Black-Scholes Model
The classic formula that turns five inputs into a fair option price — and powers the Greeks.
Definition
The Black-Scholes Model (also Black-Scholes-Merton) is a mathematical formula for pricing European-style options. It calculates a theoretical fair value from five inputs: the spot price, the strike price, time to expiry, the risk-free interest rate, and volatility. Published in 1973, it remains the foundation of modern option pricing.
Formula
Call price C = S × N(d1) − K × e^(−rT) × N(d2), where d1 = [ln(S÷K) + (r + σ²÷2)T] ÷ (σ√T), d2 = d1 − σ√T, S is spot, K is strike, T is time to expiry, r is the rate, σ is volatility, and N() is the standard normal cumulative distribution.
Why it matters
Black-Scholes is the engine behind option valuation and the Greeks — delta, gamma, theta and vega are all partial derivatives of this formula. Because four of its five inputs are observable, traders invert it to solve for the fifth, volatility, giving implied volatility.
Example
Feed the model a spot of 22,500, a strike of 22,500, 7 days to expiry, a 6.5 percent rate and 14 percent volatility, and it returns a theoretical at-the-money call value. Raise volatility to 20 percent with everything else fixed and the same call's theoretical price rises — exactly the behaviour vega measures (illustrative figures).
See it live
TradePulse uses Black-Scholes to surface live IV and Greeks for every strike on the option chain.