Black-Scholes Explained: How Options Are Priced

TL;DR. The Black-Scholes model is a closed-form formula that estimates the fair price of a European call or put option on a non-dividend-paying stock. Published in 1973 by Fischer Black and Myron Scholes, with the mathematical foundation extended by Robert Merton the same year, it became the standard reference for option valuation and earned Scholes and Merton the 1997 Nobel Prize in Economic Sciences. It is not the last word on options pricing — the assumptions break in well-known ways — but every working options trader speaks its language.

This guide explains what the formula actually says, walks through a worked example with real numbers, shows how each of the five inputs moves the price, and lays out where the model misleads in practice.

The core idea: arbitrage-free pricing

Before Black-Scholes, options were priced by intuition and rules of thumb. The breakthrough was an arbitrage argument: imagine continuously rebalancing a portfolio of the stock plus a borrowed cash position so that, second by second, it replicates the payoff of an option. If you can perfectly replicate the option, the option must cost exactly what that replicating portfolio costs — otherwise you could buy one, sell the other, and harvest a risk-free profit.

That replicating-portfolio argument leads, through some calculus and a lot of assumptions, to a partial differential equation that the option value must satisfy. The Black-Scholes equation is:

∂V/∂t + ½σ²S² ∂²V/∂S² + rS ∂V/∂S − rV = 0

Solve that PDE for a European call payoff at expiration and you get the famous closed-form result.

The formula

For a European call option on a stock paying no dividends, the price C at time zero is:

C = S · N(d₁) − K · e^−rT · N(d₂)

with

d₁ = [ ln(S/K) + (r + σ²/2)·T ] / (σ√T)
d₂ = d₁ − σ√T

Here N(·) is the cumulative distribution function of the standard normal distribution — the “area-under-the-bell-curve” function that any spreadsheet can compute as NORM.S.DIST(x, TRUE). The corresponding put price P follows from put-call parity:

P = K · e^−rT · N(−d₂) − S · N(−d₁)

The formula is published verbatim in the Wikipedia reference for the Black-Scholes model, which is in turn a faithful summary of the original 1973 paper by Black and Scholes in the Journal of Political Economy and Merton’s companion paper in the Bell Journal of Economics and Management Science.

The five inputs — what each one means

S — the current stock price. Observable.
K — the strike price specified in the option contract. Observable.
T — time to expiration, in years. Observable (count the days, divide by 365 or 252 depending on convention).
r — the continuously compounded risk-free interest rate over the life of the option. Observable from Treasury yields.
σ — the volatility of the stock’s log-returns, annualized. Not observable. This is the one input the market argues about — and the one that contains all the information.

Four inputs are pinned down by the contract and the market tape. The fifth, volatility, is what traders are actually pricing when they buy and sell options. Quote an option price, and you have implicitly quoted a volatility. The number you back out is called the implied volatility.

A worked example with real numbers

Let’s price a 3-month at-the-money call on a $100 stock when 3-month rates are 4% and we estimate annual volatility at 25%.

S = $100
K = $100
T = 0.25 years (about 91 days)
r = 0.04
σ = 0.25

Plug into the formulas:

d₁ = [ ln(1) + (0.04 + 0.0625/2) × 0.25 ] / (0.25 × √0.25)
     = (0.07125 × 0.25) / 0.125
     = 0.0178125 / 0.125
     ≈ 0.1425

d₂ = 0.1425 − 0.125 = 0.0175

N(0.1425) ≈ 0.5567
N(0.0175) ≈ 0.5070
e^{−0.04 × 0.25} = e^−0.01 ≈ 0.9901

C ≈ 100 × 0.5567 − 100 × 0.9901 × 0.5070
= 55.67 − 50.20
≈ $5.47

That $5.47 is the model’s fair value for the 3-month at-the-money call. A market that bids the same option at $6.00 is pricing in a higher volatility than the 25% we assumed; one that offers it at $5.00 is pricing in a lower one. The difference between the model price and the market price is the implied-volatility signal in disguise.

How each input moves the price — the sensitivity table

Below is the call price recomputed from the same base case after changing one input at a time. The model is mechanical; the formula does not care about your view, only about the numbers you give it.

Scenario	Inputs that change	Call price	Δ vs base
Base case	S=100, K=100, T=0.25, r=4%, σ=25%	$5.47	—
Volatility up	σ = 35%	$7.44	+$1.97
More time	T = 0.50 years	$8.01	+$2.54
Stock rallies	S = $110 (10% in the money)	$12.45	+$6.98
Higher strike	K = $110 (10% out of the money)	$1.93	−$3.54
Rates up	r = 8%	$5.99	+$0.52

Author calculations using the Black-Scholes call formula as published in Wikipedia: Black-Scholes model. Base case: S=$100, K=$100, T=0.25y, r=4%, σ=25%.

