Stress Testing Volatility Models (GARCH vs. Stochastic Volatility) under Flash Crash Scenarios

Table of Contents

A. Executive Summary / Abstract

Accurate volatility modeling is paramount for risk management, option pricing, and dynamic margining during market panics. This paper presents a rigorous stress-testing framework comparing traditional Generalized Autoregressive Conditional Heteroskedasticity (GARCH) models against Stochastic Volatility (SV) models under severe flash crash scenarios. We examine conditional variance persistence, leverage effects, parameter stability, and out-of-sample forecasting degradation when models are subjected to discontinuous jump shocks and extreme liquidity vacuums.

B. Technical Introduction

Volatility models form the backbone of quantitative risk architecture, determining Value-at-Risk (VaR), Expected Shortfall (ES), and initial margin requirements. However, standard parametric models are calibrated under normal market conditions and frequently fail during black swan events. GARCH models treat volatility as a deterministic function of past squared residuals and past conditional variances, making them highly sensitive to sudden outlier shocks that inflate persistence parameters ($\alpha + \beta \to 1$). Conversely, Stochastic Volatility models treat variance as an unobservable latent stochastic process driven by its own independent Brownian motion. This paper evaluates how both frameworks behave when stressed under synthetic flash crash scenarios characterized by instant price gaps and explosive variance spikes.

C. Formal Mathematical Framework (LaTeX)

GARCH(1,1) vs. Stochastic Volatility (SV) Specifications

In a standard GARCH(1,1) model, the conditional variance $\sigma_t^2$ is deterministic given the information set up to time $t-1$:

$$\sigma_t^2 = \omega + \alpha \epsilon_{t-1}^2 + \beta \sigma_{t-1}^2$$

Where $\epsilon_t = \sigma_t z_t$ and $z_t \sim i.i.d. N(0,1)$. To capture heavy tails and asymmetry, standardized Student-t or GJR-GARCH extensions are applied.

In contrast, the baseline Stochastic Volatility (SV) model introduces a secondary error term for the latent log-variance process $h_t = \ln(\sigma_t^2)$:

$$r_t = \exp\left(\frac{h_t}{2}\right) z_t$$ $$h_{t} = \mu + \phi (h_{t-1} – \mu) + \sigma_v \eta_t$$

Where $\eta_t \sim i.i.d. N(0,1)$ and $\text{Cov}(z_t, \eta_t) = \rho$, capturing the leverage effect where negative returns correlate with subsequent increases in volatility.

D. Step-by-Step Volatility Model Stress Testing Methodology

1. Model Calibration & Parameter Estimation

Maximum Likelihood Estimation (MLE): Fitting GARCH parameters using quasi-maximum likelihood estimation with robust standard errors.
Markov Chain Monte Carlo (MCMC): Estimating latent Stochastic Volatility parameters via Bayesian MCMC (Particle Filtering) due to the unobservable nature of volatility.
Innovation Distribution Tuning: Comparing Gaussian innovations against heavy-tailed Student-t and Generalized Error Distributions (GED) to capture kurtosis.

2. Flash Crash Scenario Design & Shock Injection

Synthetic Price Dislocation: Injecting instantaneous multi-standard deviation price drops ($-10\%$ to $-20\%$ within 5 minutes) into historical return series.
Variance Burst Simulation: Simulating exponential variance jumps where $\sigma_t$ increases by $400\%$ to $800\%$ over rolling intraday windows.
Persistence Shock Analysis: Measuring the half-life of volatility shocks to determine how long GARCH vs. SV models take to mean-revert following a crash.

3. Forecasting Performance & Loss Metrics

Evaluating out-of-sample volatility forecasts using robust statistical loss functions:

Mean Squared Error (MSE) & QLIKE Loss: Assessing absolute and relative forecasting error against proxy volatility measures (realized volatility from high-frequency intraday data).
VaR Breach Ratio: Testing whether the models under-forecast risk, leading to excessive exception ratios during stress backtesting.

4. Model Resilience & Stabilization Controls

Implementing parameter regularization, volatility capping, and hybrid jump-diffusion extensions to prevent filter divergence and explosions in conditional variance during prolonged market panics.

E. Visual Component 1: Volatility Stress Testing & Model Pipeline

Volatility Stress Testing & Model Comparison Pipeline

Deterministic Workflow from Parameter Estimation to Flash Crash Stress Simulation

Stage 01 — Calibration

GARCH vs. SV Estimation

Fits deterministic GARCH via MLE and latent Stochastic Volatility via MCMC.

