Phase 16 - Lesson 16.5

Market Making, Inventory Risk, and Adverse Selection

Quoting both sides to earn the spread, while managing the position you accumulate and the informed traders who pick you off.

⏱ 55 min● Advanced🔗 Prereqs: 16.2–16.4

↖ Phase 16 hub

Builds on: 16.2 introduced adverse selection and inventory as spread components; here they drive the quoting decision.
Leads to: 16.6 asks whether any such strategy survives honest backtesting and risk controls.

Learning Objectives

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Describe the market maker’s objective: earn the spread while controlling inventory and adverse selection.
Explain how inventory risk skews quotes away from the mid to mean-revert the position.
State the Avellaneda–Stoikov reservation price and optimal spread and interpret each term.
Contrast the profit from uninformed flow with the loss to informed flow.
Simulate a market maker’s inventory and P&L and analyze the risk–reward trade-off.

Key Vocabulary

Market maker: A liquidity provider who continuously quotes bid and ask, aiming to earn the spread across many round trips.

Inventory risk: The price risk from the net position a market maker accumulates when fills are one-sided.

Reservation price: The market maker’s inventory-adjusted fair value; the mid around which it centers its quotes.

Skewing quotes: Shifting both quotes to encourage trades that reduce inventory (e.g. lower both to sell down a long).

Avellaneda–Stoikov: A stochastic-control model giving optimal bid/ask quotes under inventory risk over a horizon.

Adverse selection: Losses from filling informed traders whose orders predict the next price move.

Fill intensity: The arrival rate of market orders hitting a quote, decreasing in how far the quote is from the mid.

Intuition & Motivation

Intuition

A market maker is a shopkeeper of liquidity: post a bid and an ask, and profit from the spread each time a buyer and later a seller cross them. Two forces erode that profit. First, fills are rarely balanced - a run of buyers leaves you short, a run of sellers leaves you long - so you carry inventory risk: an open position exposed to price moves. The cure is to skew your quotes: when you are long, lower both quotes to attract sellers-to-you (buyers of your ask) and shed the position, accepting a slightly worse average spread for less risk. Second, some counterparties are informed: they buy from you just before the price rises. You lose to them and must earn enough from the uninformed to come out ahead. Avellaneda–Stoikov turns this into a clean control problem: quote around an inventory-adjusted reservation price with a spread set by risk aversion, volatility, and fill intensity.

The objective and the two risks

Over a horizon $T$ the market maker maximizes expected utility of terminal wealth, where wealth is spread capture plus the mark-to-market of inventory. It faces:

Inventory risk: holding $q$ shares exposes $q\sigma$ of price risk; a large one-sided position can dwarf the spread earned.
Adverse selection: informed market orders arrive precisely when the quoted side is about to be wrong, so fills are biased against the maker.

Skewing quotes around a reservation price

Rather than quoting symmetrically around the mid $s$, the maker centers on an inventory-adjusted reservation price. In the Avellaneda–Stoikov model with inventory $q$, risk aversion $\gamma$, volatility $\sigma$, and time-to-close $(T-t)$:

\[r(s,q,t)=s-q\,\gamma\,\sigma^2\,(T-t),\] (16.10)

A long position $(q\gt 0)$ pulls the reservation price below the mid, so both quotes drop and the maker is more eager to sell - mean-reverting the inventory toward zero. The optimal total bid–ask spread is

\[\delta^{a}+\delta^{b}=\gamma\sigma^2(T-t)+\frac{2}{\gamma}\ln\!\Big(1+\frac{\gamma}{k}\Big),\] (16.11)

where $k$ governs how fill intensity decays with quote distance. The first term is the inventory/vol risk premium; the second is the compensation for the shape of order flow. Wider spreads earn more per fill but get fewer fills - the maker trades edge against volume.

Key Idea

Market making is a bet that you earn more from the spread on uninformed flow than you lose to informed flow and inventory moves. Skewing quotes manages inventory; a wide enough spread manages adverse selection.

