Avellaneda-Stoikov Market Maker Simulator

An interactive, high-fidelity implementation of the seminal Avellaneda-Stoikov (2008) model for algorithmic market making. This simulator explores the stochastic control problem of managing inventory risk while providing liquidity in a limit order book.

In-depth essay: here

Overview

The simulator models a market maker (MM) who provides liquidity by simultaneously posting bid and ask quotes. The MM's goal is to capture the spread while managing Inventory Risk (the risk of price movement against a held position) and Adverse Selection (the risk of trading against informed participants).

The system runs a real-time loop (160ms ticks) simulating a continuous market environment using Geometric Brownian Motion for price action and Poisson processes for trade arrivals.

---

Theoretical Foundation

The simulator is built on three core mathematical pillars:

1. The Reservation Price ($r$)

Unlike a naive trader who quotes around the mid-price $s$, the Avellaneda-Stoikov MM calculates a "fair value" adjusted for their current inventory $q$.

$$r(s, q, t) = s - q \gamma \sigma^2 (T - t)$$

• $s$: Current Mid-Price.
• $q$: Inventory position (positive for long, negative for short).
• $\gamma$: Risk aversion parameter.
• $\sigma$: Market volatility.
• $(T-t)$: Remaining time horizon.

Logic: If the MM is "Long" ($q > 0$), the reservation price drops below the mid-price. This shifts both bid and ask quotes downward, making it more likely to sell (liquidate) and less likely to buy more.

2. The Optimal Spread ($\delta$)

The model calculates the optimal distance from the reservation price to place the bid and ask quotes:

$$\delta = \gamma \sigma^2 (T - t) + \frac{2}{\gamma} \ln\left(1 + \frac{\gamma}{k}\right)$$

The total spread is comprised of an Inventory Risk Premium and an Adverse Selection Premium. As volatility ($\sigma$) or risk aversion ($\gamma$) increases, the spread widens to compensate the MM for the higher risk.

3. Trade Arrival (Poisson Process)

The probability of a quote being "hit" (a fill) is modeled as a decaying exponential function of the distance from the mid-price:

$$P(\text{fill}) = A \cdot \exp(-k \cdot \text{dist}) \cdot \Delta t$$

• $A$: Intensity of overall market trades (frequency).
• $k$: Market "depth" or price sensitivity. High $k$ means the market is very sensitive; if your price is too far from the mid, you will never get filled.

---

Software Architecture

The application is a single-file web tool (index.html) using Chart.js for visualization and vanilla JavaScript for the simulation engine.

Key Components:

• Price Engine (randn function): Uses the Box-Muller Transform to generate Gaussian noise for the Geometric Brownian Motion (GBM) update.
• Quote Engine (asQuotes function): The mathematical "brain" that calculates $r$ and $\delta$ based on current state and slider inputs.
• Fill Engine (poissonProb function): Determines fills by comparing a random float against the calculated Poisson intensity for both the bid and ask sides.
• State Management: Tracks PnL (Mark-to-Market), Cash, Inventory, and Fills in a centralized loop.

---

Parameter Guide

Parameter	Symbol	Impact on Behavior
Risk Aversion	$\gamma$	Higher values lead to wider spreads and more aggressive "skewing" to keep inventory at zero.
Volatility	$\sigma$	Increases the "drift" of the price and widens spreads to account for larger potential price swings.
Order Depth	$k$	Represents how "thick" the order book is. Higher $k$ means you must quote very close to the mid-price to get filled.
Order Frequency	$A$	Controls the total volume of trades in the market.
Manual Skew	—	A heuristic override to force the MM to lean long or short regardless of the model's math.

---

Analysis & Insights

Based on the research provided in the accompanying essay:

1. The Inventory Leash: Pure spread-capture is impossible without an inventory limit. The reservation price acts as a "leash" that pulls the MM back to a neutral position.
2. Regime Awareness: In high-volatility regimes, the MM must increase $\gamma$ to survive. In low-volatility, "mean-reverting" regimes, the MM can lower $\gamma$ to capture more volume.
3. Adverse Selection: When $k$ is high, the MM is effectively competing in a "winner-take-all" price environment, requiring tighter spreads and faster reaction times.

---

How to Run

1. Download the index.html file or open this link: here
2. Open it in any modern web browser (Chrome, Firefox, or Edge recommended).
3. Adjust the sliders to set your market conditions.

---

References

• Avellaneda, M., & Stoikov, S. (2008). High-frequency trading in a limit order book.
• Implementation and documentation by Handiko Gesang.