High-frequency Trading in a Limit Order Book

Overview

Avellaneda & Stoikov (2008) is the foundational paper for rigorous market making theory. It formulates the market maker's problem as a stochastic optimal control problem: the dealer continuously posts bid and ask quotes to maximise terminal wealth while managing the risk of holding an undesirable inventory position. The result is an elegant closed-form solution that every serious quant in market microstructure needs to understand.

The Setup

State Variables

$S_t$ — mid-price of the asset, modelled as arithmetic Brownian motion: $dS_t = \sigma \, dW_t$
$q_t \in \mathbb{Z}$ — the market maker's inventory (shares held, positive = long, negative = short)
$x_t$ — cash (wealth excluding the position value)

Total wealth: $W_t = x_t + q_t S_t$

Control Variables

The dealer controls the spreads around the mid-price:

\begin{aligned} \delta_t^a &= S_t^a - S_t \quad \text{(ask spread)} \\ \delta_t^b &= S_t - S_t^b \quad \text{(bid spread)} \end{aligned}

Order Arrival Model

Limit orders are filled by incoming market orders modelled as Poisson processes. The arrival intensity of buy (hitting the ask) and sell (hitting the bid) market orders is:

\begin{aligned} \lambda^a(\delta^a) = A e^{-k\delta^a} \\ \lambda^b(\delta^b) = A e^{-k\delta^b} \end{aligned}

where:

$A > 0$ — baseline arrival rate
$k > 0$ — sensitivity of arrival intensity to the spread (empirically estimated from order book data)

This captures the trade-off: tighter spreads attract more order flow but reduce per-trade profit.

The Optimisation Problem

The dealer solves a finite-horizon utility maximisation:

\begin{aligned} \max_{\delta^a, \delta^b} \mathbb{E}\!\left[-e^{-\gamma W_T}\right] \end{aligned}

subject to the inventory and wealth dynamics, where $\gamma > 0$ is the absolute risk aversion parameter.

The HJB Equation

Define the value function $u(t, x, q, S)$ . By the dynamic programming principle, it satisfies the Hamilton-Jacobi-Bellman (HJB) PDE:

\frac{\partial u}{\partial t} + \frac{\sigma^2}{2} \frac{\partial^2 u}{\partial S^2} + \max_{\delta^a}\!\left[\lambda^a(\delta^a)\left(u(t, x+S+\delta^a, q-1, S) - u\right)\right] + \max_{\delta^b}\!\left[\lambda^b(\delta^b)\left(u(t, x-S+\delta^b, q+1, S) - u\right)\right] = 0

Terminal condition: $u(T, x, q, S) = -e^{-\gamma(x + qS)}$

The Solution: Closed-Form Quote Schedules

Ansatz

Conjecture the form:

\begin{aligned} u(t, x, q, S) = -e^{-\gamma(x + q S + \theta(t, q))} \end{aligned}

Substituting into the HJB and solving the resulting ODE system yields the reservation price (the market maker's subjective mid-price adjusted for inventory risk):

\begin{aligned} r(t, q) = S - q \gamma \sigma^2 (T - t) \end{aligned}

The term $-q \gamma \sigma^2 (T-t)$ is the inventory penalty: a long position ( $q > 0$ ) causes the dealer to shade their quotes downward to offload inventory.

Optimal Spreads

The optimal bid-ask half-spread is:

\begin{aligned} \delta^* = \frac{1}{2}\left(\frac{\gamma \sigma^2(T-t)}{1 + \frac{\gamma}{k}}\right) + \frac{1}{\gamma} \ln\!\left(1 + \frac{\gamma}{k}\right) \end{aligned}

The total optimal spread posted around the reservation price is:

\begin{aligned} \delta^* = \frac{1}{2}\left(\frac{\gamma \sigma^2(T-t)}{1 + \frac{\gamma}{k}}\right) + \frac{1}{\gamma} \ln\!\left(1 + \frac{\gamma}{k}\right) \end{aligned}

\begin{aligned} \text{Spread} = \delta^a + \delta^b = \gamma \sigma^2 (T-t) + \frac{2}{\gamma}\ln\!\left(1 + \frac{\gamma}{k}\right) \end{aligned}

Notably, the spread has two components:

$\gamma \sigma^2 (T-t)$ — adverse selection / inventory risk component (shrinks as $T$ approaches)
$\frac{2}{\gamma}\ln\!\left(1 + \frac{\gamma}{k}\right)$ — intrinsic spread (independent of time, driven by market depth $k$ )