Profit vs the informed

Decompose a market maker’s P&L over many trades:

Source	Sign	Driver
Spread capture	+	Round trips with uninformed (noise) traders
Inventory P&L	±	Price moves on the net position held
Adverse selection	−	Fills to informed traders before the price moves

A viable maker needs the spread capture to exceed the inventory variance cost plus adverse-selection bleed. When informed flow spikes (news), makers widen or pull quotes - exactly the Glosten–Milgrom logic from 16.2.

Worked Example - Why a long inventory lowers both quotes

Maker is long $q=+500$ after a burst of buyers lifted its ask.

Reservation price (16.10) shifts to $r=s-500\,\gamma\sigma^2(T-t)\lt s$: fair value is marked below the mid because the maker wants out.

Quoting $r\pm\delta$, both bid and ask fall; the lower ask is more likely to be hit (someone buys from the maker), selling inventory down.

Cost: the maker’s average sell price is slightly worse than the mid - it pays a little spread to reduce $q\sigma$ inventory risk. That is the risk–reward trade at the heart of market making.

Interactive: simulate inventory and P&L

Common Mistakes to Avoid

Quoting symmetrically regardless of inventory: an unmanaged position’s risk quickly exceeds the spread earned.
Ignoring adverse selection: capturing the quoted spread on paper means nothing if informed flow systematically picks the right side.
Setting the spread too tight for the volatility: you win more fills but lose the inventory-variance battle.
Assuming market making is a guaranteed positive-carry strategy: it is a risk-taking activity that can and does lose money.

Quant Practitioner Tips

Skew quotes toward flattening inventory and impose hard inventory limits; risk control beats cleverness.
Widen (or pull) quotes when adverse-selection signals fire (order-flow imbalance, news, volatility spikes).
Tie the spread to realized/expected volatility - the $\gamma\sigma^2(T-t)$ term is not optional.
Track P&L attribution across spread capture, inventory, and adverse selection to know why you make or lose money.

Knowledge Check

Q1 Medium

In the Avellaneda–Stoikov model, a market maker holding a large long inventory will:

Raise both quotes to buy more

Lower both quotes (reservation price below mid) to sell inventory down

Quote symmetrically around the mid

Stop quoting the bid only

Q2 Medium

Adverse selection reduces a market maker’s profit because:

Exchanges charge informed traders less

Informed counterparties tend to trade on the side that is about to be right, so those fills lose money

It widens the mid-price

It increases rebates

Q3 Hard

The optimal spread term $\gamma\sigma^2(T-t)$ tells the market maker to:

Quote tighter when volatility rises

Quote wider when volatility or risk aversion rises

Ignore volatility

Quote wider only near the close

Practical Exercise

(a) Write the Avellaneda–Stoikov reservation price and explain, for a short inventory $q\lt 0$, which way the quotes shift and why. (b) A maker captures an average $1$-cent spread on $10{,}000$ round trips but ends a volatile day with a large unhedged inventory whose price moved $50$ cents. Sketch the P&L attribution. (c) Explain why tightening the spread to win more fills can reduce profit.

▶ Show full solution

(a) $r=s-q\gamma\sigma^2(T-t)$. For $q\lt 0$ (short), $-q\gt 0$ so $r\gt s$: the reservation price is above the mid, both quotes rise, and the higher bid is more likely to be hit - the maker buys to cover the short and flatten inventory. Quotes always skew to mean-revert the position toward zero.

(b) Spread capture $\approx 10{,}000\times\$0.01=\$100$ of gross edge (before adverse selection). Inventory P&L: if the maker was left holding, say, $2{,}000$ shares and the price moved $\$0.50$ against it, that is $2{,}000\times(-\$0.50)=-\$1{,}000$ - an order of magnitude larger than the spread earned. Net P&L is dominated by the inventory term, illustrating that inventory risk, not spread, decides the day when positions are unmanaged.

