Bachelier from zero
1/5Dollars, not percentages
Black-Scholes says "a 10% move." Bachelier says "a $10 move." That is the entire philosophical split between the two oldest option pricing models.
Louis Bachelier published his model in 1900 -- 73 years before Black and Scholes. His idea was dead simple: price changes are additive and normally distributed. The model is one equation:
If normal vol is $20, the model predicts the price can move about $20 in a year. Whether the price starts at $40 or $400, the wiggle is the same size in dollar terms. That is what "additive" means -- the noise does not scale with the price level.
Compare that with Black-Scholes, where the noise is multiplicative: dS = S·σ·dW. The same 30% vol produces a $30 move on a $100 stock but a $150 move on a $500 stock. The ruler stretches.
Drag the price slider. The Bachelier ruler keeps its tick marks at fixed dollar intervals. The BS ruler stretches or shrinks because each tick is a fixed percentage of the current price.
The additive model can produce negative prices. That is a bug if you are pricing equity options. But it is a feature for interest rates (which went negative in EUR, JPY, CHF) and for spreads (which are naturally signed). Bachelier was 73 years ahead of his time -- his "defect" became the industry standard for rates options.
The formula is simpler than you think
The Bachelier call price has fewer moving parts than Black-Scholes. No logarithms. No discount factor drama. Just subtraction, a ratio, and two normal distribution lookups.
Split the formula into two pieces and it is easy to remember:
Piece 1: (S − K)·Φ(d) -- the intrinsic payoff, probability-weighted. If the call ends in the money, you get S − K. Φ(d) is the probability that happens.
Piece 2: σn√T·φ(d) -- the time-value cushion. Even if spot is near the strike, uncertainty gives the option a chance. More vol or more time increases this term.
Compare with Black-Scholes: C = S·Φ(d₁) − K·e−rT·Φ(d₂). BS uses ln(S/K) where Bachelier uses S−K. That log is the entire difference. Near ATM, they agree.
Move the strike away from spot and watch the two prices diverge. Near ATM they are almost identical because the linear and log approximations agree locally. Far OTM, the models disagree because Bachelier allows negative prices and BS does not.
Normal vol vs BS vol
The translation between the two is simple near ATM: σn ≈ S · σBS. A flat normal smile maps to a skewed BS smile because the same dollar move is a different percentage at each strike.
If spot is $100 and BS vol is 30%, normal vol is roughly $30. If spot drops to $50, the same $30 of normal vol becomes 60% in BS terms. Nothing changed in the Bachelier world -- but BS vol doubled.
This is why a perfectly flat Bachelier smile (one normal vol for all strikes) produces a skewed BS smile. For low strikes, the same dollar move represents a bigger percentage. For high strikes, it represents a smaller percentage. The BS implied vol curve tilts downward from left to right.
The interactive below shows both views of the same market. Bachelier says one vol. BS says a curve. Neither is wrong -- they are different coordinate systems for the same set of option prices.
When Bachelier is the right model
Bachelier is the industry standard for rates options, spread options, and any product where the underlying can go negative. It is not the right default for crypto spot -- but it is perfect for basis and funding rate products.
Interest rates: When the ECB pushed rates negative in 2014, Black-Scholes broke. You cannot take the log of a negative number. Rates desks worldwide switched from lognormal to normal quoting overnight. Swaption vol is now quoted in basis points of normal vol, not percent of lognormal vol.
Spreads: The difference between two prices is naturally additive. A calendar spread, a basis trade, or a cross-currency spread can be positive or negative. Bachelier handles that without hacks.
Funding products: Crypto funding rates fluctuate around zero and can go negative. If you are pricing options on funding rates, Bachelier is the natural language.
Crypto spot: Prices are positive and exhibit leverage effects (vol rises when price drops). The lognormal framework is more natural here. Use BS for spot, Bachelier for rates and spreads.
The left panel shows Bachelier paths: additive noise, symmetric, and some cross zero. The right panel shows BS paths: multiplicative noise, always positive, and the distribution has a long right tail. Add paths and watch how many Bachelier paths go negative -- that is the "bug" that is actually a feature for rates.
The fake skew problem
If you quote a Bachelier market in Black-Scholes terms, you see skew that does not exist. The "skew" is just a coordinate transformation. This is the single most important lesson of this page.
Imagine a market maker who prices options with a flat normal vol. Every strike gets $20 of normal vol. No skew. No smile. One number.
Now a trader converts those prices to BS implied vol using a standard IV solver. The low-strike options show higher BS vol. The high-strike options show lower BS vol. The trader sees put skew and thinks the market is pricing crash risk.
But there is no crash risk in this market. The skew is an artifact of forcing a normal world through a lognormal lens. A $20 move on a $80 underlying is 25% in BS terms. The same $20 move on a $120 underlying is only 16.7%. Different percentages, same dollar move.
This matters in practice because:
You can misdiagnose skew. If a rates desk quotes in normal vol and you convert to BS, you will see skew that is 100% an artifact. Do not trade it.
The SABR connection. SABR's beta parameter controls where you sit on the Bachelier-to-BS spectrum. Beta = 0 is full Bachelier (normal). Beta = 1 is full BS (lognormal). Most of the "skew" you see at beta = 0 in BS terms is the same coordinate artifact.
The golden rule: Before you trade a skew, ask whether it is a market feature or a model feature. Flat in one coordinate system can look skewed in another.
Where to go next:
Black-Scholes -- the lognormal counterpart
SABR Model -- uses beta to choose the normal-lognormal spectrum
CEV Model -- bridges normal and lognormal via the beta parameter
Skew -- separating model artifacts from market features