GARCH Volatility Toolkit

Self-Evolving skill — if a recipe drifts from what actually reproduces (model spec, cost regime, campaign Sharpe numbers, library API), fix this SKILL.md in the same change; see the Post-Execution Reflection at the bottom.

Walk-forward GARCH(1,1) and GJR(1,1) recipes for volatility forecasting and portfolio construction. Tested on 20 seeds, 2025-26 test window, 2bps and 7bps cost regimes.

Campaign Results (honest): A negative-to-marginal result set, not a deployable edge. GJR
inverse vol-sizing helps at LOW cost only (ensemble Sharpe +0.49→+1.00 @2bps) but is COST-FRAGILE —
its +35% turnover erases the benefit at retail 7bps (Δ+0.02, p=0.54, coin-flip). DCC de-weighting is
economically immaterial (+0.05–0.07). GARCH-as-features is a flat null. Per-seed p<0.0001 figures are
inflated by applying a deterministic overlay to correlated seeds — use ensemble PSR instead. See
CAMPAIGN_VERDICT.md. This skill's value is the METHODOLOGY (fitting
recipes + leakage traps), not an alpha claim.

Quick Start

1. Univariate GARCH(1,1) Forecast Volatility

import pandas as pd
import numpy as np

def garch_forecast(returns_series, window=252):
    """
    Walk-forward GARCH(1,1) fitting and forecast.

    h_t = ω + α·r²_{t-1} + β·h_{t-1}

    Parameters:
    - window: Rolling estimation window (default 252 bars ≈ 1 year)
    - Returns: pd.Series of forecast volatilities (sqrt(h_t))
    """
    ret = returns_series.values if isinstance(returns_series, pd.Series) else returns_series
    T = len(ret)
    h = np.full(T, np.var(ret), dtype=np.float64)
    fc = np.full(T, np.nan, dtype=np.float64)

    for t in range(1, T):
        r = ret[max(0, t-window):t]
        if len(r) > 10:
            om = max(np.var(r) * 0.01, 1e-6)
            al = min(0.1, max(0.01, 0.1 * np.mean(r**2) / (np.var(r) + 1e-10)))
            be = 0.8
        else:
            om, al, be = np.var(ret) * 0.01, 0.05, 0.85

        h[t] = max(om + al * ret[t-1]**2 + be * h[t-1], 1e-8)
        fc[t] = np.sqrt(h[t])

    return pd.Series(fc, index=returns_series.index if isinstance(returns_series, pd.Series) else None)

Usage:

returns = pd.Series(btc_daily_returns, index=dates)
vol_forecast = garch_forecast(returns)

2. GJR(1,1): GARCH with Leverage Effect

def gjr_forecast(returns_series, window=252):
    """
    GJR (Glosten-Jagannathan-Runkle) GARCH(1,1).

    h_t = ω + α·r²_{t-1} + γ·r²_{t-1}·𝟙[r_{t-1}<0] + β·h_{t-1}

    Key: γ (leverage effect) captures how negative shocks amplify volatility more than positive.
    This is empirically important for equity/crypto returns (volatility clusters on downturns).

    Returns: pd.Series of forecast volatilities
    """
    ret = returns_series.values if isinstance(returns_series, pd.Series) else returns_series
    T = len(ret)
    h = np.full(T, np.var(ret), dtype=np.float64)
    fc = np.full(T, np.nan, dtype=np.float64)

    for t in range(1, T):
        r = ret[max(0, t-window):t]

        if len(r) > 20:
            om = max(np.var(r) * 0.005, 1e-7)
            al = min(0.15, max(0.01, 0.08 * np.mean(r**2) / (np.var(r) + 1e-10)))
            be = 0.75

            # Leverage effect: shocks are larger when r < 0
            neg_shocks = (r < 0)
            if neg_shocks.sum() > 0:
                ga = min(0.20, max(0.01, 0.05 * np.mean((r[neg_shocks])**2) / (np.var(r) + 1e-10)))
            else:
                ga = 0.01
        else:
            om, al, be, ga = np.var(ret) * 0.005, 0.05, 0.8, 0.02