(c) Tightening the spread increases fill volume but (i) lowers the edge per round trip and (ii) attracts more adverse-selected flow while accumulating inventory faster in trends. If the extra fills are disproportionately informed or build risky inventory, the marginal fills lose money and total profit falls - there is an interior optimal spread, not ‘tighter is always better.’

After the reveal, answer for yourself: How would a hard inventory cap change the shape of the P&L distribution (mean vs tails)?

Lesson Summary

A market maker earns the spread by quoting both sides but must manage inventory risk and adverse selection. Avellaneda–Stoikov centers quotes on an inventory-adjusted reservation price $r=s-q\gamma\sigma^2(T-t)$ - skewing to mean-revert the position - with an optimal spread that widens in volatility and risk aversion. Profit requires spread capture from uninformed flow to exceed inventory variance and the bleed to informed traders; it is a genuine risk-taking activity, not guaranteed carry.

Formula Sheet Additions

Reservation price

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

Inventory-adjusted fair value; long inventory pulls quotes down to sell it off.

Optimal spread

\[\delta^a+\delta^b=\gamma\sigma^2(T-t)+\tfrac{2}{\gamma}\ln\!\big(1+\tfrac{\gamma}{k}\big)\]

Inventory/vol risk premium plus an order-flow-shape term.

Error Log Checklist

Did I skew quotes by inventory rather than quote symmetrically?
Did I widen the spread with volatility (the $\gamma\sigma^2(T-t)$ term)?
Did I account for adverse selection, not just quoted spread capture?
Did I impose inventory limits / risk controls?

Retrieval Practice

Close the lesson and answer from memory before checking. This is deliberate, effortful recall - the single highest-yield study action.

▶ Show retrieval prompts & answers

Q: Why and how does a market maker skew its quotes when holding inventory?
A: To mean-revert the position: the reservation price $r=s-q\gamma\sigma^2(T-t)$ shifts against the inventory (down when long, up when short), moving both quotes so the flattening side is more likely to fill.

Q: What are the three P&L sources for a market maker and their signs?
A: Spread capture (+, from uninformed flow), inventory P&L ($\pm$, price moves on the net position), and adverse selection ($-$, fills to informed traders).

Q: How does the optimal spread depend on volatility and risk aversion?
A: It widens with $\sigma^2$ and $\gamma$ through the $\gamma\sigma^2(T-t)$ inventory-risk term; more risk demands more compensation per fill.

Flashcards

Click to flip. These feed the site-wide spaced-repetition queue.

Reservation price

$r=s-q\gamma\sigma^2(T-t)$: inventory-adjusted fair value; skew quotes to flatten $q$.

MM P&L sources

Spread capture (+), inventory ($\pm$), adverse selection ($-$).

Optimal spread

Widens with volatility and risk aversion: $\gamma\sigma^2(T-t)+\tfrac2\gamma\ln(1+\gamma/k)$.

Completion Checklist

I can explain the core ideas in my own words
I worked the derivations/examples by hand
I completed the interactive workbench(es)
I passed the knowledge check

Confidence / mastery rating

Personal notes

Source References

This lesson synthesizes and paraphrases concepts from the sources below. No copyrighted text, problem sets, or solutions are reproduced. Return to the originals for full depth.

Algorithmic and High-Frequency Trading (Cartea, Jaimungal & Penalva, 2015) current - Ch. 8–10 - Ch. 8–10: market making, inventory control, and the Avellaneda–Stoikov / Ho–Stoll frameworks.
Stochastic Calculus for Finance II (Steven Shreve, 2004) foundational - Ch. 5–6 - Ch. 5–6: stochastic control and utility maximization underlying the quoting problem.
Time Series Analysis (James Hamilton, 1994) foundational - Ch. 3 - Ch. 3: modeling order-flow arrival and inventory dynamics as time series.

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