Black–Scholes Model

By Sitraka Forler · Lecturer, Durham Business SchoolUpdated 13 September 2026 About this site

Closed-form price for European options.

Black–Scholes prices an option from five inputs: spot, strike, time, rate, and volatility. Volatility is the only unobservable input, so traders invert the formula to quote 'implied volatility'. It earned a Nobel Prize and remains the lingua franca of options.

The price of optionality, melting as the clock runs down

Black-Scholes call value C(S) for strike K = 100 and r = 4%, at three times to expiry.

C(S = K, 6m) = 100 × 0.5840 − 98.02 × 0.5282 = €6.63

volatility σ20.0%

Drag σ and every curve fattens above the payoff line: the gap is time value, what optionality is worth before the die is cast. It melts to zero at expiry, slowest near the strike where the uncertainty lives.

The intuition

The Black-Scholes model prices a European option by pricing the hedge that neutralises it. If you sell a call and continuously hold just the right amount of the underlying share, called delta, the combined position becomes riskless, so it must earn the risk-free rate. That single no-arbitrage argument pins down the option's fair value from five inputs: spot price, strike, time to expiry, the risk-free rate and volatility. Remarkably, the share's expected return never appears: the hedge removes the direction of the market from the problem, leaving only how much the price wobbles.

In the chart above, the dashed hockey stick is what a call pays at expiry: nothing below the strike, one for one above it. Every smooth curve floats above that line, and the gap is time value. It exists because of asymmetry: while time remains, the share can rise without limit but the option's loss is capped at zero. More time or more volatility means more chances for that asymmetry to pay off, which is why the two-year curve sits highest and why dragging the volatility slider fattens all three curves at once.

The formula itself, C = S·N(d₁) − K·e^(−rT)·N(d₂), reads as a weighted bet. The second term is the discounted strike times the risk-neutral probability that the option finishes in the money; the first is what you expect to receive from the share when it does. As expiry approaches, σ√T shrinks, the probabilities collapse towards 0 or 1, and the curve melts onto the payoff: the one-week curve already hugs the hockey stick everywhere except near the strike, where the last of the uncertainty lives.

Formula / theory

C = S·N(d₁) − K·e^(−rT)·N(d₂)
d₁ = [ln(S/K) + (r + σ²/2)T] / (σ√T),  d₂ = d₁ − σ√T

In Python

from scipy.stats import norm
import numpy as np
d1 = (np.log(S/K) + (r + 0.5*sigma**2)*T) / (sigma*np.sqrt(T))
d2 = d1 - sigma*np.sqrt(T)
call = S*norm.cdf(d1) - K*np.exp(-r*T)*norm.cdf(d2)

Pricing a six-month at-the-money call

Set the inputs: S = 100, K = 100, r = 4%, σ = 20%, T = 0.5 years. First compute σ√T = 0.20 × √0.5 = 0.20 × 0.70711 = 0.14142.
d₁ = (ln(S/K) + (r + σ²/2)T) / (σ√T) = (ln(1) + (0.04 + 0.02) × 0.5) / 0.14142 = 0.03 / 0.14142 = 0.21213.
d₂ = d₁ − σ√T = 0.21213 − 0.14142 = 0.07071.
Look up the standard normal CDF: N(d₁) = N(0.21213) = 0.5840 and N(d₂) = N(0.07071) = 0.5282.
Discount the strike back to today: K·e^(−rT) = 100 × e^(−0.02) = 98.02.
Assemble the price: C = 100 × 0.5840 − 98.02 × 0.5282 = 58.40 − 51.77 = 6.63.

The call trades at 6.63 with zero intrinsic value, so every cent of it is time value: the market price of six months of optionality.

Common pitfalls

Mixing time units: σ must be annualised and T expressed in years. Feeding a daily volatility with T in years (or days with an annual σ) scales the price wildly wrong; annualise daily volatility by multiplying by √252.
Using historical volatility as if it were the market's: traded option prices embed implied volatility, which usually differs from trailing realised volatility, so a Black-Scholes price built on historical σ is a model view, not the market price.
Applying the vanilla formula to American options or dividend-paying shares: it prices European exercise on non-dividend stocks, and both early exercise and dividends need adjustments or a different model entirely.
Reading N(d₂) as the real-world probability of exercise: it is the risk-neutral probability, computed with a drift of r rather than the share's actual expected return.

Frequently asked questions

What does the Black-Scholes model actually calculate?

It calculates the fair value of a European option as the cost of replicating it with a continuously rebalanced portfolio of the underlying share and risk-free borrowing. Because that hedge eliminates market risk, the value depends on just five inputs: spot price, strike, time to expiry, the risk-free rate and volatility, and not on the share's expected return.

What is implied volatility in the Black-Scholes model?

Implied volatility is the value of σ that makes the Black-Scholes formula reproduce an option's observed market price. Traders find it by inverting the formula: every other input is observable, so quoting an option price is effectively quoting a volatility. Plotting implied volatility across strikes reveals the smile and skew that the model itself cannot explain.

Why is volatility the most important input in Black-Scholes?

Because it is the only input you cannot observe directly. Spot, strike, time to expiry and the risk-free rate are all known, so the entire disagreement between traders about an option's worth compresses into one number, σ. Higher volatility raises the value of calls and puts alike, since it widens the range of outcomes while the option's downside stays capped at zero.

What are the main limitations of the Black-Scholes model?

The model assumes constant volatility, lognormally distributed prices with no jumps, frictionless continuous hedging and European exercise. Real markets violate all four: implied volatility varies by strike and maturity (the smile and skew), prices gap on news, hedging costs money, and many listed options are American. Practitioners therefore treat Black-Scholes as a quoting convention and a first approximation, not a literal description.

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