Category: Market Microstructure

Reading time: ~8 min

If you’ve spent any time in the market microstructure literature, you’ve probably come across the Avellaneda-Stoikov (A-S) model. Published in 2008, it’s one of those rare papers that gives you a clean, closed-form answer to a genuinely hard problem: where should a market maker post its bid and ask quotes?

I recently built a full simulation of the model — with several extensions drawn from Fushimi, González Rojas & Herman (2018) — and ran it across three volatility regimes to see how well the theory holds up in practice. Here’s the short version of what I found. The full technical article is available on my research page.

The Core Idea

A market maker sits in the order book all day, posting a bid and an ask. Every time both sides fill, it earns the spread. The risk is inventory: if the market moves against your accumulated position before you can unwind it, those spread gains disappear fast.

A-S solves this by introducing the reservation price: instead of quoting symmetrically around the mid-price [math]S_t[/math], the market maker shifts its quotes based on its current inventory [math]q_t[/math]:

r_t = S_t – q_t \, \gamma \, \sigma^2 (T – t)

where [math]\gamma[/math] is the risk-aversion coefficient, [math]\sigma[/math] is volatility, and [math]T – t[/math] is time remaining. If you’re long (positive [math]q[/math]), your reservation price drops — you quote a lower ask to encourage selling. If you’re short, the opposite happens. It’s inventory management baked directly into the pricing.

The optimal spread around [math]r_t[/math] is:

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

Two things to notice: the spread is wider when volatility is high (you need more compensation for the risk), and it narrows over time (near market close, you get aggressive to unwind). Both are intuitive.

What I Added

The vanilla A-S model is elegant but unrealistic for a few reasons. I implemented three extensions from Fushimi et al. (2018), plus one of my own:

1. Intraday volume pattern. Real order flow isn’t constant — it’s high at open, low at noon, high again at close (the classic U-shape). Following Cartea, Jaimungal & Penalva (2015), I model fill probability as [math]\alpha(t) \cdot e^{-k\delta}[/math] where [math]\alpha(t)[/math] follows:

\alpha(t) = \alpha_{\min} + (\alpha_{\max} – \alpha_{\min}) \left(\frac{2t}{T} – 1\right)^2

2. Three-state order book algorithm. Real market makers don’t just quote continuously — they track whether they have zero, one, or two live orders. Fushimi et al. (2018) formalise this as a state machine: if one side fills and the other doesn’t within a timeout, cancel and requote. I implement this with a 5-second timeout.

3. Dynamic order sizing. When your inventory is large, you should trade smaller on the side that makes it worse. The size model (from Fushimi et al. 2018):

\varphi^{\text{bid}}_t = \varphi^{\max} \cdot e^{-\eta \, q_t} \quad \text{if } q_t > 0

…and symmetrically for the ask when [math]q_t < 0[/math]. This exponentially reduces size in the direction of excess inventory.

4. Volatility-adjusted size cap (my extension). Fushimi et al. fix [math]\varphi^{\max} = 100[/math] for all stocks. In a cross-volatility study, this is a problem — at [math]\sigma = 5[/math], large positions create massive mark-to-market swings. I set:

\varphi^{\max} = \max\left(1,\; \frac{5}{\sigma}\right)

Without this, the optimal strategy actually loses money on average at high volatility (Sharpe [math]-0.66[/math]). With it, it recovers to [math]+0.91[/math]. That single change is arguably the most impactful thing in the whole paper.

The Experiment

I ran 1,000 Monte Carlo simulations per scenario, each simulating a full 6.5-hour trading day at 5-second resolution ([math]N = 4{,}680[/math] steps). Three volatility regimes:

Low ([math]\sigma = 0.5[/math]): think stable ETF
Medium ([math]\sigma = 2.0[/math]): typical large-cap
High ([math]\sigma = 5.0[/math]): volatile event-driven stock

I compared the extended A-S strategy against a naive fixed-spread baseline that always quotes at the time-independent component of the spread, with no inventory adjustment.

