A desk should focus on the paper's $10,460 a day. Even that figure comes from a policy given the rest of the day's order flow in advance.
Chávez Casillas and Gan take up a question that market-making research largely passes over. Since Ho and Stoll, and later Avellaneda and Stoikov, the literature has concentrated on where to quote. This paper asks how many shares execute at the chosen depth. It models one second's executed volume by multiplying three factors.
Average market-order size supplies the first factor: 203.6 shares for buys and 206.9 for sells. Next comes an exponential demand curve, P_N(L) = A e^(−BL) + C. It assigns a fill probability at depth L within a given bin of market-order counts N. The final factor, a multiplier f, takes the form of a two-term Prony series. This is a weighted sum of exponentials, here using depth L, count N and order-flow intensity λ.
A state-dependent Hawkes model supplies λ. Past events excite future events with a strength that varies according to queue imbalance at the best quotes. The state thresholds are −0.6, −0.2, 0.2 and 0.6. There are 12 event types, comprising six count bins on each side, and 1,452 parameters.
The 16 Prony coefficients are chosen by CMA-ES, a derivative-free evolution strategy, to maximise the initial value of a backward dynamic programme. The programme chooses bid and ask depths from {1,...,5}². It books depth times volume as spread capture, assumes a martingale mid and imposes a hard inventory limit of 10,000 shares.
The sample contains about 5.4 million LOBSTER MSFT messages and 130,000 market orders from November 3 to 7, 2014. Each day is divided into a one-second grid of 23,400 intervals. November 3 to 6 supplies the training data; November 7 is the test day.
What counts as an execution?
Posted shares and executed volume are treated as the same quantity. Q represents a modelled execution. The paper's demand-curve target assumes a representative order at the front of the queue, rather than measuring an observed fill. To calculate fill probability, it subtracts opposite-side depth through level L−1 from aggregate market-order size. It then scales the result by average order size and caps it at one. We found no treatment of queue position, adverse selection or fees on the market maker's own orders.
The fitted parameters press against their constraints. In every bin, on both sides, amplitude  reaches its upper bound of 1.0000. On the buy side, the decay rate declines from 0.7234 in the 1 to 3 count bin to 0.4231 in the 13-plus bin.
The approximation theorem establishes density through Stone-Weierstrass: finite Prony sums can approximate any continuous f over the state domain. It guarantees that some finite sum can get arbitrarily close. It leaves open whether two terms suffice and whether the calibrated f is optimal. The paper makes an empirical claim that one or two terms come within a few percent, although we did not find an M = 1 run among its results. The authors say they postulate the form and describe (8) as an ansatz. Their introduction and conclusion nevertheless claim to derive, for the first time, a closed-form expression for expected executed volume.
Hindsight enters the quote
The authors state the information problem directly: "Since the continuation values depend on the entire realised sequence, this policy incorporates order-flow observations subsequent to each decision time." Their test is retrospective and conditional on the observed daily path. The abstract retains that qualification, describing results "in retrospective evaluations conditional on realised order-flow paths".
The conclusion omits it. There, the paper reports "$119,707 of out-of-sample value per trading day" and an "order-of-magnitude improvement" without mentioning hindsight. Training fixes the Prony coefficients, but the recursion is solved again on the realised November 7 path. The resulting policy still lacks causality.
The information problem has another layer.
Proposition 2 assumes that executed volume is known at t_k, and the paper claims its closed form meets that condition. Yet Q includes N, the count of market orders arriving in [t_k, t_k+1), after the quote has been set. Under Assumption 1, only N's conditional distribution is known through Hawkes prediction. The recursion instead takes the realised counts.
Two calibrations, one flat multiplier
The headline comparison gives S_opt $119,707 on November 7, versus $8,559 for the trivial multiplier f ≡ 1. Its inventory stays inside the bound: S_opt peaks at 9,988 shares and closes at −16.5 shares.
The calibration section describes a different fit. S_opt's coefficients were trained without the hard inventory bound; a no-post action was added afterwards to keep the policy admissible under that bound. S_retr, the calibration actually trained with the hard bound in the calibration section, earned $10,460.
The authors acknowledge the fitting problem. They write that the buy-sell asymmetry in S_retr's coefficients "suggests sensitivity to local optima." The optimiser trace gives that concern substance. All of the first five CMA-ES generations were infeasible. At generation 6 the best value reached $7,462.80, then stayed there for the remainder of a run permitted up to 25 generations.
The soft penalty contributed nothing. The paper explicitly attributes inventory control to the hard bound. Penalties of 0.01, 0.05 and 0.10 produced identical coefficients, and their training values differed by no more than $0.39.
S_retr against S0 is therefore the defensible comparison. On one day, S_retr's $10,460 exceeds S0's $8,559 by about 22%. The in-sample advantage was larger: $7,462.80 versus $4,338.39, a factor of 1.72.
The paper's decomposition makes its conclusion harder to sustain. According to the abstract, the comparisons distinguish "the contribution of trading revenues from that of inventory control". The authors describe S_opt as the only strategy in the comparison set combining aggressive-policy revenue with inventory discipline. Yet its reported intraday reward of $100,277 falls marginally below the greedy policy's $103,512.
The residual raises the larger question. S_opt's total value is about $19.4k above its own trading reward (119,707 minus 100,277). Inventory reaches 9,988 shares. Against that exposure, the residual looks like position P&L from a single day.
The greedy policy loses $20.3 million, and the fixed L = 1 rule loses $24.7 million. Their peak inventories reach 204,802 and 232,369 shares, respectively. Both policies ignore inventory. Their losses show the value of a bound that neither baseline has.
Event data required
We could not backtest this. Reproduction requires timestamped event messages carrying trade direction and order size, together with bid and ask depths through level 5. Those data are needed to construct the fill targets and queue imbalance. We have minute bars and no book. Minute bars cannot reconstruct one-second counts or establish whether a passive order would have filled.
For anyone with LOBSTER-grade data, two tasks stand between this result and an interpretable 22%. First, solve the policy forward with only the information available at t_k. Then validate simulated fills against actual queue outcomes rather than the front-of-queue target. Until both are done, the formula remains a parameterisation worth trying within an existing quoting rule, an application the authors themselves propose.