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# Chapter 11: Market Microstructure and Liquidity

*Part III: Asset Markets — Financial Economics: Claims, Prices, and Holders*

***

## Opening Episode: 2:32 P.M., May 6, 2010

It had already been an ugly day. Greek sovereign spreads were widening, European equity markets had fallen, and volatility in US index products had been elevated since the open. Then, at about 2:32 in the afternoon, a large asset manager began selling.

The order was not disguised and it was not clever. It was an instruction to sell seventy-five thousand E-mini S\&P 500 futures contracts — roughly four billion dollars of notional exposure — executed by an algorithm told to participate at nine percent of the trailing minute's trading volume, with no instruction about price and no instruction about time. The algorithm did what it was told. As the market's volume rose, the algorithm sold faster. As it sold faster, volume rose.

What happened next is the CFTC-SEC staff's account, published on September 30, 2010, and the numbers below are approximate and theirs. Between roughly 2:41 and 2:45 the E-mini contract fell about three percent. The Dow Jones Industrial Average dropped something on the order of six hundred points in about five minutes — on top of an already substantial decline, making an intraday fall of nearly a thousand points, close to nine percent — and then recovered those six hundred points within roughly twenty minutes. Over twenty thousand trades in more than three hundred securities executed at prices more than sixty percent away from where the same securities had traded moments earlier. Some of those trades printed at one cent per share. Some printed at one hundred thousand dollars per share.

Those two prices are the chapter in miniature. Neither was a valuation. Both were **stub quotes** — placeholder bids and offers, posted absurdly far from the market, that a market maker leaves standing to satisfy a nominal obligation to quote while intending never to trade on them. When the real quotes vanished, the stubs were what remained, and the incoming market orders executed against them. A market order is an instruction to trade at the best available price. The instruction does not check whether the best available price is a price anybody meant.

The staff report's central finding is not about the algorithm. It is about who was on the other side. Market makers in the E-mini and in individual equities, faced with a sustained one-directional flow they could not interpret, did three things in quick succession: they bought, accumulating inventory; they turned around and sold what they had bought, to each other, generating enormous volume with almost no net absorption — twenty-seven thousand contracts changed hands in fourteen seconds at one point, close to half of all trading volume, against net buying of about two hundred contracts; and then a number of them stopped. They paused their systems, withdrew their quotes, or widened to stubs. Buy-side depth in the E-mini fell to something like one percent of its level that morning. Demand for immediacy spiked, and the supply of immediacy went away.

That conjunction is the thesis of this chapter and the reason it sits in Part III rather than in an appendix on trading mechanics. Every chapter so far has treated the price of a claim as the answer to a question about payoffs and discount rates. It is also the answer to a question about who will take the other side of your trade right now, at what size, and at what price — and the entities that answer that question are holders, with balance sheets, inventory limits, and the legal right to stop. Liquidity is not a property of a security. It is a service, produced by identifiable people using scarce capital, and like every other service it can be withdrawn.

Sections 11.1 through 11.4 build the machinery: what liquidity is, the two canonical models of how informed trading makes a spread and moves a price, and how the resulting quantities are measured. Section 11.5 is the chapter's center — liquidity as something investors are paid to bear and something that constrains what they can hold. Section 11.6 returns to 2:32 p.m. with the tools.

***

## 11.1 What Liquidity Is

A liquid market is one in which you can trade a lot, quickly, at a price close to the one you saw, without moving it much. That sentence contains four distinct properties, and the literature since Kyle (1985) has organized them into three dimensions plus a caveat.

**Tightness** is the cost of a small round trip: buy and immediately sell, and the loss is the bid-ask spread. It is the dimension everyone quotes because it is the one that is posted.

**Depth** is the size that can be traded at the quoted prices, and more generally the quantity required to move the price by a given amount. A market can be tight and shallow — a penny wide for a hundred shares and nothing behind it — which is a fair description of much of the US equity market's displayed order book.

**Resiliency** is the speed at which prices return after a shock unrelated to information. A resilient market absorbs a large uninformed sale and comes back; a brittle one takes the sale as news and stays down.

