howard-lynn-ye Market-making models
Avellaneda-Stoikov answers one question — how do I quote so that inventory does not kill me — and it is routinely quoted as if it answered a second one it never touches: how do I quote so that informed traders do not. The formulas are right. The risk they price is not the only risk there is.
Everything marked ✅ Measured comes from scripts/market_making.py — numpy only, seed 20260909,
4,000 paths, 3 s. Everything marked ✅ source-verified was read by this library in the
published article (Quantitative Finance 8(3), 217–224); equation numbers are as printed there
and the full transcription is in the script's docstring.
1. The two formulas, verified
✅ source-verified:
(8) r(s, q, t) = s - q*gamma*sigma^2*(T - t) reservation price
(30) delta^a + delta^b = gamma*sigma^2*(T - t) + (2/gamma)*ln(1 + gamma/k) TOTAL spread
(12) lambda(delta) = A * exp(-k * delta) arrival intensity
eq. (8) is the average of the reservation ask (6) and bid (7); eq. (29) shows the trading
agent gets the same price under an expansion in q (22) and a linear approximation of the
arrival term (26).
🔑 eq. (30) is the total spread, not a half-spread, and it is quoted around the reservation
price: p^a = r + spread/2, p^b = r − spread/2. Halving it once more is the most common
implementation error, and it doubles your fill rate silently.
✅ Measured — the sharpest available check on eq. (30) is the paper's own "Average spread"
column, because the time-average of eq. (30) over [0, T] is closed-form
γσ²T/2 + (2/γ)ln(1 + γ/k):
| γ | this formula | the paper prints | in |
|---|---|---|---|
| 0.1 | 1.4908 | 1.49 | Table 1 |
| 0.01 | 1.3489 | 1.35 | Table 2 |
| 1 | 3.0217 | 3.02 | Table 3 |
Three independent numbers, three of the paper's own tables, one closed form. If your
ln(1 + γ/k) is wrong, this catches it.
✅ Measured — eq. (8) at s=100, γ=0.1, σ=2, T−t=1: q=0 → 100.00, q=+5 → 98.00, and at
t=T → 100.00. Long inventory pushes the quote pair down, so the ask is likelier to be hit.
🚨 The spread does not depend on q. ✅ source-verified — AS: "the bid-ask spread in (25)
is independent of the inventory. This follows from our assumption of exponential arrival
rates." All the inventory control is in where the spread is centred. A market maker who
widens when long is doing something the model does not say — possibly sensible, but it is not
Avellaneda-Stoikov and it does not inherit the optimality.
2. The paper's own experiment, reproduced
✅ source-verified, §3.3 verbatim: s=100, T=1, σ=2, dt=0.005, q=0, γ=0.1, k=1.5, A=140, 1,000
simulations; the "symmetric" benchmark "uses the average bid/ask spread of the inventory
strategy over the time period, but centres it around the mid-price."
✅ Measured — 4,000 paths, an independent implementation, a different RNG:
| γ | strategy | spread | PnL | std | q | q std | the paper |
|---|---|---|---|---|---|---|---|
| 0.1 | inventory | 1.49 | 64.8 | 6.7 | 0.04 | 2.8 | 1.49 / 65.0 / 6.6 / 2.9 |
| 0.1 | symmetric | 1.49 | 67.9 | 13.7 | −0.15 | 8.4 | 1.49 / 68.4 / 12.7 / 8.4 |
| 0.01 | inventory | 1.35 | 68.1 | 9.1 | −0.06 | 5.1 | 1.35 / 68.6 / 8.7 / 5.1 |
| 0.01 | symmetric | 1.35 | 68.4 | 13.8 | −0.11 | 8.7 | 1.35 / 68.8 / 12.8 / 8.7 |
| 1 | inventory | 3.03 | 31.5 | 4.9 | 0.02 | 1.6 | 3.02 / 31.4 / 5.0 / 1.7 |
| 1 | symmetric | 3.02 | 43.6 | 10.7 | 0.00 | 5.1 | 3.02 / 44.0 / 11.0 / 5.1 |
All six rows of Tables 1, 2 and 3, within about 1%.
