Pricing & Mechanics
Black-Scholes Model
Also called: Black-Scholes-Merton, BSM
Black-Scholes is the foundational model for pricing European options from five inputs: underlying price, strike, time to expiration, risk-free rate and volatility (plus dividend yield in the Merton extension). Every Greek and every implied volatility is defined relative to it or its descendants.
The model assumes the underlying follows a lognormal random walk with constant volatility and continuous, frictionless trading. Under those assumptions a riskless hedge exists, and the option's price is the discounted expected payoff under the risk-neutral distribution. Its closed form makes IV, delta, gamma, theta and vega computable instantly.
Real markets violate the assumptions, and the violations are visible as volatility skew and term structure: the market assigns different IVs to different strikes and dates because it does not believe in one constant volatility. Practitioners still use Black-Scholes as the quoting language and correct for reality through the IV surface and, for American-style equity options, through binomial or numerical methods that handle early exercise.
Formula
Call = S·N(d1) − K·e^(−rT)·N(d2), where d1 = [ln(S/K) + (r + σ²/2)T] / (σ√T) and d2 = d1 − σ√T
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Frequently asked questions
Does Black-Scholes work for American options?
It prices European exercise. For American options, early exercise (mainly deep in-the-money puts, and calls before dividends) adds value that binomial trees or other numerical methods capture.
Why do traders still use Black-Scholes if its assumptions are wrong?
Because it is a consistent translation between price and volatility. Quoting in IV lets traders compare options across strikes and expirations, and the model's Greeks remain good local hedging ratios.