Introduction to market making: inventory, spread and adverse selection (Avellaneda-Stoikov).

The business of quoting both sides

A market maker keeps, at the same time, a buy limit order below the mid price and a sell limit order above it. If a market order arrives on each side, it buys at its bid, sells at its ask and keeps the difference: the bid-ask spread. It does not bet on the direction of the price; it sells immediacy to those who need it (lesson 1.1.1).

flowchart TB
  M[Mid price] --> C[Quote bid and ask]
  C --> F{Fill?}
  F -- one side --> I[Inventory]
  F -- both --> H[Spread earned]
  I --> R[Price risk]
  R --> C
  H --> C

The problem is in the left branch. Orders do not arrive in pairs: if five sells arrive in a row, the market maker buys five times and ends up long, exposed to a falling price. And not all orders are equal: some come from those who know the price is about to move, and those fill the market maker just before the move makes it lose. These are the two sources of loss this lesson measures:

The order flow of a session

mm2_flujo.parquet has every market order of the session: when it arrives, on which side (compra takes the ask, venta the bid), its size and its reach, the distance from the mid down to which it sweeps the book. A sell quote at δ from the mid is filled by a buy if that buy's reach is at least δ. mm2_medio.parquet is the mid price at the end of each second.

import numpy as np
import pandas as pd

flujo = pd.read_parquet("mm2_flujo.parquet")
medio = pd.read_parquet("mm2_medio.parquet")
inicio = pd.Timestamp("2025-03-12 13:30:00", tz="UTC")
T = len(medio)                   # seconds in the session
print(len(flujo), "orders,", T, "seconds")
print(flujo.groupby("lado")["tamano"].agg(["size", "mean"]).round(0))
print(f"informed: {flujo['informada'].mean():.1%}")
flujo.groupby("informada")["alcance"].describe()[
    ["count", "mean", "50%", "max"]].round(4)

42092 orders, 23400 seconds
         size   mean
lado
compra  20889  300.0
venta   21203  302.0
informed: 10.0%
             count    mean    50%    max
informada
False      37903.0  0.0250  0.017  0.281
True        4189.0  0.0492  0.034  0.691

In 6.5 hours, 42,092 market orders arrive, almost the same on each side (0.89 and 0.91 per second), of about 300 shares on average. 10.0% are informed, and their reach is double: an average of 4.9 cents versus 2.5 for uninformed orders. Those who know the price is about to move are willing to sweep more levels of the book to get filled now, and that has a direct consequence for the market maker: the farther from the mid it quotes, the larger the share of informed traders among the orders that reach it.

Fill intensity as a function of distance from the mid

The Avellaneda and Stoikov (2008) model assumes that a quote at distance δ from the mid is filled as a Poisson process with intensity

$A$ is the intensity of a quote at the mid itself and $k$ says how quickly it falls as the quote moves away. It is estimated by counting, for each δ, how many orders on one side have a reach of at least δ and dividing by the length of the session; on a log scale the relationship is a straight line with slope $-k$.

deltas = np.arange(0, 0.1001, 0.005)

def intensidad(ordenes):
    lam = np.array([(ordenes["alcance"] >= d).sum() / T
                    for d in deltas])
    pend, ordenada = np.polyfit(deltas, np.log(lam), 1)
    return np.exp(ordenada), -pend, lam

filas = {}
for lado in ["compra", "venta"]:
    g = flujo[flujo["lado"] == lado]
    for nombre, sub in [("all", g), ("uninformed",
                                     g[~g["informada"]])]:
        A, k, _ = intensidad(sub)
        filas[(lado, nombre)] = {"A_por_s": A, "k_por_usd": k}
est = pd.DataFrame(filas).T
print(est.round(3))
A = est.xs("all", level=1)["A_por_s"].mean()
k = est.xs("all", level=1)["k_por_usd"].mean()
print(f"\nmarket maker parameters: A = {A:.2f} per s, "
      f"k = {k:.1f} per USD")
print(f"half spread that maximizes δ·λ(δ): "
      f"1/k = {100 / k:.2f} cents")

