volatility-modeling Checked

### 5. EWMA (Exponentially Weighted Moving Average)

RiskMetrics approach — no parameters to estimate beyond λ.

lam = 0.94  # RiskMetrics default for daily
ewma_var = np.zeros(len(returns))
ewma_var[0] = returns[0] ** 2
for t in range(1, len(returns)):
    ewma_var[t] = lam * ewma_var[t - 1] + (1 - lam) * returns[t - 1] ** 2
vol_ewma = np.sqrt(ewma_var) * np.sqrt(365)

• λ = 0.94 for daily data (RiskMetrics).
• λ = 0.97 for weekly data.
• Higher λ → smoother, slower reaction to new information.

### 6. GARCH(1,1)

The workhorse autoregressive volatility model. Captures volatility clustering.

σ²_t = ω + α · r²_{t-1} + β · σ²_{t-1}

• **ω**: long-run variance weight.
• **α**: reaction to recent shock (typically 0.05–0.15 for crypto).
• **β**: persistence (typically 0.80–0.90 for crypto).
• **α + β < 1**: stationarity constraint.
• **Long-run variance**: ω / (1 − α − β).

Estimated via maximum likelihood. See references/estimators.md for details.

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Volatility Cones

Volatility cones show the percentile distribution of realized volatility at

different lookback windows, revealing whether current vol is historically

high or low.

### Construction

1. Get 1+ years of daily data.

2. For each lookback window (5, 10, 20, 60, 120 days):

• Compute rolling realized volatility.
• Extract percentiles: 5th, 25th, 50th, 75th, 95th.

3. Plot percentiles vs window length — the "cone" shape.

4. Overlay current realized vol at each window.

### Interpretation

• **Current vol > 75th percentile**: historically elevated — expect mean reversion.
• **Current vol < 25th percentile**: historically compressed — expect expansion.
• **Cone narrowing at longer windows**: vol mean-reverts over longer horizons.

See references/volatility_cones.md for full methodology and worked examples.

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Crypto Volatility Characteristics

Crypto vol differs from traditional assets in important ways:

| Characteristic | Detail |

|---|---|

| **Level** | 50–150% annualized is typical; TradFi equities are 15–25% |

| **Clustering** | Strong — high-vol days cluster together |

| **Weekday patterns** | Weekend vol often lower but weekend gaps can be large |

| **Volume correlation** | Vol and volume are positively correlated |

| **Regime dependence** | Bull market vol ≠ bear market vol; ranges are different |

| **Mean reversion** | Vol mean-reverts more reliably than price |

| **Tail risk** | Fat tails — more extreme moves than normal distribution predicts |

### Regime Classification by Volatility

| Regime | Annualized Vol Range | Characteristics |

|---|---|---|

| Low vol | < 40% | Range-bound, mean reversion works |

| Normal vol | 40–80% | Trending possible, balanced strategies |

| High vol | 80–120% | Strong trends or sharp reversals |

| Crisis vol | > 120% | Liquidation cascades, reduced position size |

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Volatility Forecasting

### EWMA Forecast

Simple and effective. The current EWMA variance estimate *is* the 1-step

forecast. Multi-step forecasts are flat (same as 1-step).

### GARCH Forecast

GARCH produces a term structure of variance forecasts:

σ²_{t+h} = V_L + (α + β)^h · (σ²_t − V_L)

Where V_L = ω / (1 − α − β) is the long-run variance.

• Short-horizon forecasts reflect current conditions.
• Long-horizon forecasts converge to long-run variance.
• The speed of convergence depends on α + β (persistence).

See scripts/vol_forecast.py for a working implementation.

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Practical Applications

### Position Sizing with Volatility

# Vol-target position sizing
target_vol = 0.02  # 2% daily portfolio vol target
current_vol = 0.05  # 5% daily asset vol (annualized ~95%)
weight = target_vol / current_vol  # = 0.40 → 40% allocation

See the position-sizing skill for complete integration.

### ATR-Based Stop Placement

atr_14 = talib.ATR(highs, lows, closes, timeperiod=14)
stop_distance = 2.0 * atr_14[-1]  # 2x ATR stop
stop_price = entry_price - stop_distance  # for longs

### Vol-Regime Strategy Selection

vol_percentile = current_vol_percentile(token, window=30)
if vol_percentile < 25:
    strategy = "mean_reversion"
elif vol_percentile > 75:
    strategy = "momentum_breakout"
else:
    strategy = "balanced"

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Files

### References

| File | Description |

|---|---|

| references/estimators.md | Full derivations and details for all volatility estimators |

| references/volatility_cones.md | Cone construction methodology and interpretation guide |

### Scripts

| File | Description |

|---|---|

| scripts/estimate_volatility.py | Multi-estimator volatility computation with cone analysis |

| scripts/vol_forecast.py | EWMA and GARCH forecasting with term structure output |

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Related Skills

• **regime-detection** — Classify market regimes using volatility as a key input.
• **position-sizing** — Scale positions inversely with volatility.
• **risk-management** — Portfolio-level vol targeting and risk budgets.
• **pandas-ta** — ATR and Bollinger Bands are volatility-based indicators.
• **custom-indicators** — Build crypto-specific volatility indicators.

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Dependencies