The pattern is the one every options trader carries in their head: vega (sensitivity to volatility) and theta (sensitivity to time) move the price the most for at-the-money options; delta dominates once the option goes meaningfully in or out of the money. The five sensitivities have their own names — the option Greeks — and each is a partial derivative of this same formula.

Visualizing the call price

$0 $10 $20 $30 $40

$60 $80 $100 $120 $140 Stock price S Call price C

K = $100

Intrinsic value

$5.47 at the money Black-Scholes call

Source: author calculations using the Black-Scholes call formula. The gap between the solid and dashed curves at any S is the option’s time value — the premium for optionality on top of intrinsic worth.

The Black-Scholes curve sits strictly above the kinked intrinsic-value line, asymptotically approaching it deep in the money and deep out of the money. The gap is the time value — the premium a buyer pays for the right, but not the obligation, to exercise later.

$0 $4 $8 $12 $16

0% 20% 40% 60% 80% Volatility σ (annualized) Call price C

σ = 25% (base)

$5.47 More vol → more premium

Source: author calculations using the Black-Scholes call formula. For at-the-money options, price rises almost linearly with volatility — this is the intuition behind “vega.”

The assumptions — and where they break

Black-Scholes is a model, not a law of nature. As the standard reference spells out, the closed-form solution relies on the following:

The stock’s log-returns follow a geometric Brownian motion with constant drift and constant volatility — no jumps, no regime changes.
The risk-free rate is constant over the life of the option.
The stock pays no dividends during the option’s life (the baseline version — Merton extended it to a continuous dividend yield).
Markets are frictionless: no transaction costs, no taxes, you can borrow and lend at the risk-free rate, you can short any quantity, and trading is continuous.
The option is European — it can be exercised only at expiration, not before.
There are no arbitrage opportunities.

Every one of these is an idealization. Real-world breakdowns the model cannot capture include:

Jumps and crashes. Stock prices jump on earnings, deals, and macro shocks — geometric Brownian motion is too smooth to represent that.
Non-constant volatility. Vol clusters, mean-reverts, and spikes when prices fall. That is why GARCH-family models and stochastic-volatility models like Heston exist.
Early exercise. American options on dividend-paying stocks can rationally be exercised before expiration; pricing them requires numerical methods like the binomial tree or finite-difference solvers.
Liquidity and frictions. Bid-ask spreads, margin requirements, and balance-sheet costs mean the replicating portfolio is never literally free to rebalance.

The single most famous empirical breakdown is the volatility smile (or skew). If Black-Scholes were correct, the implied volatility you back out of the market would be the same for every strike at the same maturity — one stock, one volatility number. In practice, out-of-the-money puts trade at materially higher implied vols than at-the-money options, especially in equity index markets. That pattern, visible in every options board since the 1987 crash, is the market’s way of saying it does not believe assumption (1).

What Black-Scholes still gets right

Despite all of the above, the formula remains the lingua franca of options markets for three reasons.

First, it is a clean arbitrage relationship. Put-call parity, the upper and lower bounds on option prices, and the conservation laws between strikes hold under Black-Scholes and survive its empirical violations.

Second, it is the standard quoting convention. Whether on an OCC-cleared equity option or an OTC FX option, dealers quote in implied volatility — the number that, plugged into Black-Scholes, would produce the market price. Traders argue about vol, not premium, because Black-Scholes lets them compare across strikes and maturities on a common scale.

Third, it is differentiable. Every option Greek — delta, gamma, theta, vega, rho — is a partial derivative of the formula. Risk managers rely on those Greeks to size hedges and stress portfolios, and they exist because the closed-form solution exists.

The intellectual contribution earned Robert Merton and Myron Scholes the 1997 Nobel Prize in Economic Sciences; Fischer Black would have shared the award but died in 1995, and the Nobel is not awarded posthumously.

Common mistakes when using the model

Using historical volatility for σ. Black-Scholes wants future volatility over the life of the option. Past realized vol is one estimate; the market’s implied vol is another, and usually the better one for pricing.
Ignoring dividends. The baseline formula assumes none. For dividend-paying stocks, use the Merton extension that replaces S with S · e^−qT where q is the continuous dividend yield.
Applying it directly to American options. American puts and American calls on dividend-paying stocks can have early-exercise value Black-Scholes does not capture. Use a binomial tree or numerical solver instead.
Treating implied vol as a forecast. Implied vol is a price, not a prediction. It can stay above realized for years (the variance risk premium) without being “wrong.”
Quoting one vol across the surface. Real markets have a smile or skew, and pricing an out-of-the-money put with the at-the-money vol will understate its market price.