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Derivation of Black Scholes Price and Greeks

We present the derivation of the formulae for the Price and the most common Greeks (derivatives with respect to inputs) of the European options under the Black-Scholes assumptions. For the derivation of all Greeks up to the third order, please see our book which is available on Amazon.

Summary of Black Scholes Price and Greeks Formula

We first list the Black-Scholes price formula and its Greeks for a Call option:

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

Note the symbols \(r_f\) and \(r_d\) are meant in the FX sense - i.e., foreign and domestic interest rates, respectively. In case of stocks, \(r_f\) should be interpreted as the dividend yield rate and \(r_d\) as the interest/discount rate.

And for a Put option:

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

Where,

$$ d_1 =\frac{1}{\sigma \sqrt{\tau}} \left(\ln{\left (\frac{S}{K} \right ) + \left(r_{d} - r_{f} + \frac{\sigma^{2}}{2}\right) \tau}\right)$$ $$ N\left( y \right)= \int_{-\infty}^{y}{\frac{1}{\sqrt{2\pi}} {e^{-\frac{x^{2}}{2}}}dx}={\text {Cumulative Standard Normal}}$$ $$ \frac{d N\left( y \right)}{d y}=\frac{1}{\sqrt{2\pi}} {e^{-\frac{y^{2}}{2}}}=n\left( y \right)={\text {Standard Normal Density}} $$