GARCH Volatility Modeling & Value-at-Risk

Model the time-varying volatility of financial returns with a GARCH(1,1) process, forecast future volatility, and turn it into a Value-at-Risk (VaR) estimate that is then backtested for calibration. Covers the time-series and risk-management toolkit central to a quant analyst.

⚠️ Academic / personal research project. Not investment advice.

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🧠 Why GARCH

Returns exhibit volatility clustering — calm and turbulent periods come in runs. GARCH(1,1) models the conditional variance as:

sigma2_t = omega + alpha * eps2_{t-1} + beta * sigma2_{t-1}

• alpha (ARCH) — reaction to the latest shock
• beta (GARCH) — persistence of volatility
• alpha + beta — persistence; < 1 means variance mean-reverts
• long-run variance = omega / (1 - alpha - beta)

📊 Example output (series simulated from a known GARCH process)

omega        = 2.60e-06
alpha (ARCH) = 0.0765
beta  (GARCH)= 0.8949
persistence  = 0.9714        (true value 0.98)
long-run vol = 15.16% annualized

VaR backtest (95%):
expected breaches = 5.00%
actual breaches   = 4.87%   (73/1499)   -> well-calibrated

The estimator recovers the true persistence, and the VaR breach rate matches the 95% level — evidence the volatility model is sound.

🛡️ What's done carefully

• Parameter recovery is tested. The model is fit to data simulated from a known GARCH process; tests assert it recovers the true persistence.
• VaR is backtested, not just computed. A Kupiec-style coverage check uses the prior day's volatility (genuinely one-step-ahead), so the calibration test is honest.
• Graceful fallback. Uses the arch package if installed; otherwise a self-contained maximum-likelihood estimator keeps the project runnable.
• Mean-reverting forecasts. Multi-step forecasts decay toward long-run vol.

⚙️ Install & run

pip install -r requirements.txt

# Synthetic series from a true GARCH process (no network)
python examples/run_garch.py

# Real series, e.g. the S&P 500 ETF
python examples/run_garch.py --ticker SPY --start 2015-01-01 --end 2024-12-31

from garch import fit_garch, forecast_volatility
from risk import value_at_risk, var_backtest

fit = fit_garch(returns)
print(fit.summary())

var = value_at_risk(fit.conditional_vol, confidence=0.95)
print(var_backtest(returns, var, confidence=0.95).summary())

✅ Validation

python tests/test_garch.py

Confirms the estimator recovers known persistence, detects volatility clustering, produces a well-calibrated 95% VaR, and yields mean-reverting forecasts.

🗂️ Project layout

src/
  garch.py        # GARCH(1,1) fit (arch + MLE fallback) and vol forecast
  risk.py         # parametric VaR + calibration backtest
  data_loader.py  # Yahoo! Finance + synthetic GARCH simulator
examples/
  run_garch.py    # fit -> forecast -> VaR -> backtest
tests/
  test_garch.py   # parameter recovery + VaR calibration checks

🔭 Roadmap

• EGARCH / GJR-GARCH to capture the leverage effect (asymmetric vol)
• Student-t innovations for fatter tails in VaR
• Expected Shortfall (CVaR) alongside VaR

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Author: Dennis Chen (陳雲皓) · LinkedIn · GitHub