                   A_por_s  k_por_usd
compra all           0.866     35.327
       uninformed    0.824     39.969
venta  all           0.883     35.433
       uninformed    0.836     40.230

market maker parameters: A = 0.87 per s, k = 35.4 per USD
half spread that maximizes δ·λ(δ): 1/k = 2.83 cents

With uninformed orders, the estimate recovers the planted parameters almost exactly: $A$ between 0.82 and 0.84 per second (truth 0.81) and $k$ between 40.0 and 40.2 per USD (truth 40). With all the flow, which is all a real market maker sees, $A$ rises to 0.87 and $k$ falls to 35.4: informed orders, with more reach, fatten the tail. The market maker will use these parameters, the observable ones. The half spread that maximizes one side's revenue with no risk, $1/k$, is 2.83 cents.

The figure shows the measured intensity at each distance, on a log scale, with the fitted line: the points fall almost on it. One cent from the mid, a sell quote is filled about 0.6 times per second; at 5 cents, 0.15; at 10, less than 0.03. Each extra cent of spread costs 30% of the fills.

Adverse selection: what happens after each order

The markout of an order is how much the mid moves in its direction over the next $h$ seconds: $\text{sign} \times (m_{t+h} - m_t)$. For the market maker that fills it, it is what it loses because of the information the order carried. Glosten and Milgrom (1985) explain the spread with this mechanism: the market maker cannot tell the informed apart and charges everyone enough to make up for what it loses to them.

m = np.r_[medio["medio"].iloc[0], medio["medio"].to_numpy()]
# m[s] = mid at the start of second s (the last one known)
seg = np.floor((flujo["ts"] - inicio).dt.total_seconds()).astype(int)
seg = seg.clip(0, T - 1).to_numpy()
signo = np.where(flujo["lado"] == "compra", 1, -1)
informada = flujo["informada"].to_numpy()

markout = {}
for h in [1, 5, 10, 20, 30, 45, 60]:
    mo = 100 * signo * (m[np.minimum(seg + h, T)] - m[seg])
    markout[h] = {"informadas_cent": mo[informada].mean(),
                  "no_informadas_cent": mo[~informada].mean()}
pd.DataFrame(markout).T.round(3)

    informadas_cent  no_informadas_cent
1             0.066               0.002
5             0.311               0.004
10            0.616               0.007
20            1.323              -0.010
30            1.998              -0.008
45            1.959              -0.001
60            1.945               0.009

After an informed order, the mid moves in its direction almost linearly for 30 seconds, up to 2.0 cents, and then stays there: it is the planted impact. After an uninformed one it does not move at all (from −0.010 to +0.009 cents). A fill against an informed trader costs the market maker, on average, 2 cents per share of adverse move, almost the same as it earns in half spread. The figure plots the two curves.

A market maker simulator

The simulator walks through the session second by second. At the start of each second, the market maker sets its bid and ask from the last known mid and its inventory, rounded to the tick (the bid down and the ask up). Each order that arrives in that second fills 100 shares if its reach gets to the quote. The simplifications matter and are worth stating: its quote is the only one in the book at that price, it is always filled in full, it pays no commissions and it updates the quote with no latency.

The accounting separates the three parts of the result. The spread is what it earns on each fill relative to the mid at that moment; the inventory, what the position gains or loses with changes in the mid, $\sum_t q_t\,(m_{t+1} - m_t)$; the sum of the two is exactly the final result marked at the mid. Adverse selection is measured separately, with the 30-second markout of fills against informed orders: it is the part of inventory lost because of them.

