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Bates Model

Bates (1996) adds Merton jumps to Heston. Stochastic vol covers the smooth daily dynamics; jumps cover gap risk. Most exotic desks run either SLV or Bates.

Neither Heston nor Merton alone fits crypto -- sticky vol regimes and sudden gaps need separate mechanisms.

💡
Heston alone cannot produce steep short-dated wings

Crypto can gap 15% in a single block. No amount of vol-of-vol produces the steep short-dated wings you see after a liquidation cascade. Jumps fill that gap. Bates = Heston for normal times + Merton for crashes.

Explore the Parameters​

Toggle "Show Heston only" to see the dashed Heston curve. The gap between the dashed line and the solid Bates curve is the pure jump contribution.

Bates Smile Explorer

Frequent gaps plus high vol-of-vol. Wide smile with steep put wing from both stochastic vol and jumps.
35%53%71%758595ATM105115125StrikeImplied Vol (%)Bates (full)Heston only
Vol of vol0.70
Heston side: controls smile curvature
ρ (spot-vol corr)-0.60
Negative = put skew from stochastic vol
Jump frequency2.00
Expected jumps per year. 0 = pure Heston.
Jump skew-0.12
Avg jump direction. Negative = crash bias.

Toggle "Show Heston only" to see the dashed stochastic-vol-only curve. The gap between the two lines is the jump contribution.

What each parameter does​

  • Vol of vol: The Heston side. Controls smile curvature from random vol. Higher = wider smile, even without jumps.
  • Spot-vol correlation: Controls skew from the stochastic vol component. More negative = steeper put wing from the vol-price link.
  • Jump frequency: How many gap events per year. Zero = pure Heston. Two = roughly one gap every six months. This lifts both wings.
  • Jump skew: Average jump direction. Negative = crashes dominate. This steepens the put wing on top of whatever skew rho already provides.

Two sources of skew​

Bates has two independent skew sources:

Source
Parameter
What it does
Where it matters
Stochastic vol
ρ (spot-vol corr)
Vol rises when price drops → put skew
All maturities, especially long-dated
Jumps
μⱼ (jump skew)
Crashes are bigger than rallies → put skew
Short-dated (jumps are sudden events)

Jump skew dominates at the short end (weeklies, 7-day expiries), while the vanna-driven stochastic vol skew dominates at the long end. Bates lets you capture both term-structure regimes.

ℹ️
Decomposition trick

If you toggle the Heston overlay on and off while adjusting jump frequency, you can see exactly how much skew comes from each mechanism. On a production desk, this decomposition tells you whether your skew P&L is coming from vol dynamics or event risk.

When Each Component Matters​

Maturity
Dominant mechanism
Why
1-7 days
Jumps
Not enough time for vol dynamics. A single gap event sets the wing price.
7-30 days
Both
Jumps and stochastic vol both contribute. This is where Bates shines vs. either alone.
30-90 days
Stochastic vol
Multiple vol regime changes average out the jump contribution. Rho dominates skew.
90+ days
Stochastic vol
Central limit theorem. Jumps look like extra diffusion. Mean reversion drives everything.
💡
When to reach for Bates

Cross-maturity pricing where short-dated wings come from jumps and long-dated skew comes from stochastic vol. One model, two regimes.

Strengths and Limitations​

Strength
What it means for you
Handles both smooth and gap moves
The only way to fit crypto short-dated and long-dated smiles simultaneously with one model.
Still has semi-closed-form pricing
Fourier inversion works just like Heston. Fast enough for real-time Greeks.
Decomposable skew
You can attribute skew P&L to the jump component vs. the stochastic vol component.
Production-grade
Used by serious exotic desks for barriers, Bermudans, and auto-callables.
Limitation
What it means for you
7+ parameters
Heston (5) + jump (3) = too many knobs. Calibration is tricky and can be unstable.
Parameter identifiability
Stochastic vol and jumps can produce similar smiles. Hard to separate them from smile data alone.
Jumps are i.i.d.
Real crashes cluster. Bates jumps are independent events -- no contagion or momentum.
Overkill for vanillas
If you just need to fit the current smile, SVI is simpler and more stable.
⚠️
Calibration warning

With 7+ free parameters, you can fit almost anything -- including noise. Desks fix several parameters (kappa, theta) from the implied vol term structure and only calibrate vol-of-vol, rho, and the jump parameters daily. Never let all parameters float simultaneously.

Bates vs. Alternatives​

Bates
Heston
Merton
SLV
Short-dated wings
Steep (jumps)
Too flat
Steep (jumps)
Exact fit
Long-dated smile
Smooth (stoch vol)
Smooth (stoch vol)
Fades out
Exact fit
Pricing speed
Fast (Fourier)
Fast (Fourier)
Fast (series)
Slow (PDE/MC)
Parameters
7-8
5
4
Many (local vol grid)
Calibration
Tricky
Moderate
Simple
Complex
Best for
Cross-maturity exotics
Long-dated vol trading
Short-dated events
Exact fit needed

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 Bates fit short-dated crypto smiles better than Heston alone?
Q: Bates has two independent sources of put skew. What are they, and when does each dominate?
Q: What is the main practical problem with calibrating Bates?
Q: When would you choose Bates over SLV for pricing exotics?

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

Building mathematical intuition​

Learn Bates from scratchInteractive lesson · no prerequisites

This lesson frames Bates as Heston plus jumps, then shows which parameters control smooth skew, which ones control gap risk, and why the short end changes first.

See also:

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