Black-Scholes Option Price
Closed-form European call/put pricing under the geometric Brownian motion assumption.
Details
- category: Derivatives
- formula: C = S × N(d1) - K × e^(-rT) × N(d2) P = K × e^(-rT) × N(-d2) - S × N(-d1)
- formula latex: C = S\,N(d_1) - K e^{-rT} N(d_2),\quad d_1 = \dfrac{\ln(S/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}},\quad d_2 = d_1 - \sigma\sqrt{T}
- assumptions: Constant volatility, No dividends (or constant continuous yield), Continuous, frictionless trading, Lognormal terminal stock price, Constant risk-free rate
- limitations: Volatility smile/skew shows real markets violate constant-vol assumption, European-only (no early exercise)
- related terms: black-scholes, implied-volatility, delta-hedging, greeks, european-option
Formula
C = S × N(d1) - K × e^(-rT) × N(d2) P = K × e^(-rT) × N(-d2) - S × N(-d1)