TICK, LOTE = 0.01, 100
por_segundo = [[] for _ in range(T)]
for i, s in enumerate(seg):
    por_segundo[s].append(i)
alcance = flujo["alcance"].to_numpy()

def simular(cotizar):
    q, caja, horquilla, inventario = 0, 0.0, 0.0, 0.0
    ejec, q_seg = [], np.zeros(T)
    for s in range(T):
        bid, ask = cotizar(m[s], q, s)
        bid = np.floor(bid / TICK + 1e-9) * TICK
        ask = np.ceil(ask / TICK - 1e-9) * TICK
        for i in por_segundo[s]:
            if signo[i] == 1 and alcance[i] >= ask - m[s] - 1e-9:
                q -= LOTE; caja += ask * LOTE
                horquilla += (ask - m[s]) * LOTE
                ejec.append((s, -1, informada[i]))
            elif signo[i] == -1 and alcance[i] >= m[s] - bid - 1e-9:
                q += LOTE; caja -= bid * LOTE
                horquilla += (m[s] - bid) * LOTE
                ejec.append((s, 1, informada[i]))
        inventario += q * (m[s + 1] - m[s])
        q_seg[s] = q
    e = np.array(ejec)
    s_e, lado_e, inf_e = e[:, 0].astype(int), e[:, 1], e[:, 2] == 1
    mo = lado_e * (m[np.minimum(s_e + 30, T)] - m[s_e]) * LOTE
    res = {"resultado": caja + q * m[T], "horquilla": horquilla,
           "inventario": inventario,
           "sel_adversa_30s": mo[inf_e].sum(),
           "ejecuciones": len(e), "pct_informadas": 100 * inf_e.mean(),
           "q_desv": q_seg.std(), "q_max_abs": np.abs(q_seg).max()}
    return res, q_seg

The first strategy quotes symmetrically around the mid, at the half spread $1/k$ that maximizes each side's revenue, without looking at inventory:

def simetrica(medio_s, q, s):
    return medio_s - 1 / k, medio_s + 1 / k

res_sim, q_sim = simular(simetrica)
pd.Series(res_sim).round(1)

resultado          26169.0
horquilla          41382.5
inventario        -15213.5
sel_adversa_30s    -4709.5
ejecuciones        12820.0
pct_informadas        17.1
q_desv              3553.9
q_max_abs          15500.0
dtype: float64

The captured spread is 41,383 USD over 12,820 fills of 100 shares: about 3.2 cents per share, because the 2.83-cent quote is rounded outward to the tick and the mid is a multiple of half a cent. Inventory takes 15,214 USD: without control, the position drifts like a random walk and reaches 15,500 shares in absolute value, with a standard deviation of 3,554. The final result, 26,169 USD, depends mainly on which way the price moved while inventory was large. Adverse selection, measured separately, is −4,710 USD: 17.1% of fills are against informed traders, more than their 10% of the flow, because they reach farther.

Avellaneda-Stoikov: reservation price and optimal spread

Avellaneda and Stoikov solve the problem of a market maker with risk aversion $\gamma$, inventory $q$ and a horizon $T$ at which it closes the position. Their approximate solution has two pieces. The first is the reservation price, the price at which the market maker would be indifferent between buying or selling one more unit:

With positive inventory, $r$ sits below the mid $s$; with negative inventory, above it. The second is the optimal spread around $r$:

The first term is a premium for inventory risk, larger the more of the session remains; the second depends on the shape of demand ($k$) and, with small $\gamma$, tends to $2/k$, the symmetric spread from before. The difference between the two strategies lies mainly in the inventory shift.

flowchart TB
  Q[Inventory q] --> R[Reservation price]
  S[Mid s] --> R
  V[σ, γ, T - t] --> R
  R --> B[Bid = r - δ]
  R --> A[Ask = r + δ]
  K[k, γ] --> D[Half spread δ]
  D --> B
  D --> A

The volatility of the mid is 0.0065 USD per √s (the generator's). With $q$ in shares, $\gamma$ has units of 1/USD and small values: with $\gamma = 10^{-4}$ and 100 shares, the shift at the start of the session is one cent.

