Black Scholes

Black Scholes' First Order Greeks' Formulae

Please select a variable:

The formula for the first order derivative of Black Scholes with respect to the above variable is as follows:

$$ \begin{align}\frac{\partial P}{\partial S} =&{\color{blue} \phi} \, \,{ e^{- q \tau} }\, N\left ({\color{blue} \phi}d_{1} \right )\end{align}$$

Where \(S\) represents the spot price of the underlying, \(K\) the strike price, \(\sigma\) the volatility, \(t\) the time, \(r\) the discount rate, \(q\) the dividend yield, and \(\phi\) is a dummy variable, taking value 1 for a call option and -1 for a put option.

For the derivation, please see our book which is available on Amazon, and contains the derivations of all Greeks up to the third order. You can also find the derivations of some of the most common Greeks on this page.