_{f} $$

Now for the second term,

$$K \frac {\partial } {\partial t} \left( e^{- r_{d} \tau} N{\left (d_{1} - \sigma \sqrt{\tau} \right )} \right)$$ $$=K\left( { e^{- r_{d} \tau} } \frac{\partial } {\partial t} { N\left (d_{1} - \sigma \sqrt{\tau} \right ) } + { N\left (d_{1} - \sigma \sqrt{\tau} \right ) } \frac{\partial } {\partial t} { e^{- r_{d} \tau} } \right) $$ $$=K { e^{- r_{d} \tau} } n\left (d_{1} - \sigma \sqrt{\tau} \right) \left( \frac{\partial d_{1} }{\partial t}-\frac{\partial\sigma \sqrt{\tau} }{\partial t} \right)+ K{ N\left (d_{1} - \sigma \sqrt{\tau} \right ) } { e^{- r_{d} \tau} } r_{d} $$ $$=K { e^{- r_{d} \tau} } n\left (d_{1} - \sigma \sqrt{\tau} \right) \left( \frac{\partial d_{1} }{\partial t}+\frac{\sigma }{ 2\sqrt{\tau} } \right)+K { N\left (d_{1} - \sigma \sqrt{\tau} \right ) } { e^{- r_{d} \tau} } r_{d} $$ $$=K { e^{- r_{d} \tau} } n \left( d_{1}\right) \frac{S}{K} e^{ \left (r_{d} - r_{f}\right) \tau} \left( \frac{\partial d_{1} }{\partial t}+\frac{\sigma }{ 2\sqrt{\tau} } \right)+ K{ N\left (d_{1} - \sigma \sqrt{\tau} \right ) } { e^{- r_{d} \tau} } r_{d} $$ $$=S { e^{- r_{f} \tau} } n \left( d_{1}\right) \left( \frac{\partial d_{1} }{\partial t}+\frac{\sigma }{ 2\sqrt{\tau} } \right)+ K{ N\left (d_{1} - \sigma \sqrt{\tau} \right ) } { e^{- r_{d} \tau} } r_{d} $$ $$=S { e^{- r_{f} \tau} } n \left( d_{1}\right) \frac{\partial d_{1} }{\partial t}+S { e^{- r_{f} \tau} } n \left( d_{1}\right) \frac{\sigma }{ 2\sqrt{\tau} }+ K{ N\left (d_{1} - \sigma \sqrt{\tau} \right ) } { e^{- r_{d} \tau} } r_{d} $$

Plugging the components into the formula and canceling the first terms, we get

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

And we finally repeat the above procedure for the Theta of a Put option :

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

We continue with the differentiations of the two components. For the first component:

$$ S\frac {\partial } {\partial t} \left( e^{- r_{f} \tau} N{\left (-d_{1} \right )} \right) $$ $$=S\left( { e^{- r_{f} \tau} } \frac{\partial } {\partial t} { N\left(-d_{1} \right ) } + { N\left (-d_{1} \right ) } \frac{\partial } {\partial t} { e^{- r_{f} \tau} } \right) $$ $$=S { e^{- r_{f} \tau} } n\left (-d_{1} \right) \frac{\partial \left(-d_{1} \right) }{\partial t}+ S{ N\left (-d_{1} \right ) } { e^{- r_{f} \tau} } r_{f} $$ $$=-S { e^{- r_{f} \tau} } n\left (d_{1} \right) \frac{\partial d_{1}}{\partial t}+ S{ N\left (-d_{1} \right ) } { e^{- r_{f} \tau} } r_{f} $$

Now for the second term,

$$K \frac {\partial } {\partial t} \left( e^{- r_{d} \tau} N{\left(-d_{1}+ \sigma \sqrt{\tau} \right )} \right) $$ $$=K\left( { e^{- r_{d} \tau} } \frac{\partial } {\partial t} { N\left (-d_{1} + \sigma \sqrt{\tau} \right ) } + { N\left(-d_{1} + \sigma \sqrt{\tau} \right ) } \frac{\partial } {\partial t} { e^{- r_{d} \tau} } \right) $$ $$=K { e^{- r_{d} \tau} } n\left (-d_{1} + \sigma \sqrt{\tau} \right) \left( -\frac{\partial d_{1} }{\partial t}+\frac{\partial\sigma \sqrt{\tau} }{\partial t} \right)+ K{ N\left (-d_{1} + \sigma \sqrt{\tau} \right ) } { e^{- r_{d} \tau} } r_{d} $$ $$=K { e^{- r_{d} \tau} } n\left (-d_{1} + \sigma \sqrt{\tau} \right) \left(- \frac{\partial d_{1} }{\partial t}-\frac{\sigma }{ 2\sqrt{\tau} } \right)+K { N\left (-d_{1} + \sigma \sqrt{\tau} \right ) } { e^{- r_{d} \tau} } r_{d} $$ $$=K { e^{- r_{d} \tau} } n \left( d_{1}\right) \frac{S}{K} e^{ \left (r_{d} - r_{f}\right) \tau} \left(- \frac{\partial d_{1} }{\partial t}-\frac{\sigma }{ 2\sqrt{\tau} } \right)+ K{ N\left (-d_{1}+ \sigma \sqrt{\tau} \right ) } { e^{- r_{d} \tau} } r_{d} $$ $$=-S { e^{- r_{f} \tau} } n \left( d_{1}\right) \left( \frac{\partial d_{1} }{\partial t}+\frac{\sigma }{ 2\sqrt{\tau} } \right)+ K{ N\left (-d_{1} +\sigma \sqrt{\tau} \right ) } { e^{- r_{d} \tau} } r_{d} $$ $$=-S { e^{- r_{f} \tau} } n \left( d_{1}\right) \frac{\partial d_{1} }{\partial t}-S { e^{- r_{f} \tau} } n \left( d_{1}\right) \frac{\sigma }{ 2\sqrt{\tau} }+ K{ N\left (-d_{1}+ \sigma \sqrt{\tau} \right ) } { e^{- r_{d} \tau} } r_{d} $$

Plugging the components into the formula and canceling the first terms, we get

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