Hawkes process model with a time-dependent background rate and its application to high-frequency financial data

A Hawkes process model with a time-varying background rate is developed for analyzing the high-frequency financial data. In our model, the logarithm of the background rate is modeled by a linear model with a relatively large number of variable-width basis functions, and the parameters are estimated by a Bayesian method. Our model can capture not only the slow time variation, such as in the intraday seasonality, but also the rapid one, which follows a macroeconomic news announcement. By analyzing the tick data of the Nikkei 225 mini, we find that (i) our model is better fitted to the data than the Hawkes models with a constant background rate or a slowly varying background rate, which have been commonly used in the field of quantitative finance; (ii) the improvement in the goodness-of-fit to the data by our model is significant especially for sessions where considerable fluctuation of the background rate is present; and (iii) our model is statistically consistent with the data. The branching ratio, which quantifies the level of the endogeneity of markets, estimated by our model is 0.41, suggesting the relative importance of exogenous factors in the market dynamics. We also demonstrate that it is critically important to appropriately model the time-dependent background rate for the branching ratio estimation.

Figure 1

Example of the transaction price change (black dots) and the filtered transactions (red arrows; see text for the detail). The time series of the filtered transactions is analyzed in this study.

Figure 2

Schematic view of generating variable-width basis functions. See Appendix pp2 for the detail method. (a) A cubic B-spline base. (b) We first place the events in an equidistant manner, and put the cubic B-spline bases with equal width represented by colored curves (the top panel). This time coordinate is called natural time. The vertical dotted lines represent knots (see Appendixes pp1 and pp2). We then make a coordinate change from the natural time to the actual time to obtain basis functions with variable width (the bottom panel).

Figure 3

Simulation study. One-hundred synthetic sequences are simulated by using the Hawkes process with a given background rate $\mu \left(t\right)$ (black curve). Each red curve represents the estimate of the background rate obtained by applying our model to each sequence. Two different background rate functions are considered in (a) and (b).

Figure 4

The cumulative distribution of the $p$ value obtained in the KS test for each session for the CONST, ${\text{PL}}_{2\mathrm{h}} (M=3), {\text{PL}}_{30\min } (M=2)$, and BCB $(M=2)$ models. If a distribution is in a gray zone, the distribution passes the KS test of the uniformity with $p=95\%$.

Figure 5

Examples of the estimation of the background rate from the data of the Nikkei 225 mini for the sessions on (a) February 01, 2016 and (b) April 28, 2016. The estimates from the BCB ($M=2$), ${\text{PL}}_{2\mathrm{h}}$ ($M=3$), and ${\text{PL}}_{30\min }$ ($M=2$) models are shown, where $M$ represents the number of exponentials in the triggering kernel. The fitting scores (${\text{PL}}_{2\mathrm{h}},{\text{PL}}_{30\min }$, BCB) are, respectively, $-5.2$, 0.0, 0.0 for (a) and $-303.1, -299.5$, 0.0 for (b), where each value is relative to the BCB model. The branching ratios are, respectively, 0.62, 0.49, 0.46 for (a) and 0.91, 0.87, 0.48 for (b).

Figure 6

Examples of the estimation of the background rate from the data of the Nikkei 225 mini for the sessions on (a) January 29, 2016, when there was an announcement by Bank of Japan, and (b) June 24, 2016, when the United Kingdom European Union membership referendum resulted in the UK voting to leave the EU. The estimates from the ${\text{PL}}_{30\min }$ ($M=2$) and BCB ($M=2$) models are shown. The fitting scores (${\text{PL}}_{30\min }$, BCB) are, respectively, $-342.2$, 0.0 for (a) and $-77.1$, 0.0 for (b), where each value is relative to the BCB model. The branching ratios are, respectively, 0.89, 0.54 for (a) and 0.84, 0.47 for (b). In a gray shaded zone in (b), the trading was stopped due to the circuit breaker.

Figure 7

The day-to-day distribution of the background rate from January 04, 2016 to June 30, 2016, estimated by the BCB ($M=2$) model.