SIGMA = 0.0065

def avellaneda_stoikov(gamma):
    def cotizar(medio_s, q, s):
        resto = T - s
        r = medio_s - q * gamma * SIGMA**2 * resto
        semi = 0.5 * (gamma * SIGMA**2 * resto
                      + 2 / gamma * np.log(1 + gamma / k))
        return r - semi, r + semi
    return cotizar

res_as, q_as = simular(avellaneda_stoikov(1e-4))
pd.DataFrame({"symmetric": res_sim,
              "AS γ=1e-4": res_as}).round(1)

                 symmetric  AS γ=1e-4
resultado          26169.0    38055.0
horquilla          41382.5    39647.5
inventario        -15213.5    -1592.5
sel_adversa_30s    -4709.5    -4453.0
ejecuciones        12820.0    13165.0
pct_informadas        17.1       16.6
q_desv              3553.9      225.7
q_max_abs          15500.0     1700.0

With the same session and almost the same captured spread (39,648 versus 41,383 USD), the Avellaneda-Stoikov market maker cuts the inventory loss from 15,214 to 1,593 USD and its result rises to 38,055. The standard deviation of inventory drops from 3,554 to 226 shares and the maximum, from 15,500 to 1,700. Adverse selection barely changes (−4,453 versus −4,710): inventory control does not protect against the informed, who keep arriving and keep moving the price. It is a loss that can only be reduced by charging more spread or by detecting toxic flow.

The header figure shows the difference: the symmetric strategy's inventory wanders across thousands of shares for the whole session; Avellaneda-Stoikov's, highlighted, oscillates near zero, because each buy lowers the reservation price and makes the next sell more likely. The figure below breaks down the result of both.

barrido = {g: simular(avellaneda_stoikov(g))[0]
           for g in [2e-5, 1e-4, 5e-4, 2e-3]}
pd.DataFrame(barrido).T[["resultado", "horquilla", "inventario",
                         "sel_adversa_30s", "q_desv"]].round(0)

         resultado  horquilla  inventario  sel_adversa_30s  q_desv
0.00002    37652.0    40854.0     -3202.0          -4615.0   563.0
0.00010    38055.0    39647.0     -1592.0          -4453.0   226.0
0.00050    22447.0    22923.0      -475.0          -4457.0   136.0
0.00200   -76819.0   -76500.0      -319.0          -4224.0   105.0

With $\gamma = 2 \times 10^{-5}$ control is looser (a deviation of 563 shares) and the result similar; with $5 \times 10^{-4}$, inventory barely moves (136), but the captured spread halves: the reservation price moves so far from the mid, with little inventory, that the market maker gives away part of the spread to get rid of it. With $2 \times 10^{-3}$ the spread is negative (−76,500 USD): with 100 shares of inventory, the shift exceeds the half spread and one of the quotes crosses the mid. The market maker buys above the mid or sells below it on every fill. $\gamma$ is not a parameter that can be chosen by looking at risk alone.

Inventory risk over many sessions

A single session does not say how much the result varies. With the estimated parameters many can be simulated: the mid as a random walk with volatility σ per second and, in each second, a fill on each side with probability $1 - e^{-\lambda(\delta)}$ (at least one order reaching the quote). These sessions have no informed orders, so they isolate inventory risk. 200 sessions are simulated at once, vectorized with NumPy, with a fixed seed.

def sesiones(gamma, n=200, semilla=7):
    rng = np.random.default_rng(semilla)
    s_m = np.full(n, 48.0)
    q, caja = np.zeros(n), np.zeros(n)
    for s in range(T):
        resto = T - s
        if gamma == 0:
            r, semi = s_m, 1 / k
        else:
            r = s_m - q * gamma * SIGMA**2 * resto
            semi = 0.5 * (gamma * SIGMA**2 * resto
                          + 2 / gamma * np.log(1 + gamma / k))
        bid = np.floor((r - semi) / TICK + 1e-9) * TICK
        ask = np.ceil((r + semi) / TICK - 1e-9) * TICK
        p_ask = 1 - np.exp(-A * np.exp(-k * np.maximum(ask - s_m, 0)))
        p_bid = 1 - np.exp(-A * np.exp(-k * np.maximum(s_m - bid, 0)))
        vende = rng.random(n) < p_ask
        compra = rng.random(n) < p_bid
        q += LOTE * (compra.astype(float) - vende)
        caja += LOTE * (ask * vende - bid * compra)
        s_m = s_m + SIGMA * rng.standard_normal(n)
    return caja + q * s_m, q

