Bachelier model
Quantitative Finance glossary · Quantitative Finance
Definition
Arithmetic Brownian motion for the price level: dS_t = μ,dt + σ_N,dW_t, so S_T is normally distributed (additive, not log-additive). Yields a closed-form call price C = (S - K)Phi(d) + σ_N√(T),φ(d) with d = (S-K)/(σ_N√(T)). Permits negative prices, which is exactly why it became the post-2020 standard for SOFR caps/floors and oil options after WTI futures printed -$37 in April 2020. Bachelier vol σ_N and Black vol σ_B are related by σ_N ≈ σ_B · F ATM (where F is the forward).
Bachelier model in plain English
The Bachelier model is one of the oldest ways to describe how a price wanders over time, and its quirk is that it lets the price move up or down by fixed dollar amounts — which means, unlike newer models, it allows the price to go *negative*. That sounds absurd until you remember April 2020, when the price of oil futures actually crashed below zero to about $-37 because traders were desperate to avoid taking delivery of barrels they had nowhere to store. Picture a coin-flip walk where each step is the same size in dollars, so the total can drift anywhere on the number line, including below zero. Because it handles negatives naturally, this once-old-fashioned model became the go-to standard for pricing options on things like oil and modern interest rates.
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