CGMY (Tempered Stable) Lévy Process

Lévy

Generalizes VG (Y=0) and approximates NIG

The most general tempered stable Lévy process, introduced by Carr, Geman, Madan, and Yor (2002). CGMY generalizes both VG (Y=0) and approximates NIG. The activity index Y controls the jump behavior: Y<0 gives finite activity, 0<Y<1 gives infinite activity with finite variation, and 1≤Y<2 gives infinite activity with infinite variation.

Mathematical Formulation

Parameters

SymbolDescriptionConstraint
Overall activity level (jump intensity scale)
Decay rate for positive jumps
Decay rate for negative jumps (M > 1 for martingale)
Activity index (controls small jump behavior)
Lévy density (jump measure)

Key Assumptions

  • •Y < 0: Finite activity (compound Poisson-like)
  • •Y = 0: Variance Gamma process (finite activity)
  • •0 < Y < 1: Infinite activity, finite variation
  • •1 ≤ Y < 2: Infinite activity, infinite variation
  • •Tempered stable: power-law small jumps with exponential tempering for large jumps
  • •Generalizes VG and approximates NIG

Reference

Carr, P., Geman, H., Madan, D., Yor, M. (2002). "The Fine Structure of Asset Returns: An Empirical Investigation." Journal of Business.

Model Hierarchy

Built on simpler models
SimplerStandard

VG

Lévy
SimplerBaseline

GBM

Diffusion

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

Empirical Context (SPY Returns)

How CGMY relates to observed return properties

SPY returns require flexible tail modeling. CGMY generalizes VG with activity index Y, allowing fine-tuned control over small vs large jump contributions. The G/M asymmetry captures skewness. Being pure-jump, it lacks the volatility persistence mechanism.

What CGMY captures

  • Flexible tail behavior
  • Tunable activity level (Y parameter)
  • Asymmetric decay rates

What it cannot capture

  • Volatility dynamics
  • No diffusion smoothing
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

VG

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 comp...

Compare with VG →

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 →
View all stochastic models

Run Simulation

Watch the CGMY (Tempered Stable) 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

Log returns will appear here

Model

General tempered stable Lévy process. Nests VG (Y=0) and approximates NIG. Activity index Y controls jump behavior: Y<0 finite activity, 0<Y<1 infinite activity/finite variation, 1≤Y<2 infinite variation. Based on Carr-Geman-Madan-Yor (2002).

Category: levy

Parameters

Starting asset price

Expected annual return (e.g., 0.05 = 5%)

Overall jump intensity. Higher = more jumps

Tempering rate for upward jumps. Higher = smaller upward jumps

Tempering rate for downward jumps. Must be > 1 for martingale. Higher = smaller downward jumps

Controls jump activity. Y<0: finite activity, 0<Y<1: infinite activity/finite variation, 1≤Y<2: infinite variation

Simulation period in years

Total simulation steps

Small jump cutoff for Gaussian approximation. Smaller = more accurate but slower

Simulation

0.25x

Adjust speed before or during simulation

Status:idle

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.