6.2.3 The Avellaneda–Stoikov Model

Basic Prerequisites: 02 · The Market-Maker Problem.

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1. Intuition & Practical Objective

This is the complete closed-form lookup page for the AS model. The practical objective: one master recipe for the optimal bid and ask quotes, with the derivation of the asymptotic solution the paper actually uses. Everything below is AS 2008 §3, cross-checked numerically against the paper’s own tables.

The solution is a two-step procedure:

1. Compute the reservation price $r(s,q,t)$ - solve the PDE for $\theta$ (or use the asymptotic closed form) to get the dealer’s inventory-adjusted fair value.
2. Calibrate quotes to the book - solve the implicit first-order conditions for the distances ${\delta }^a,{\delta }^b$, which for exponential intensities gives the closed-form spread.

The headline result is beautifully compact:

$$\boxed{p^{\text{ask}}=r+\frac{1}{2}\psi ,p^{\text{bid}}=r-\frac{1}{2}\psi ,\psi =\gamma {\sigma }^2(T-t)+\frac{2}{\gamma }\ln \left(1+\frac{\gamma }{k}\right)}$$

with $r=s-q\gamma {\sigma }^2(T-t)$. The quotes are the reservation price plus/minus half the spread - and the spread itself does not depend on inventory (a consequence of exponential intensities). All inventory dependence lives in the skew of $r$.

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2. Mathematical Ground Truth & Derivations

2.1 The reduced equation for $\theta$

From 02, the ansatz $u=-\exp (-\gamma x)\exp (-\gamma \theta (s,q,t))$ turns the HJB into (AS eq. 3.3)

$${\theta }_t+\frac{1}{2}{\sigma }^2{\theta }_{ss}-\frac{1}{2}{\sigma }^2\gamma {\theta }_s^2+\underset{{\delta }^b}{\max }\frac{{\lambda }^b({\delta }^b)}{\gamma }\left[1-e^{+\gamma (s-{\delta }^b-r^b)}\right]+\underset{{\delta }^a}{\max }\frac{{\lambda }^a({\delta }^a)}{\gamma }\left[1-e^{-\gamma (s+{\delta }^a-r^a)}\right]=0,$$

with $\theta (s,q,T)=qs$. The first-order conditions for the two maxes give the implicit quote distances (AS eq. 3.6–3.7). For exponential intensities ${\lambda }^a(\delta )={\lambda }^b(\delta )=A{e}^{-k\delta }$ they solve explicitly.

2.2 Asymptotic expansion in inventory $q$

Expand $\theta (q,s,t)={\theta }_0+q{\theta }_1+\frac{1}{2}{q}^2{\theta }_2+\dots$ (3.10). The indifference relations (3.4)–(3.5) give

$$r^a={\theta }_1+(1-2q){\theta }_2+\dots ,r^b={\theta }_1+(-1-2q){\theta }_2+\dots ,$$

so that $r=\frac{r^a+r^b}{2}={\theta }_1-2q{\theta }_2$ and the spread is ${\delta }^a+{\delta }^b=2{\theta }_2+\frac{2}{\gamma }\ln (1+\frac{\gamma }{k})$. Grouping orders of $q$:

• Order 1: ${\theta }_t^1+\frac{1}{2}{\sigma }^2{\theta }_{ss}^1=0$, ${\theta }^1(s,T)=s$ ⇒ ${\theta }_1(s,t)=s$ - the reservation price at zero inventory is simply the mid-price.
• Order $q^2$: ${\theta }_t^2+\frac{1}{2}{\sigma }^2{\theta }_{ss}^2-\frac{1}{2}{\sigma }^2\gamma ({\theta }_s^1{)}^2=0$, ${\theta }^2(s,T)=0$ ⇒

$${\theta }_2(s,t)=\frac{1}{2}{\sigma }^2\gamma (T-t).$$

Hence $r={\theta }_1-2q{\theta }_2=s-q\gamma {\sigma }^2(T-t)$, recovering the frozen-inventory reservation price, and the spread is

$${\delta }^a+{\delta }^b=2{\theta }_2+\frac{2}{\gamma }\ln \left(1+\frac{\gamma }{k}\right)=\gamma {\sigma }^2(T-t)+\frac{2}{\gamma }\ln \left(1+\frac{\gamma }{k}\right).(3.18)$$

Reading the spread. It has two economics:

• $\frac{2}{\gamma }\ln (1+\frac{\gamma }{k})$ - the adverse-selection/book component: wider when risk aversion $\gamma$ is high or the book is thin (small $k$). In the risk-neutral limit $\gamma \to 0$ this tends to $2/k$ (finite).
• $\gamma {\sigma }^2(T-t)$ - the inventory-risk component: wider when the position can hurt (high $\gamma$, high ${\sigma }^2$, far from $T$). It vanishes at $t=T$.

2.3 The two-component interpretation of $\theta$

• ${\theta }_1$ = reservation price at zero inventory.
• ${\theta }_2=\frac{1}{2}\gamma {\sigma }^2\tau$ = sensitivity of the quotes to inventory. If ${\theta }_2$ is large, accumulating $q>0$ drives the quotes aggressively down. The skew per unit inventory is exactly $2{\theta }_2$ (the two terms are intentionally asymmetric in sign)… - see 04 · Inventory & Risk Aversion.

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3. Computational Implementation - the AS quote engine + paper verification

The engine below reproduces the paper’s own spread column exactly. NumPy only.

The stationary spread column reproduces the paper’s Tables 1–3 ($\gamma =0.1\Rightarrow 1.29$, $\gamma =0.01\Rightarrow 1.33$, $\gamma =0.5\Rightarrow 1.15$) to their published precision. Note the skew: at $q=+10$ both quotes sit $2 below the mid (bid $95.15$, ask $96.85$) - the dealer is actively trying to sell.

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4. Failure Modes & First-Principles Breakdowns

1. Quote crossing / marketable quotes. If the skew $q\gamma {\sigma }^2\tau$ exceeds half the spread, the “bid” rises above the “ask” of the unskewed market, or the quote becomes marketable (crosses the spread). Production engines clamp ${\delta }^a,{\delta }^b\ge 0$ and cap $|q|$; the raw formula does not.
2. Inventory-independent spread is an artefact. ${\delta }^a+{\delta }^b$ is independent of $q$ only because intensities are exponential and the expansion is linear. A quadratic approximation of the arrival term makes ${\theta }_2$ solve a nonlinear PDE (AS §3.2) and restores inventory dependence - don’t over-claim the closed form.
3. The linear $\theta$ expansion is only good near $q=0$. Guéant et al. (2013) show the approximation degrades for large inventories, where the true quotes must be clamped by an inventory limit. The AS formula is a local (small-$q$) result.
4. Spread is quoted around $r$, not $s$. A common bug is to quote the spread symmetrically about the mid and then add the skew - double-counting inventory. The correct construction: skew the centre to $r$ first, then add $\pm \psi /2$.

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5. References

• Avellaneda & Stoikov (2008), Quantitative Finance 8(3)
• Guéant, Lehalle & Fernandez-Tapia (2013), Math. & Financial Econ. 7(4)
• Cartea, Jaimungal & Penalva (2015), Algorithmic and High-Frequency Trading

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6. Connected Graph Bridges

• Back: 02 · The Market-Maker Problem
• Forward: 04 · Inventory & Risk Aversion · Index Hub
• Sibling: Inventory Management & Quote Skewing · Spread Decomposition & the Roll Model