The caveat is that all three are conditional on how fast you insist on going. Almost any position can be liquidated at close to its mid-price if you are willing to take a month. **Immediacy** is the thing actually being bought, and Demsetz's (1968) formulation remains the cleanest: the bid-ask spread is the price of immediacy, paid by whoever is unwilling to wait for a natural counterparty and collected by whoever is willing to stand between two arrivals. Somebody must sell that service, which means somebody must hold the claim in the interval — and that person's willingness is where holder constraints enter.

Why is the spread positive? Three reasons, and it is worth stating them before the models that formalize them, because the models each isolate one.

**Order processing costs.** Exchange fees, clearing and settlement, technology, the market maker's own capital and labor. These are real but small in modern equity markets, and they are the part of the spread competition has crushed.

**Inventory costs.** A market maker who buys from a seller is left holding a position she did not choose, exposed to price risk until she can lay it off. She will quote asymmetrically to encourage the trades that flatten her book — shading both quotes down when she is long — and she will widen when her inventory or her risk limit is close to binding. This is the component that connects directly to Chapter 19: inventory capacity is balance-sheet capacity.

**Adverse selection.** Some of the people trading against her know something she does not. She loses to them systematically, by construction, because they buy from her precisely when the claim is worth more than she thinks. She cannot identify them trade by trade, so she recovers the loss from everyone, by quoting a spread. This is the component that survives perfect competition, zero costs, and infinite capital, and it is the one Sections 11.2 and 11.3 model.

The third component is the deep one, and it is Chapter 7 §7.2's economics wearing different clothes. Grossman and Stiglitz showed that a price cannot fully reveal information because the informed must be compensated; here we ask who pays them. The answer is: the uninformed, through a spread that a price-setting intermediary charges to break even against a population she cannot sort.

***

## 11.2 Glosten-Milgrom: Adverse Selection Makes the Spread

Glosten and Milgrom (1985) strip the problem to its skeleton. Traders arrive one at a time and trade one unit each. A **market maker**, competitive and risk-neutral, posts a bid and an ask before each arrival and must trade at her posted prices with whoever shows up. Competition drives her expected profit on each trade to zero, so her quotes must be conditional expectations of the claim's value given the trade she is about to do.

That last sentence is the entire model. Write $$x$$ for the claim's eventual payoff, taking two values $$x\_H$$ and $$x\_L$$. A fraction $$n$$ of arriving traders are **informed** — they have seen $$x$$ — and the remaining $$1-n$$ are **liquidity traders** who buy or sell with probability one half each for reasons unrelated to value. The informed buy when $$x = x\_H$$ and sell when $$x = x\_L$$. Then

$$
\text{ask} = E\big\[x \mid \text{buy}\big], \qquad \text{bid} = E\big\[x \mid \text{sell}\big]
$$

and the spread is $$s = E\[x \mid \text{buy}] - E\[x \mid \text{sell}]$$.

**A worked case.** Let $$x\_H = 110$$ and $$x\_L = 90$$, each with prior probability one half, and let $$n = 0.20$$. Conditional on the high value, a buy arrives with probability $$n + (1-n)/2 = 0.60$$; conditional on the low value, with probability $$(1-n)/2 = 0.40$$. Bayes' rule gives a posterior on the high state of $$0.60$$ after a buy and $$0.40$$ after a sell, so

$$
\text{ask} = 0.60(110) + 0.40(90) = 102, \qquad \text{bid} = 0.40(110)+0.60(90) = 98
$$

The spread is 4, on a pre-trade expected value of 100 — four percent of the price, produced by nothing but the possibility that the counterparty knows something. In this symmetric two-state case the algebra collapses to a formula worth remembering:

$$
s = n(x\_H - x\_L)
$$

The spread is the probability of facing an informed trader multiplied by the amount that trader knows. Both halves matter, and they are the two levers every real market maker actually pulls. Set $$n = 0$$ and the spread is zero even though the claim is enormously risky: risk alone does not make a spread in this model, and neither do costs, which are assumed away. Set $$n = 1$$ and the spread is the entire range of possible values — the market maker will quote 90 bid, 110 offered, which is a way of saying she will not trade at all. A market in which everyone is informed does not function, which is the Grossman-Stiglitz impossibility result restated in quotes rather than in demand curves.