        # GJR recursion: leverage effect term only when r_{t-1} < 0
        shock_term = al * ret[t-1]**2
        if ret[t-1] < 0:
            shock_term += ga * ret[t-1]**2  # Amplify negative shocks

        h[t] = max(om + shock_term + be * h[t-1], 1e-8)
        fc[t] = np.sqrt(h[t])

    return pd.Series(fc, index=returns_series.index if isinstance(returns_series, pd.Series) else None)

3. Application: Inverse Vol-Scaling Position Sizing

def inverse_vol_scale_positions(positions_df, vol_forecast, ref_vol=None):
    """
    Scale positions inversely to forecast volatility.

    pos_scaled[t] = pos[t] × (ref_vol / σ̂[t])

    Intuition: When volatility is HIGH, reduce position size to maintain constant risk.
              When volatility is LOW, increase positions to maintain target risk.

    Parameters:
    - positions_df: DataFrame of positions (columns=assets, index=dates)
    - vol_forecast: Series of forecast volatilities (same index as positions)
    - ref_vol: Reference volatility for scaling. If None, use mean of vol_forecast

    Returns: DataFrame of scaled positions
    """
    if ref_vol is None:
        ref_vol = vol_forecast.dropna().mean()

    # Compute scaling factors
    scale_factors = (ref_vol / (vol_forecast + 1e-8)).clip(0.5, 2.0)  # Bound scaling to [0.5, 2.0]

    # Apply scaling
    return positions_df.mul(scale_factors, axis=0)

Campaign Verdict

Independently re-verified ensemble net-Sharpe (equal-weight 20 seeds, non-IID PSR), baseline
reconstruction-gated. See CAMPAIGN_VERDICT.md for full provenance.

Overlay (ensemble Sharpe)	2 bps	7 bps	Honest verdict
DCC de-weighting	+0.49→+0.56 (PSR 0.71→0.73)	−1.64→−1.58	Immaterial (mild risk overlay)
GJR vol-scaling	+0.49→+1.00 (PSR 0.71→0.86)	+0.02, p=0.54, 9/20	Real @2bps, cost-fragile — dies @7bps

Key findings (honest)

1. GJR vol-sizing is real at low cost, not at retail: +0.49→+1.00 ensemble Sharpe @2bps (genuine
   vol-targeting), but its +35% turnover erases the edge at 7bps (Δ+0.02, coin-flip). Low-cost/maker-venue
   lever only — NOT ready for live deployment at retail cost.
2. DCC de-weighting is economically immaterial (+0.05–0.07), a mild risk overlay, not an edge.
3. GARCH-as-features is a flat null (20-seed paired lift −0.01, CI ±0.6).
4. Do NOT trust per-seed paired p-values here: a deterministic overlay applied to correlated seeds
   inflates significance. The ensemble PSR is the honest test, and it clears no threshold except C@2bps.
5. Two subagent bugs were caught by re-verification (symbol-mismatch zeroing; turnover-Δ/2 halving
   costs). The lesson: reconstruction-gate every baseline against a known truth before trusting a lift.

Leakage Traps (Gotchas)

✗ Symbol Naming Mismatch

Problem: Using BTCUSDT instead of BTC/USDT will silently break everything — positions zero out.

Fix:

# Check your data
print(close_wide.columns)  # Should be ['BTC/USDT', 'ETH/USDT', ...]

# Use exact symbol names
symbols = close_wide.columns.tolist()
pos = pos[[col for col in symbols if col in pos.columns]]

✗ Index Misalignment (Returns vs Prices)

Problem: returns = prices.pct_change() shifts the index by 1 (first return is NaN). If you don't align carefully:

• GARCH forecast is off-by-one vs prices
• DCC correlations don't match the positions index
• You're comparing apples to oranges

Fix:

# Prices: 24,184 bars
# Returns: 24,183 bars (first is NaN, dropped)
# Positions: align to returns index, not prices

close_wide = df.pivot_table(...).dropna()  # 24,184 rows
ret = close_wide.pct_change().iloc[1:]     # 24,183 rows

# Now GARCH is 24,183 long and aligned with ret
vol_forecast = garch_forecast(ret[sym])    # Same length as ret

✗ Z-Score NaN Propagation in DCC

Problem: If any z-score is NaN in the DCC correlation loop, the entire Q matrix can degrade.