🔑 And the paper's own reading reproduces: the symmetric strategy earns MORE (it sits on the mid and takes more volume) with about twice the PnL dispersion and several times the inventory dispersion. ✅ source-verified — AS: "in the limit as γ → 0 the two strategies are identical", which the γ=0.01 rows show. The inventory strategy is not a way to make more money. It is a way to make slightly less of it with a much thinner tail.
3. 🚨 "With probability λ·dt" is not a probability here
Everyone reimplementing AS reaches this fork, and most do not notice they took it.
✅ Measured — at the paper's own A = 140, dt = 0.005:
| δ | λ(δ) | λ·dt | 1 − e^(−λ·dt) |
|---|---|---|---|
| −0.5 | 296.4 | 1.000 (clipped) | 0.773 |
| 0.0 | 140.0 | 0.700 | 0.503 |
| 0.5 | 66.1 | 0.331 | 0.282 |
| 1.0 | 31.2 | 0.156 | 0.145 |
A·dt = 0.70. This is not a small-probability regime, and λ·dt exceeds 1 whenever a
large inventory pushes a quote through the mid — which eq. (8) does not forbid.
✅ Measured — the same six rows, both readings:
| γ | strategy | PnL (λ·dt) |
PnL (Poisson) | gap |
|---|---|---|---|---|
| 0.1 | inventory | 64.8 | 57.1 | −12% |
| 0.1 | symmetric | 67.9 | 60.8 | −11% |
| 0.01 | inventory | 68.1 | 60.1 | −12% |
| 0.01 | symmetric | 68.4 | 60.4 | −12% |
| 1 | inventory | 31.5 | 28.2 | −10% |
| 1 | symmetric | 43.6 | 42.1 | −4% |
🔑 The reproduction in §2 only works with λ·dt read literally. An implementer who
"corrects" it to the exact Poisson probability 1 − e^{−λ dt} reports 10–12% less profit on
the same strategy and stops matching the paper. Neither reading is wrong. Publishing a
number without saying which one you used is.
4. 🚨 The risk the model has no term for
Every counterparty in AS is uninformed — arrivals depend only on distance from the mid. The entire risk in the model is inventory risk.
✅ Measured — a fraction φ of arrivals know the sign of the next mid move and take only the
side that pays (φ = 0 is the paper's model exactly):
| φ | strategy | PnL | PnL std | fills | PnL/fill | picked off |
|---|---|---|---|---|---|---|
| 0% | inventory | 64.8 | 6.7 | 86 | 0.750 | 0.0 |
| 0% | symmetric | 67.9 | 13.7 | 81 | 0.839 | 0.0 |
| 10% | inventory | 60.9 | 6.3 | 83 | 0.738 | 4.9 |
| 25% | inventory | 54.9 | 6.1 | 77 | 0.713 | 12.1 |
| 50% | inventory | 45.0 | 5.7 | 68 | 0.666 | 24.3 |
| 50% | symmetric | 48.0 | 12.2 | 63 | 0.757 | 22.8 |
🚨 −31% of mean PnL, on quotes that are optimal by eq. (30) at every single step. Nothing in the model is wrong; the model simply has no informed trader, so the spread it prescribes compensates for inventory risk and for nothing else.
🔑 Both strategies degrade almost identically — the inventory strategy is no defence, because the reservation-price skew is an inventory control, not a toxicity one. It knows how much you hold. It does not know why you were filled.
5. So how much wider? Less than the story says
The Glosten-Milgrom argument — the spread must cover the expected loss to informed flow — arrived at by simulation rather than by their equilibrium.
