Volatility Forecasting Benchmarks: GARCH, HAR, and ML on Equity Indices

1. Target Definition: Realised Volatility

For an index with daily returns r_t, define the 1-day realised volatility target as:

σ_{t+1}^{real} = √(Σ_{i=0}^{h-1} r_{t+1+i}^2)

for horizon h (often 1 or 5 days). Alternatively, use high-frequency-based realised measures where available.

2. Model Classes

2.1 GARCH Models

Standard GARCH(1,1):

r_t = σ_t ε_t, ε_t ~ N(0,1)
σ_t² = ω + α r_{t-1}² + β σ_{t-1}²

Variants: EGARCH, GJR-GARCH (asymmetry), long-memory variants, different error distributions.

2.2 HAR Model

HAR-RV model represents volatility as:

RV_{t+1} = β₀ + β₁ RV_t^{(d)} + β₂ RV_t^{(w)} + β₃ RV_t^{(m)} + ε_t

where RV_t^{(d)} is daily, RV_t^{(w)} is weekly average, RV_t^{(m)} is monthly average. This captures multi-horizon volatility dynamics in a simple linear framework.

2.3 ML Models

Input features may include:

lagged realised vol, RV d/w/m
lagged returns
volatility-of-volatility measures
macro proxies, VIX, term structure of implied vol (if available)

Models: Ridge / ElasticNet, Random Forest, Gradient Boosted Trees (XGBoost/LightGBM), MLP / small LSTM on volatility features.

Targets: next-day or next-5-day realised volatility (or log-vol to stabilise).

3. Benchmark Design

To avoid typical pitfalls:

Strict time-based splits
Rolling or expanding re-estimation (walk-forward)
Non-overlapping evaluation windows where feasible
Multi-criteria evaluation: RMSE / MAE on volatility, R² on log-vol, accuracy of volatility buckets, performance of strategies that use the forecasts

4. From Vol Forecast to Trading Performance

One useful research question: "Does a better volatility forecast translate into better risk-adjusted returns when used for position sizing?"

Example: vol targeting strategy w_t = τ / σ̂_t where τ is target risk, σ̂_t is forecast vol.

We can compare performance using GARCH-based vol, HAR vol, and ML vol. This closes the loop from forecasting metrics to trading metrics.

5. How volarixs Hosts Volatility Benchmarks

In volarixs:

the target is defined as realised volatility series per asset
model templates exist for GARCH-type models, HAR, and ML regression templates
evaluation pipelines compute both pure forecast metrics (RMSE, MAE, R², calibration plots) and strategy metrics when the forecasts are used for vol targeting
results are stored as benchmark runs, comparable across assets, horizons, and models

The goal is to make "serious volatility forecasting research" as easy to run as a basic backtest.

Volatility

GARCH

HAR

Benchmarks

Forecasting