  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.
