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.

$$C = (F-K)\,N(d) + \sigma\sqrt{T}\,\phi(d), \qquad d = \frac{F-K}{\sigma\sqrt{T}}$$

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.