Model Fit

Stage 02 — Shock Injection

Flash Crash Dislocation

Injects instant price gaps and exponential variance bursts into backtest series.

Shock Test

Stage 03 — Evaluation

QLike Loss & Persistence Analysis

Measures forecasting error and shock half-life decay rates.

Loss Metric

Stage 04 — Stabilization

Regularization & Jump Guards

Deploys robust filter controls to prevent variance explosion during panics.

Risk Guardrail

Volatility Invariance Rule: Volatility stress testing models must account for latent stochastic shocks and leverage effects to prevent severe underestimation of tail risk during flash crashes.

F. Visual Component 2: GARCH vs. Stochastic Volatility Stress Benchmark Matrix

GARCH vs. Stochastic Volatility Stress Benchmark Matrix

Comparative Matrix of Volatility Specifications, Estimation Complexity, and Crash Resilience

Model Dimension	GARCH(1,1) Framework	Stochastic Volatility (SV) Framework	Critical Failure Risk	Audit Mitigation Protocol
Variance Specification	Deterministic function of past residuals	Latent stochastic process with own error	Overly reactive to single outliers	Implement robust Student-t innovations.
Estimation Complexity	Fast Maximum Likelihood (MLE)	Computationally intensive MCMC / Particle Filter	Filter divergence in live stream	Optimized C++ MCMC routines.
Flash Crash Shock Response	Spikes instantly, slow decay ($\alpha+\beta \to 1$)	Smooth latent adaptation with leverage	Explosive variance persistence	Jump-diffusion SV extensions.
Leverage Effect Handling	Requires asymmetric extensions (GJR/EGARCH)	Built-in via error correlation $\rho$	Ignoring asymmetric panic skew	Enforce nonzero error covariance.
Out-of-Sample QLIKE Loss	Higher error during structural breaks	Superior fit across regime shifts	Systemic VaR exception breaches	Multi-model ensemble forecasting.

Source: AuditQuant Volatility Research Standards.

G. Volatility Model Audit Diagnostic Matrix

Metric / Dimension	Benchmark Standard	Warning Threshold	Critical Failure State	Audit Action Protocol
GARCH Persistence Sum ($\alpha+\beta$)	≤ 0.985	> 0.995	≥ 1.0 (Integrated GARCH)	Recalibrate with component GARCH or switch to stochastic volatility.
VaR Exception Rate (99% confidence)	≤ 1.0% of observations	1.5% to 3.0% exceptions	> 3.0% exceptions (Kupiec fail)	Increase initial margin multipliers and adopt heavy-tailed distributions.
QLike Forecast Error	Minimized relative to benchmark	Elevated error during shifts	Severe divergence from realized vol	Incorporate high-frequency realized volatility into model calibration.
Model Calibration Frequency	Daily rolling window	Monthly recalibration	Static parameters in crisis	Automate daily particle filter updating for SV models.

H. Institutional Monetization & Testing Lab Access

TStress test your volatility models and evaluate GARCH vs. Stochastic Volatility performance under simulated flash crash scenarios using our specialized environments:

Volatility Testing Lab: Access high-performance simulation portals at lab.auditquant.com to run MCMC particle filters and flash crash shock tests.

I. Academic & Institutional References

Bollerslev, T. (1986). Generalized Autoregressive Conditional Heteroskedasticity. Journal of Econometrics, 31(3), 307-327.
Taylor, S. J. (1986). Modelling Financial Time Series. John Wiley & Sons.
Ghysels, E., Harvey, A. C., & Renault, E. (1996). Stochastic Volatility. Handbook of Statistics, 14, 119-191.
Patton, A. J. (2011). Volatility Forecast Evaluation. Volatility and Time Series Econometrics, Oxford University Press.

J. Compliance & Legal Disclaimer (CFTC Rule 4.41 / NFA)

CFTC RULE 4.41 / NFA DISCLAIMER: Hypothetical or simulated performance results entail inherent limitations. Unlike actual performance records, simulated results do not represent actual trading and may not be impacted by brokerage commissions, execution slippage, and liquidity constraints. No representation is being made that any account will or is likely to achieve profits or losses similar to those shown. AuditQuant.com is an independent institutional audit research portal and does not provide regulated financial or investment advisory services.