The break-even rule earns exactly zero on the Gaussian surrogate: trade the moment the gap between mid and efficient price covers the spread. In the exact jump model, the same impatient policy retains only overshoot income, somewhere between minus half a tick and a tick per lot. Every unit of profit comes from the option value of trades that the break-even rule would already have taken.
That zero is the result to remember from Rabechini Amaral's paper. Whether it describes a tradeable book depends on everything that follows.
Six counting processes, plus parity
The efficient price follows dX = sigma_X dZ. It is exogenous and driftless, untouched by the order book. The mid occupies the half-tick grid and jumps according to six counting processes: full-tick quote slides, half-tick spread openings and half-tick spread closings. Each intensity combines a positive baseline with a one-sided linear ramp in G = M - X, the gap between mid and efficient price.
That ramp pulls the book toward value. When the gap is positive, an upward jump increases the downward ramp. When it is negative, the same move releases the upward ramp. Crossing zero activates both effects.
The paper's structural device is the parity lock. A liquid large-tick asset has two possible spreads, one tick or two. On the half-tick grid, spread is encoded exactly by the mid's parity: a half-integer means tight, while an integer means open. Spread contributes no separate state information. The model therefore needs one continuous coordinate, the gap, and a bit. Its assumed state space excludes spreads of three ticks or more, as well as small-tick and illiquid assets.
One balanced-response condition does the heavy lifting: 2alpha_s + alpha_o = alpha_c = alpha. It makes the corrective drift equal across both parities. Mean reversion then follows as a theorem rather than an empirical fit, with E[G_{t+h} | F_t] = G_t e^{-alphah} exactly at every horizon and from every state. Stationary variance is s_G² = (sigma_X² + sigma_M²)/2alpha, and autocovariance is s_G² e^{-alphah}. The paths remain discontinuous. Ornstein-Uhlenbeck describes only the first two moments.
The trader maximises long-run average profit, paying only the half-spread phi(S) = S/2.
There is no empirical work of any kind.
The paper labels its two sample paths as using "illustrative parameters; nothing is calibrated". Its optimum sweep is also Monte Carlo from that uncalibrated model, shown with one-standard-error bands. Gamma moves through the ramp slopes while baseline intensities remain fixed.
Patience has a closed-form price
Buy at -theta and sell at +theta. Hold inside the band and never rest flat. The optimum satisfies theta(theta - phi) = s_G², with rate R = alpha * s_G * sqrt(2/pi) * e^{-theta²/2 s_G²}. Expressed in gap units, the answer depends on one dimensionless quantity: gamma = phi/s_G, the half-spread divided by the typical gap. Then u* = (gamma + sqrt(gamma² + 4))/2.
The extra margin beyond break-even equals s_G²/theta*. Set theta = phi and the Gaussian surrogate, matched to the gap's first two moments, produces exactly zero rate. The formula therefore assigns a value to waiting.
The policy trades gap timing. Its wealth decomposition makes this explicit: each completed round trip earns the gap traverse minus both half-spreads, plus a zero-mean martingale generated by holding inventory against X.
A further reduction holds exactly for the jump process. Fractional inventory anywhere in [-1, +1] adds nothing under linear reward and proportional costs. The layer argument works path by path, so positions in {-1, 0, +1} are sufficient.
Which gap can the trader see?
The abstract says plainly that G is observable. In the conclusion, the author identifies three assumptions behind the result: balanced response, a small trader and observability of the gap. He concedes the objection and supplies an answer.
The model forms a state-space system, with a Brownian state observed through point-process intensities that vary linearly with that state. This implies a point-process filter, while Kalman recursions apply to the surrogate. Quiet periods also contain information because total event intensity increases with the dislocation. For the reduced pair, alpha is estimated from the decay of the gap's autocovariance, sigma_X from two-scale realised variance, and the permanent component from Hasbrouck's decomposition. The author is explicit: "Estimation costs performance relative to the full-information benchmark solved here; how much is a filtering question we leave open."
For an implementer, that missing figure decides the strategy. Both trading legs trigger on the gap, the sole state variable. Phi and s_G determine the threshold through theta* = (phi + sqrt(phi² + 4 s_G²))/2.
The paper's forgiveness result addresses threshold error. Because the rate curve is flat around its maximum, an error of epsilon in the threshold costs only O(epsilon²). The author correctly limits this claim to imprecision in alpha and s_G. Bias or noise in the estimate of G_t leaves the stated threshold unchanged while shifting the points where both legs fill. No second-order protection applies to those shifts.
The swept gammas strain the small parameter
The reward error has a rigorous relative bound of order delta/(theta - phi), using the overshoot inequality theta <= E|G_F| < theta + delta. The paper explicitly places the result in the regime where delta is much smaller than theta - phi.
The following arithmetic is ours and appears nowhere in the paper. Gamma ranges from roughly 0.28 to 0.47 in the Monte Carlo sweep, with cost fixed at phi = delta/2. Since gamma = phi/s_G, the implied s_G runs between about 1.1 and 1.8 ticks, while u ranges from 1.15 to 1.26. The resulting margin, theta - phi = s_G/u*, lies between about 0.8 and 1.6 ticks. Delta/(theta - phi) therefore falls between roughly 0.6 and 1.2.
The validation reaches the boundary of the paper's own asymptotic regime. Its kernel discussion already puts s_G "of the order of the tick". An implementer must first choose s_G in ticks, and that choice is consequential.
Theorems give way to a surrogate
The jump process supports exact theorems for the reversion identity, ergodicity, moment identities and inventory reduction. Passage times are then calculated on the Gaussian diffusion uniquely matched to those two moments. Diffusion limit theorems supply the relative timing error of order delta/theta as a heuristic. No proof is given.
Optimality of the band over all admissible strategies is established on the surrogate using results from the switching literature. For the exact jump process, the author calls it a conjecture and leaves it there.
Even theta(theta - phi) = s_G² is the large-threshold asymptote of an exact first-order condition involving the Dawson function. Once gamma exceeds about 0.4, the roots differ by at most 6% in threshold and 2.3% in rate, with the worst case near gamma around 1.7. At gamma = 0.05, the simple root gives up about 9% of rate. Below gamma about 0.28, the exact optimum falls inside one standard deviation, outside the range where the Kramers escape interpretation applies.
What the sweep actually checks
In simulations of the exact jump model, the rate-maximising band lies about a fifth inside the surrogate optimum theta_D throughout gamma from roughly 0.28 to 0.47. Discrete-tick overshoot draws the optimum inward.
The cost of that displacement is small. Using theta_D sacrifices 3 to 4% of rate, while theta* sacrifices 5 to 6%. Almost the entire rate returns for any threshold in [0.8, 0.9] times theta_D.
The sweep evaluates the surrogate's preferred threshold within the band class. As the paper acknowledges, it leaves the class itself untested. The exercise also includes no estimation error, filtering error, queue position, fill probability, adverse selection, latency or impact. The trader is small by assumption and limited to one lot.
Risk neutrality defines the objective. The author observes that replacing it with a Sharpe or utility criterion would restore a flat zone around zero, permit optimal interior positions and shift the thresholds.
We could not test any of this. Our finest data consists of one-minute OHLCV bars, without bid, ask or spread state. The model defines the gap against a latent price over seconds on the half-tick grid. Quote-level data is required to filter that state and charge the half-spread against it.
The paper does something unusual for a closed form: it identifies the missing piece. One result would change my view, the realised rate of a filtered band on the same simulated book compared with the full-information R. Until that ratio is available, theta(theta* - phi) = s_G² remains a clean result for a state nobody observes.