A market maker that held its quotes two ticks from the mid finished $278,287 ahead of the adaptive strategy derived by Chávez Casillas and Gan on five days of Microsoft order flow. The adaptive strategy earned $3,165,081. Fixed two-tick quotes earned $3,443,368, or 8.8% more. The adaptive strategy did beat fixed one-tick ($2,549,852) and fixed three-tick ($2,695,842).

These four figures measure terminal wealth after each day's inventory is liquidated at the closing mid. They exclude the quadratic inventory penalty that the model optimizes. The authors acknowledge the two-tick defeat in the abstract, adding that their results "distinguish terminal-wealth comparisons from evaluation under the inventory-penalized objective." The distinction is fair in principle. But they never run the penalized comparison. The paper says that calculation would require terminal inventories for every strategy; its benchmark table gives wealth alone.

The introduction claims the strategy "outperforms fixed-spread benchmarks on a risk-adjusted basis." No risk-adjusted figure supports that claim. Later, the paper acknowledges that the comparison cannot establish superiority under the penalized objective.

Inside the quoting loop

Every second, the market maker chooses bid and ask offsets for the next interval. A linear demand rule determines fills: a hypothetical 205-share order, the average market-order size, at level L fills a fraction a_N + b_N·L. The coefficients vary with the order-count block (1 to 10, 11 to 20, 21 to 30, 31 to 40, 41+). Seconds without orders produce no fills.

The objective combines terminal wealth and marked inventory, less λI², with λ = 1.0. From a quadratic value function, the authors obtain quotes affine in inventory and calculate their coefficients by backward recursion. Spread capture supplies the income. The penalty draws inventory toward zero.

The paper claims two innovations: feeding Hawkes arrivals into that recursion and fitting linear demand separately for each order-count block. A state-dependent Hawkes process supplies the block probabilities, with excitation that varies with book state, including queue imbalance. The probabilities come from a companion paper. The data are LOBSTER MSFT for November 3 to 7, 2014, covering 117,000 one-second intervals and 130,287 market orders. The same five days serve for calibration and evaluation.

What does Hawkes contribute?

Less than the title suggests. Demand moments assign each nonzero block a weight of p_j/(1−p_0), cancelling the zero-arrival probability. The recursion takes arrival probabilities separately, estimating them from whether orders arrived during the preceding three seconds.

The Hawkes model therefore supplies the allocation across nonzero blocks. About 97.4% of its buy weight and 97.6% of its sell weight fall in the 1 to 10 block. In the paper's buy-side count table, that block contains 9,523 of 11,205 active intervals, about 85%.

The zero-arrival fit fares worse. For buys, the calibrated model gives p_0 = 0.4766, implying at least one market buy in roughly 52% of seconds. In the observations, 90.42% of seconds contain none. Anyone reproducing the strategy must choose between the companion calibration and empirical frequencies. We did not find an ablation showing how much that choice matters.

Dollar profits need a unit check

Offsets are measured in ticks, yet the cash update adds L directly to the price S. Across the five-day sample, there are 11,205 buy-active and 10,755 sell-active seconds, about 4,390 per day. At level 2, the calibrated mean fill is 145.22 − 35.85×2 ≈ 73 shares. With a one-cent tick, two ticks per share imply roughly $6,500 in daily spread capture. Reported daily wealth ranges from $563,110 to $776,807. Those amounts are close to what L = 2 produces when entered into the cash update as two dollars.

Inventory marking affects wealth too. This arithmetic proves nothing, but the units need settling before a replicator compares profit magnitudes. A scale error would alter the balance between spread income and inventory P&L. It would also change the DPP quotes, since those depend on λ expressed in the same units.

Four layers separate theory from posted quotes

The verification theorem applies to the raw controls. The authors acknowledge that their implementation departs from those controls. Reproducing it requires adopting their choices or supplying alternatives:

The demand model adds two more choices. Cross-side moments rely on a product approximation, explicitly identified by the authors as an extra assumption. The authors also acknowledge that fills are left untruncated outside [0, 205 shares]. For the 1 to 10 block, a⁺ = 0.707 and b⁺ = −0.175 imply a negative fill at level 5. The sell 40+ intercept of 1.275 produces an overfill at level 1.

The model assumes conditional independence between price and order counts, ruling out adverse selection. It also assumes front-of-queue priority for fills. We raised the same gap for Hawkes-filtered quotes in an earlier simulator.

Inventory escapes the threshold

The inventory results do not demonstrate control. Inventory remained within ±5,000 shares for just 47.09% of seconds and reached 6,311.2. Its mean was −2,469.5 shares, with a standard deviation of 4,039.6. On November 7, the strategy entered the close holding +4,193 shares.

With an initial inventory coefficient of α₀ ≈ −1.0×10⁻⁶, the model's own inventory term has little weight early in the day. We therefore suspect the soft rule accounts for most of the quote asymmetry: it was active in 52.91% of seconds, with 20.96% of asks versus 9.38% of bids at level 3. The paper acknowledges that aggregate frequencies cannot distinguish the effects of the rule and the model's inventory term.

The count tables also conflict. One reports 106,245 sell zero-intervals; another reports 105,184. For the 1 to 10 block, the regression counts give 9,103 buy intervals, against 9,523 in the frequency table.

Minute bars cannot test this model

We could not run it.

Calibrating sdHawkes requires timestamped order events, five-level depth and queue imbalance. The fill model also needs book depth at every level. Our equity data extends only to one-minute OHLCV bars, from which passive fills cannot be reconstructed.

The authors already suggest calculating penalized payoffs for every strategy, applying the penalty to pre-liquidation inventory. They also propose evaluation across other securities and periods. Before calling this an edge, we would add a test they do not propose: replace the Hawkes block weights with empirical weights to isolate the arrival model's contribution. The evidence currently covers five in-sample days, on which the paper's strategy finishes second to constant two-tick quotes.