Simple & Foundation Models

The building blocks. Black-Scholes: constant vol, no smile. CEV adds one parameter to get skew. Displaced Diffusion shifts the price axis to handle negative rates. Too simple for production smile fitting, but every complex model extends one of these.

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Every complex model extends a simple one

SABR needs CEV (its backbone). Heston needs Black-Scholes (its special case). Start here.

At a Glance

Model

Parameters

Produces a smile?

Key idea

<a href="/docs/reference/black-scholes">Black-Scholes</a>

1

No

Constant vol. The baseline that everything improves on.

<a href="/docs/reference/cev">CEV</a>

2

Skew only

Vol scales with price. The backbone inside SABR.

<a href="/docs/reference/displaced-diffusion">Displaced Diffusion</a>

2

Skew only

Shifted Black-Scholes. Handles negative rates.

<a href="/docs/reference/bachelier">Bachelier</a>

1

No (flat by definition)

Normal dynamics. Prices can go negative.

What they share

All four models describe a single diffusion process for the underlying price. None of them have stochastic vol, jumps, or any second source of randomness. They differ in what dynamics they assume for the price.

Model

Price dynamics

Produces skew?

Key limitation

Black-Scholes

Geometric Brownian motion (lognormal)

No

Flat smile -- no skew, no curvature

CEV

Power-law vol: sigma * S^(beta-1)

Yes

Skew only, no independent curvature control

Displaced Diffusion

Shifted lognormal: d(S + d)

Yes

Skew only, equivalent to CEV for small shifts

Bachelier

Arithmetic Brownian motion (normal)

No

Flat smile, prices can go negative

How they relate to each other

Black-Scholes is the baseline: constant vol, lognormal price, no smile. CEV generalizes it by making vol scale with the price level (sigma times S to the power beta minus one), which produces skew. This is the backbone of SABR -- when SABR sets its local vol component, it uses CEV. Displaced Diffusion takes a different route: it shifts the price axis (model S + d instead of S), which also produces skew and lets you handle negative rates or prices. For small shifts it behaves similarly to CEV. Bachelier is the additive version of Black-Scholes: prices follow a normal distribution instead of lognormal. It produces a flat smile (in normal-vol terms) and naturally allows negative prices, which is why it became the standard for interest rate options when rates went negative.

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Models in this section:

• Black-Scholes — The foundation
• CEV Model — Vol scales with price
• Displaced Diffusion — Shifted lognormal
• Bachelier — Normal dynamics, additive vol