**Prices discover.** The quotes are not static. Each trade is evidence, so the posterior moves and the next pair of quotes moves with it. Table 11.1 follows three arrivals and Figure 11.1 draws them.

**Table 11.1: Quote revision in the Glosten-Milgrom model** ($$x\_H = 110$$, $$x\_L = 90$$, $$n = 0.20$$)

| Trade | Prior on high state | Pre-trade expected value | Bid    | Ask    | Spread | Trade that arrives |
| ----- | ------------------- | ------------------------ | ------ | ------ | ------ | ------------------ |
| 1     | 0.5000              | 100.00                   | 98.00  | 102.00 | 4.00   | Buy at 102.00      |
| 2     | 0.6000              | 102.00                   | 100.00 | 103.85 | 3.85   | Buy at 103.85      |
| 3     | 0.6923              | 103.85                   | 102.00 | 105.43 | 3.43   | Sell at 102.00     |
| —     | 0.6000              | 102.00                   | —      | —      | —      | —                  |

*Source: author's calculation from the model in the text; posteriors by Bayes' rule, quotes as conditional expectations. Entries rounded to two decimals.*

![Figure 11.1: Glosten-Milgrom updating](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-e3be7a62b82301d5ad82b9ad38a22c92afd4df1d%2Ffig_11_01_glosten_milgrom_updating.png?alt=media)

**Figure 11.1: Glosten-Milgrom updating.** Panel (a): Table 11.1's three arrivals. Each vertical rule is a quote pair — ask above, bid below, and the pre-trade expected value between them — and the marker is the price the arriving trader actually pays or receives. The dashed arrows carry the point the table makes twice: the ask before a buy *is* the expected value after it, so executing at the quote moves the market maker's belief to the price she just traded at, and the transaction series bounces around a martingale. The spread narrows from 4.00 to 3.85 to 3.43 as her beliefs move away from the middle and there is less left for her to lose. Panel (b): where the opening spread comes from. In the symmetric two-state case it is exactly the probability of facing an informed trader multiplied by what that trader knows. At n = 0 there is no spread however risky the claim is — risk alone does not make a spread in this model. At n = 1 the quote is 90 bid, 110 offered, which is a way of declining to trade: Grossman-Stiglitz's impossibility result, written in quotes rather than in demand curves. *Source: Author's calculation from the model of §11.2 and Table 11.1.*

Three features of the table are general, and each is a fact about real markets.

The **ask before a buy equals the expected value after it**. Look at row 1's ask, 102.00, and row 2's pre-trade expected value, 102.00. This is not a coincidence of the parameters: the ask *is* $$E\[x \mid \text{buy}]$$, so executing at it moves the market maker's belief exactly to the price she just traded at. Prices follow a martingale with respect to the market maker's information, and the transaction price series bounces between bid and ask around a random walk. That structure is what Section 11.4's Roll estimator exploits.

The **spread narrows as beliefs move away from the middle**: 4.00, then 3.85, then 3.43. The spread tracks how much the market maker still has to lose, and by row 3 she has already learned a good deal. Spreads are wide when uncertainty about value is high, which is why they widen around earnings announcements and — the point Section 11.6 needs — in the middle of an unexplained selloff.

And **the market maker breaks even in aggregate while losing every informed trade**. At the row 1 quotes she loses eight per informed trade — selling at 102 what is worth 110, or buying at 98 what is worth 90 — and makes two per liquidity trade. With $$n = 0.20$$ the two legs are $$-1.60$$ and $$+1.60$$. The spread is a transfer from liquidity traders to informed traders, intermediated by someone who nets to zero. Nobody in this model is exploiting anybody; the uninformed are paying for the privilege of trading in a market where prices are informative.

