Limit Order Book Models

Limit-order-book models

The order book is a queueing system, and the two quantities a queueing model gives you that a tape does not are the direction of the next mid move and the fill probability of your own passive order. Both depend on the queue sizes and your position; neither depends on volume, and volume is what most fill models are built from.

../../../fin-core/skills/intraday-microstructure/SKILL.md measures the book you have (spreads, Kyle's lambda, Amihud, order-flow imbalance, MBO reconstruction). This skill is the model — what the arrival rates imply about events that have not happened yet.

Every number marked ✅ Measured is printed by scripts/lob_models.py (numpy 2.2.6 + scipy 1.13.0, seed 20260909, about 10 s). ✅ source-verified means it was read in Cont, Stoikov & Talreja (2010), A stochastic model for order book dynamics, Operations Research 58(3), 549–563, in the authors' preprint at www.columbia.edu/~ww2040/orderbook.pdf; equation, table and proposition numbers are as printed there.

1. The model, and the one thing that makes it tractable

✅ source-verified — CST §2.2, for a level i ticks from the opposite best quote:

limit orders    arrive at rate  lambda(i)
market orders   arrive at rate  mu
cancellations   arrive at rate  theta(i) * x      when x orders rest at that level

🔑 The cancellation rate is proportional to the depth. That single choice is what makes the queue ergodic for any lambda (CST Proposition 1) and gives a closed-form stationary law, pi_x ∝ lambda^x / prod_{k=1..x}(mu + k*theta). A constant cancellation rate does not, and a model with one will blow up when lambda > mu + theta.

✅ source-verified — Table 2, estimated from Tokyo Stock Exchange data for Sky Perfect Communications, in units of the average limit-order size and per minute:

✅ Measured — the stationary mean depth those numbers imply at the touch is 1.62 units. Any statement below about a 20-deep queue is a statement about the mechanics, not about that stock.

2. ✅ The direction of the next move, reproduced

CST Proposition 3 (eqs. 9–11) gives P[mid up before down] as an inverse Laplace transform of a continued fraction. You do not need to implement that. ✅ source-verified — the proposition's own proof says the price "changes for the first time exactly when one of the two independent birth-death processes X~_A and X~_B reaches the state 0 for the first time … the quantity (8) is given by P[sigma_A < sigma_B]."

That is a first-passage race between two independent birth-death chains, and it is an absorption probability of the two-dimensional chain — one sparse linear solve, no Laplace inversion, no continued fractions, exact.

✅ Measured — prob_mid_up(a, b) against the paper's own Table 3 (bottom panel, its Laplace-transform values), rows b = bid depth, columns a = ask depth:

Worst |difference| over all 25 cells: 8.27e-04 — every cell agrees to the three decimals Table 3 is printed in. ✅ Two independent Monte Carlo checks: a=1, b=5 → 0.8117 ± 0.0011 against the exact 0.8122; a=5, b=1 → 0.1880 ± 0.0011 against 0.1878.

🔑 What the table says: the direction of the next mid move is a function of two integers you can read off the top of the book. b=5 against a=1 is a 0.812 chance of an up move. The diagonal is exactly 0.500 — no drift, no price history, no signal beyond the queues. This is the model behind every "queue imbalance predicts the next tick" result; the measurement side of it (order-flow imbalance, and the Cont-Kukanov-Stoikov OFI regression) is ../../../fin-core/skills/intraday-microstructure/SKILL.md §4.

3. ⚠️ Proposition 5 did not reproduce — reported, not hidden

CST Proposition 5 (eqs. 14–16) is the probability that a limit order at the bid executes before the mid moves. ✅ source-verified — eq. (14) makes the order's waiting time epsilon_B a sum of independent exponentials with rates mu + theta*(i-1), i = 1..b (so b counts you: b=1 is the front of the queue, and the last transition is at rate mu because your own order never cancels), and eq. (16) makes the event P[epsilon_B < sigma_A].

⚠️ Implementing exactly that gives, against the paper's Table 4 (bottom panel):

Worst |difference| 0.214. ✅ An independent Monte Carlo agrees with the left column of each pair to three decimals, and ✅ best_lambda_for_table4() sweeps the limit-order rate over (0, 6] and finds no value that reproduces Table 4 — the best is lambda = 1, still 0.074 off, and Table 3 needs 1.85. So the arithmetic is not the problem; the reading of the event is. Use Proposition 3, which does reproduce, and measure fill probability directly — §4.

⚠️ If you are checking your own implementation against this paper, check it against Table 3.

