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.
The formula, term by term
- S × N(d1) — what you expect to receive: the current stock value weighted by the option's delta. N(d1) is the call's delta.
- K × e^(−rT) × N(d2) — what you expect to pay: the strike, discounted back from expiry at the risk-free rate, weighted by N(d2), the risk-neutral probability the option finishes in the money.
- σ√T — the total uncertainty between now and expiry. Volatility only ever appears multiplied by the square root of time, which is why short-dated options are so cheap in premium yet so sensitive near the money.
- The put price follows from put-call parity: P = C − S + K×e^(−rT).
Assumptions and limitations
The model assumes volatility is constant, prices move continuously without gaps, rates are fixed, there are no transaction costs and the option cannot be exercised early. Every one of those is violated in practice — indices gap on news, and implied volatility differs across strikes (the skew). That does not make the model useless; it makes it a convention. Traders quote and compare options through the implied volatility that Black-Scholes backs out, the same way distances are quoted in kilometres regardless of the road.
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). You can reproduce this yourself in the free Black-Scholes calculator — change one input at a time and watch which Greek picks it up.
Black-Scholes or binomial?
Black-Scholes is the closed-form answer for European options; the binomial option pricing model reaches the same value through a discrete tree of up and down moves, and additionally handles American-style early exercise and discrete dividends. For NIFTY, Bank Nifty and Sensex options — all European, all cash settled — Black-Scholes-type formulas are the market standard, and they are what TradePulse uses to surface IV and Greeks on the live option chain.
See it live
TradePulse uses Black-Scholes to surface live IV and Greeks for every strike on the option chain.