**The link to Chapter 7.** Section 7.2's Grossman-Stiglitz economy has traders submitting demand schedules to an auctioneer who clears them at a price that is a noisy signal of the informed traders' information. This model has the same two populations, the same noise, and the same conclusion — the informed are compensated, the uninformed pay, and the price becomes informative without becoming fully revealing. What changes is the institution. There is no auctioneer. There is a firm that quotes, and the compensation to the informed is a spread rather than a favorable position in a Walrasian allocation. Chapter 7's price informativeness $$\Psi$$ becomes the speed at which the posterior converges: run the table long enough and quotes reach the true value, at a rate governed by $$n$$. Information gets into prices through orders, and the spread is the toll.

***

## 11.3 Kyle: Price Impact and Depth

Glosten and Milgrom's informed trader is passive — he trades one unit and takes what he gets. Kyle (1985) makes him strategic, and the result is the single most used object in empirical microstructure.

**The setting.** One risky claim with terminal payoff $$x \sim N(p\_0, \sigma\_x^2)$$, where $$p\_0$$ is the pre-trade price. There is one informed trader, who observes $$x$$ exactly and chooses a signed quantity $$q$$. **Liquidity traders** submit a net quantity $$z \sim N(0, \sigma\_z^2)$$, independent of $$x$$ — the same noise that Chapter 7 §7.2 requires, here measured in shares rather than in per-capita supply. Orders are not executed one at a time. They are pooled into a **batch auction**: a competitive market maker observes only the net order flow

$$
\mathrm{OF} = q + z
$$

and sets a single price at which everything clears. She cannot see whose order is whose. That is the whole friction, and it is exactly the friction of Chapter 7: a high price could mean good news or it could mean that the liquidity traders happened to buy.

**The equilibrium.** Guess that the market maker uses a linear rule and that the informed trader responds linearly. Then the equilibrium is

$$
p = p\_0 + \Lambda\_K\mathrm{OF}, \qquad q = \frac{x - p\_0}{2\Lambda\_K}, \qquad \Lambda\_K = \frac{\sigma\_x}{2\sigma\_z}
$$

**Kyle's lambda**, $$\Lambda\_K$$, is the price impact coefficient: the price change per unit of net order flow. Its reciprocal, $$1/\Lambda\_K = 2\sigma\_z/\sigma\_x$$, is **market depth** — the quantity of order imbalance the market can absorb per unit of price movement, and the formal version of the second dimension of Section 11.1. (The symbol is capitalized and subscripted deliberately. This book's $$\lambda\_k$$ is a price of risk in a beta representation, and the two objects have nothing to do with each other; $$\Lambda\_K$$ is never written as a bare lambda here.)

Read the formula for depth before doing anything else with it. Depth rises with $$\sigma\_z$$, the volume of uninformed trading, and falls with $$\sigma\_x$$, the amount of private information there is to have. A market is deep when there is a lot of noise to hide in and little worth knowing. Nothing in the expression refers to the market maker's capital, her risk aversion, or her costs — she is risk-neutral and competitive by assumption, so the entire impact is adverse selection. Kyle's lambda measures the information content of trading, and only derivatively anyone's capacity to bear risk. Sections 11.5 and 11.6 explain why the two get conflated in practice, and why that conflation made May 2010 possible.

**A worked case.** Take a mid-cap stock trading at fifty dollars a share. Suppose the standard deviation of its terminal value is five dollars — ten percent uncertainty — and that liquidity traders' daily net imbalance has a standard deviation of one million shares. Then

$$
\Lambda\_K = \frac{5}{2 \times 1{,}000{,}000} = 2.5 \times 10^{-6} \ \text{per share}
$$

so a hundred thousand share buy imbalance moves the price by twenty-five cents, or fifty basis points, and depth is four hundred thousand shares per dollar of price movement. Now give the informed trader a signal: he learns that the claim is worth fifty-six dollars, six above the pre-trade price. He does *not* buy as much as he can. He buys

$$
q = \frac{56 - 50}{2 \times (2.5\times 10^{-6})} = 1{,}200{,}000 \ \text{shares}
$$

and if the liquidity traders happen to net to zero the price ends at fifty-three dollars — exactly halfway to the truth, which is what Figure 11.2 draws. That is not an artifact of the numbers. In this equilibrium the market maker's posterior variance is always $$\sigma\_x^2/2$$: the informed trader reveals precisely half of what he knows, whatever the parameters, because trading twice as aggressively would move the price against him faster than the extra size is worth. His expected profit, at that signal, is 3.6 million dollars; averaged over signals it is $$\sigma\_x\sigma\_z/2$$, or 2.5 million.