4. 🚨 The trap: fill probability from volume, ignoring queue position

The rule almost every backtest uses is "I was filled once the tape printed more size at my price than was ahead of me." It has one input, volume, and it cannot see where in the queue you were.

✅ Measured — a symmetric book, both queues 20 units deep, one passive buy order per row, sweeping only its position (0 = front). Same level, same tape, same printed volume:

• 🚨 At position 10 the volume-only rule says 0.006 and the truth is 0.714 — 110× too low, because 76% of the places you gain are gained by cancellations ahead of you, and a cancellation never prints on the tape. The level only ever trades 4.11 times per episode; it clears mostly by people leaving.
• 🚨 The same level supports fill probabilities from 0.533 to 0.977. Volume sees one number.
• ✅ The same shape inside CST's own calibrated depth range (both queues 5 deep): pos 0 → 0.862, 1 → 0.740, 2 → 0.637, 3 → 0.547, 4 → 0.467.
• 🔑 The back-of-queue number is structural, not a parameter. From the back of a symmetric queue, being filled is very nearly the same event as the bid queue emptying — which is the mid falling. 0.533 against P[mid down] = 0.501. No depth and no volume changes that; it is what "the back of the queue" means.

⚠️ The direction of the volume-only error flips with the cancel-to-trade ratio: at the front it is exact, and in a book where the queue clears by trading rather than cancelling it would overstate. Measure your own ratio before trusting either sign. hftbacktest ships an L3/L2 queue-position model for exactly this reason — ../../../fin-core/skills/intraday-microstructure/SKILL.md §6 has the data requirements (you need order IDs; with L2 you assume the queue, you do not replay it).

5. 🚨 The fills you get are the ones you did not want — with no informed traders anywhere

You buy at the bid = mid − 0.5 ticks. Mark to the mid at the end of the episode: +1.0 tick if the mid then goes up, 0.0 if it goes down. "Earning the half spread" means 0.5.

✅ Measured, same runs. The book is symmetric, so P[mid up] over all episodes must be 0.500 exactly — the run reports 0.499, which is the measurement checking itself:

🚨 Every counterparty in this model is uninformed and anonymous. There is no news, no signal, no toxic flow — and a back-of-queue fill still gives back 31% of the half-spread. Conditioning on having been filled is itself the information: a fill is evidence the bid queue is being consumed, and a consumed bid queue is a mid about to fall.

🔑 Queue position buys fill probability and fill quality at the same time. From the front, a fill is one market sell among many and says almost nothing (0.491 ≈ 0.5). From the back it is almost the down-move itself (0.345). A fill model that gets the probability right and prices every fill at half the spread is wrong in the second decimal at the front and by a third at the back.

🔑 This is on top of, not instead of, informed-flow adverse selection. ../../../fin-strategies/skills/market-making-models/SKILL.md §4 measures −31% of mean PnL from a 50% informed arrival rate in Avellaneda-Stoikov, whose §6 table lists "queue position" as something AS does not model at all — quotes there fill at intensity lambda(delta) with no queue. The two costs stack. Avellaneda-Stoikov tells you how wide; this tells you what a fill at that width is actually worth.

6. What the script gives you

scripts/lob_models.py — numpy + scipy only, importable, no file writes, no network:

⚠️ What this model does not have: order sizes (everything is one unit — the paper's unit is the average limit-order size), hidden and iceberg liquidity, a spread wider than one tick in §§4–5, latency, self-impact, and any dependence of arrival rates on recent price moves. CST's own conclusion names order-size heterogeneity and order-flow/price correlation as the two extensions it leaves out.

Where this sits

• ../../../fin-core/skills/intraday-microstructure/SKILL.md — measuring the book you have: effective vs realized spread, Kyle's lambda, Amihud, OFI, trade classification, and what MBO data you need before queue position is even computable. Read that first if you have data.
• ../../../fin-strategies/skills/market-making-models/SKILL.md — Avellaneda-Stoikov: where to centre the quotes and how wide, and the informed-flow adverse selection the model has no term for. This skill is the queue underneath its lambda(delta).
• ../../../fin-strategies/skills/execution-algorithms/SKILL.md — the other side: working a parent order by taking liquidity, and Almgren-Chriss.
• ../../../fin-core/skills/execution-cost-analysis/SKILL.md — what a set of fills actually cost after the fact. §5's mark-to-mid is the ex-ante version of that measurement.
• ../hawkes-processes/SKILL.md — CST's arrivals are Poisson. Real order flow clusters, and Poisson standard errors on a clustered arrival rate are too narrow by a measured factor.
• ../monte-carlo-methods/SKILL.md — the standard errors above are binomial; §4's ± 0.004 is an error bar on the estimator, not on the model.