The mathematical model formulated and used in this study was the Hawkes Self-Exciting Process. The benefit of using this model is it takes into account the cascade behavior of trading and also includes exponential decay terms that is appropriate for how trades arrive.

An example of how the arrival rate (intensity) of trades looks on 1/2/18 for small day trades is shown below:

When compared to the realization of a Hawkes Process (pictured below on bottom), they look strikingly similar which makes the Hawkes Process a perfect model for trading data:
Mathematically, the Hawkes Process takes the following form:
Where,

• 𝜆(𝑡) = Trade arrival intensity
• 𝜇 = Baseline trade arrival intensity
• 𝛼 = The jump in arrival intensity after a trade arrive at time 𝑡𝑖
• 𝛽 = The rate at which the arrival intensity decays for a given arrival

Thus, in this study we are trying to:

• Estimate parameters 𝜇, 𝛼, and 𝛽 for small/large for day/night trades
• Use these parameters in simulation to estimate the Hawkes fitted intensity to compare to the real intensity
• Develop a method to predict these parameters based on historical parameters