Black-Scholes Option Pricing

The Black–Scholes model or Black–Scholes-Merton is a mathematical model of a financial market containing certain derivative investment instruments. From the model, one can deduce the Black–Scholes formula, which gives the price of European-style options. The formula led to a boom in options trading and the creation of the Chicago Board Options Exchange. lt is widely used by options market participants.

The black scholes option pricing model uses the following variables:

Variable	Option
$X$	Exercise price
$S$	Stock price
$t$	Time to experation
$\sigma ^{2}$	Variance of returns on stock
$r$	Risk free rate of return

The value of a call option for a non-dividend paying underlying stock in terms of the Black–Scholes parameters is:

$C(S,t)$ = $N(d_{1})~S-N(d_{2})~Ke^{-r(T-t)}\,$

with $d_{1}$=${\frac {\ln({\frac {S}{K}})+(r+{\frac {\sigma ^{2}}{2}})(T-t)}{\sigma {\sqrt {T-t}}}}$ $d_{2}$=${\frac {\ln({\frac {S}{K}})+(r-{\frac {\sigma ^{2}}{2}})(T-t)}{\sigma {\sqrt {T-t}}}}=d_{1}-\sigma {\sqrt {T-t}}$