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.

Where Black 76 is used#

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.

Lognormal vs normal volatility#

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.

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.

Sources#

• Wikipedia, Black model
• Wikipedia, Bachelier model