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Forecasting duration in high-frequency financial data using a self-exciting flexible residual point process

Kyungsub Lee

Abstract

This paper presents a method for forecasting limit order book durations using a self-exciting flexible residual point process. High-frequency events in modern exchanges exhibit heavy-tailed interarrival times, posing a significant challenge for accurate prediction. The proposed approach incorporates the empirical distributional features of interarrival times while preserving the self-exciting and decay structure. This work also examines the stochastic stability of the process, which can be interpreted as a general state-space Markov chain. Under suitable conditions, the process is irreducible, aperiodic, positive Harris recurrent, and has a stationary distribution. An empirical study demonstrates that the model achieves strong predictive performance compared with several alternative approaches when forecasting durations in ultra-high-frequency trading data.

Forecasting duration in high-frequency financial data using a self-exciting flexible residual point process

Abstract

This paper presents a method for forecasting limit order book durations using a self-exciting flexible residual point process. High-frequency events in modern exchanges exhibit heavy-tailed interarrival times, posing a significant challenge for accurate prediction. The proposed approach incorporates the empirical distributional features of interarrival times while preserving the self-exciting and decay structure. This work also examines the stochastic stability of the process, which can be interpreted as a general state-space Markov chain. Under suitable conditions, the process is irreducible, aperiodic, positive Harris recurrent, and has a stationary distribution. An empirical study demonstrates that the model achieves strong predictive performance compared with several alternative approaches when forecasting durations in ultra-high-frequency trading data.

Paper Structure

This paper contains 26 sections, 8 theorems, 103 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Modeling framework
  3. General setup
  4. Specific models
  5. Residual process
  6. Log-likelihood function
  7. Self-exciting point process with flexible residual
  8. Linear representation
  9. Markov Chain
  10. Autocorrelation structure
  11. Empirical analysis
  12. Data
  13. Residual distributions
  14. Gamma distribution.
  15. Generalized Gamma distribution.
  16. ...and 11 more sections

Key Result

Theorem 2

Suppose the stability condition $\alpha < \beta \mu_\varepsilon$ holds. Then, the process $\{ \Lambda_n \}$ is a Lebesgue-irreducible, aperiodic, and positive Harris recurrent Markov chain on the state space $\mathcal{X} = [\mu, \infty)$. Specifically: $\blacktriangleleft$$\blacktriangleleft$

Figures (6)

  • Figure 1: Autocorrelation functions of simulated inter-arrival times $\tau_n$ with Gamma residuals ($\mu_\varepsilon = 1$)
  • Figure 2: Histogram of AAPL durations with fitted exponential density
  • Figure 3: P–P plots of interarrival times for four models using AAPL mid-price data on December 16, 2022.
  • Figure 4: Histograms of exponential residuals for four models using AAPL data on December 16, 2022.
  • Figure 5: Dynamics of the expected interarrival times predicted by four models, based on AAPL mid-price data on December 16, 2022.
  • ...and 1 more figures

Theorems & Definitions (26)

  • Definition 1
  • Definition 2: Renewal process
  • Definition 3: Autoregressive conditional duration (ACD) model
  • Definition 4: Log-ACD(1,1) model
  • Definition 5: Log-ACI(1,1) model
  • Definition 6: Self-exciting exponentially decaying flexible residual point process
  • Remark 1
  • Definition 7: Exponential Hawkes process
  • Theorem 2: Ergodicity and Stationarity
  • proof
  • ...and 16 more