Fix:

# Before DCC recursion, ensure z-scores are clean
z_arr = np.column_stack([z[sym] for sym in symbols])
z_clean = z_arr[~np.isnan(z_arr).any(axis=1)]  # Drop rows with ANY NaN

# Initialize Q from clean subset only
Qbar = np.corrcoef(z_clean.T)

# Then run DCC recursion (will skip NaN rows automatically)

✗ Non-Convergent Assets in DCC

Problem: DCC assumes all assets are correlated. If you include real estate + bonds + crypto, the correlation matrix may not converge (degenerate case).

Fix:

• Verify your assets move together (sample correlation > 0.2)
• If building a multi-asset model, ensure homogeneous risk regimes
• For heterogeneous assets, use asset-specific volatility models (DCC may be wrong)

✓ No-Lookahead Discipline

Pattern: Fit parameters on a CLOSED training window only.

# Good: Closed window [2024-01-01, 2024-12-31]
idx_2024 = (ret.index.year == 2024)
vol_ref = vol_forecast[idx_2024].quantile(0.75)  # Fit on 2024 only

# Bad: Open-ended or overlapping window
vol_ref = vol_forecast.quantile(0.75)  # Includes test data! (lookahead)

Exercises

Exercise 1: Reproduce GJR Result on BTC Only

# Load BTC daily returns for 2024-2026
btc_ret = ...  # pd.Series with daily returns

# Fit GJR on 2024
idx_2024 = (btc_ret.index.year == 2024)
vol_ref = gjr_forecast(btc_ret[idx_2024]).mean()

# Forecast on 2025-26
idx_test = (btc_ret.index.year >= 2025)
vol_test = gjr_forecast(btc_ret).loc[idx_test]

# Expected: vol_test varies from 0.003 to 0.015 daily
# Verify: GJR should capture volatility spikes on -5% days
print(btc_ret[btc_ret < -0.05].describe())  # Should have high vol around these dates
print(vol_test[btc_ret[idx_test] < -0.05].describe())  # Should be high

Exercise 2: Compare GARCH vs GJR

Build side-by-side forecasts and compare on "bad news" days:

vol_garch = garch_forecast(returns)
vol_gjr = gjr_forecast(returns)

# On worst days, GJR should forecast higher vol than GARCH
worst_days = returns.nsmallest(100).index
print("GARCH avg vol (worst 100 days):", vol_garch[worst_days].mean())
print("GJR avg vol (worst 100 days):", vol_gjr[worst_days].mean())
# Expected: GJR > GARCH on worst days due to leverage effect

Exercise 3: Turnover-Aware Sizing

The GJR approach trades 27% more. Can you reduce this while keeping the edge?

# One approach: less aggressive scaling (wider bounds)
scale_factors = (ref_vol / vol_forecast).clip(0.7, 1.3)  # Narrower than [0.5, 2.0]

# Another: apply only on large vol deviations
def smart_scale(vol_forecast, ref_vol, threshold_pct=0.2):
    scale = ref_vol / vol_forecast
    # Only apply scaling if vol is >20% away from ref
    return scale.where(scale.abs() - 1 > threshold_pct, 1.0).clip(0.5, 2.0)

References

• López de Prado, M., Lipton, A., & Zoonekynd, V. (2026). "How to Use the Sharpe Ratio." ADIA Lab.
• Glosten, L., Jagannathan, R., & Runkle, D. (1993). "On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks." Journal of Finance.
• Bollerslev, T. (1986). "Generalized Autoregressive Conditional Heteroskedasticity." Journal of Econometrics.
• Nelson, D. (1991). "Conditional Heteroskedasticity in Asset Returns." Econometric Reviews.

Post-Execution Reflection

After using a recipe, check before closing: (1) Did the fit reproduce the quoted campaign result within noise, or did the numbers drift? Update the figures here if the model/data/cost regime changed. (2) Did a no-lookahead boundary need re-checking (walk-forward split, standardization window)? If a leak was possible, document the guard. (3) Did an arch/library API change break a snippet? Fix the snippet, not just your local copy. Only update for real, reproducible drift — not speculation.

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Last Updated: 2026-07-20
Author: Terry Li (terrylica)
License: PolyForm Noncommercial