Variance Gamma (VG) Lévy Process

Lévy

A pure-jump Lévy process constructed via time-changed Brownian motion. Unlike Merton/Kou which add finite jumps to diffusion, VG has no diffusion component - all price movement comes from infinitely many small jumps. The process evaluates Brownian motion at random "business time" controlled by a Gamma subordinator. VG is a special case of CGMY with Y=0.

Mathematical Formulation

Parameters

SymbolDescriptionConstraint
Gamma subordinator ("business time")
Standard Brownian motion
Deterministic drift rate
Skewness parameter
Volatility parameter
Variance rate (controls kurtosis)

Key Assumptions

  • •Pure-jump process (no continuous diffusion component)
  • •Infinite activity: infinitely many small jumps per unit time
  • •Finite variation despite infinite jumps
  • •Independent control of skewness (θ) and kurtosis (ν)
  • •As ν → 0, VG converges to GBM
  • •Special case of CGMY with Y = 0

Reference

Madan, D., Carr, P., Chang, E. (1998). "The Variance Gamma Process and Option Pricing." European Finance Review.

Model Hierarchy

Built on simpler models
SimplerBaseline

GBM

Diffusion
Extended by more complex models
More ComplexAdvanced

CGMY

Lévy

Complexity increases from left to right. More complex models capture additional market phenomena but require more parameters and may be harder to estimate.

When VG Fits Better

VG fits better when return distributions have heavier tails than normal and exhibit excess kurtosis. The pure-jump nature captures the "infinite activity" of small price movements. Captures excess kurtosis via time-changed Brownian motion.

Recommended Use Cases

  • •High-frequency return modeling
  • •Assets with heavy tails and infinite activity
  • •When normal distribution assumptions fail dramatically
  • •Capturing excess kurtosis in return distributions

Empirical Context (SPY Returns)

How VG relates to observed return properties

SPY returns have heavier tails than a normal distribution. VG reproduces this through infinitely many small jumps (no diffusion), with independent control of skewness and kurtosis. However, as a Lévy process, it cannot capture the volatility persistence visible in autocorrelation of squared returns.

What VG captures

  • Heavy tails via infinite activity
  • Excess kurtosis
  • Skewness through θ parameter

What it cannot capture

  • Volatility clustering
  • Time-varying volatility dynamics
View full empirical analysis of SPY returns

Estimation Data

Select ticker for P-measure estimation

SPY

Real-World Parameter Estimates

P-measure estimation from historical SPY returns (10y)

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Related Models

NIG

Lévy

A pure-jump Lévy process constructed by subordinating Brownian motion with an Inverse Gaussian process. NIG provides semi-heavy tails and is closed un...

Compare with NIG →

CGMY

Lévy

The most general tempered stable Lévy process, introduced by Carr, Geman, Madan, and Yor (2002). CGMY generalizes both VG (Y=0) and approximates NIG. ...

Compare with CGMY →
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Run Simulation

Watch the Variance Gamma (VG) Lévy Process in action. Adjust parameters and observe how the price path evolves in real-time.

Charts:

Price Path Simulation

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Log Returns

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Disclaimer: These simulations are for educational purposes only. They demonstrate the behavior of mathematical models and should not be used for trading decisions. Real market dynamics are significantly more complex than any single stochastic model can capture.