The Avellaneda–Stoikov Model

The previous lesson handed you the market maker’s P&L and told you the whole job is “quote wide enough and skew cleverly enough.” Lovely advice. How wide? Skew how much? In 2008 Marco Avellaneda and Sasha Stoikov wrote down the answer — the model every modern quoting engine still cribs from. It is the F = ma of market making: two short formulas that take your inventory, your nerves, the volatility, and the clock, and spit out exactly where to put your two prices.

By the end you’ll be able to compute a bid and an ask from scratch, explain why a long maker quotes both prices below the mid, and recite the model’s blind spots so you don’t trade it like gospel.

The setup: a utility-maximizing maker

Analogy. Picture a market stall owner who must keep buying and selling all day, can’t go home until closing time, and hates being caught holding a pile of unsold stock when the bell rings. Every price tag is a bet: too generous and nobody trades with you, too greedy and you go home with a warehouse full of inventory and a heart full of regret. AS formalizes exactly that anxiety.

The precise model. A single maker quotes a bid and an ask over a finite horizon $[0,T]$ and wants to maximize the expected exponential utility of terminal wealth:

$\max\ \mathbb{E}\!\left[-\exp(-\gamma\, X_T)\right]$

where $X_T$ is wealth (cash plus inventory marked at the mid) at time $T$ and $\gamma>0$ is the risk-aversion coefficient. Exponential (CARA) utility is the trick that makes the math close: it punishes losses more than it rewards equal gains, which is precisely the fear of holding inventory.

Three modeling assumptions do the heavy lifting:

1. The mid follows arithmetic Brownian motion with volatility $\sigma$: $\,dS_t = \sigma\, dW_t$. No drift — the maker has no directional view, it only fears variance.
2. Orders arrive as a Poisson process whose intensity decays with distance from the mid. If your quote sits a distance $\delta$ from the mid, fills arrive at rate $\lambda(\delta) = A\,e^{-k\delta}.$ Quote close to the mid (small $\delta$) and you get hit often but earn a thin edge; quote far (large $\delta$) and fills dry up. $k$ controls how fast that trade-off bites.
3. Two control knobs: the bid distance $\delta_b$ and ask distance $\delta_a$ from a center the maker gets to choose.

The reservation (indifference) price

Analogy. The mid is the market’s fair value. The reservation price is your fair value once you account for the inventory you’re already carrying. If you’re already long a pile of stock, you secretly value it a little lower than the market does — you’d love an excuse to sell — so your personal fair value sits below the mid. Short the stock? Your fair value sits above the mid because you’re itching to buy it back.

The formula. The reservation price is

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

where $s$ is the current mid, $q$ is your net inventory (positive = long), $\gamma$ risk aversion, $\sigma^2$ variance, and $(T-t)$ time remaining. It is the price at which you’d be indifferent between holding your current inventory and trading one more unit — hence “indifference price.”

Worked example. Take $s=100$, $\sigma=2$ (so $\sigma^2=4$), $\gamma=0.1$, $(T-t)=1$, and inventory $q=5$ (you’re long 5):

$r = 100 - 5\cdot 0.1\cdot 4\cdot 1 = 100 - 2 = 98.$

Your personal fair value is 98, a full 2 below the market’s mid of 100 — because you’re long and want out. Now crank the knobs:

More time on the clock and more risk aversion both pull $r$ further from the mid: the longer you must carry the position and the more you fear it, the harder you lean to shed it.

The optimal spread

Analogy. The spread is your price for taking the other side. Charge too little and informed flow and variance eat you; charge too much and you never trade. AS shows the optimal width is the sum of two rents: an inventory-risk rent you charge because every fill saddles you with risk, and a liquidity/competition rent that depends on how quickly fills vanish as you back away from the mid.

The formula. The total optimal spread (ask distance plus bid distance) is

$\delta_a + \delta_b = \gamma\,\sigma^2\,(T-t) + \frac{2}{\gamma}\ln\!\left(1 + \frac{\gamma}{k}\right).$

• First term, $\gamma\sigma^2(T-t)$ — the inventory/risk term. Pure compensation for the variance you’ll carry: scales with risk aversion, variance, and time left. This is the same quantity (per unit inventory) that skewed the reservation price.
• Second term, $\frac{2}{\gamma}\ln(1+\gamma/k)$ — the market/liquidity term. Comes entirely from the order-arrival model. The faster fills decay with distance (large $k$ = lots of competition, liquid name), the smaller this term — you can’t afford to quote wide when rivals will undercut you.

Worked example. Use $\gamma=0.1$, $\sigma^2=4$, $T-t=1$, $k=1.5$:

So the total optimal spread is about 1.69, and the half-spread is $\delta/2 \approx 0.845$.

Placing the actual quotes

Analogy. You’ve found your personal fair value (the reservation price) and how far apart to set the two tags (the spread). Now just wrap the spread symmetrically around the reservation price — not around the mid. That last word is the whole game.

