# Black-76 and Bachelier Models

> Black 76 prices options on futures and forwards; Bachelier assumes normal price changes and handles negative prices. Learn the formulas, uses and differences.

Source: https://learn.tradelabsai.com/options/black-76-and-bachelier-models/  
Track: Options · Level: Advanced · Updated: 2026-10-03  
Publisher: TradeLabs AI (https://tradelabsai.com). Education, not financial advice.  
Cite as: TradeLabs Learn, "Black-76 and Bachelier Models", https://learn.tradelabsai.com/options/black-76-and-bachelier-models/

The Black Scholes model was built for options on stocks. Two close relatives cover other important markets. The Black 76 model, published by Fischer Black in 1976, prices options on futures and forwards and is standard for commodity options, interest rate caps, floors and swaptions. The Bachelier model, first proposed by Louis Bachelier in 1900, assumes that prices change by normally distributed amounts rather than percentages, which makes it suitable for interest rates near or below zero and for spreads that can be negative.

## Black 76

Black 76 replaces the spot price with the futures (or forward) price F and discounts the whole payoff, since a futures contract requires no upfront payment for the underlying.

```
call = e^(-rT) × [F × N(d1) - K × N(d2)]
put = e^(-rT) × [K × N(-d2) - F × N(-d1)]
d1 = [ln(F / K) + σ² × T / 2] / (σ × √T)
d2 = d1 - σ × √T
```

Compared with Black Scholes, the interest rate no longer appears inside d1, because the cost of carry is already built into the futures price. See [How Futures Contracts Work](https://learn.tradelabsai.com/futures/how-futures-contracts-work/).

### Where Black 76 is used

| Market | Underlying in the formula |
|---|---|
| Commodity options (crude oil, gold, corn) | The relevant futures contract |
| Interest rate caps and floors | Forward interest rates |
| Swaptions | Forward swap rates. See [Interest Rate Swaps](https://learn.tradelabsai.com/bonds-credit/interest-rate-swaps/) |
| Bond options | Forward bond prices |

**Example: Pricing a crude oil option**
Crude oil futures trade at $80. A 3 month $85 call, with implied volatility of 35% and a 5% interest rate:

- σ√T = 0.35 × 0.5 = 0.175
- d1 = (ln(80 / 85) + 0.35² × 0.25 / 2) / 0.175 = (minus 0.0606 + 0.0153) / 0.175 ≈ minus 0.259
- d2 ≈ minus 0.259 minus 0.175 = minus 0.434
- N(d1) ≈ 0.398, N(d2) ≈ 0.332
- Call ≈ e^(minus 0.0125) × (80 × 0.398 minus 85 × 0.332) ≈ 0.9876 × (31.83 minus 28.23) ≈ $3.55 per barrel

On a 1,000 barrel contract, that is about $3,550.

## Bachelier (normal) model

Bachelier's 1900 thesis, "The Theory of Speculation", modelled prices as moving by normally distributed amounts, decades before Black Scholes. In this model, volatility is expressed in price units (such as basis points per year) rather than as a percentage.

```
call = e^(-rT) × [(F - K) × N(d) + σn × √T × n(d)]
d = (F - K) / (σn × √T)
```

Here σn is the normal (absolute) volatility and n(d) is the standard normal density.

### Why it came back

Lognormal models like Black 76 cannot handle zero or negative prices. When interest rates in Europe, Japan and Switzerland went negative in the 2010s, Black 76 broke for many rate options, and markets widely switched to quoting in normal volatility using Bachelier. In April 2020, when the front month WTI crude oil futures contract settled at about minus $37 a barrel, CME switched some energy options to the Bachelier model so they could be priced and margined with negative prices. See [Crude Oil](https://learn.tradelabsai.com/commodities/crude-oil/).

## Lognormal vs normal volatility

| | Black 76 (lognormal) | Bachelier (normal) |
|---|---|---|
| Price changes | Proportional to price | Absolute amounts |
| Negative prices | Impossible | Possible |
| Volatility units | Percent per year | Price units per year (e.g. basis points) |
| Typical use | Commodities, positive rates | Low or negative rates, spreads |

For an at the money option, normal volatility is roughly the lognormal volatility times the forward price. A 20% lognormal volatility on a 3% rate is roughly 60 basis points of normal volatility.

## Shifted models and SABR

Traders also use "shifted lognormal" models, adding a constant to rates so they stay positive, and the SABR model, which can blend normal and lognormal behaviour and fit volatility smiles. See [SABR Model](https://learn.tradelabsai.com/options/sabr-model/).

## Frequently asked questions

### What is the Black 76 model used for?

Pricing options on futures and forwards, including commodity options, interest rate caps and floors, swaptions and bond options.

### What is the Bachelier model?

An option pricing model that assumes prices change by normally distributed amounts, allowing negative prices and quoting volatility in absolute terms.

### Why did markets switch to the Bachelier model?

Because lognormal models cannot handle negative prices, which appeared in interest rates in the 2010s and in WTI crude oil futures in April 2020.

Next, learn a flexible numerical approach in [Binomial and Trinomial Trees](https://learn.tradelabsai.com/options/binomial-and-trinomial-trees/).

## Sources

- Wikipedia, [Black model](https://en.wikipedia.org/wiki/Black_model)
- Wikipedia, [Bachelier model](https://en.wikipedia.org/wiki/Bachelier_model)

## Continue learning

- Next lesson: [Binomial and Trinomial Trees](https://learn.tradelabsai.com/options/binomial-and-trinomial-trees/)
- Previous lesson: [Black-Scholes Model](https://learn.tradelabsai.com/options/black-scholes-model/)
- Related: [Black-Scholes Model](https://learn.tradelabsai.com/options/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.
- Related: [How Futures Contracts Work](https://learn.tradelabsai.com/futures/how-futures-contracts-work/): A futures contract is a standardised agreement to buy or sell an asset at a set price on a future date. Learn how futures trade, margin, daily settlement and expiry.
- Related: [Interest Rate Swaps](https://learn.tradelabsai.com/bonds-credit/interest-rate-swaps/): An interest rate swap exchanges fixed interest payments for floating ones on a notional amount. Learn how swaps work, SOFR, swap rates, valuation, uses and risks.
- Related: [Normal Distribution](https://learn.tradelabsai.com/math/normal-distribution/): The normal distribution is the bell curve behind many financial models. Learn its properties, the 68 95 99.7 rule, where traders use it and why markets break it.
- Related: [SABR Model](https://learn.tradelabsai.com/options/sabr-model/): The SABR model describes how forward prices and volatility move together and fits smiles with four parameters. Learn the model, each parameter and its uses in rates.