![Figure 11.2: Kyle's lambda](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-6021561dbdc959dc570a7ee16b1faa9ae06e587c%2Ffig_11_02_kyles_lambda.png?alt=media)

**Figure 11.2: Kyle's lambda.** The worked case of this section: a mid-cap stock at fifty dollars, a terminal-value standard deviation of five dollars, and a liquidity-trader imbalance with a standard deviation of a million shares. Panel (a): the market maker's rule. She sees only the net order flow and prices it linearly, at 2.5 cents per hundred thousand shares — a depth of four hundred thousand shares to the dollar. Panel (b): why the informed trader buys 1.2 million shares and not 2.4 million. His expected profit is a concave quadratic in his own order, because he is choosing a quantity against a residual supply curve he himself creates; a price-taker who ignored that would buy twice as much and push the entire six dollars of mispricing into the price, earning nothing. Stopping at half leaves the price at fifty-three — exactly halfway to the truth — and three dollars a share of expected profit. The halfway result is not an artifact of these numbers: in this equilibrium the market maker's posterior variance is half the prior's whatever the parameters are. *Source: Author's calculation from the worked case of §11.3.*

The scaling result is the one to carry away. **The informed trader's optimal size is proportional to the amount of noise trading**, at $$\sigma\_z/\sigma\_x$$ shares per dollar of mispricing. He camouflages himself in the liquidity traders' flow, taking exactly as much as the crowd will hide. Double the noise and he doubles his position, doubles his profit, and leaves price informativeness unchanged — which is precisely Chapter 7's comparative static in $$\sigma\_z$$, reproduced in a model with no auctioneer. More liquidity trading supports more informed trading and buys no additional price accuracy.

### ★ Where the equilibrium comes from

*Starred. A reader who skips this keeps the three formulas and the intuition and loses the fixed point.*

The equilibrium is a pair of best responses solved together, because each side's linear coefficient is the other side's parameter.

*The informed trader's side.* Take the market maker's rule $$p = p\_0 + \Lambda\_K \mathrm{OF}$$ as given and note that the informed trader's expected profit, conditional on $$x$$ and averaging over $$z$$, is

$$
E\big\[q(x-p)\big] = q\big(x - p\_0 - \Lambda\_K q\big)
$$

since $$E\[z] = 0$$. This is a concave quadratic in $$q$$; the first-order condition $$x - p\_0 - 2\Lambda\_K q = 0$$ gives $$q = (x-p\_0)/(2\Lambda\_K)$$. He trades half of what a price-taker would, because he internalizes his own impact. Note the monopoly structure: he is choosing a quantity against a residual supply curve of his own making.

*The market maker's side.* Take the informed trader's rule as given. Order flow is $$\mathrm{OF} = (x-p\_0)/(2\Lambda\_K) + z$$, a normal variable, and $$x$$ is normal, so the conditional expectation is linear:

$$
E\[x \mid \mathrm{OF}] = p\_0 + \frac{\mathrm{Cov}(x, \mathrm{OF})}{\mathrm{Var}(\mathrm{OF})}\mathrm{OF}
$$

