6.9.1 From Zero

Basic Prerequisites: none - this page assumes only that you know a bid is the price at which someone will buy and an ask is the price at which someone will sell.

---

1. Intuition & Practical Objective

Imagine a currency-exchange booth at an airport. It posts buy €1 at $1.08 and sell €1 at $1.10. If a tourist sells €100, the booth pays $108. If the next tourist buys €100, the booth receives $110. The booth ends the day flat in euros and $2 richer. That $2 is the spread - the booth’s gross revenue. This is exactly what a market maker does, one hundred million times a day and one cent at a time.

The intuition a beginner must absorb first is that the spread is a price for a service, not a gift. The booth is selling immediacy: it guarantees that any tourist can trade right now, at a known price, without waiting for another tourist to appear. That service has a cost, and the whole economics of market making is the accounting of that cost:

Gross spread − adverse selection − inventory cost − fees + rebate = true profit.

Almost every intuition failure in this topic comes from looking only at the first term. The quoted spread is the headline, and every other term is a deduction from it. This page gives you the five-term picture; the rest of the folder makes each term precise.

The one-sentence essence. “A market maker is a shop that sells immediacy; its revenue is the spread, its cost of goods sold is adverse selection and inventory, and the exchange’s rebates are a supplier discount - profit is what’s left.”

---

2. Mathematical Ground Truth & Derivations

Work per share. Let the maker post a two-sided quote around the mid-price $m$: bid $m-h$, ask $m+h$, so the quoted spread is $S=2h$ and $h$ is the half-spread the maker captures on each fill (assuming the mid does not move between the two legs).

Term 1 - Gross spread capture. Each fill earns the maker $h$ per share.

$$\text{spread revenue}=h.$$

Term 2 - Adverse selection $\lambda$. Some of the counterparties know something. When an informed buyer lifts your ask, the mid is about to rise; you sold $h$ above the pre-trade mid but the mid itself jumps. If the mid moves against your new position by $\lambda$ on average, you lose $\lambda$ per share. (This is the whole content of Glosten–Milgrom: the winner’s curse.)

Term 3 - Inventory cost $c_{\text{inv}}$. Fills are one-sided in bursts. The inventory you accumulate must be financed (repo/capital), hedged, and carried through volatility - and it is a non-diversifiable risk you must be paid for (Stoll 1978; Grossman & Miller 1988). Write it as a per-share penalty $c_{\text{inv}}$.

Term 4 - Fees and rebates. If you post a resting order and it is hit, you are a maker and the exchange pays you a rebate $r\ge 0$ (equivalently a make fee $f_m=-r<0$). If you cross the spread, you are a taker and pay $f_t\ge 0$.

The master equation (per share, maker side):

$$\boxed{\pi =\underset{\text{spread}}{\underbrace{h}}+\underset{\text{rebate}}{\underbrace{r}}-\underset{\text{adverse selection}}{\underbrace{\lambda }}-\underset{\text{inventory}}{\underbrace{c_{\text{inv}}}}-\underset{\text{access fee}}{\underbrace{f_{\text{take}}\cdot 1[\text{taker}]}}}$$

Break-even half-spread. Set $\pi =0$ and solve for the quote that just covers costs:

$$h^{\star }=\lambda +c_{\text{inv}}-r.$$

Two immediate readings, both central to this folder:

1. The rebate lowers the break-even spread. A maker receiving $r$ can quote $r$ tighter and still break even - this is the mechanism by which exchanges believe rebates improve liquidity (see 03 · Maker-Taker Fees & Rebates).
2. Cash terms are the only tradable lever. $h$ is set by competition, $\lambda$ by the counterparties you cannot avoid, $c_{\text{inv}}$ by your capital and risk appetite. The rebate is the one line an exchange can hand you directly - which is exactly why it became a central piece of market-structure policy.

---

3. Computational Implementation - the five terms, in dollars

Standard library only. This is the mental model made arithmetic: a desk that captures a penny of half-spread and pays for it in adverse selection and inventory, with and without the exchange rebate.

Read the last two lines carefully. The rebate is 40% of net profit when it is present, and removing it cuts the desk’s profit by 40% ($500k\to$300k). This is the empirical reason the maker-taker debate is so bitter: a payment that looks like a rounding error on the spread is a first-order term in the margin. That is the subject of the rest of this folder.

---

4. Failure Modes & First-Principles Breakdowns

1. Reading the quoted spread as revenue. The quoted pencil-and-paper spread $S=2h$ is gross. The maker keeps $h+r-\lambda -c_{\text{inv}}$. In the example above, the headline spread is $0.020$ and the true take is $0.005$ - the spread overstates profit by 4×. This single error is the most common one in casual commentary.
2. Treating the rebate as free money. It is not: $r$ is financed by the taker’s $f_t$, and under competition it is largely passed through into a tighter quote (Colliard & Foucault 2012). Ignoring the pass-through double-counts the subsidy.
3. Ignoring adverse selection because it is invisible on the confirm screen. Every fill prints a price; no blotter line says “you were picked off.” $\lambda$ is an expectation, measurable only statistically (→ VPIN, 02 · Market-Maker P&L).
4. Assuming both legs fill. The two-tourist example is the ideal: bid and ask both filled at the same mid. Real spreads are earned when two-sided flow arrives; when only one side arrives you are left holding inventory, which is Term 3.

---

5. References

• Stoll, Hans R. (1978), The supply of dealer services in securities markets, JF 33(4)
• Grossman & Miller (1988), Liquidity and market structure, JF 43(3)
• Hasbrouck (2007)
• Demsetz (1968), The cost of transacting, QJE 82

---

6. Connected Graph Bridges

• Next: 02 · Market-Maker P&L - the decomposition made precise and simulated.
• Model inputs: Adverse Selection & Glosten–Milgrom (Term 2) · Inventory Management & Quote Skewing (Term 3).
• Hub: Market-Maker Economics & Rebates - Index Hub