Python Implementation

import numpy as np
from dataclasses import dataclass
 
@dataclass
class AvellanedaStoikov:
    """
    Optimal market making quotes under the Avellaneda-Stoikov (2008) model.
    """
    sigma: float   # Mid-price volatility (per unit time)
    gamma: float   # Market maker's absolute risk aversion
    k: float       # Order arrival decay (sensitivity to spread)
    T: float       # Total horizon (seconds, minutes, or days)
 
    def reservation_price(self, S: float, q: int, t: float) -> float:
        """Inventory-adjusted mid price (dealer's fair value)."""
        time_remaining = self.T - t
        return S - q * self.gamma * self.sigma**2 * time_remaining
 
    def optimal_spread(self, t: float) -> float:
        """Total optimal bid-ask spread."""
        time_remaining = self.T - t
        inventory_component = self.gamma * self.sigma**2 * time_remaining
        intrinsic_component = (2 / self.gamma) * np.log(1 + self.gamma / self.k)
        return inventory_component + intrinsic_component
 
    def quotes(self, S: float, q: int, t: float) -> tuple[float, float]:
        """Return (bid_price, ask_price) optimal quotes."""
        r = self.reservation_price(S, q, t)
        half_spread = self.optimal_spread(t) / 2
        return r - half_spread, r + half_spread

Let's simulate a market making session:

import matplotlib.pyplot as plt
 
mm = AvellanedaStoikov(sigma=2.0, gamma=0.1, k=1.5, T=1.0)
 
inventories = [-5, -2, 0, 2, 5]
times = np.linspace(0, 0.9, 100)
S = 100.0
 
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 4))
 
# Reservation price shift vs inventory
for q in inventories:
    r_prices = [mm.reservation_price(S, q, t) for t in times]
    ax1.plot(times, [r - S for r in r_prices], label=f"q={q}")
ax1.set_xlabel("Time (fraction of T)")
ax1.set_ylabel("Reservation price shift (r - S)")
ax1.set_title("Inventory bias in reservation price")
ax1.legend(fontsize=8)
ax1.axhline(0, color="gray", lw=0.5, ls="--")
 
# Optimal spread over time
spreads = [mm.optimal_spread(t) for t in times]
ax2.plot(times, spreads)
ax2.set_xlabel("Time (fraction of T)")
ax2.set_ylabel("Total bid-ask spread")
ax2.set_title("Optimal spread (decreases near expiry)")
plt.tight_layout()

At $t = 0$ , $q = 0$ (neutral inventory):

#S=100, q=0, t=0, T=1
bid, ask = mm.quotes(S=100.0, q=0, t=0.0)
print(f"Reservation price: {mm.reservation_price(S=100.0, q=0, t=0.0):.4f}")
print(f"Optimal spread:    {mm.optimal_spread(t=0.0):.4f}")
print(f"Bid: {bid:.4f}  |  Ask: {ask:.4f}")

Reservation price: 100.0000
Optimal spread:    1.6908
Bid: 99.1546  |  Ask: 100.8454

Key Insights

1. Inventory skews the fair value

When the dealer is long ( $q > 0$ ), their reservation price is below mid. They want to sell — so they shade their ask downward, and their bid downward. This creates asymmetric quotes that correct the inventory imbalance without market impact.

2. Spread widens with volatility

Higher $\sigma$ → wider spread. This is consistent with empirical observation: dealers widen quotes in volatile markets to compensate for the increased adverse-selection risk.

3. Risk aversion controls spread scaling

$\gamma \to 0$ : the dealer is risk-neutral and only posts the intrinsic spread $\frac{2}{k}$ .
$\gamma \to \infty$ : the dealer is extremely risk-averse, quotes extremely wide to avoid any inventory accumulation.

4. Spread narrows near horizon

As $t \to T$ , the $\gamma\sigma^2(T-t)$ component vanishes. Near end-of-day, the inventory risk matters less (there's no time left to get hurt), so the dealer tightens quotes.

Limitations and Extensions

Limitation	Extension
Symmetric Poisson arrival (same $\lambda$ for buy/sell)	Guéant, Lehalle & Fernandez-Tapia (2013) — general intensity functions
No adverse selection (informed traders)	Cartea, Jaimungal & Penalva (2015) — incorporating toxic flow
Continuous quoting (no discrete tick sizes)	Fodra & Pham (2015) — discrete LOB models
Single asset	Guéant (2017) — multi-asset market making
No queue dynamics	Laruelle, Lehalle & Pagès (2011) — queue-reactive models

Avellaneda-Stoikov remains the canonical starting point. Every serious HFT market making system either uses this model directly or can trace its intellectual lineage back to it.

Implementation Notes

In practice, the parameters require careful estimation:

$\sigma$ : estimate from realized variance over a short rolling window (e.g., 5-minute bars)
$k$ : estimate from LOB data by regressing $\log(\text{fill rate})$ on spread-to-mid distance
$\gamma$ : often calibrated via Sharpe ratio targeting or set by risk limits on intra-day inventory exposure
$T$ : usually the trading session length (6.5 hours for US equities) or a rolling window