Results

Scenario	Strategy	Avg PnL	Std PnL	Sharpe
Low vol ([math]\sigma=0.5[/math])	Optimal	34.40	15.61	2.20
	Baseline	35.15	16.49	2.13
Medium vol ([math]\sigma=2.0[/math])	Optimal	7.68	4.99	1.54
	Baseline	8.77	8.53	1.03
High vol ([math]\sigma=5.0[/math])	Optimal	2.54	2.81	0.91
	Baseline	4.30	8.45	0.51

A few things stand out:

At low volatility, it barely matters. Both strategies are nearly identical. When price moves are tiny, inventory risk is negligible, and dynamic quoting adds almost nothing.

At medium volatility, variance control kicks in. The baseline actually earns more on average ($8.77 vs $7.68) — but at nearly twice the standard deviation. The Sharpe improvement is 50%: 1.54 vs 1.03.

At high volatility, the gap is enormous. The baseline blows up its P&L variance with undiscriminating quoting — it keeps filling at the fixed spread even as inventory accumulates dangerously. The optimal strategy widens its spread (it quotes 3–4× wider at [math]\sigma = 5[/math]) and trades less often (3.8 vs 6.8 trades/day). The Sharpe improvement is 78%: 0.91 vs 0.51.

The key insight: A-S doesn’t maximise expected P&L — the baseline often earns more per trade. It maximises risk-adjusted returns, trading a small reduction in mean for a large reduction in variance. That’s the whole point of the CARA utility framework.

What This Tells Us About Market Making

The model reveals a hierarchy of what matters:

Volatility-adjusted sizing is the most critical piece. Without it, the strategy fails at high vol.
Dynamic spread adjustment (the reservation price) matters most precisely when markets are most dangerous — high volatility, trending conditions.
The U-shaped intensity and Gamma fills add realism but don’t change the qualitative ordering.

This maps directly to what professional market makers do in practice: quote tighter on quiet large-caps, wider on volatile small-caps, and always smaller when you’re sitting on a large position.

Fig.V

Want the Full Details?

The complete technical write-up covers all the mathematics, the three-state algorithm pseudocode, the full parameter table, and eight figures including P&L paths, inventory distributions, and conditional P&L analysis. You can find it on my research page.

The Python simulation code is available in the repository linked.

References: Avellaneda & Stoikov (2008), Quantitative Finance 8(3). — Fushimi, González Rojas & Herman (2018), Stanford Technical Report. — Cartea, Jaimungal & Penalva (2015), Algorithmic and High-Frequency Trading, Cambridge UP.

Market Making with Avellaneda-Stoikov: Managing the Inventory

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Mar 5

Reading time: ~3 min
What Is a Market Maker?

When we talk about market microstructure, price formation is usually the first thing that comes up. In a simple market with few participants, prices emerge through direct negotiation — like bargaining at a market stall. If many participants compete for the same good, prices are determined through an auction, where the highest bid wins.

In financial markets, things are significantly more complex. There are many buyers, many sellers, and prices adjust continuously based on supply and demand. Unlike traditional goods markets, financial instruments trade extremely fast and continuously. In this setting, sellers provide the security while buyers provide the cash — and a third special participant enters the picture: the market maker.

A market maker is a unique agent that supplies what microstructure economists call immediacy — the ability for any participant to buy or sell very quickly, without having to wait for a natural counterparty. The market maker is a financial institution with enough liquidity to buy from sellers and resell instantly to buyers, acting as an intermediary.

At first glance, this role might seem redundant. Why not just let buyers and sellers find each other? In reality, market makers play a crucial role precisely when it matters most: when new information hits the market and participants want to transact fast. They absorb shocks, reduce transaction costs, and keep markets functioning.

How Do Market Makers Earn?

In an apparently simple way: they buy at the Bid price and resell at the Ask price, where Ask > Bid always. The difference — the Bid–Ask Spread — is their main source of income.

But this raises an immediate and non-trivial question: how does a market maker actually choose those prices? Set them too tight and you get picked off by informed traders who know more than you. Set them too wide and nobody trades with you. This is exactly the tension that the model I want to introduce captures so cleanly.

The Glosten–Milgrom Model (1985)

The model proposed by Glosten and Milgrom in their 1985 paper is a “toy model” — simple enough to solve by hand, rich enough to explain something fundamental about how spreads emerge. The central idea is that the spread exists not as pure profit for the market maker, but as protection against adverse selection: the risk of systematically trading against someone who knows more than you.

