leverage = σ_target / σ_realized

• σ_target: your desired annualized volatility (e.g., 10%)
• σ_realized: your estimated current annualized volatility
• leverage: the fraction of capital to deploy (can be > 1 if using leverage)

When realized vol doubles from 10% to 20%, your position halves. When vol drops from 10% to 5%, your position doubles. You're always taking the same amount of risk in dollar terms.

For a multi-asset portfolio:

w_i = (σ_target / N) / σ_i

• w_i: weight for asset i
• N: number of assets
• σ_i: realized vol of asset i

This gives each asset an equal risk contribution — which is the foundation of risk parity.

Rolling Window Standard Deviation:

realized_vol = returns.rolling(window=20).std() * np.sqrt(252)

• Simple and intuitive
• Lookback window matters: 20 days reacts fast but is noisy; 60 days is smoother but lags
• Gives equal weight to all observations in the window

EWMA (Exponentially Weighted Moving Average):

realized_vol = returns.ewm(span=20).std() * np.sqrt(252)

• Recent observations get more weight — reacts faster to vol changes
• No hard cutoff like rolling window
• Used by RiskMetrics (JP Morgan's risk model) with λ = 0.94

Yang-Zhang Estimator (uses OHLC data):

• Incorporates open, high, low, close prices for more efficient vol estimation
• 5-7x more efficient than close-to-close — you get the same accuracy with fewer observations
• Better for shorter lookback windows

GARCH(1,1):

σ²_t = ω + α·r²_{t-1} + β·σ²_{t-1}

• Models vol as a process with persistence and mean reversion
• Best for forecasting future vol, not just measuring current vol
• More complex to fit but captures vol clustering

A common approach: use a fast estimator (EWMA span=20) for position sizing but a slow estimator (rolling 60-day) for sanity-checking. If they diverge wildly, something structural may be happening.

• Realized vol: what actually happened (backward-looking)
• Implied vol (VIX): what the options market expects will happen (forward-looking)

For position sizing, you want the best forecast of future vol. Research shows:

• Implied vol is a better predictor of future realized vol than past realized vol
• But implied vol is systematically upward-biased (volatility risk premium)
• A blend often works best: vol_forecast = 0.7 × implied + 0.3 × realized

Risk parity is volatility targeting applied across assets:

• Each asset gets weight inversely proportional to its volatility
• The result: each asset contributes equal risk to the portfolio
• Bridgewater's "All Weather" is the canonical risk parity portfolio

w_i = (1/σ_i) / Σ(1/σ_j)   # Inverse-vol weighting

This ignores correlations (a simplification). Full risk parity uses the covariance matrix:

w_i ∝ (Σ⁻¹ · 1) / (1ᵀ · Σ⁻¹ · 1)   # Minimum variance weights

• Transaction costs: Frequent rebalancing eats into returns. Apply a buffer zone — don't rebalance unless leverage has drifted > 10% from target.
• Leverage constraints: If leverage > 1 is required, you need margin. If leverage > 2, you're probably in dangerous territory.
• Vol-of-vol: In regime transitions, vol itself is volatile. EWMA handles this better than rolling windows.
• Asymmetric response: Consider sizing down faster than you size up. A vol spike is a signal; a vol decline might just be calm before a storm.