Jump & Fat-Tail Models

The market gaps. A protocol exploit, a surprise Fed decision, a liquidation cascade. Stochastic vol models struggle with sudden jumps. Jump models handle them directly: the price teleports to a new level at random times.

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Two ways to get fat tails

Stochastic vol (Heston, SABR) makes vol random. Jump models make the price itself jump. Both effects are present in real markets -- production systems often combine them.

At a Glance

Model

Key idea

Best for

<a href="/docs/reference/merton-jump-diffusion">Merton Jump-Diffusion</a>

Black-Scholes + random jumps. The original jump model.

Understanding crash risk, short-dated smile steepness

<a href="/docs/reference/kou-jump-diffusion">Kou</a>

Asymmetric jumps. Crashes bigger than rallies.

Independent wing fitting

<a href="/docs/reference/variance-gamma">Variance Gamma</a>

Pure jumps, no diffusion. Returns driven by a random clock.

Fat tails without stochastic vol. Academic benchmark.

What they share

All three models explain fat tails and steep short-dated smiles by allowing the price to jump. They differ in the jump distribution and whether a continuous diffusion component is present.

Model

Jump distribution

Has diffusion?

Closed-form?

Wing behavior

Merton

Lognormal (symmetric)

Yes

Yes (series)

Symmetric fattening

Kou

Double exponential (asymmetric)

Yes

Yes

Independent left/right tails

Variance Gamma

Gamma-subordinated Brownian motion

No

Yes

Controlled by skew and kurtosis params

How they relate to each other

Merton is the original: take Black-Scholes and add random jumps drawn from a lognormal distribution. The jumps are symmetric, so the model fattens both tails equally. Kou fixes this by replacing the lognormal jump with a double exponential, giving separate parameters for upward and downward jumps -- crashes can be bigger than rallies. Variance Gamma takes a different path: it removes the diffusion entirely and models returns as a Brownian motion running on a random clock (a gamma process). All movement comes from jumps. This makes it a pure-jump process where the kurtosis and skew parameters directly control tail shape.

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Models in this section:

• Merton Jump-Diffusion — The original jump model
• Kou Jump-Diffusion — Asymmetric double-exponential jumps
• Variance Gamma — Pure-jump process with a random clock