Realized Volatility

Realized volatility is computed by summing the squared log-returns of an asset sampled at high intraday frequencies—typically 5-minute or 1-minute intervals. As the sampling frequency increases toward the continuous limit, the estimator converges to the integrated variance of the underlying price process. This property makes it a consistent estimator of true latent volatility under the assumption of a frictionless semimartingale.

• Formula: RV = Σ r²ᵢ where rᵢ are intraday returns
• Convergence: RV → Integrated Variance as sampling interval → 0
• Common frequencies: 1-min, 5-min, 10-min, 30-min intervals

Sampling at ultra-high frequencies introduces market microstructure noise—spurious price variations caused by bid-ask bounce, discrete price grids, and order flow fragmentation. This noise biases the realized volatility estimator upward. The optimal sampling frequency balances the reduction in discretization error against the amplification of noise contamination.

• Bid-ask bounce: Creates negative autocorrelation in observed returns
• Optimal frequency: Typically 5-30 minutes for liquid equities
• Signature plot: Graphs RV against sampling frequency to identify the noise floor

A signature plot graphs average realized volatility against the sampling interval. In a noise-free world, the plot would be flat. In practice, RV rises sharply at very high frequencies due to microstructure noise, then stabilizes at moderate frequencies. The point where the curve flattens indicates the optimal sampling frequency for that asset.

• Interpretation: Flat region = noise-minimized estimate
• Upward slope at high freq: Indicates noise contamination
• Asset-specific: Optimal frequency varies by liquidity and tick size

Realized volatility can be decomposed into its continuous and jump components using bipower variation. The continuous component captures smooth price diffusion, while the jump component identifies significant discontinuous price moves. This decomposition is critical for risk management and option pricing, as jumps carry different risk premia than diffusive volatility.

• Bipower variation: Robust to jumps, estimates continuous component
• Jump test statistic: Significant when RV exceeds bipower variation
• Jump days: Often coincide with earnings announcements or macro news

Realized kernels are robust estimators that correct for microstructure noise using weighted autocovariances of high-frequency returns. Unlike simple RV, kernel estimators apply a Bartlett, Parzen, or Tukey-Hanning weighting scheme to lagged autocovariances, producing a consistent estimate even when sampling at the highest available frequency.

• Kernel types: Bartlett, Parzen, Cubic, Tukey-Hanning
• Bandwidth selection: Determines how many lags to include
• Advantage: Uses all available tick data without downsampling

The Heterogeneous Autoregressive (HAR) model exploits the long-memory property of realized volatility by regressing future RV on past RV computed over daily, weekly, and monthly horizons. Despite its simplicity, HAR consistently outperforms more complex models in out-of-sample forecasting, capturing the volatility persistence observed across different time scales.

• HAR-RV: Regresses tomorrow's RV on daily, weekly, monthly lags
• HAR-RV-J: Adds jump component for improved crisis-period forecasts
• Long memory: Volatility shocks decay hyperbolically, not exponentially

The foundational data structure for computing realized volatility. A LOB is an electronic record of all outstanding buy and sell orders for a specific asset, organized by price level and updated in real-time.

• Bid Side: Queue of buy orders ranked from highest to lowest price.
• Ask Side: Queue of sell orders ranked from lowest to highest price.
• Depth: The cumulative volume available at each price level.
• Realized volatility is calculated from the mid-price derived from the best bid and ask, making LOB data the raw input for the sum of squared returns.

The high-frequency random variation in asset prices caused by operational frictions, not fundamental value changes. This noise is the primary obstacle to accurate realized volatility estimation.

• Bid-Ask Bounce: Price oscillates between bid and ask without a true trend.
• Discreteness: Prices can only move in minimum tick increments.
• Inventory Effects: Market makers adjust quotes to manage risk.
• At very high sampling frequencies (e.g., tick-by-tick), microstructure noise dominates the signal, biasing realized volatility upward. The optimal sampling frequency balances noise reduction with information capture.

A data sampling technique that creates bars not by fixed time or volume intervals, but when the amount of new information arriving in the market reaches a threshold. This is a modern alternative to fixed-interval sampling for realized volatility.

• Tick Imbalance Bars: Sample when the imbalance between buy and sell ticks exceeds a threshold.
• Volume Imbalance Bars: Sample when the imbalance between buy and sell volume exceeds a threshold.
• Dollar Imbalance Bars: Sample when the dollar value imbalance exceeds a threshold.
• These bars produce more statistically desirable return series, reducing the impact of market microstructure noise and heteroskedasticity.

A diagnostic tool that plots the average realized volatility against the sampling frequency. It visually reveals the optimal interval for estimation.

• X-axis: Sampling interval (seconds, minutes, ticks).
• Y-axis: Average realized volatility.
• Signature Shape: Typically high at very high frequencies due to microstructure noise, then stabilizes at a moderate frequency, and may decline at low frequencies due to insufficient data.
• The volatility signature helps practitioners select the sampling frequency where the plot flattens, indicating noise has dissipated but information is preserved.

The phenomenon where the statistical properties of a target variable change over time in unforeseen ways. Realized volatility regimes are a classic example of concept drift in financial time series.

• Sudden Drift: A volatility spike from an earnings surprise or geopolitical event.
• Incremental Drift: A gradual shift from a low-volatility to a high-volatility regime.
• Recurring Drift: Seasonal patterns like increased volatility at market open and close.
• Forecasting models trained on historical realized volatility must incorporate regime-switching mechanisms or frequent retraining to adapt to this drift.

A self-exciting point process where the occurrence of an event increases the probability of future events in the near term. It is used to model the clustering of trades and volatility.

• Intensity Function: The instantaneous rate of event arrivals, which jumps after each event and decays exponentially back to a baseline.
• Self-Excitation: A large price move triggers a cascade of subsequent trades, creating volatility clustering.
• Mutual Excitation: Volatility in one asset can excite volatility in a correlated asset.
• Hawkes processes provide a mathematical framework for understanding why realized volatility exhibits autocorrelation and clustering, distinct from the assumptions of standard Brownian motion.