# Black-Scholes Model

> The Black Scholes model prices European options from five inputs. Learn the formula, its assumptions, a step by step example and where the model breaks down.

Source: https://learn.tradelabsai.com/options/black-scholes-model/  
Track: Options · Level: Advanced · Updated: 2026-10-03  
Publisher: TradeLabs AI (https://tradelabsai.com). Education, not financial advice.  
Cite as: TradeLabs Learn, "Black-Scholes Model", https://learn.tradelabsai.com/options/black-scholes-model/

The Black Scholes model is the most famous formula in finance. Published in 1973 by Fischer Black and Myron Scholes, with key contributions from Robert Merton, it gives a price for a European option from five inputs. It transformed options trading by giving traders a common language for value and risk. Scholes and Merton received the Nobel Memorial Prize in Economic Sciences in 1997; Black had died in 1995. Today the model is used less to find "the" price and more to translate prices into implied volatility and to calculate the Greeks.

## The five inputs

| Input | Symbol | Observable? |
|---|---|---|
| Underlying price | S | Yes |
| Strike price | K | Yes |
| Time to expiration (years) | T | Yes |
| Risk free interest rate | r | Yes (approximately) |
| Volatility | σ | No: must be estimated or implied |

Because volatility is the only input you cannot see, traders run the formula backwards: given the market price, they solve for the volatility that produces it. That is implied volatility. See [Implied Volatility (IV)](https://learn.tradelabsai.com/volatility/implied-volatility/).

## The formula

For a European call on a non dividend paying stock:

```
C = S × N(d1) - K × e^(-rT) × N(d2)
d1 = [ln(S / K) + (r + σ² / 2) × T] / (σ × √T)
d2 = d1 - σ × √T
```

For a European put:

```
P = K × e^(-rT) × N(-d2) - S × N(-d1)
```

N(x) is the cumulative standard normal distribution: the probability that a standard normal variable is below x. See [Normal Distribution](https://learn.tradelabsai.com/math/normal-distribution/).

## Reading the formula

- **S × N(d1):** the expected value of receiving the stock if the option finishes in the money, in present value terms. N(d1) is also the call's delta. See [Delta](https://learn.tradelabsai.com/options/delta/).
- **K × e^(minus rT) × N(d2):** the present value of paying the strike, weighted by N(d2), the risk neutral probability that the call finishes in the money.

## Worked example

**Example: Pricing a one year call**
S = $100, K = $100, T = 1, r = 5%, σ = 20%.

1. ln(S / K) = ln(1) = 0.
2. d1 = (0 + (0.05 + 0.02) × 1) / (0.20 × 1) = 0.07 / 0.20 = 0.35.
3. d2 = 0.35 minus 0.20 = 0.15.
4. N(0.35) ≈ 0.6368 and N(0.15) ≈ 0.5596.
5. e^(minus 0.05) ≈ 0.9512.
6. C = 100 × 0.6368 minus 100 × 0.9512 × 0.5596 ≈ 63.68 minus 53.23 = $10.45.

The put, from put call parity: P = C minus S + K × e^(minus rT) = 10.45 minus 100 + 95.12 ≈ $5.57. Try other inputs in the [Black-Scholes and Greeks Calculator](https://learn.tradelabsai.com/tools/black-scholes-calculator/).

## The assumptions

| Assumption | Reality |
|---|---|
| Prices follow geometric Brownian motion (lognormal returns) | Real returns have fat tails and jumps. See [Fat Tails](https://learn.tradelabsai.com/math/fat-tails/) |
| Volatility is constant | Volatility changes over time and differs by strike. See [Volatility Smile and Skew](https://learn.tradelabsai.com/volatility/volatility-smile-and-skew/) |
| Interest rates are constant | Rates change |
| No transaction costs, continuous trading | Costs exist; markets close and gap |
| European exercise only | Many options are American. See [American Option Pricing](https://learn.tradelabsai.com/options/american-option-pricing/) |
| No dividends (in the basic form) | Extensions handle dividends |

See [Lognormal Distribution](https://learn.tradelabsai.com/math/lognormal-distribution/).

