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Project Summary

HKUST IEDA4000E — Statistical Modelling for Financial Engineering

Quantifying VIX Tail Risk: Volatility Clustering and Jump Processes

1. Data Overview

  • Source: Yahoo Finance VIX closes, business-day frequency
  • Period: 2010-01-05 to 2025-11-27 (4,148 observations)
  • Pre-processing: Forward-filled gaps, 0.1% winsorization, log-level and daily log-change features
  • Train/Test Split: 75% training (2010–2021), 25% test (2022–2025)

2. Volatility Models

Model Comparison

Model Distribution AIC BIC Persistence Half-life
GARCH(1,1) GED 27,531 27,569 0.852 4.3 days
EGARCH(1,1) GED 27,395 27,439 0.934 10.2 days

Key Findings

  • Best Model: EGARCH provides lowest AIC/BIC
  • Longer Memory: EGARCH captures ~10-day half-life for volatility shocks (vs ~4 days for GARCH)
  • Leverage Effect: EGARCH γ term confirms asymmetric response (negative shocks increase volatility more)
  • Distribution: GED chosen automatically via PIT KS-statistic (captures fat tails)

3. Compound Poisson Process

Model Formulation

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

Models both shock timing (Poisson arrivals) AND magnitude (jump sizes).

Jump Distribution Selection

Distribution AIC KS Statistic KS p-value Selected
Exponential 412.3 0.142 0.003 No
Gamma 385.7 0.089 0.085 No
Lognormal 391.2 0.098 0.052 No
Pareto 378.4 0.061 0.42 Yes
Weibull 388.9 0.095 0.068 No

Fitted Parameters (Full Sample)

Parameter Value Interpretation
λ (arrival rate) 12.64/year Expected shocks per year
α (Pareto shape) 2.50 Tail index
x_min (Pareto scale) 0.127 Minimum shock size
E[J] (mean jump) 0.211 21.1% average log-move
Std[J] 0.189 Jump size volatility
E[S] = λ × E[J] 2.67/year Expected annual impact
VaR (95%) 4.24 95th percentile annual impact
CVaR (95%) 5.01 Expected Shortfall

4. Regime Analysis

CPP Parameters by Regime

Regime Period λ/Year E[J] E[S]/Year VaR 95% CVaR 95%
Pre-Crisis 2010–2019 12.3 0.209 2.57 4.15 4.92
COVID 2020 17.3 0.262 4.53 7.44 9.65
Post-COVID 2021–2023 11.6 0.188 2.19 3.44 3.85
Recent 2024–2025 13.6 0.216 2.95 4.70 5.63

Key Finding: COVID period shows:

  • 41% higher arrival rate (λ = 17.3 vs 12.3)
  • 25% larger mean jumps (E[J] = 0.262 vs 0.209)
  • 76% higher expected annual impact (E[S] = 4.53 vs 2.57)
  • Nearly double VaR (7.44 vs 4.15)

5. CPP Out-of-Sample Evaluation

Train/Test Split

  • Training: 3,111 observations (2010–2021)
  • Test: 1,037 observations (2022–2025)

Results

Metric Value Notes
Trained Parameters
λ (trained) 0.050/day 12.6 shocks/year
Jump Distribution Pareto α = 2.50
E[J] (trained) 0.211 Mean jump size
Test Period Results
Actual Shocks 63 Observed
Predicted Shocks 51.8 λ × 1,036 days
Shock Count Error -17.8% Underforecast
Actual Impact 13.4 Cumulative
Predicted Impact 10.9 λ × E[J] × T
Impact Error -18.5% Underforecast
Risk Validation
Scaled VaR 95% 15.2 For test period
VaR Exceeded? No Actual < VaR ✓

Interpretation:

  • ~18% underforecast is acceptable given unusual test period (2022 Fed rate hikes, 2024 volatility spike)
  • VaR bounds not exceeded → risk measure is appropriately conservative
  • Actual outcome at 72nd percentile of simulated distribution → model is well-calibrated

6. Generated Figures

Figures are organized into subfolders by category for better organization:

  • 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:

Figure Location Description
vix_series.png vix_data/ VIX time series with shock markers
news_impact.png model_evaluation/ EGARCH asymmetric response curve
qq.png model_evaluation/ Q-Q plot (GED captures fat tails)
jump_distribution.png cpp_model/ Pareto fit to shock magnitudes
cpp_paths.png cpp_model/ Monte Carlo simulated paths
cpp_var.png cpp_model/ VaR/CVaR distribution
cpp_regime.png cpp_model/ Regime-specific CPP parameters
cpp_forecast.png cpp_model/ Out-of-sample evaluation
regime_comparison.png model_evaluation/ Shock rates by regime

7. Key Conclusions

  1. EGARCH provides best in-sample fit; captures ~10-day half-life for volatility shocks
  2. Leverage effect confirmed: negative shocks increase volatility more than positive
  3. Compound Poisson Process with Pareto jumps quantifies aggregate shock risk:
    • VaR 95% = 4.24/year
    • CVaR 95% = 5.01/year
  4. COVID regime shows 76% higher expected annual impact than pre-crisis baseline
  5. CPP out-of-sample: ~18% forecast error; VaR bounds respected; well-calibrated predictions

8. Future Directions

  1. Hawkes processes for self-exciting shock arrivals (clustering)
  2. Hybrid models combining GARCH with jump processes
  3. High-frequency data for improved jump detection
  4. Multivariate extensions for cross-asset contagion
  5. Machine learning for regime detection