✅ Measured — φ = 25%, sweeping a multiplier on eq. (30)'s spread. "jump" is the extra mid
displacement an informed fill causes; the ratio is (jump + σ√dt) / half-spread:
| jump | adverse / half-spread | ×1 | ×1.25 | ×1.5 | ×2 | ×3 | best |
|---|---|---|---|---|---|---|---|
| 0.0 | 0.19 | 54.9 | 53.5 | 49.6 | 38.9 | 19.5 | ×1 |
| 0.5 | 0.86 | 52.1 | 51.2 | 47.8 | 37.7 | 19.1 | ×1 |
| 1.0 | 1.53 | 49.2 | 48.9 | 46.0 | 36.6 | 18.6 | ×1 |
| 2.0 | 2.87 | 43.6 | 44.3 | 42.4 | 34.3 | 17.7 | ×1.25 |
| 4.0 | 5.56 | 32.2 | 35.1 | 35.2 | 29.8 | 15.8 | ×1.5 |
🔑 Column 2 is the whole answer. Widening pays only once the adverse move per informed fill
is well above the half-spread being earned. ⚠️ And at AS's own parameters it is nowhere near
— the prescribed half-spread is 0.745 against a mid step of σ√dt = 0.141, i.e. 5.3× one
step — so an AS quoter is genuinely hard to pick off there, and widening mostly starves the
flow.
🔑 The governing quantity is the ratio, not your risk aversion. eq. (30) prices γ. It does
not price φ. Measure your own ratio — the post-fill price move at a horizon, against your
half-spread — before deciding whether to widen. The measurement tools are in
../../../fin-core/skills/intraday-microstructure/SKILL.md (effective vs realized spread and
the price-impact decomposition are exactly this quantity).
⚠️ A, k and the size of an informed move are the three numbers this depends on, and AS
calibrates none of them. Read the shape of this table, not its levels.
6. What else the model leaves to you
| Not in AS | Why it matters |
|---|---|
| an inventory bound | ✅ source-verified: the paper states no q_max. eq. (8) skews the quotes but nothing stops q growing; every production implementation adds a cap, and that cap is not optimal in the paper's sense |
| a tick grid | quotes are continuous reals; rounding to a tick changes the fill rate non-linearly |
| queue position | a limit order at a price is assumed to fill at intensity λ(δ); in a real book you are behind a queue |
calibration of A, k |
eq. (12) derives the form from a power-law order-size distribution and logarithmic impact; the constants are chosen, not fitted |
the terminal time T |
the whole inventory skew scales with T − t. There is no natural T for a maker who quotes every day; picking it is a modelling decision the paper does not make for you |
⚠️ A note on the arrival law that is often garbled: eq. (12) is λ(δ) = A·e^{−kδ}, derived
from a power-law order-size density f^Q(x) ∝ x^{−1−α} combined with logarithmic impact.
✅ source-verified — the paper's power-law intensity, from combining that density with
Δp ∝ Q^β instead, is λ(δ) = B·δ^{−α/β} (unnumbered) — not A·δ^{−α}, which is a common
misquote.
7. Scripts and where this sits
scripts/market_making.py — reservation_price (eq. 8), optimal_spread (eq. 30),
mean_optimal_spread (its time-average, the check against the paper's tables),
arrival_intensity (eq. 12), fill_probability (both readings), the vectorized simulate
with informed_frac / informed_jump, and best_spread_multiplier. numpy only, seed
20260909, 3 s.
- Effective and realized spreads, price impact, trade classification, order-flow imbalance —
../../../fin-core/skills/intraday-microstructure/SKILL.md. That is where theφof §5 is measured on real data. - Working a parent order by taking liquidity, and Almgren-Chriss —
../execution-algorithms/SKILL.md. - What a set of fills actually cost, after the fact —
../../../fin-core/skills/execution-cost-analysis/SKILL.md. - Order types, venue connectivity, and not pointing this at a funded account —
../../../fin-core/skills/broker-execution-apis/SKILL.mdand its../../../fin-core/skills/broker-execution-apis/scripts/paper_account_guard.py. 🚨 A quoting strategy is two-sided and always in the market; it is the worst possible thing to run untested against a live account. - How much capital to put behind the quotes —
../position-sizing-kelly/SKILL.md. - Whether a backtested quoting P&L is a discovery —
../../../fin-core/skills/backtest-validation/SKILL.md.