Setup

Assume a single stock with a binomial fundamental value: tomorrow it will be worth either:

$V_{Low} = 90 \quad \text{or} \quad V_{High} = 110$

There are two types of traders:

Informed traders know the true value with certainty. If [math]V = V_{High}[/math], they buy at the Ask (they’re getting a bargain). If [math]V = V_{Low}[/math], they sell at the Bid (or short-sell). Their order flow perfectly reveals private information.

Uninformed (noise) traders trade for exogenous reasons — liquidity needs, rebalancing, whatever. They buy with probability [math]\gamma[/math] and sell with probability [math]1 – \gamma[/math], regardless of the true value.

The market maker knows none of this at the individual level. They only observe whether the incoming order is a Buy or a Sell. They know:
- Any order is informed with probability [math]\alpha[/math]
- Any order is uninformed with probability [math]1 – \alpha[/math]
Their prior on the fundamental value is:

$\pi_0 = \mathbb{P}(V = V_{High})$

So their fair-value estimate before observing any order is:

$\mathbb{E}_0[V] = \pi_0 \cdot V_{High} + (1 – \pi_0) \cdot V_{Low}$

Deriving the Ask Price

The market maker sets the Ask price [math]A[/math] so that its expected profit on a Buy order equals zero — they don’t speculate, they just cover themselves.

On a Buy order, two things can happen:

1. The trader is informed (probability [math]\alpha[/math]). This only happens when [math]V = V_{High}[/math] — the informed only buy when the asset is cheap relative to its true value. The market maker sells at [math]A[/math] an asset worth [math]V_{High}[/math], losing:

$\alpha \cdot (A – V_{High}) < 0$

2. The trader is uninformed (probability [math](1-\alpha)\gamma[/math]). The market maker sells at [math]A[/math] an asset they value at [math]\mathbb{E}_0[V][/math], earning:

$(1 – \alpha)\gamma \cdot (A – \mathbb{E}_0[V]) > 0[$

Setting total expected profit to zero:

$\alpha(A – V_{High}) + (1 – \alpha)\gamma(A – \mathbb{E}_0[V]) = 0$

Solving for [math]A[/math]:

$\boxed{A = \frac{\alpha \cdot V_{High} + (1-\alpha)\gamma \cdot \mathbb{E}_0[V]}{\alpha + (1-\alpha)\gamma}}$

This is a weighted average of [math]V_{High}[/math] and [math]\mathbb{E}_0[V][/math], with weights proportional to the expected composition of buyers. By the same logic, the Bid price is:

$\boxed{B = \frac{\alpha \cdot V_{Low} + (1-\alpha)(1-\gamma) \cdot \mathbb{E}_0[V]}{\alpha + (1-\alpha)(1-\gamma)}}$

What the Formulas Tell Us

Fig. I

The figure above plots Ask, Bid, and Spread as functions of [math]\alpha[/math], with [math]\pi_0 = 0.5[/math] (symmetric prior), [math]V_L = 90[/math], [math]V_H = 110[/math], [math]\gamma = 0.5[/math].

A few things jump out immediately:

When [math]\alpha = 0[/math] (no informed traders), the spread collapses to zero. If all order flow is pure noise, there’s nothing to protect against — Ask = Bid = [math]\mathbb{E}_0[V] = 100[/math]. The market maker can quote at fair value with no risk.

As [math]\alpha[/math] grows, the spread widens monotonically. More informed traders in the market means the market maker is more likely to be on the wrong side of a trade. They protect themselves by quoting a higher Ask and a lower Bid.

At [math]\alpha = 1[/math] (all traders are informed), Ask = [math]V_{High}[/math] and Bid = [math]V_{Low}[/math]. The spread spans the entire range of uncertainty — the market maker fully passes the informational risk back to the counterparty.

The spread here is not a tax or a rent — it’s the equilibrium price of adverse selection risk.

The Spread as a Heatmap

Fig. II

The spread also depends on the prior [math]\pi_0[/math]. The heatmap shows the full surface: spread as a function of both [math]\alpha[/math] and [math]\pi_0[/math]. The spread is maximised when [math]\alpha[/math] is high (many informed traders) and [math]\pi_0 = 0.5[/math] (maximum uncertainty about the fundamental value). When [math]\pi_0[/math] approaches 0 or 1, the market maker is already fairly certain of the value, so adverse selection matters less even at high [math]\alpha[/math].