mc = {}
for nombre, g in [("symmetric", 0), ("AS γ=2e-5", 2e-5),
                  ("AS γ=1e-4", 1e-4), ("AS γ=5e-4", 5e-4)]:
    pyg, q_fin = sesiones(g)
    mc[nombre] = {"media": pyg.mean(), "desv": pyg.std(),
                  "peor": pyg.min(),
                  "q_final_abs": np.abs(q_fin).mean()}
pd.DataFrame(mc).T.round(0)

             media    desv     peor  q_final_abs
symmetric  36890.0  6385.0   6774.0       6732.0
AS γ=2e-5  36362.0   564.0  35028.0       1248.0
AS γ=1e-4  35344.0   398.0  34230.0        777.0
AS γ=5e-4  27902.0   309.0  27095.0        538.0

Over 200 sessions without informed traders, the symmetric strategy earns on average 36,890 USD with a standard deviation of 6,385, and its worst session leaves 6,774; the average inventory at the close is 6,732 shares. Avellaneda-Stoikov with $\gamma = 10^{-4}$ earns somewhat less on average (35,344), but with a standard deviation 16 times smaller (398) and a worst session of 34,230. With $\gamma = 5 \times 10^{-4}$ dispersion keeps falling and the mean drops to 27,902. It is the trade-off the model formalizes: less inventory risk in exchange for less spread.

It is striking that inventory at the close is not zero with Avellaneda-Stoikov (777 shares on average with $\gamma = 10^{-4}$). The formula multiplies inventory by $T - t$, which tends to zero at the end of the session: in the last minutes the market maker stops caring about the position, just when it has to close it. It is a known flaw of the finite-horizon version.

From the model to the market

The Avellaneda and Stoikov model is the starting point of the market making literature, not a ready-made strategy. Guéant, Lehalle and Fernandez-Tapia (2013) solve the same problem with an inventory limit and obtain approximations of the optimal quotes that, far from the end of the horizon, do not depend on time, which avoids the formula's flaw near the close. A real market also has pieces that the simulation ignores:

• Competition and queue. Other market makers quote at the same level and price-time priority decides who gets filled (lessons 1.1.3 and 2.4.8). Competition narrows the spread until the expected profit is small.
• Latency. A quote that is not updated in time is the one the informed pick off: real adverse selection depends heavily on speed.
• Fees and rebates. Exchanges charge for taking liquidity and often pay a rebate for providing it; for a market maker, that rebate can be a large part of the margin.
• Overnight inventory and jumps. The position left at the close is exposed to the opening gap (lesson 2.3.2), and a news item can move the price many spreads at once.

Exercises

1. Repeat the symmetric strategy with half spreads of 1, 2, 3 and 5 cents in the MMQ session. Which half spread gives the most captured spread and which the best result?

def fija(d):
    return lambda medio_s, q, s: (medio_s - d, medio_s + d)

filas = {f"{100 * d:.0f} cent": simular(fija(d))[0]
         for d in [0.01, 0.02, 0.03, 0.05]}
pd.DataFrame(filas).T[["resultado", "horquilla", "inventario",
                       "ejecuciones", "pct_informadas"]].round(0)

        resultado  horquilla  inventario  ejecuciones  pct_informadas
1 cent    14146.0    32910.0    -18764.0      26842.0            12.0
2 cent    20662.0    40980.0    -20317.0      18403.0            14.0
3 cent    26169.0    41382.0    -15213.0      12820.0            17.0
5 cent    22838.0    32584.0     -9745.0       6227.0            23.0

2. Check the decomposition: for the Avellaneda-Stoikov market maker, spread plus inventory must add up to the final result. What part of inventory does adverse selection explain?

print(f"spread + inventory = "
      f"{res_as['horquilla'] + res_as['inventario']:.1f}")
print(f"result             = {res_as['resultado']:.1f}")
print(f"adverse selection (30 s, informed) = "
      f"{res_as['sel_adversa_30s']:.1f}")

spread + inventory = 38055.0
result             = 38055.0
adverse selection (30 s, informed) = -4453.0

3. A colleague proposes “removing” adverse selection from the simulator by never quoting in the 30 seconds after an informed order. What is wrong with the proposal?