6.2 Avellaneda–Stoikov & Optimal Quoting

Basic Prerequisites: Stochastic Calculus & Itô’s Lemma and Probability & Stochastic Control. (these are the folder-level prerequisites for pages 02–06; page 01 states its own, smaller, entry requirements)

1. Intuition & Practical Objective

The Avellaneda–Stoikov (AS) model is the zero point of optimal high-frequency market making, the way Black–Scholes is the zero point of derivative pricing. Its claim is sharp: a dealer who quotes a two-sided market and maximizes the expected exponential utility of his terminal wealth should shade his quotes off the mid-price by a reservation price that depends on his inventory, and should quote a spread that is fixed by risk aversion and the liquidity of the book.

This folder is the topic-folder for Pillar 6 of the Kwant-Atlas. It is a hub: it (a) gives you the fast formula lookup below (job #1 - the reservation price and the optimal spread), and (b) routes you to six sub-pages that walk you from raw intuition through the market-maker’s problem, the solution, risk aversion, the failure modes, and the extensions.

The one-sentence essence. “A market maker with inventory $q$ and risk aversion $\gamma$ should treat the reservation (indifference) price $r=s-q\gamma {\sigma }^2(T-t)$ - not the mid-price $s$ - as his personal fair value, and quote a spread $\psi =\gamma {\sigma }^2(T-t)+\frac{2}{\gamma }\ln (1+\frac{\gamma }{k})$ around it.”

2. Mathematical Ground Truth & Derivations

Quick-Reference Lookup (job #1). All formulas below are transcribed from Avellaneda & Stoikov (2008), High-frequency trading in a limit order book, Quantitative Finance 8(3) - the canonical paper - and cross-checked against Guéant, Lehalle & Fernandez-Tapia (2013) and Cartea, Jaimungal & Penalva (2015). The numbers in the check column were re-executed and reproduced exactly (see §3).

Notation: $s$ mid-price, $q\in \mathbb{Z}$ inventory (shares), $t$ time, $T$ terminal horizon, $\tau =T-t$ remaining time, $\sigma$ mid-price volatility (per unit time), $\gamma >0$ absolute risk aversion, $k$ book liquidity density, $A$ order-arrival scale, ${\delta }^b,{\delta }^a$ distances of the bid/ask from the mid.

Critical scaling caveat. The paper’s Tables 1–3 report a spread of $1.29/1.33/1.15$ for $\gamma =0.1/0.01/0.5$ - those figures are the stationary $k$-component $\frac{2}{\gamma }\ln (1+\gamma /k)$ alone. The full spread at $t=0$ adds the inventory-risk term $\gamma {\sigma }^2(T-t)$. Confusing which component is which is the single most common lookup error in this model.

3. Computational Implementation - the AS quote engine

This runs on NumPy and reproduces every checked number above, including the paper’s own spread column. It is the same engine the sub-pages use.

The spread column reproduces the paper’s $1.29/1.33/1.15$ exactly (to their two decimals). Note that long inventory shifts both quotes down by $q\gamma {\sigma }^2\tau$: at $q=+10$ the MM is willing to sell below the mid-price because carrying the position is expensive.

4. Failure Modes & First-Principles Breakdowns

Hub signposts - the folder’s failure-mode analysis lives in 05 · Failure Modes & Practice. In one line each:

1. Adverse selection is not in the model. AS prices pure inventory risk over a Poisson flow of uninformed orders. Informed flow (pick-off risk) is invisible to it - our simulation shows the AS P&L only turning negative once roughly 60% of fills are informed (it stays positive to ~40%).
2. Parameter estimation kills it. The arrival parameters $A$ and $k$ must be estimated from the book, and $\sigma ,\gamma$ chosen; the strategy is only as good as those numbers. $k$ collapses in news, exactly when quotes matter most.
3. The terminal penalty vanishes near $T$. As $T-t\to 0$ the reservation skew $q\gamma {\sigma }^2(T-t)\to 0$, so the model abandons inventory protection just when liquidity evaporates (the reason production systems add a hard inventory cap - see 06 · Advanced Extensions).

5. References

• Avellaneda, Marco & Stoikov, Sasha: High-frequency trading in a limit order book, Quantitative Finance 8(3), 217–224 (2008). The canonical paper for this folder; every formula above transcribed and numerically verified.
• Guéant, Olivier, Lehalle, Charles-Albert & Fernandez-Tapia, Joaquin: Dealing with the inventory risk: a solution to the market making problem, Mathematics and Financial Economics 7(4), 477–507 (2013). *Rigorous HJB treatment
• Cartea, Álvaro, Jaimungal, Sebastian & Penalva, José: Algorithmic and High-Frequency Trading, Cambridge University Press (2015). The modern textbook; the A–S family, adverse selection, and execution in one voice.
• Ho, Thomas & Stoll, Hans: Optimal dealer pricing under transactions and return uncertainty, Journal of Financial Economics 9(1), 47–73 (1981). The intellectual ancestor: dealer pricing under inventory risk.
• Cartea, Álvaro & Jaimungal, Sebastian: Risk metrics and fine tuning of high-frequency trading strategies, Mathematical Finance 25(3), 576–611 (2015). *Generalizes A–S beyond exponential utility

6. Connected Graph Bridges

• Foundational base: Stochastic Calculus & Itô · Probability & Stochastic Control
• Sibling topic (in-pillar): Inventory Management & Quote Skewing · Adverse Selection & Glosten–Milgrom
• Sub-pages (in-folder): 01 From Zero · 02 The Market-Maker Problem · 03 The AS Model · 04 Inventory & Risk Aversion · 05 Failure Modes · 06 Advanced Extensions
• Cross-pillar: Optimal Execution & Almgren–Chriss

Beginner: start at 01 · Practitioner: start at 05