The formulas. With total spread $\delta = \delta_a+\delta_b$:

$\text{ask} = r + \frac{\delta}{2}, \qquad \text{bid} = r - \frac{\delta}{2}.$

Worked example (carrying our numbers). We had $r = 98$ and $\delta \approx 1.691$, so $\delta/2 \approx 0.845$:

Look closely: the mid is 100, yet both quotes — bid 97.155 and ask 98.845 — sit below the mid. The maker is long, so it has shoved its entire price ladder down to make its ask attractive (sell quickly!) and its bid unattractive (please stop selling me more!). That asymmetry around the mid is inventory skew, and it fell straight out of two formulas.

Drag the inventory, risk-aversion, and time sliders below and watch the reservation price and both quotes slide off the mid in real time:

Comparative statics: which knob does what

Analogy. Think of four dials on the quoting machine. Two outputs you care about: the skew $|r-s| = |q|\gamma\sigma^2(T-t)$ (how far your quotes lean off the mid) and the spread $\delta$ (how wide they are). Each dial pushes these two in characteristic directions.

| Turn up… | Effect on skew $|r-s|$ | Effect on spread $\delta$ | |---|---|---| | Risk aversion $\gamma\uparrow$ | More skew (lean harder to flatten) | Wider (inventory term up; liquidity term down — inventory usually dominates near close) | | Volatility $\sigma\uparrow$ | More skew (variance is scarier) | Wider (inventory term grows with $\sigma^2$) | | Time-to-close $(T-t)\uparrow$ | More skew (longer to carry risk) | Wider (inventory term grows) | | Liquidity/competition $k\uparrow$ | No direct effect ($k$ isn’t in $r$) | Tighter (the log liquidity term shrinks) |

The clean takeaway: inventory pressure ($\gamma,\sigma,T-t$) drives both skew and width, while competition ($k$) only tightens the market term and never touches your centering.

The end-of-horizon effect

Analogy. A trader who must be flat by the closing bell stops fearing inventory as the bell approaches — there’s no overnight gap to dread when there’s no overnight. AS bakes this in through the $(T-t)$ factor.

The mechanics. As $t\to T$, the term $(T-t)\to 0$, so:

• the skew $q\gamma\sigma^2(T-t)\to 0$ — the reservation price re-centers on the mid;
• the inventory term of the spread $\gamma\sigma^2(T-t)\to 0$ — that half of the spread vanishes.

What’s left at the very end is just the liquidity term $\frac{2}{\gamma}\ln(1+\gamma/k)$: quotes snap back symmetric around the mid and tighten to their competitive floor.

Worked example. Keep $q=5,\gamma=0.1,\sigma^2=4,k=1.5$ and watch $(T-t)$ run down:

By the bell, the maker quotes a symmetric $\pm 0.6455$ around 100 regardless of inventory. Drag the time slider to the far end of the simulator to feel the same collapse:

From the formulas back to inventory skew

You may have noticed AS never has a rule that says “if long, shade quotes down.” That behavior is emergent. The reservation price subtracts $q\gamma\sigma^2(T-t)$ — proportional to inventory — and the spread wraps symmetrically around it. Combine the two and the algebra produces the skew automatically:

$\text{bid} = s - \underbrace{q\gamma\sigma^2(T-t)}_{\text{skew}} - \frac{\delta}{2}, \qquad \text{ask} = s - \underbrace{q\gamma\sigma^2(T-t)}_{\text{skew}} + \frac{\delta}{2}.$

Both quotes ride the same skew off the mid; the spread just sets how far apart they sit. Get long, both slide down; get short, both slide up. This turns a wandering inventory into a mean-reverting one, because the skewed quotes make the flattening side of every trade more likely. The next lesson, Inventory Risk & Quote Skewing, zooms entirely into this behavior — but now you know exactly which line of algebra creates it.

Assumptions & limitations

AS is a model, not a money printer. Know where it bends:

• Constant volatility $\sigma$. Real volatility clusters and spikes; a fixed $\sigma$ under-quotes right when you most need width (around news).
• No explicit adverse selection. Order arrivals are assumed independent of the true value — fills don’t carry information. Reality disagrees: the trader lifting your ask often knows the price is about to rise. That’s the Glosten–Milgrom world, a separate lesson, and AS simply doesn’t price it.
• Symmetric half-spreads in the simple form. The closed-form approximation splits $\delta$ evenly ($\delta_a=\delta_b=\delta/2$). The exact solution lets the halves differ with inventory; practitioners often re-introduce that asymmetry.
• Finite vs. infinite horizon. The clean $(T-t)$ formulas assume a fixed terminal time. A continuously running maker has no natural “close,” so practice uses the Guéant–Lehalle–Fernandez-Tapia infinite-horizon closed forms, which replace $(T-t)$ with a stationary risk term and keep quotes well-behaved indefinitely.

Recap

Key Takeaways