Bayesian semi-parametric estimation of compound inhomogeneous Poisson processes for ultra-high frequency financial transaction data

Figures & data

Figure 1. Simulated CIPP and daily close. (A) Simulated price process; (B) Returns based on daily close; (C) Histogram of returns from daily close; (D) ACF of squared returns from daily close.

Figure 2. Covariance functions. (A) Spectral covariance function τ = 2 and γ = 1; (B) B-spline covariance function τ = 2 and 13 knots.

Table 1. Estimated parametric components for duration models. All of the coefficients are significant at 0.05 according to the Bayesian p-value^a except Tuesday and Wednesday for CITI in the mixture model. TST are time sensitive trades. TIT are time insensitive trades.

Number of Trades	CITI	PG	UGI	WMT
Total for Estimation	1,242,560	1,003,464	91,193	1,177,086
Regular Hours Trading	1,165,358	1,001,970	91,122	1,175,578
Pre-Open Trading	55,019	1157	4	916
Post-Close Trading	22,183	337	67	592
Homogeneous	CITI	PG	UGI	WMT
Monday	−0.2496	−0.5574	−1.2451	−0.4462
Tuesday	−0.1786	−0.5122	−1.3153	−0.3318
Wednesday	−0.1800	−0.5101	−1.3582	−0.3153
Thursday	−0.2451	−0.4599	−1.3544	−0.3365
Friday	−0.2167	−0.5159	−1.5052	−0.3752
Lag Duration	−0.1508	−0.0741	−0.0074	−0.0960
Mixture: TST	CITI	PG	UGI	WMT
Monday	0.0234	0.6310	−0.4511	1.4692
Tuesday	0.0011	0.8134	−0.4049	−0.2448
Wednesday	−0.0483	0.7832	0.4971	0.5749
Thursday	−0.2181	0.5139	0.5140	1.2445
Friday	−0.9907	1.2802	0.8205	−0.0675
Mixture: TIT	CITI	PG	UGI	WMT
Constant	0.9755	−1.5892	−1.6550	−1.8171
Lag Duration	−0.4544	−0.0365	−0.0052	−0.0437
Logit P(TIT)	CITI	PG	UGI	WMT
Intercept	−0.0701	−1.5936	0.9257	−1.7713
Return	0.3241	1.1007	0.7962	−2.3495
Log Order Size	0.1820	0.0133	−0.2140	0.0090
Proportion of TIT Trades	0.7331	0.1784	0.4724	0.1512

^a The coefficients β is significant at 0.05 if at least 95% of the posterior distribution is shifted away from zero: P(β >0|Data) > 0.025 or P(β <0|Data)<0.025.

Figure 3. Price process for PG. (A) Prices for 2011-04-01; (B) Average number of arrivals per 15 min; (C) Return histogram all trades; (D) Return histogram 30 min sampling; (E) ACF of return squared all trades; (F) ACF of returned squared 30 min sampling.

Figure 4. Cosine series and B-spline for duration model of UGI. (A) 30 term cosine series of lagged percent return. (B) 30 term cosine series of time of day. (C) B-splines with knots at 1 percentile points (35 internal knots) and customised (12 internal knots) for lagged per cent return. (D) B-splines with knots at 1% points (99 internal knots) and 12 min intervals (32 internal knots) for time of day.

Figure 5. Nonparametric estimates of lagged return in duration intensity function. Posterior means and 2.5 and 97.5 posterior percentiles.

Figure 6. Nonparametric estimates of time of day in duration intensity function. Posterior means and 2.5 and 97.5 posterior percentiles.

Figure 7. Return distributions. Posterior means and 2.5% and 97.5% percentiles.

Table 2. Estimated off-hour trading models.

	Posterior Mean				Posterior SD
Pre-Open Trading	CITI	PG	UGI	WMT	CITI	PG	UGI	WMT
P(Pre-Open Trades)^a	0.9878	0.9390	0.0488	0.8780	0.0122	0.0265	0.0239	0.0364
Lag: Mean (Hours)^b	3.0580	4.2428	4.7493	4.4616	0.0016	0.0079	0.1098	0.0080
Lag: SD (Hours)^b	0.1281	0.6050	0.4390	0.5772	0.0097	0.0467	0.0946	0.0460
Poisson Arrival Rate^c	0.1035	0.0043	0.0002	0.0041	0.0004	0.0001	0.0002	0.0001
Post-Close Trading	CITI	PG	UGI	WMT	CITI	PG	UGI	WMT
Poisson Mean Trades^d	369.12	5.4077	0.2000	9.7671	2.4783	0.3000	0.0476	0.4031
Poisson Arrival Rate^e	0.0257	0.0009	0.0023	0.0011	0.2620	0.0004	0.0009	0.0004

^a P(Pre-Open Trades) is the probability that a day has pre-open trades.

^b The lag between 4am and the first trade has a normal distribution.

^c Trades after the first trade arrival according to a homogeneous Poisson process.

^d The daily total of post-close trades have a Poisson distribution.

^e Given the total number of daily post-close trades, trades arrive according to a homogeneous Poisson process.

Figure 8. Simulated CIPP in validation sample and option prices for UGI. The colour bands are intervals that contain 5% of the simulated values from 5% to 95%. The outmost limits are 2.5% and 97.5%. (A) Simulated price process in validation sample; (B) Simulated price process in day 61; (C) Option prices for simulated CCIP; (D) Ratio of CCIP to black-sholes option prices.