Modeling microstructure price dynamics with symmetric Hawkes and diffusion model using ultra-high-frequency stock data

Abstract

This study examine the theoretical and empirical perspectives of the symmetric Hawkes model of the price tick structure. Combined with the maximum likelihood estimation, the model provides a proper method of volatility estimation specialized in ultra-high-frequency analysis. Empirical studies based on the model using the ultra-high-frequency data of stocks in the S&P 500 are performed. The performance of the volatility measure, intraday estimation, and the dynamics of the parameters are discussed. A new approach of diffusion analogy to the symmetric Hawkes model is proposed with the distributional properties very close to the Hawkes model. As a diffusion process, the model provides more analytical simplicity when computing the variance formula, incorporating skewness and examining the probabilistic property. An estimation of the diffusion model is performed using the simulated maximum likelihood method and shows similar patterns to the Hawkes model. integer. This paper only considers the simple counting measure, i.e., k i = 1 for all i. A point process N can be regarded as a stochastic process by letting N (t, ω) = N ((−∞, t], ω). Consider a filtered probability space (Ω, {F t }, P), −∞ < t ≤ T , where the σ-field F t is generated by N (t). The Hawkes process is an orderly stationary point process N constructed by modeling the conditional intensity, λ. The conditional intensity function is represented as an adapted process toFor an M -dimensional Hawkes process (N 1 , . . . , N M is normally a deterministic function and called kernel. The integration of the r.h.s. is the stochastic integration defined pathwise. To apply the stochastic integration theory in the later, the Hawkes and intensity processes are considered to be right continuous processes with left limits. Self and mutually excited HawkesThis subsection briefly reviews the self and mutually excited Hawkes model. Consider a two dimensional Hawkes process (N 1 , N 2 ) with exponential decay kernels in the conditional intensities with constants µ i , α ij and β ij , for 0 < t:α 12 e −β12(t−u) dN 2 (u) = µ 1 + λ 11 (0)e −β11t + λ 12 (0)e −β12t + t 0 α 11 e −β11(t−u) dN 1 (u) + t 0

Cojumps and asset allocation in international equity markets

“…We formally test jump propagation using the Hawkes process (Hawkes, 1971). 21 This process is a self-excited point process whose intensity depends on the path followed by the point process and has been extensively used in different domains such as seismology and neurology, but only recently in finance to model the dynamics of microstructure prices (Lee and Seo, 2017), trading activity, and asset jumps in financial markets (Ait-Sahalia et al, 2015;Bormetti et al, 2015). For example, Ait-Sahalia et al The univariate Hawkes process we use to capture the time clustering of intraday jumps for each of the ETFs is given by:…”

Systemic risk in market microstructure of crude oil and gasoline futures prices: A Hawkes flocking model approach

“…As a pioneering work, Bowsher (2007) introduce a bivariate Hawkes processes to model the joint dynamics of trades and mid-price changes in NYSE stocks. After that, many studies related to high-frequency finance have employed Hawkes processes (Bacry et al, 2012;Bacry et al, 2013;Da Fonseca and Zaatour, 2014;Bacry et al, 2015;Lee and Seo, 2017).…”

A switching microstructure model for stock prices

“…Bacry and Muzy (2016) Bacry and Muzy (2014) to explain the behaviour of swap rates. Lee and Seo (2017) examines the theoretical and empirical perspectives for the symmetric Hawkes model of the price tick structure.…”