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.

Put Call Parity

While we are at it, we might as well verify that the Greeks formulae satisfy the Put Call parity, which essentially states that buying a call and selling a put is equivalent to a long position in a forward contract. In symbols:

$$ \text{Call}-\text{Put} = {\text {Long Position in Forward}} $$ $$ \text{Present Value of Call}-\text{Present Value of Put} = \text {Present Value of Forward} $$ $$ BS Call Price-BS Put Price = e^{-r_d \tau}\left( S e^{\left( r_d -r_f\right) \tau}-K \right) $$ $$ BS Call Price-BS Put Price = S e^{-r_f \tau}-K e^{-r_d \tau} $$

Taking derivatives of both sides with respect to \( S, S^2, \sigma, r_d, \text{and } t \), we get:

$$ \Delta (BS Call Price)- \Delta (BS Put Price) = e^{-r_f \tau} $$ $$ \Gamma (BS Call Price)- \Gamma(BS Put Price) = 0 $$ $$ Vega (BS Call Price)- Vega(BS Put Price) = 0 $$ $$ \rho (BS Call Price)- \rho (BS Put Price) = K \tau e^{-r_d \tau} $$ $$ \Theta (BS Call Price)- \Theta(BS Put Price) = S r_f e^{-r_f \tau}- K r_d e^{-r_d \tau} $$

The equalities are easily verified, main trick being \( N(-x)=1-N(x) \):

$$ \Delta (BS Call Price)- \Delta (BS Put Price) $$ $$ = e^{- r_{f} \tau} N{\left (d_{1} \right )}+e^{- r_{f} \tau} N{\left (-d_{1} \right )}$$ $$ = e^{- r_{f} \tau} \left( N{\left (d_{1} \right )}+ N{\left (-d_{1} \right )} \right) $$ $$ = e^{- r_{f} \tau} \left( N{\left (d_{1} \right )}+ 1-N{\left (d_{1} \right )} \right) $$ $$ = e^{- r_{f} \tau} $$

$$ \Gamma (BS Call Price)- \Gamma(BS Put Price) $$ $$ = \frac{ e^{- r_{f} \tau} }{S \sigma \sqrt{\tau}} n{\left (d_{1} \right)}-\frac{ e^{- r_{f} \tau} }{S \sigma \sqrt{\tau}} n{\left (d_{1} \right)}$$ $$ = 0 $$

$$ Vega (BS Call Price)- Vega(BS Put Price) $$ $$ = S e^{- r_{f} \tau} \sqrt{\tau} n{\left (d_{1} \right)}-S e^{- r_{f} \tau} \sqrt{\tau} n{\left (d_{1} \right)}$$ $$ = 0 $$

$$ \rho (BS Call Price)- \rho (BS Put Price) $$ $$ = K { e^{- r_{d} \tau} }\tau { N\left (d_{1} - \sigma \sqrt{\tau} \right ) }+ K { e^{- r_{d} \tau} }\tau { N\left(-d_{1} + \sigma \sqrt{\tau} \right ) }$$ $$ = K { e^{- r_{d} \tau} }\tau \left( N\left (d_{1} - \sigma \sqrt{\tau} \right )+ N\left(-d_{1} + \sigma \sqrt{\tau} \right )\right) $$ $$ = K { e^{- r_{d} \tau} }\tau \left( N\left (d_{1} - \sigma \sqrt{\tau} \right )+ 1-N\left(d_{1} -\sigma \sqrt{\tau} \right )\right) $$ $$ = K { e^{- r_{d} \tau} }\tau $$

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