Normal Inverse Gaussian (NIG) Lévy Process

Lévy

A pure-jump Lévy process constructed by subordinating Brownian motion with an Inverse Gaussian process. NIG provides semi-heavy tails and is closed under convolution, making it attractive for multi-period modeling. The process has infinite activity but finite variation.

Mathematical Formulation

Parameters

SymbolDescriptionConstraint
Inverse Gaussian subordinator
Tail heaviness (steepness of density)
Skewness parameter
Scale parameter
Location/drift parameter
Derived parameter for subordinator

Key Assumptions

  • •Pure-jump process with infinite activity
  • •Semi-heavy tails (exponential decay, heavier than Gaussian)
  • •Closed under convolution: NIG(t) + NIG(s) ~ NIG(t+s)
  • •Subordinated via Inverse Gaussian process
  • •Excellent fit to financial return distributions

Reference

Barndorff-Nielsen, O.E. (1997). "Normal Inverse Gaussian Distributions and Stochastic Volatility Modelling." Scandinavian Journal of Statistics.

Model Hierarchy

Built on simpler models
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 NIG relates to observed return properties

SPY return distributions show semi-heavy tails that decay exponentially but slower than Gaussian. NIG fits this shape well and maintains convolution closure for multi-period modeling. Like VG, it cannot capture the time-varying volatility evident in the data.

What NIG captures

  • Semi-heavy tails
  • Skewness
  • Return distribution shape

What it cannot capture

  • Volatility clustering
  • Conditional heteroskedasticity
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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Model Comparison

vs 4 other models on SPY

Normal Inverse Gaussian estimation on SPY ranks 5th of 5 models by AIC. The model accurately captures extreme left-tail events (1% quantile ratio = 1.00). Compare with Merton Jump-Diffusion for the best overall fit.

5
Rank #5
Below median
AIC
1.68282474090356e+30
ΔAIC
+1.68282474090356e+30
Akaike Weight
0.0%
Best Model
Merton Jump-Diffusion
AIC: -6930.6
1.68282474090356e+30 lower than this model
0
View full model comparison

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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. ...

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View all stochastic models

Run Simulation

Watch the Normal Inverse Gaussian (NIG) 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

Pure-jump Lévy process via Inverse Gaussian subordination. Infinite activity with semi-heavy tails. Closed under convolution. Based on Barndorff-Nielsen (1997).

Category: levy

Parameters

Starting asset price

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

Controls tail decay. Higher = lighter tails. Must be > |β|

Skewness parameter. Negative = left skew, Positive = right skew. Must satisfy |β| < α

Scale parameter controlling overall volatility

Simulation period in years

Total simulation steps

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.