Black-Scholes Pricing
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Definition
The closed-form model that prices a European option as a function of underlying price S, strike K, time to expiry T, risk-free rate r, dividend yield q, and volatility sigma. Call = S*e^(-qT)*N(d1) - K*e^(-rT)*N(d2); it also defines the Greeks as its partial derivatives.
How to read it
Black-Scholes is the lingua franca of options: even traders who don't believe its assumptions quote prices via the volatility you must plug into it (implied volatility). It assumes geometric Brownian motion with constant volatility, no jumps, continuous frictionless hedging, and lognormal terminal prices. Its outputs - price and the Greeks (delta, gamma, theta, vega, rho) - are the risk language of the whole options book. The model's known FAILURE (constant vol) is exactly why the market exhibits a volatility smile/skew: real prices demand higher IV for OTM strikes than a single sigma allows.
How practitioners use it
Used as context among multiple indicators — never as a standalone signal to act.
Less common professional uses
The constant-sigma assumption is falsified by the volatility SMILE/SKEW: because real return distributions have fat tails and negative skew, OTM puts trade at higher implied vol than BSM's lognormal allows, so practitioners price each strike with its own IV and use BSM only as a quoting/interpolation device, not a belief about dynamics. For American options and dividend-paying underlyings, BSM's European assumption breaks - early exercise of American calls before ex-dividend and American puts when deep ITM requires binomial/PDE or approximations (Barone-Adesi-Whaley); using European BSM misprices assignment risk. The Greeks are model derivatives, so they inherit BSM's misspecification: delta from a flat-vol BSM ignores the skew's effect (the 'vanna' and 'volga' adjustments), and hedging with naive BSM delta leaks P&L in a skewed, stochastic-vol world - hence local-vol (Dupire) and stochastic-vol (Heston, SABR) extensions. BSM assumes continuous costless hedging; with transaction costs the replicating strategy has a bid-ask band and the true price sits in a no-arbitrage interval, not a single point (Leland's adjusted-vol correction).
Sources & provenance
Black & Scholes (1973), Journal of Political Economy; Merton (1973), Bell Journal; Hull, 'Options, Futures, and Other Derivatives'
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