Bachelier Model

What is the Bachelier model?

The Bachelier model is an option pricing model in which the underlying follows arithmetic Brownian motion, so absolute price changes are normally distributed. Its terminal forward price is centered on the current forward and spreads with normal volatility times the square root of time. Because that distribution includes negative prices, the model fits instruments such as rates and spread products whose underlying value can cross zero.

Louis Bachelier introduced the framework in 1900; modern implementations apply the contract’s forward and discounting conventions.

What is normal volatility?

Normal volatility measures absolute price dispersion. With time measured in years, its dimensional units are price per square root of a year. Multiplying annualized normal volatility by sqrt(T) gives the standard deviation in price units over the selected horizon.

At a normal volatility of 40, the one-year standard deviation is 40 price units. Over 30 days, it is 40 × sqrt(30 / 365) = 11.47. In rate markets, the analogous yield dispersion is quoted in basis points.

What is the Bachelier option formula?

Let D = e^(−rT), a = sigma_N × sqrt(T), and d = (F − K) / a. For positive a:

• Call: D × [(F − K)N(d) + a n(d)].
• Put: D × [(K − F)N(−d) + a n(d)].

F is the forward, K is strike, N is the standard normal cumulative distribution, and n is its density. At the forward-ATM strike, F equals K and both values simplify to D × a / sqrt(2π).

Bachelier model example

Set forward to $20, normal volatility to 40, time to 30 / 365 years, and the rate to zero. The ATM call is worth $4.57 per unit, so a 1,000-unit cash multiplier gives $4,570.

At a −$5 strike, the Bachelier call value is $25.06 per unit. The model assigns a 4.1% probability to a terminal price below zero.

A nonnegative lognormal underlying with the same $20 forward also prices the negative-strike call. Its payoff is underlying plus $5 in every state, so the zero-rate value is $25.

How do the Greeks work in the Bachelier model?

Using derivatives with respect to forward F and normal volatility:

• Call forward delta: D × N(d).
• Put forward delta: D × [N(d) − 1].
• Gamma: D × n(d) / a.
• Normal vega: D × sqrt(T) × n(d).

Normal vega is quoted per unit of normal volatility. Apply the contract multiplier to obtain position values, and keep these units distinct from vega per percentage point of lognormal volatility.

How do you compare normal and lognormal volatility quotes?

Near ATM with positive forward, normal volatility ≈ forward × lognormal volatility provides a first conversion. To match values precisely, solve for the volatility that reproduces the same option price under the other model.

A constant normal volatility is flat in normal-volatility coordinates. A market chain supplies separate implied values across strikes, producing its normal smile or skew. The same option prices take a different shape when expressed as lognormal implied volatility.

Where does the Bachelier model matter in trading?

CME Clearing moved specified energy products to Bachelier valuation effective April 22, 2020, supporting negative underlying prices and negative strikes. The episode demonstrates why the distribution must fit the underlying instrument.

For a trade, align the underlying, expiration, exercise convention, multiplier, forward, discounting, and volatility units before comparing prices.

Limitations of the Bachelier model

• Keep equity options on a lognormal model. Bachelier gives prices below zero real probability, 4.1% in the crude example, and a share cannot trade there.
• Read the implied normal volatility strike by strike. Market prices form a normal-volatility smile or skew, so one constant input fits a single strike, not the chain.
• Use the forward × lognormal volatility shortcut near the money only; away from the money and at longer maturities, convert by matching option prices.
• Price American-style exercise, or a rate or volatility that changes over the option’s life, with a model built for it. The closed form covers European exercise with both held constant.
• Reprice the position when two inputs move at once, because delta, gamma and vega each hold the other inputs fixed. Normal vega is per unit of normal volatility; convert units before setting it beside lognormal vega per percentage point.

Bottom line

The Bachelier model prices options from normally distributed price changes, which lets it value underlyings that trade near or below zero, such as rates, spreads and some energy futures. Quote its volatility in price units (basis points for rates), convert to lognormal volatility by matching option prices, and keep lognormal models for equities, whose prices cannot fall below zero.