## Where the model breaks down

The clearest sign is the volatility smile: if Black Scholes were exactly right, every option on the same underlying and expiry would have the same implied volatility. In reality, out of the money puts on equity indices trade at higher implied volatilities, especially since the 1987 crash. Traders handle this by using different volatilities for different strikes, or by using models such as local volatility, Heston and SABR. See [Local Volatility](https://learn.tradelabsai.com/options/local-volatility/), [Stochastic Volatility and the Heston Model](https://learn.tradelabsai.com/options/heston-model/) and [SABR Model](https://learn.tradelabsai.com/options/sabr-model/).

## The replication idea

The deep insight of Black, Scholes and Merton was that an option's payoff can be replicated by continuously adjusting a position in the stock and a risk free bond. If replication is possible, the option's price must equal the cost of the replicating portfolio, regardless of investors' views on expected returns. This is why the stock's expected return does not appear in the formula. It is also the basis of [Delta Hedging](https://learn.tradelabsai.com/options/delta-hedging/).

## Extensions

- **Merton (1973):** continuous dividend yield, by replacing S with S × e^(minus qT).
- **Black 76:** options on futures. See [Black-76 and Bachelier Models](https://learn.tradelabsai.com/options/black-76-and-bachelier-models/).
- **Garman and Kohlhagen (1983):** currency options, using domestic and foreign interest rates. See [FX Options](https://learn.tradelabsai.com/forex/fx-options/).

## Frequently asked questions

### What is the Black Scholes model?

A mathematical model, published in 1973, that prices European options from the underlying price, strike, time to expiry, interest rate and volatility.

### Why is Black Scholes still used if its assumptions are wrong?

It provides a standard way to quote options in implied volatility terms and to compute Greeks, and traders adjust for its flaws with volatility smiles and better models.

### What does N(d2) mean in Black Scholes?

It is the risk neutral probability that a call option finishes in the money at expiration.

Next, see versions for futures and negative prices in [Black-76 and Bachelier Models](https://learn.tradelabsai.com/options/black-76-and-bachelier-models/).

## Sources

- Wikipedia, [Black Scholes model](https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_model)

## Continue learning

- Next lesson: [Black-76 and Bachelier Models](https://learn.tradelabsai.com/options/black-76-and-bachelier-models/)
- Previous lesson: [Ratio Spreads](https://learn.tradelabsai.com/options/ratio-spreads/)
- Related: [Ratio Spreads](https://learn.tradelabsai.com/options/ratio-spreads/): Ratio spreads buy and sell different numbers of options at different strikes. Learn front ratios, backspreads, payoffs, uses and the risk of the extra short options.
- Related: [Implied Volatility (IV)](https://learn.tradelabsai.com/volatility/implied-volatility/): Implied volatility is the market's forecast of future movement, backed out from option prices. Learn how to read it, convert it to expected moves and use it.
- Related: [Black-Scholes and Greeks Calculator](https://learn.tradelabsai.com/tools/black-scholes-calculator/): Free Black Scholes calculator. Enter stock price, strike, days to expiry, rate, volatility and dividend yield to get call and put prices and the Greeks.
- Related: [Binomial and Trinomial Trees](https://learn.tradelabsai.com/options/binomial-and-trinomial-trees/): Binomial and trinomial trees price options by stepping prices up and down through time. Learn how the Cox Ross Rubinstein model works, with a worked example.
- Related: [The Option Greeks Explained](https://learn.tradelabsai.com/options/the-option-greeks-explained/): The option Greeks measure how an option's price responds to price, time, volatility and rates. Learn what each Greek means and how traders use them together.
- Related: [Lognormal Distribution](https://learn.tradelabsai.com/math/lognormal-distribution/): A lognormal distribution describes values whose log is normal, like prices that cannot go negative. Learn log returns, volatility drag and its option uses.