Black–Scholes

The closed-form European option price and the assumptions behind it.

The Black-Scholes formula prices a European call as

C = S\, N(d_1) - K e^{-rT}\, N(d_2)

where $d_1 = \frac{\log(S/K) + (r + \sigma^2/2)T}{\sigma \sqrt{T}}$ and $d_2 = d_1 - \sigma \sqrt{T}$ .

Underlying assumptions: GBM dynamics for $S$ , constant volatility $\sigma$ , constant risk-free rate $r$ , frictionless trading, no dividends, continuous trading. The derivation builds a self-financing portfolio that perfectly replicates the option's payoff — and the cost of that portfolio is the price.

Real markets violate every assumption: volatility moves, rates move, trading isn't continuous, jumps happen, and dividends are real. Despite that, Black-Scholes is the universal language. Quoting an option price as an implied volatility (the $\sigma$ that makes BS match the market) is how options are routinely communicated, even though everyone knows the model is wrong.