Black-Scholes Model
The foundational closed-form formula for pricing European options from spot, strike, time, interest rate, and volatility.
Formula
Call = S·N(d1) − K·e^(−rT)·N(d2), where d1 = [ln(S/K) + (r + σ²/2)T] / (σ√T), d2 = d1 − σ√T
The Black-Scholes model (Black-Scholes-Merton) is the foundational framework for pricing European-style options. Published in 1973, it produces a theoretical fair value for a call or put from five inputs: the underlying spot price, strike price, time to expiration, the risk-free interest rate, and the volatility of the underlying.
It rests on several idealized assumptions: the underlying follows a lognormal random walk with constant volatility, markets are frictionless (no transaction costs or taxes), the risk-free rate is constant, and the option is European (exercisable only at expiry). From these, the model derives a hedged, risk-neutral portfolio whose value can be solved in closed form.
Its real-world limits drive much of options trading: because actual volatility is not constant across strikes, the market quotes a volatility skew rather than the single flat number the model assumes. In practice the formula is most often run backwards — feeding in the observed market premium to solve for the implied volatility — and its partial derivatives define the option Greeks used to manage risk.
Related Terms
Implied Volatility
The market's forward-looking expectation of volatility, derived by solving the options pricing model for the volatility that matches the observed premium.
AdvancedOptions Contract
A contract giving the buyer the right — but not the obligation — to buy or sell an underlying asset at a set price before or on expiration.
BeginnerPut-Call Parity
The no-arbitrage relationship linking the prices of a European call and put at the same strike and expiry: C − P = S − K·e^(−rT).
AdvancedThe Greeks
The collective name for the sensitivity measures — delta, gamma, theta, vega, rho — that describe how an option's price responds to changes in market variables.
IntermediateVolatility Skew
The pattern in which implied volatility varies across strikes at the same expiration, usually with OTM puts pricing higher IV than OTM calls.
Advanced