Price Discovery: The Market Learns Trade by Trade

One of the model’s most elegant features is that it describes price discovery — how the market maker updates its beliefs after each trade. Every order carries information, and the market maker uses Bayes’ rule to revise [math]\pi[/math] accordingly.

After observing a Buy order:

$\pi_1^{Buy} = \frac{[\alpha + (1-\alpha)\gamma] \cdot \pi_0}{\alpha \cdot \pi_0 + (1-\alpha)\gamma}$

After observing a Sell order:

$\pi_1^{Sell} = \frac{(1-\alpha)(1-\gamma) \cdot \pi_0}{(1-\alpha)(1-\gamma)\cdot\pi_0 + \alpha(1-\pi_0) + (1-\alpha)(1-\gamma)(1-\pi_0)}$

A Buy is evidence that the informed trader might have bought — so [math]\pi[/math] goes up (we become more confident [math]V = V_{High}[/math]). A Sell pushes [math]\pi[/math] down. As beliefs update, so do Ask and Bid — the market maker continuously reprices.

Fig. III

The figure simulates 40 trades under two scenarios: the true value is [math]V_{High}[/math] (blue) or [math]V_{Low}[/math] (red) with [math]\alpha = 0.30[/math]. In both cases, the market maker’s belief [math]\pi[/math] gradually converges toward the truth through Bayesian updating. The quoted prices follow suit — Ask and Bid converge to the true value. This is the GM model’s description of price discovery: information is incorporated trade by trade through the order flow.

What the Model Captures — and What It Doesn’t

The Glosten–Milgrom model is a first-principles story. It explains why spreads are non-zero, why they widen during high-uncertainty events, and how prices incorporate private information sequentially. For these three things, it’s hard to beat.

But it’s deliberately minimal:
- No inventory risk. The market maker has no concern about accumulating a large position — they just match orders at zero profit. In reality, the risk of holding inventory overnight or through a price shock is central to how market makers think. This is what the Avellaneda–Stoikov model (and others in that tradition) address.
- No strategic traders. Informed traders behave mechanically. Real informed traders optimise how much to trade, when, and at what size to avoid revealing their information too quickly.
- No competition. There’s one market maker. With multiple competing market makers, spreads are driven down further.
- Discrete, binary value. The binomial fundamental value is a simplification. Extensions to continuous distributions (as in the original Glosten–Milgrom 1985 paper) produce richer dynamics.
The Key Takeaway

The spread exists because the market maker cannot distinguish informed from uninformed traders. It is the competitive price of providing liquidity in the presence of adverse selection. In equilibrium, the gains from trading with uninformed noise traders exactly offset the losses from trading against the informed.

This is a result with real implications. It tells us:
- Spreads will be wider in opaque markets, before earnings announcements, or in assets where insider trading is more likely.
- Spreads will be narrower in liquid, transparent markets with mostly institutional noise traders and tight regulation.
- Order flow is informative. The direction of trades tells the market maker — and the rest of the market — something about the true value of the asset.
What’s Next

The Glosten–Milgrom model is a great starting point, but it treats the market maker as a passive agent who just absorbs information. In a follow-up article, I look at the Avellaneda–Stoikov model — where the market maker actively optimises their bid and ask quotes over time, accounting for both inventory risk and the passage of time toward market close. You can also read the full technical article on my research page.

References: Glosten, L.R. & Milgrom, P.R. (1985). Bid, ask and transaction prices in a specialist market with heterogeneously informed traders. Journal of Financial Economics.
A Simple Introduction to the Glosten–Milgrom Model and Market Making

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Nov 24

Category: Market Microstructure

The Core Idea

What I Added

The Experiment

Results

What This Tells Us About Market Making

Want the Full Details?

Market Making with Avellaneda-Stoikov: Managing the Inventory

What Is a Market Maker?

How Do Market Makers Earn?

The Glosten–Milgrom Model (1985)

Setup

Deriving the Ask Price

What the Formulas Tell Us

The Spread as a Heatmap

Price Discovery: The Market Learns Trade by Trade

What the Model Captures — and What It Doesn’t

The Key Takeaway

What’s Next

A Simple Introduction to the Glosten–Milgrom Model and Market Making