Random World: Book - The Complete guide to Option Pricing Formulas

Black-Scholes-Merton formulas - The underlying assest S follows a geometric Brownian Motion

$dS = \mu Sdt + \sigma SdZ$
$\mu$ is the expected instantaneous rate of return on the underlying asset, $\sigma$ is the instantaneous volatility of the rate of return, and dz is Wiener process.

Black-Scholes Option Pricing Formulas - c and p denote European Call and Put options

$c= SN(d1) - Xe^{-rT} N(d2)$
where
$d{1} = \frac{ln(S/X) + (r + \sigma ^{2}/2) T}{\sigma \sqrt{T}}$
$d{2} = \frac{ln(S/X) + (r - \sigma ^{2}/2) T}{\sigma \sqrt{T}} = d{1} - \sigma \sqrt{T}$
S = Stock price
X = Strike price of option
r = Risk-free interest rate
T = Time of expiration in years
$\sigma$ = volatility of the relative price change of the underlying stock price N(x) = The cumulative normal distribution funciton

Black-Scholes PDE
Black-Scholes option value can be found by solving the Black-Scholes partial differentation equation (PDE). The PDE is

$\left [ \frac{\partial c}{\partial t} + \frac{1}{2} \frac{\partial ^{2}c}{\partial S^{2}}{\sigma ^{2} S^{2}} + + \frac{\partial c}{\partial S} S \right ] dt = rc$

The Black-Scholes formula is a closed-form solution to this PDE given the payoff function of a plain vanilla option

Book - The Complete guide to Option Pricing Formulas - BSM