The Black-Scholes model, explained with a worked example

By Daniel Tolenaar · Updated 1 November 2026

The Black-Scholes model is a formula for the fair price of a European option. It takes five inputs, the stock price, the strike, the time to expiry, the risk-free interest rate and the stock’s volatility, and returns the price at which neither buyer nor seller can lock in a riskless profit.

Fischer Black and Myron Scholes published it in 1973, and Robert Merton extended it the same year. Merton and Scholes received the 1997 Nobel Memorial Prize in Economic Sciences for the work (Black had died in 1995). It is still the starting point for how options are quoted and hedged, which is why it is the first thing most quant courses teach.

The Black-Scholes formula

The price of a call option C and a put option P on a stock that pays no dividends is:

N(·) is the cumulative standard normal distribution: N(x) is the chance that a standard normal draw is below x. Read the call formula as two pieces. K·e^−rT·N(d₂) is what you expect to pay at expiry, discounted to today, because N(d₂) is the risk-neutral probability that the call ends in the money. S·N(d₁) is the value of the stock you expect to receive. N(d₁) is also the option’s delta, the number of shares that hedges one call.

What each input does

Every input moves the price in a predictable direction:

Worked example

A one-year at-the-money call on a stock at 100, with a strike of 100, a 5% risk-free rate and 20% volatility, is worth 10.45, and the matching put is worth 5.57. Here is the arithmetic:

1. d₁ = [ln(100/100) + (0.05 + 0.2²/2)·1] / (0.2·√1) = 0.07 / 0.2 = 0.35
2. d₂ = 0.35 − 0.2 = 0.15
3. N(0.35) = 0.6368 and N(0.15) = 0.5596
4. K·e^−rT = 100·e^−0.05 = 95.1229
5. C = 100 × 0.6368 − 95.1229 × 0.5596 = 63.68 − 53.23 = 10.45

You can change any of these inputs and see the Greeks update in the Black-Scholes calculator.

Black-Scholes in Python

The formula fits in ten lines of Python using only the standard library. This is the code and what it printed:

from math import exp, log, sqrt
from statistics import NormalDist

N = NormalDist().cdf  # standard normal CDF


def black_scholes(S, K, T, r, sigma):
    """European call and put prices under Black-Scholes (no dividends)."""
    d1 = (log(S / K) + (r + 0.5 * sigma**2) * T) / (sigma * sqrt(T))
    d2 = d1 - sigma * sqrt(T)
    call = S * N(d1) - K * exp(-r * T) * N(d2)
    put = K * exp(-r * T) * N(-d2) - S * N(-d1)
    return call, put, d1, d2


call, put, d1, d2 = black_scholes(S=100, K=100, T=1, r=0.05, sigma=0.20)
print(f"d1 = {d1:.4f}, d2 = {d2:.4f}")
print(f"call = {call:.4f}")
print(f"put  = {put:.4f}")

d1 = 0.3500, d2 = 0.1500
call = 10.4506
put  = 5.5735

Assumptions and limits

The formula is exact only under its assumptions. The stock follows geometric Brownian motion, so its log returns are normally distributed with a constant volatility. The interest rate is constant, trading is continuous and free, there is no arbitrage, and the option can only be exercised at expiry.

• Volatility is not constant. Options with different strikes trade at different implied volatilities, the volatility smile, which the model cannot produce.
• Returns have fat tails. Large moves happen more often than a normal distribution predicts, so far out-of-the-money options tend to cost more than the formula says.
• American options can be exercised early. For those, a binomial tree is the usual tool.
• Dividends lower call prices. Merton’s version replaces S with S·e^−qT for a dividend yield q; the calculator on this site includes it.

None of this stops traders using it. They quote options in implied volatility and treat Black-Scholes as the translation between that number and a price.