A&S Model Implementation.ipynb

import numpy as np
from dataclasses import dataclass
 
@dataclass
class AvellanedaStoikov:
    """
    Optimal market making quotes under the Avellaneda-Stoikov (2008) model.
    """
    sigma: float # Mid-price volatility (per unit time)
    gamma: float # Market maker's absolute risk aversion
    k: float # Order arrival decay (sensitivity to spread)
    T: float # Total horizon (seconds, minutes, or days)
 
    def reservation_price(self, S: float, q: int, t: float) -> float:
        """Inventory-adjusted mid price (dealer's fair value)."""
        time_remaining = self.T - t
        return S - q * self.gamma * self.sigma**2 * time_remaining
 
    def optimal_spread(self, t: float) -> float:
        """Total optimal bid-ask spread."""
        time_remaining = self.T - t
        inventory_component = self.gamma * self.sigma**2 * time_remaining
        intrinsic_component = (2 / self.gamma) * np.log(1 + self.gamma / self.k)
        return inventory_component + intrinsic_component
 
    def quotes(self, S: float, q: int, t: float) -> tuple[float, float]:
        """Return (bid_price, ask_price) optimal quotes."""
        r = self.reservation_price(S, q, t)
        half_spread = self.optimal_spread(t) / 2
        return r - half_spread, r + half_spread

import matplotlib.pyplot as plt
 
mm = AvellanedaStoikov(sigma=2.0, gamma=0.1, k=1.5, T=1.0)
 
inventories = [-5, -2, 0, 2, 5]
times = np.linspace(0, 0.9, 100)
S = 100.0
 
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 4))
 
# Reservation price shift vs inventory
for q in inventories:
    r_prices = [mm.reservation_price(S, q, t) for t in times]
    ax1.plot(times, [r - S for r in r_prices], label=f"q={q}")
ax1.set_xlabel("Time (fraction of T)")
ax1.set_ylabel("Reservation price shift (r - S)")
ax1.set_title("Inventory bias in reservation price")
ax1.legend(fontsize=8)
ax1.axhline(0, color="gray", lw=0.5, ls="--")
 
# Optimal spread over time
spreads = [mm.optimal_spread(t) for t in times]
ax2.plot(times, spreads)
ax2.set_xlabel("Time (fraction of T)")
ax2.set_ylabel("Total bid-ask spread")
ax2.set_title("Optimal spread (decreases near expiry)")
plt.tight_layout()

Out [2]

At t=0, q=0 (neutral inventory):

#S=100, q=0, t=0, T=1
bid, ask = mm.quotes(S=100.0, q=0, t=0.0)
print(f"Reservation price: {mm.reservation_price(S=100.0, q=0, t=0.0):.4f}")
print(f"Optimal spread:    {mm.optimal_spread(t=0.0):.4f}")
print(f"Bid: {bid:.4f}  |  Ask: {ask:.4f}")

Out [3]

Reservation price: 100.0000
Optimal spread:    1.6908
Bid: 99.1546  |  Ask: 100.8454

# S=100, q=0, t=0, T=10
mm = AvellanedaStoikov(sigma=2.0, gamma=0.1, k=1.5, T=10.0)
bid, ask = mm.quotes(S=100.0, q=0, t=0.0)
print(f"Reservation price: {mm.reservation_price(S=100.0, q=0, t=0.0):.4f}")
print(f"Optimal spread:    {mm.optimal_spread(t=0.0):.4f}")
print(f"Bid: {bid:.4f}  |  Ask: {ask:.4f}")

Out [4]

Reservation price: 100.0000
Optimal spread:    5.2908
Bid: 97.3546  |  Ask: 102.6454

# S=100, q=0, t=5, T=10
mm = AvellanedaStoikov(sigma=2.0, gamma=0.1, k=1.5, T=10.0)
bid, ask = mm.quotes(S=100.0, q=0, t=5.0)
print(f"Reservation price: {mm.reservation_price(S=100.0, q=0, t=5.0):.4f}")
print(f"Optimal spread:    {mm.optimal_spread(t=5.0):.4f}")
print(f"Bid: {bid:.4f}  |  Ask: {ask:.4f}")

Out [5]

Reservation price: 100.0000
Optimal spread:    3.2908
Bid: 98.3546  |  Ask: 101.6454

# S=100, q=0, t=10, T=10
mm = AvellanedaStoikov(sigma=2.0, gamma=0.1, k=1.5, T=10.0)
bid, ask = mm.quotes(S=100.0, q=0, t=10.0)
print(f"Reservation price: {mm.reservation_price(S=100.0, q=0, t=10.0):.4f}")
print(f"Optimal spread:    {mm.optimal_spread(t=10.0):.4f}")
print(f"Bid: {bid:.4f}  |  Ask: {ask:.4f}")

Out [6]

Reservation price: 100.0000
Optimal spread:    1.2908
Bid: 99.3546  |  Ask: 100.6454

Live Simulator

The model described above is fully implemented below as an interactive browser simulation. The price process follows the exact discrete GBM solution with Itô correction. Order fills are modelled as two independent Poisson processes — one per side — with intensity $\lambda(\delta) = A e^{-\kappa\delta}$ and fill probability $P(\text{fill}) = 1 - e^{-\lambda \, dt}$ , exactly as in the A&S (2008) paper. All ten parameters are adjustable in real time.

▶ Interactive A&S Market Making SimulatorNew tab

Best viewed on a desktop browser. Simulation runs entirely in your browser — no server compute.