Microstructure Noise Models in High Frequency Data

Understanding microstructure noise

Market microstructure noise refers to the deviation between observed prices and the underlying efficient price in high frequency data sampling. This noise arises from various market frictions including:

• Bid-ask bounce
• Discreteness of price changes due to tick size
• Temporary price impact from trades
• Latency in price discovery
• Order processing costs

The true efficient price process $p_t$ is typically modeled as:

$p_t = p^*_t + \epsilon_t$

Where:

• $p^*_t$ is the unobserved efficient price
• $\epsilon_t$ is the microstructure noise term

Key noise models and their applications

Additive noise model

The simplest approach assumes additive independent noise:

$\epsilon_t \sim N(0, \sigma^2_\epsilon)$

This model captures basic frictions but may not account for more complex dependencies in high-frequency data.

Roll model

The Roll model specifically addresses bid-ask bounce:

$p_t = p^*_t + \frac{s}{2}q_t$

Where:

• $s$ is the bid-ask spread
• $q_t$ is an indicator for trade direction (+1 for buys, -1 for sells)

State space representation

Many microstructure noise models use state space frameworks:

This allows for:

• Filtering of the true price process
• Estimation of noise parameters
• Incorporation of multiple noise sources

Statistical properties

Autocorrelation structure

Microstructure noise typically exhibits:

$E[\epsilon_t\epsilon_{t-k}] \neq 0$ for small $k$

This negative autocorrelation is particularly evident at ultra-high frequencies.

Volatility signature plots

The relationship between sampling frequency and realized volatility reveals noise patterns:

$RV(h) = \int_0^T \sigma^2_sds + O(h^{-1})$

Where:

• $h$ is the sampling interval
• $RV(h)$ is realized volatility at frequency $h$

Applications in market analysis

Price discovery

Microstructure noise models help in:

• Estimating efficient prices
• Understanding price formation
• Measuring information flow

Trading strategy development

Models inform:

• Optimal sampling frequencies
• Signal processing techniques
• Execution timing decisions

Risk measurement

Applications include:

• High-frequency VaR estimation
• Volatility forecasting
• Liquidity risk assessment

Model selection and estimation

Maximum likelihood estimation

The likelihood function incorporates both efficient price and noise components:

$L(\theta) = \sum_{i=1}^n \log f(p_i|p_{i-1};\theta)$

Where $\theta$ represents model parameters.

Realized kernels

Realized kernels provide noise-robust volatility estimation:

$K(X)_t = \sum_{h=-H}^H k(\frac{h}{H})\gamma_h(X)$

Where:

• $k(x)$ is a kernel function
• $\gamma_h(X)$ are realized autocovariances

Impact on trading systems

Execution algorithms

Order execution algorithms must account for microstructure noise when:

• Determining trade timing
• Estimating market impact
• Calculating transaction costs

Risk controls

Algorithmic risk controls use noise models for:

• Price validation
• Abnormal market detection
• Position monitoring

Best practices for implementation

1. Select appropriate noise models based on:

   • Asset characteristics
   • Trading frequency
   • Market structure

2. Consider multiple noise sources:

   • Quote discreteness
   • Trade impact
   • Information effects

3. Validate model performance using:

   • Out-of-sample testing
   • Cross-validation
   • Sensitivity analysis