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Bachelier (Normal) Model

Bachelier (1900) was the first option pricing model -- predating Black-Scholes by 73 years. Price changes are additive and normally distributed. Instead of modeling percentage returns (lognormal), Bachelier models dollar changes (normal). The price can go negative -- a bug for equities, a feature for interest rates.

The model has exactly one parameter: normal vol, measured in absolute terms (e.g., "$50/year" instead of "30%/year"). There is no smile. If the world were Bachelier, every option across all strikes would have the same normal vol. That flat smile is the model's core prediction.

💡
Skew can be a model artifact

Bachelier produces a flat smile by construction. Convert those prices to Black-Scholes implied vol and you get a skew. That skew is not in the market -- it is a consequence of forcing lognormal math onto a world that might be normal.

Explore the Model​

The flat blue dashed line is Bachelier's view: one vol for all strikes. The green curve shows the same option prices re-expressed in Black-Scholes terms. Lower the spot price and watch the apparent BS skew steepen -- even though nothing changed in the Bachelier world.

Bachelier vs Black-Scholes Explorer

Typical setup. Bachelier smile is flat by definition. The same prices re-expressed in BS terms produce a skew.
16%22%28%828894ATM106112118StrikeImplied VolBS implied vol (%)Bachelier (normal vol)
Normal vol20
Absolute vol in $/year (not percentage)
Spot price (S)100
Lower spot = more apparent BS skew

The flat blue dashed line is Bachelier's view: one vol for all strikes. The green curve is the same option prices re-expressed in Black-Scholes terms. The "skew" is a modeling artifact, not a market feature.

What each parameter does​

  • Normal vol: The single parameter. Measured in absolute price units per year (not percentage). A normal vol of 20 means the price is expected to move $20 over one year (one standard deviation). All strikes get this same vol -- the smile is flat.
  • Spot price: Does not change the Bachelier smile (still flat). But it dramatically affects the BS-equivalent smile. At lower spot prices, the same dollar-move translates to a larger percentage move, so BS implied vol rises -- creating apparent put skew.

Why the BS "skew" appears​

What happens
Bachelier view
BS view
ATM option pricing
Normal vol applies directly
Lognormal vol is roughly normal_vol / spot
OTM put (low strike)
Same vol as ATM
Higher IV because same $ move = bigger % move at lower price
OTM call (high strike)
Same vol as ATM
Lower IV because same $ move = smaller % move at higher price
Drop the spot price
Smile stays flat
Entire curve shifts up, put wing steepens
ℹ️
SABR's beta picks the backbone

SABR's backbone (the smile with vol-of-vol turned off) depends on beta. Beta = 0: Bachelier. Beta = 1: Black-Scholes. Beta chooses where you sit on the normal-vs-lognormal spectrum.

Where Bachelier Is Used​

Market
Why Bachelier
Normal vol unit
Interest rate swaptions
Rates went negative in EUR, JPY, CHF. BS breaks at zero. Bachelier does not.
bps/year (e.g., 50 bps)
Spread options
Spreads can be negative. Additive model is natural.
$/year or bps/year
CDS options
Credit spreads are naturally modeled as additive moves.
bps/year
Crypto (niche)
Funding rate options or basis options where the underlying can go negative.
%/year (absolute)
⚠️
Not for crypto spot options

Crypto spot prices are positive and exhibit leverage effects (vol rises when price drops). The lognormal framework (Black-Scholes family) is more natural here. Bachelier is the right tool for rates, spreads, and anything that can go negative.

Bachelier vs. Black-Scholes at a Glance​

Bachelier
Black-Scholes
Price dynamics
Additive (normal)
Multiplicative (lognormal)
Vol unit
$/year (absolute)
%/year (relative)
Negative prices?
Yes (by design)
No (log of negative is undefined)
Smile shape
Flat by definition
Flat only if world is truly lognormal
Parameters
1 (normal vol)
1 (lognormal vol)
Conversion
σ_n ≈ σ_BS × S (near ATM)
σ_BS ≈ σ_n / S (near ATM)
Used for
Rates, spreads, CDS
Equities, FX, crypto spot

The conversion formula​

Near ATM, you can convert between the two:

σnormal≈σBS×S\sigma_{\text{normal}} \approx \sigma_{\text{BS}} \times S

A stock at 100with30100 with 30% BS vol has roughly 30 of normal vol. But this approximation breaks down away from ATM, which is exactly why the BS "smile" appears when you convert Bachelier prices.

💡
Flat smile by definition

Bachelier treats price changes as additive. Its smile is flat by definition. Skew that appears after converting to BS terms is an artifact of the model choice, not a market feature.

Equation Explorer​

Equation Explorer

$
$
days
%
%
Call Price
$8300
Put Price
$7890
Call Δ
0.555
d₁
0.102
Vega
$114

Test your understanding before moving on.

Q: Why does Bachelier produce a flat smile while Black-Scholes does not?
Q: If you take Bachelier option prices and convert them to BS implied vol, you get put skew. Where does that skew come from?
Q: When would you use Bachelier instead of Black-Scholes for crypto?
Q: What is the near-ATM relationship between normal vol and BS vol?

💡 Tip: Try answering each question yourself before revealing the answer.

Building mathematical intuition​

Learn Bachelier from scratchInteractive lesson · no prerequisites

This lesson starts with the plain-English mental model, then walks through normal volatility, the pricing formula, and why a flat normal smile can show up as skew after you translate it into Black-Scholes terms.

See also:

  • Black-Scholes -- The lognormal counterpart
  • CEV Model -- Bridges normal and lognormal via the beta parameter
  • SABR Model -- Uses beta to choose the normal-lognormal spectrum
  • Displaced Diffusion -- Another way to handle near-zero underlyings
  • Implied Volatility -- The concept that depends on which model you choose
  • Skew -- Separating model artifacts from market features
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