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Kou Double-Exponential Jump-Diffusion

Merton models jumps as a single normal distribution -- up-jumps and down-jumps have the same shape. Wrong. Crashes are sharper than rallies. A -20% gap happens in minutes; a +20% rally takes weeks. Kou (2002) fixes this by giving up-jumps and down-jumps different sizes.

The mechanism: exponential distributions instead of normal. Down-jumps get one exponential (typically with a larger mean), up-jumps get another (typically with a smaller mean). Steepen the put wing without touching the call wing, and vice versa.

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Each wing has its own parameter

In Merton, steepening the put wing (via negative mean jump) also affects the call wing. In Kou, each wing is independent. Down-jump size steepens the put wing. Up-jump size steepens the call wing. This matches crypto smiles.

Explore the Parameters​

Toggle "Show Merton equiv" to see how a symmetric (Merton) model compares to Kou's asymmetric wings. Try the "Crypto crashes" preset to see the steep put wing with a gentle call wing.

Kou Double-Exponential Smile Explorer

Down-jumps dominate: 70% of jumps are downward and 4x larger than up-jumps. Steep put wing.
33%42%51%758595ATM105115125StrikeImplied Vol (%)Kou (asymmetric)Merton (symmetric)
Jump frequency2.00
Expected jumps per year. 0 = flat (BS).
Up-jump probability0.30
Fraction of jumps that go up. Low = crash bias.
Up-jump size0.05
Avg up-jump magnitude (e.g. 0.08 = 8%)
Down-jump size0.20
Avg down-jump magnitude (e.g. 0.15 = 15%)

Toggle "Show Merton equiv" to compare asymmetric (Kou) vs symmetric (Merton) jumps. Notice how Kou can steepen one wing independently.

What each parameter does​

  • Jump frequency (lambda): How many jumps per year. Zero = Black-Scholes (flat smile). Higher lambda lifts both wings because any jump -- up or down -- makes OTM options more valuable.
  • Up-jump probability (p): What fraction of jumps go up. Low p means most jumps are crashes. This shifts the skew balance.
  • Up-jump size: Average magnitude of upward gaps. Larger = steeper call wing.
  • Down-jump size: Average magnitude of downward gaps. Larger = steeper put wing. In crypto, this is typically 2-4x the up-jump size.

How Kou shapes the wings​

Parameter change
Put wing effect
Call wing effect
Intuition
Increase down-jump size
Steepens
Minimal change
Bigger crashes = more expensive put protection
Increase up-jump size
Minimal change
Steepens
Bigger rallies = more expensive call wing
Decrease up-jump probability
Steepens
Flattens
More jumps are downward = crash bias
Increase jump frequency
Lifts
Lifts
More total events = more tail risk in both directions
ℹ️
Independent wing control

In Merton, steepening the put wing via a negative mean jump also affects the call wing (normal distribution is symmetric around the mean). In Kou, down-jump size controls the put wing and up-jump size controls the call wing. Toggle "Show Merton equiv" to see the difference.

Kou vs. Merton​

Kou
Merton
Jump distribution
Double exponential (asymmetric)
Normal (symmetric around mean)
Wing independence
Put and call wings controlled separately
Changing skew affects both wings
Tail decay
Exponential tails (heavier than normal)
Gaussian tails (thinner)
Parameters
5 (σ, λ, p, η₁, η₂)
4 (σ, λ, μ_J, σ_J)
Barrier/lookback pricing
Closed-form available
No closed-form (requires MC)
Crypto fit
Better (asymmetric wings match reality)
Decent (but struggles with wing independence)

Why Crypto Traders Should Care​

Crypto gap risk is deeply asymmetric:

Event type
Typical size
Speed
Kou parameter
Liquidation cascade
-10% to -30%
Minutes
Down-jump size (large)
Exchange outage gap
Either direction, -20% to +10%
Instant
Both jump sizes + probability
ETF approval rally
+5% to +15%
Hours
Up-jump size (moderate)
Stablecoin depeg
-5% to -50%
Blocks
Down-jump size (very large)

Notice the pattern: down-moves are faster and larger than up-moves. Merton cannot capture this asymmetry cleanly -- you can shift the mean negative, but the normal distribution's symmetry around that mean still bleeds into the call wing. Kou's double exponential naturally separates the two.

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The jump model for independent wing fitting

Kou separates the put and call wings. Down-jump size is the crash parameter. Up-jump size is the rally parameter. They do not interfere. If you trade OTM puts and calls as separate books -- and in crypto, you should -- Kou matches that structure.

Equation Explorer​

Equation Explorer

w = σ2 × Ttotal variance = IV2 × time
%
The implied volatility
days
Calendar days to expiration
Total Variance (w)
0.022225
Annualized Variance (σ²)
0.2704
Round-trip IV
52.00%
Total variance is what SVI and other models fit. It scales with time, so a 50% vol for 30 days has less total variance than 50% vol for 90 days.

Test your understanding before moving on.

Q: What is the key advantage of Kou over Merton for fitting vol smiles?
Q: Why are exponential tails more realistic than Gaussian tails for crypto jump sizes?
Q: If you increase the down-jump size from 10% to 25%, what happens to the call wing?
Q: What practical advantage does Kou have over Merton for exotic pricing?

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

Building mathematical intuition​

Learn Kou from scratchInteractive lesson · no prerequisites

This lesson explains the model as separate upside and downside jump engines, then walks through the double-exponential intuition and why it gives cleaner wing control than Merton.

See also:

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