# Derivation of Black Scholes Price and Greeks

We derive the formulae for the Price and Greeks (derivatives with respect to inputs) of the European options under the Black-Scholes assumptions.

### 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,

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