2.6.3 The Fill Model

Basic Prerequisites: 02 · Why Execution Backtests Lie and Queue Position: 03 · Fill-Probability Models.

---

1. Intuition & Practical Objective

The fill model is the component that separates an execution backtest from a signal backtest. It answers one question - did my order fill, how much, and when - and it is the only part of the simulator that cannot be lifted from the price series. This page builds the fill model from first principles and implements a working LOB replay / fill simulator, validated against the exact closed form.

Three mechanisms must be in the model, and each is a distinct source of error if missing:

1. Queue position - under FIFO you fill when the cumulative outflow ahead $\xi$ passes your position $x$.
2. Trade-through - when the market trades through your price, price priority guarantees you fill regardless of queue (the sweep consumed every better price first).
3. Adverse selection - the fills you get are correlated with the price moving against you; the model must charge the post-fill drift.

The one-sentence essence. “Simulate the fill as $\text{Filled}(x,L,\xi )=(\xi -x{)}^{+}-(\xi -x-L{)}^{+}$ with $\xi$ the cumulative trades and cancels ahead, add a hard fill when the tape trades through your price, and subtract the adverse post-fill drift - anything less is a price backtest wearing an execution costume.”

---

2. Mathematical Ground Truth & Derivations

2.1 The fill function

Under price-time priority, a resting order of size $L$ at queue position $x$ (shares ahead) receives, from a cumulative outflow $\xi$ (trades $+$ cancels ahead),

$$\boxed{\text{Filled}(x,L,\xi )=(\xi -x{)}^{+}-(\xi -x-L{)}^{+},(z{)}^{+}=\max (z,0)}$$

You start filling once $\xi >x$ and finish once $\xi >x+L$. The fill fraction is $\mathbb{E}[\text{Filled}]/L$, and the fill probability (any fill) is $\mathbb{P}(\xi \ge x)$.

2.2 Fill probability and fill time

• Poisson trades at rate $\mu$ (share volume), $N_S=\mu T$: $\xi (T)\sim \text{Poisson}(\mu T)$, so the fill probability is the Poisson upper tail

$$\mathbb{P}(\xi (T)\ge x)=1-\sum _{k<x}\frac{(\mu T{)}^k{e}^{-\mu T}}{k!}.$$

• Binomial trades (per-tick probability $p$, trade size $s$): $\xi =s\cdot \text{Bin}(T,p)$; the exact expected fill is a finite sum

$$\mathbb{E}[\text{Filled}]=\sum _{k=0}^T\left(\genfrac{}{}{0ex}{}{T}{k}\right)p^k(1-p{)}^{T-k}\min ((ks-x{)}^{+},L).$$

• Fill time at the front (unit size): the waiting time to the $x$-th trade is negative-binomial, mean $x/\mu$.

2.3 Exponential-outflow closed form (Cont–Kukanov)

If the outflow over the horizon is exponential, $\xi \sim \text{Exp}(\text{mean }m)$, then

$$\mathbb{E}[(\xi -Q{)}^{+}]=m{e}^{-Q/m},\mathbb{E}[\text{filled}]=m(e^{-Q/m}-e^{-(Q+L)/m}),$$

the tractable primitive a Monte Carlo fill engine is built on.

2.4 Trade-through (price priority)

For a resting buy limit at price $b$: any trade printed at a price $p<b$ means the matching engine had already exhausted every bid at price $\ge p$, hence your bid at $b$ was hit. Formally,

$$(\exists \text{ trade at price }p<b)⟹\text{filled with probability }1.$$

So trade-through is a guaranteed-fill lower bound. For a sweep of size $S$ through a level of total depth $D$, $\mathbb{P}(\text{guaranteed fill})=\mathbb{P}(S\ge D)$ - a heavy-tailed quantity that matters exactly in the fast markets a naive model handles worst.

2.5 Adverse selection at the fill

Conditioned on a passive fill, the subsequent mid move is negative:

$$\mathbb{E}[\Delta M_T\mid \text{filled}]<0<\mathbb{E}[\Delta M_T\mid \text{not filled}],$$

so the true per-fill edge is $\text{edge}=\frac{1}{2}s-\text{AS}-(\text{fees}+\text{impact})$.

---

3. Computational Implementation - a working LOB replay / fill simulator

Stdlib only. First the simulator, validated against the exact binomial closed form; then the Cont–Kukanov closed form vs Monte Carlo, the optimistic-vs-conservative comparison, and the trade-through guarantee.

Block A - the replay simulator, checked against the closed form.

The simulator reproduces the exact closed form to the third decimal - so the model is correct, and the only remaining errors are modelling errors (wrong $\xi$, wrong $x$, wrong $L$), not implementation errors.

Block B - Cont–Kukanov closed form, optimistic vs conservative, and trade-through.

Three lessons. (A) The binomial closed form and its Monte Carlo agree to ~3 decimals - the fill model is simulable, which is what makes execution backtesting a Monte Carlo discipline. (C) The optimistic rule is benign at the front of the queue ($1.02\times$) and disastrous behind it ($4.47\times$ at $x=1000$) - the error is concentrated exactly where a strategy’s edge is thinnest. (D) Trade-through is rare at deep levels ($1.1\%$ at depth $5000$) but common at shallow ones ($24.5\%$ at depth $1000$) - so the guaranteed-fill channel must be modelled, and it is strongest in the same shallow, volatile books where queue models are least reliable.

---

4. Failure Modes & First-Principles Breakdowns

1. FIFO assumed when the venue is pro-rata. The fill function is priority-rule-specific. On a pro-rata venue (many futures markets) your allocated share depends on your size relative to the level, not your arrival order - using the FIFO formula mis-books fills in both directions.
2. Cancel-ahead ignored (double error). Cancels are a large part of $\xi$; ignoring them under-fills. But cancels are also correlated with sweeps - the queue ahead of you vanishes exactly when a sweep arrives (phantom liquidity), so a constant cancel rate under-states fills in fast markets and over-states them in calm ones.
3. Trade-through mis-used. A trade printed at your price is not a trade-through; a trade printed through it is. Confusing the two re-introduces the standing-queue delusion on the up-side.
4. Stale queue position. $x$ is not constant - cancels, other participants’ fills, and your own repricing move it. A simulator that fixes $x$ at submission mis-prices any strategy that reprices.
5. Partial fills treated as binary. The fill function returns a quantity; backtests that record fill/no-fill as $0/1$ (the optimistic rule) throw away the partial-fill structure that dominates real execution.
6. Adverse selection omitted. Even a correct fill model that stops at the fill and ignores $\mathbb{E}[\Delta M\mid \text{filled}]<0$ still overstates edge by the whole adverse-selection drift (Queue Position: 05 · Failure Modes).

---

5. References

• Cont, Rama; Kukanov, Arseniy - “Optimal order placement in limit order markets,” Quantitative Finance 17(4), 553–571 (2017)
• Cont, Stoikov & Talreja - “A stochastic model for order book dynamics,” Operations Research 58(3) (2010)
• Gould et al. - “Limit order books,” Quantitative Finance 13(11) (2013)
• Hasbrouck, Joel - Empirical Market Microstructure (2007)
• Lo, MacKinlay & Zhang - “Econometric models of limit-order executions,” JFE 65(1) (2002)
• Abergel et al. - Limit Order Books (Cambridge, 2016)

---

6. Connected Graph Bridges

• Back: 02 · Why Execution Backtests Lie · Index Hub
• Forward: 04 · Market Replay vs Monte Carlo
• Depth: Queue Position & Fill Probability (the full queue theory) · Limit Order Book Mechanics