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Merton Jump-Diffusion

Black-Scholes assumes prices move smoothly -- no gaps, no sudden crashes. Merton (1976) adds jumps. The price can suddenly teleport up or down, not just diffuse. The market gaps overnight. A stablecoin depegs in one block.

Fat tails and steep short-dated smiles follow directly. More jump risk = steeper wings on the vol surface.

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Why jumps matter for options

An OTM put that expires in 2 days is nearly worthless under Black-Scholes -- there is not enough time for diffusion to reach the strike. But if the market can jump 15% overnight, that put has real value. Jump models capture this. That is why short-dated smiles are so steep.

Explore the Parameters​

Start with "No jumps" to see flat Black-Scholes. Then switch to "Crash risk" and watch the put wing steepen.

Merton Jump-Diffusion Smile Explorer

One expected crash per year, -15% average. Steep put skew from downside jump risk.
31%37%44%758595ATM105115125StrikeImplied Vol (%)
Jump intensity1.00
Expected jumps per year. 0 = Black-Scholes.
Mean jump size-0.15
Negative = crash bias. -0.10 means -10% average jump.
Jump volatility0.20
How variable each jump is. Higher = steeper wings.
Base vol0.20
Diffusion volatility (between jumps).

Start with "No jumps" to see flat Black-Scholes, then switch to "Crash risk" to see how jumps create the skew.

What each parameter does​

  • Lambda (jump intensity): How many jumps per year you expect. Zero = Black-Scholes. One = roughly one crash-sized event per year. In crypto, this can be 2-3.
  • Mean jump size: The average direction of a jump. Negative = crashes are more common than spikes. This is what creates put skew.
  • Jump volatility: How variable each jump is. Even if the mean jump is zero, high jump vol creates fat tails (both wings lift).
  • Base vol (sigma): The normal diffusion volatility between jumps. This sets the overall level.

How jumps shape the smile​

Parameter change
Effect on smile
Intuition
Increase lambda
Both wings lift
More jumps = more tail risk = OTM options worth more
More negative mean jump
Put wing steepens
Crashes are more likely than spikes, so puts get more expensive
Increase jump vol
Wings get steeper
Each jump is more unpredictable, so extreme moves become more likely
Increase base vol
Entire smile shifts up
More diffusion volatility raises all option prices

The Jump Smile vs. the Stochastic Vol Smile​

Merton and Heston (stochastic vol) both produce smiles, but they do it differently. The distinction matters for trading.

Merton (jumps)
Heston (stoch vol)
What creates the smile?
Sudden price gaps
Random volatility
Short-dated behavior
Steep smile (jump risk dominates)
Mild smile (not enough time for vol to move)
Long-dated behavior
Smile flattens (jumps average out)
Smile persists (vol randomness accumulates)
Tail shape
Fat tails from discrete jumps
Fat tails from vol clustering
Best for
Short-dated options, event risk
Longer-dated options, vol trading
ℹ️
Short-dated vs. long-dated

Merton's model is most useful for short-dated options where jump risk dominates. For longer maturities, the central limit theorem kicks in -- many small jumps look like diffusion, and the smile from jumps alone fades. Stochastic vol takes over at the long end of the term structure.

Merton in Crypto​

Crypto is arguably where Merton matters most. Markets trade 24/7 but liquidity gaps are common -- exchange outages, oracle failures, sudden liquidation cascades. These are jumps. The ATM level may not change much, but the wings steepen dramatically.

Crypto event
Jump character
Smile impact
Flash crash / liquidation cascade
Large negative jump
Steep put skew, especially short-dated
Stablecoin depeg
Negative jump with high vol
Extreme put wing, elevated call wing
Positive catalyst (ETF approval, etc.)
Positive jump
Call wing lifts, temporary skew reversal
Exchange outage during volatility
Gap in either direction
Both wings elevated (pure kurtosis)
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Simplest model that prices gap risk

Merton explains why short-dated OTM options are more expensive than Black-Scholes predicts. If you trade weeklies or short-dated crypto options, jump risk is what you are really pricing. Delta hedging under Merton differs from Black-Scholes because the jump component is unhedgeable -- only the diffusion part can be replicated. Vega exposure is structurally higher.

Equation Explorer​

Convert between implied vol, total variance, log-moneyness, and option prices.

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: Why does Black-Scholes underprice short-dated OTM options?
Q: What happens to the Merton smile as maturity increases?
Q: If mean jump size is zero but jump vol is high, what does the smile look like?

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

Building mathematical intuition​

Learn Merton jumps from scratchInteractive lesson · no prerequisites

This lesson starts with the simple question "what if price can teleport?" and then builds the full intuition for jump intensity, jump size, and why short-dated wings get expensive.

See also:

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