Bachelier Normal Model

Black-Scholes assumes a price moves by a percentage of itself each instant, which is a quiet way of assuming it can never cross zero, a stock down 100% is worthless, not negative. That assumption is fine for equities. It fails outright for interest rates that can go negative, and it failed spectacularly on April 20, 2020, when the front-month WTI crude oil futures contract traded at minus $37.63 a barrel. Louis Bachelier, writing in 1900, five years before Einstein's own paper on Brownian motion, built a model that never assumed a floor at zero in the first place.

A ruler instead of a percentage gauge

Black-Scholes measures moves like a percentage gauge on a fuel tank: it can read "half full," but never "negative fuel." Bachelier measures moves like an ordinary ruler laid along a number line: a move of $5 is a move of $5, whether the price starts at $100 or at $2, and the ruler keeps working straight through zero into negative territory. That's the entire difference, the underlying moves by dollars, not by percent.

In plain English: $F$ and $K$ are the forward price and strike in dollars, $\sigma$ here is a dollar volatility (not a percentage), $N$ and $\phi$ are the normal CDF and density, and $d$ measures how many "dollar-vol units" the forward sits above the strike. There's no logarithm anywhere in the formula, that's the tell that this is arithmetic, not geometric, Brownian motion underneath.

Worked example 1: pricing when the forward is negative. Take a call on a futures contract with $F=-10$ (minus $10), $K=0$, dollar volatility $\sigma=15$ ($15 per $\sqrt{\text{year}}$), and $T=1$ month $=0.0833$. Then $\sigma\sqrt{T}=15\times0.2887=4.330$, and $d=(-10-0)/4.330=-2.309$. Using $N(-2.309)\approx0.0105$ and $\phi(-2.309)\approx0.0270$: $C=(-10)(0.0105)+4.330(0.0270)=-0.105+0.117=0.012$, i.e. $0.012. A tiny but strictly positive price, exactly right, since there's still some chance the negative forward recovers above zero before expiry, and Black-Scholes couldn't even have accepted $F=-10$ as an input to try.

Slide the underlying below zero in the payoff above, a strike of $0 and a starting forward already below it is exactly the shape Black-Scholes cannot draw at all, since it has no way to plot a negative price.

Worked example 2: recovering ordinary volatility intuition. A 3-month at-the-money option ($F=K$, so $d=0$) simplifies to $C=\sigma\sqrt{T}\,\phi(0)=\sigma\sqrt{T}\times0.3989$. With a $5-per-share dollar volatility and $T=0.25$: $C=5\times0.5\times0.3989=0.997$, essentially $1.00. Compare a Black-Scholes ATM call, $C\approx0.4\,S\sigma\sqrt{T}$ using percentage $\sigma$, the two formulas converge to nearly the same shape at the money, which is why the normal model and the log-normal model barely disagree for short-dated, near-the-money options; they diverge only once the underlying gets close to (or below) zero.

Compare this to a GBM path: an arithmetic Brownian path is free to wander below zero, while a geometric one is mathematically walled off from ever reaching it, drag the volatility up and watch the arithmetic path cross the axis.

What this means in practice

Rates desks priced swaptions and caps with Bachelier-style normal volatility for years after 2015, once European and Japanese short rates went negative and log-normal models simply couldn't quote a strike below zero. Commodity desks reached for the same tool the day WTI went negative, because no percentage-based model could express "the forward is worth minus $37 a barrel" in the first place.