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Variance Gamma

Variance Gamma (VG): no diffusion at all. Prices do not move smoothly between jumps -- every move is a jump. The jumps happen on a random clock. Time moves fast during high activity and slow during quiet periods. This random clock produces fat tails without needing a "jump size distribution" like Merton. The resulting vol surface can match real-market skew and kurtosis simultaneously.

Three parameters control everything: vol (sigma), skew (theta), kurtosis (nu).

💡
The random clock idea

The market has its own internal clock that runs at a random speed. Busy days: the clock ticks fast, prices move a lot. Quiet days: the clock barely moves. VG = Black-Scholes on a random clock. Fat tails and a natural smile fall out, without any assumptions about crashes or jump sizes.

Explore the Parameters​

Try "Thin tails" first to see near-Black-Scholes. Then crank up nu (kurtosis) to watch the wings lift.

Variance Gamma Smile Explorer

Negative skew plus heavy tails. The classic crypto smile: steep put wing, elevated call wing.
46%53%60%758595ATM105115125StrikeImplied Vol (%)
Volatility0.45
Overall vol level. Higher = everything more expensive.
θ (skew)-0.15
Negative = put skew. Controls which side of the smile is steeper.
ν (kurtosis)0.30
Controls tail fatness. Higher = more extreme moves, steeper wings.

Try "Thin tails" to see near-flat Black-Scholes, then crank up nu to watch the wings lift from excess kurtosis.

What each parameter does​

  • Sigma (volatility): The baseline vol when the clock ticks at normal speed. This is the overall level -- like ATM vol.
  • Theta (skew): The drift of the process. Negative theta means the market tends to move down more than up in a given time step. This creates put skew -- the left wing is steeper than the right.
  • Nu (kurtosis): Controls how "random" the clock is. Low nu = the clock ticks steadily (thin tails, close to Black-Scholes). High nu = the clock is very erratic (fat tails, steep wings). OTM options become significantly more expensive.
Parameter
Controls
Smile effect
σ (sigma)
Vol level
Shifts entire smile up or down
θ (theta)
Skew / asymmetry
Negative = steep put wing. Zero = symmetric.
ν (nu)
Tail fatness
Higher = both wings lift. Zero = no excess kurtosis (Black-Scholes).

Why Pure-Jump?​

Black-Scholes and even Merton assume a continuous diffusion component -- prices move smoothly most of the time, with occasional jumps. VG says: maybe all price movement is discontinuous. At the tick level, prices jump from one level to the next. No smooth path between trades. Delta hedging is imperfect by construction -- you cannot replicate the payoff continuously.

A good description of how crypto markets actually work -- especially on low-liquidity pairs where the order book is thin and prices gap from one level to another.

ℹ️
Three parameters, three moments

VG is elegant because each parameter maps directly to a statistical property of returns. Sigma controls variance (second moment), theta controls skewness (third moment), and nu controls excess kurtosis (fourth moment). No redundancy, no parameter correlation headaches.

VG vs. Other Models​

Variance Gamma
Merton
Black-Scholes
Price path
Pure jumps (random clock)
Diffusion + occasional jumps
Smooth diffusion only
Tail behavior
Fat tails from clock randomness
Fat tails from discrete jumps
Thin (Gaussian) tails
Parameters
3 (sigma, theta, nu)
4 (sigma, lambda, mu_J, sigma_J)
1 (sigma)
Smile shape
Smooth, controlled by 3 knobs
Steep short-dated, fades long-dated
Flat (no smile)
Best for
General smile fitting, thin liquidity
Event risk, short-dated options
Quick and dirty, liquid markets

VG in Practice​

VG is less common than Heston or SABR on traditional desks, but it has a niche in crypto and credit:

Use case
Why VG
Crypto options on illiquid pairs
Pure-jump nature matches gappy price action. No need to fake a continuous diffusion.
Credit derivatives
Default is a jump event. VG naturally handles discontinuous payoffs.
Quick 3-parameter smile fit
Fewer params than Heston (5) or Merton (4). Each param has a clear meaning.
Moment matching
Direct control over variance, skewness, and kurtosis makes calibration intuitive.
💡
One parameter per moment

Each VG parameter controls exactly one statistical property of returns. Cleanest separation of skew and tail fatness in any smile model. Vega exposure under VG differs from Black-Scholes because the implied vol smile is not flat. If you want more than Black-Scholes but do not need Heston or SLV complexity, VG fits.

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: What is the 'random clock' in Variance Gamma, and why does it produce fat tails?
Q: If theta is zero and nu is high, what does the smile look like?
Q: Why might VG be a better fit than Merton for crypto options on illiquid pairs?

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

Building mathematical intuition​

Learn Variance Gamma from scratchInteractive lesson · no prerequisites

This lesson teaches Variance Gamma through the random-clock mental model, then shows how theta, sigma, and nu control skew, ordinary move size, and tail thickness.

See also:

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