For a trader in a one-cent-tick stock, the fleeting two-tick spread is the event this model needs to catch. It opens and closes within milliseconds. A constrained Hawkes model of the best quotes struggles with that snap-back; Lee and Lee's cross-side term lifts the INTC narrowing count from 116 to 408 of 884. The order-sizing example is less persuasive. It uses day-averaged frequencies, and fitted intensities shift its median size by two shares.
How the cross-side term works
The starting point is Zheng, Roueff and Abergel. Four counting processes record upward and downward moves at the best ask and the best bid. A hard gate forces the two narrowing intensities, ask down and bid up, to zero at the one-tick floor. The spread therefore cannot cross. An exponential kernel Φ captures excitation from earlier moves on the same side.
Lee and Lee add Ψ, a cross-side kernel the authors call flocking after Jang, Lee and Lee. A move on one side can then excite moves on the other. At one tick, its active rows feed widening; above one tick, they feed narrowing. The kernels are nonnegative, and the state enters through 0/1 switches. An ordinary linear Hawkes process can therefore dominate this model, giving non-explosion on any finite horizon without a spectral-radius condition. The likelihood takes O(N) to evaluate and has 12 parameters. The simulation recovers them: β2 bias drops from 0.1106 at T=500 to 0.0059 at T=10000, versus a true value of 1.2.
The evidence comes from LOBSTER quotes for INTC and MSFT on 2012-06-21, from 10:00 to 15:30. The samples contain 1,768 and 2,600 events. Both stocks spend more than 99% of the time at one tick. AMZN was checked and dropped because its spread was one tick only 0.16% of the time.
The fit gain is concentrated in snap-back
Adding Ψ takes INTC's log-likelihood from −5269.4 to −572.5. MSFT moves from −6867.3 to 107.5. The LR statistic exceeds 9,300 on 4 degrees of freedom for each stock.
A shared baseline for widening and narrowing could disadvantage the null, so the authors repeat the fit with four separate baselines. The LR statistics fall to 5,154.3 for INTC and 7,085.9 for MSFT; both still reject. They caution that the χ² reference is inexact because the null lies on the boundary.
INTC records 884 narrowing events during the day. The Ψ=0 compensator accounts for 116, compared with 408 under the full model. The null has no other mechanism to close the spread quickly after a widening move. The coefficients tell a similar story. INTC's own-side self-excitation, α1s, is 0.052 with a standard error of 5.9; its mutual term α1c is 80.23, and cross-side widening term α1w is 300.55. Decay rates of 439 and 679 per second for INTC, and 550 and 840 for MSFT, place the action on a millisecond clock.
This reverses the familiar Hawkes emphasis. Self-excitation is indistinguishable from zero, while cross-side widening coefficients range from 300 to 592 across the two stocks.
Does it time the events?
The residuals still fail. Kolmogorov-Smirnov tests reject Exponential(1) rescaled residuals for every event type under both models (p<0.05). For INTC, the full model generates 1,337 widening events where 884 were observed, and captures only 408 of 884 narrowing events. The authors acknowledge that neither model is correctly specified. Their likelihood claim is about relative fit: the AIC gaps are 9,385.8 for INTC (1169.0 against 10554.8) and 13,938.0 for MSFT (−187.4 against 13750.6).
The search for the optimum raises another concern, one the abstract flags. Nominally converged starts span 769.8 log-likelihood units on INTC (30 starts) and 1,472.2 on MSFT (24 starts). The authors attribute that multimodality to optimisation rather than curvature around the best fit. In scaled coordinates, the observed information there has condition numbers of 486 for INTC and 449 for MSFT. Local curvature cannot establish that the search found the highest peak. In one restricted test, with self-excitation set to zero, it returned a negative LR statistic because the search for the full model had missed its own optimum.
The paper fits one trading day. It estimates that about 18 trading days would put every standard error within 10% of its scale. A longer sample would improve precision; the paper gives no reason to think it would resolve the search problem.
An 85-share example
The application chooses the size of a single mean-variance limit order at the best or second-best quote, assessed at the next price move. Quadratic utility makes the optimum closed form. For INTC at one tick, with rebate r=$0.0003, η=1 and zero inventory, it comes to about 85 shares and utility around $0.17.
Those inputs deserve a close look. Next-move probabilities of 0.455 for up-ask and 0.545 for down-bid come from same-day observed frequencies, 402 and 481 of 883. The fill probability q1a=0.7 is illustrative. Substituting fitted intensities at each of the 883 one-tick moments gives a median of 83 shares; 86% of evaluations are within 5 shares of that size. At the median, the fitted model changes the order by two shares. The remaining 14% follow dense same-side clusters and run higher. Three exceed 200 shares, with a maximum of 7,297.
The authors place 85 shares below 1% of the roughly 11,900-share median displayed size at the INTC best ask. They identify fill probabilities as the binding input. Risk aversion works mechanically: halving η doubles the size to about 169. There is also a mismatch in where Ψ acts. It drives narrowing intensities in the two-tick state, while the one-tick sizing example receives it only through widening intensities.
Why we did not replay the order
We have one-minute bars and no equity quotes. The model follows best-quote changes clustered at millisecond scale, while the sizing rule requires fill probabilities at particular queue levels. Minute bars provide neither. A replay with those data would test something else.
Nor does the paper report P&L, costs beyond the flat rebate, or anything out of sample.
Against the constrained Hawkes null, Ψ adds 4,696.9 log-likelihood units on INTC. Across one 2012 day for INTC and MSFT, self-excitation is indistinguishable from zero and cross-side widening terms run from 300 to 592. The authors point toward estimation over more days, about 18 to bring every standard error within 10% of scale, and fill probabilities estimated from queue position. Those steps would make the result more usable.