Quantifying VIX Tail Risk: Volatility Clustering and Jump Processes

HKUST IEDA4000E — Statistical Modelling for Financial Engineering

Course Project by: Vittorio Prana CHANDREAN, CHONG Tin Tak, CHOI Man Hou

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Objective

This project compares two complementary frameworks for modeling VIX dynamics:

1. GARCH-family models (GARCH, EGARCH) — capturing volatility clustering and persistence
2. Compound Poisson Process (CPP) — modeling shock timing AND magnitude for risk quantification

We analyze 15+ years of daily VIX data spanning calm and crisis regimes, demonstrating how these approaches enable VaR/CVaR computation for risk management.

Key Features

• Volatility Models: GARCH(1,1) and EGARCH(1,1) with automatic distribution selection
• Compound Poisson Process: Models shock arrivals (Poisson) and magnitudes (Pareto) for VaR/CVaR
• Regime Analysis: Pre-crisis, COVID, post-COVID, and recent period comparison
• Out-of-Sample Evaluation: Train/test split with CPP forecast validation
• Comprehensive Visualization: Jump distributions, simulated paths, regime comparisons

Project Structure

Shock-Persistence-and-Shock-Frequency-in-VIX/
├── runall.py                  # Main pipeline driver
├── src/                       # Core modules
│   ├── config.py              # Shared parameters
│   ├── data_pipeline.py       # Data loading and cleaning
│   ├── volatility_models.py   # GARCH/EGARCH implementation
│   ├── shock_modeling.py      # CPP and shock definitions
│   ├── forecast_evaluation.py # OOS evaluation utilities
│   └── visualization.py       # Plotting functions
├── notebooks/                 # Analysis notebooks
│   ├── 01_data_and_volatility.ipynb
│   ├── 02_shock_arrivals.ipynb
│   └── 03_forecast_evaluation.ipynb
├── figures/                   # Generated plots (organized by category)
│   ├── vix_data/              # VIX time series and ACF plots
│   ├── volatility_models/     # GARCH/EGARCH volatility outputs
│   ├── shock_analysis/        # Shock detection and analysis
│   ├── cpp_model/             # Compound Poisson Process figures
│   └── model_evaluation/       # Model diagnostics and comparisons
├── zz slides/                 # LaTeX documents
│   ├── report.tex             # Main CPP report (renamed from CPP_VIX_Shock_Modeling_Report.tex)
│   ├── slides.tex             # Beamer presentation (renamed from main.tex)
│   └── references.bib         # Bibliography file
├── tests/                     # Unit tests
└── requirements.txt

Quick Start

# Setup environment
python -m venv .venv
source .venv/bin/activate
pip install -r requirements.txt

# Run full pipeline
python runall.py

# Run tests
pytest tests/test_models.py -v

Methodology

GARCH Models

GARCH(1,1): $$\sigma_t^2 = \omega + \alpha \varepsilon_{t-1}^2 + \beta \sigma_{t-1}^2$$

• Captures volatility clustering: large shocks increase future variance
• Persistence governed by α + β (values near 1 imply long memory)

EGARCH(1,1): $$\ln \sigma_t^2 = \omega + \beta \ln \sigma_{t-1}^2 + \alpha(|\epsilon_{t-1}| - \mathbb{E}|\epsilon|) + \gamma \epsilon_{t-1}$$

• Captures leverage effect: negative shocks impact volatility more than positive
• Ensures positive variance without parameter constraints

Compound Poisson Process

$$S(T) = \sum_{i=1}^{N(T)} J_i, \quad N(T) \sim \text{Poisson}(\lambda T)$$

• N(t): Shock count (arrival rate λ)
• Jᵢ: Shock magnitude (fitted to Pareto distribution)
• S(T): Cumulative shock impact for VaR/CVaR computation

Regime Analysis

Regime	Period	Key Characteristic
Pre-Crisis	2010–2019	Baseline volatility
COVID	2020	76% higher expected annual impact
Post-COVID	2021–2023	Recovery period
Recent	2024–2025	Current conditions

Key Results

Volatility Models

Model	Distribution	AIC	Persistence	Half-life
GARCH(1,1)	GED	27,531	0.852	4.3 days
EGARCH(1,1)	GED	27,395	0.934	10.2 days

• EGARCH provides best fit (lowest AIC)
• Longer memory: shocks decay over ~10 days

Compound Poisson Process

Parameter	Value	Interpretation
λ	12.64/year	Shock arrival rate
E[J]	0.211	Mean jump size (21% log-move)
VaR (95%)	4.24	95th percentile annual impact
CVaR (95%)	5.01	Expected Shortfall

CPP Out-of-Sample (2022–2025)

Metric	Value
Predicted Shocks	51.8
Actual Shocks	63
Forecast Error	-17.8%
VaR Exceeded?	No ✓

Generated Figures

Figures are organized into subfolders by category:

• vix_data/: VIX time series, visualization, and autocorrelation plots
• volatility_models/: GARCH/EGARCH conditional volatility overlays
• shock_analysis/: Shock detection, counts, magnitudes, and interarrival analysis
• cpp_model/: Compound Poisson Process figures (paths, VaR, regime analysis, forecasts)
• model_evaluation/: Model diagnostics (Q-Q plots, PIT, ACF, comparisons)

Key figures:

• vix_data/vix_series.png - VIX time series with shock markers
• model_evaluation/news_impact.png - EGARCH asymmetric response
• model_evaluation/qq.png - Q-Q plot (GED fit validation)
• cpp_model/jump_distribution.png - Pareto fit to shock magnitudes
• cpp_model/cpp_paths.png - Monte Carlo simulated paths
• cpp_model/cpp_var.png - VaR/CVaR distribution
• cpp_model/cpp_regime.png - Regime-specific CPP parameters
• cpp_model/cpp_forecast.png - Out-of-sample evaluation

References

• Bollerslev, T. (1986). Generalized Autoregressive Conditional Heteroskedasticity. J. Econometrics.
• Nelson, D. B. (1991). Conditional Heteroskedasticity in Asset Returns. Econometrica.
• Cont, R., & Tankov, P. (2004). Financial Modelling with Jump Processes. Chapman & Hall/CRC.
• McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative Risk Management. Princeton University Press.