Writing the informed trader's intensity as $$\iota = 1/(2\Lambda\_K)$$, we have $$\mathrm{Cov}(x,\mathrm{OF}) = \iota\sigma\_x^2$$ and $$\mathrm{Var}(\mathrm{OF}) = \iota^2\sigma\_x^2 + \sigma\_z^2$$, so zero expected profit for the market maker requires

$$
\Lambda\_K = \frac{\iota\sigma\_x^2}{\iota^2\sigma\_x^2 + \sigma\_z^2}
$$

*The fixed point.* Substituting $$\iota = 1/(2\Lambda\_K)$$ and solving yields $$\iota = \sigma\_z/\sigma\_x$$ and hence $$\Lambda\_K = \sigma\_x/(2\sigma\_z)$$. Feeding that back gives $$\mathrm{Var}(x\mid \mathrm{OF}) = \sigma\_x^2 - (\iota\sigma\_x^2)^2/(\iota^2\sigma\_x^2+\sigma\_z^2) = \sigma\_x^2/2$$, the half-revelation result. Problem 6 works the extension to $$N$$ competing informed traders, where $$\Lambda\_K = (\sigma\_x/\sigma\_z)\sqrt{N}/(N+1)$$ and the residual variance is $$\sigma\_x^2/(N+1)$$: competition among the informed makes the market deeper and prices more informative, and in the limit destroys the informational rent entirely.

**Lambda as an empirical object.** Kyle's model licensed an entire measurement program. Regress price changes over some interval on signed order flow over the same interval,

$$
\Delta p\_t = \Lambda\_K \mathrm{OF}\_t + \text{error}
$$

and the slope is an estimate of price impact for that security in that period. Signing the flow requires knowing which side initiated each trade, which is why the literature depends on trade-and-quote data and on algorithms that classify a trade as buyer-initiated when it prints above the prevailing midpoint. Estimated lambdas are used to price the execution of a large order, to compare venues, to detect informed trading around corporate events, and — the use Section 11.5 cares about — as a state variable whose *movements* describe aggregate liquidity conditions. What they are not is a structural constant. A lambda estimated in a calm week and applied to a stressed one will understate the cost of trading by an order of magnitude, for the reason Section 11.6 gives.

***

## 11.4 Measuring Liquidity

Four families of measures, in ascending order of data requirements and descending order of availability.

**Quoted and effective spreads.** The **quoted spread** is ask minus bid, usually expressed relative to the midpoint. It overstates trading costs whenever executions happen inside the quotes, which they routinely do. The **effective spread** measures what a trade actually cost:

$$
\text{effective spread} = 2\big|p\_{\text{trade}} - m\_{\text{pre}}\big|
$$

twice the distance from the prevailing midpoint $$m\_{\text{pre}}$$ to the execution price, doubled to put it on a round-trip footing. The effective spread splits, and the split is the empirical counterpart of Section 11.1's components. Let $$m\_{+}$$ be the midpoint some interval after the trade — five minutes is the convention. Then the **price impact** is $$2(m\_{+} - m\_{\text{pre}})$$ for a buy, and the **realized spread** is $$2(p\_{\text{trade}} - m\_{+})$$, so that

$$
\text{effective spread} = \text{realized spread} + \text{price impact}
$$

Price impact is the adverse-selection component: the permanent revision in the market's belief caused by the trade. The realized spread is what the liquidity supplier keeps after that revision — her compensation for order processing and inventory risk. Effective spreads on large US stocks are now a very few basis points, and their realized-spread portion is often near zero — a statement about how thin the market-making business has become.

Figure 11.3 shows why the two measures differ, on a book of the kind §11.1 described as tight and shallow.

![Figure 11.3: Spread, depth, and the book](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-c0de8d0a8ab21ec73b1c5adb1ab08a04845d2bd5%2Ffig_11_03_spread_depth_and_the_book.png?alt=media)

**Figure 11.3: Spread, depth, and the book.** Panel (a) is a limit order book on a forty-dollar stock with a penny tick: a hundred shares at the touch on each side, and progressively more resting further away. The quoted spread is the whole of what a screen displays, and here it is one tick — two and a half basis points — for the first hundred shares a side. Panel (b) walks a five-thousand-share market buy up that ladder. The order fills a hundred shares at forty dollars, then four hundred a penny higher, and so on until the last share pays 40.04; the average fill is 40.0302, and the shaded area between the staircase and the pre-trade midpoint is what the order paid for immediacy — 176 dollars, an effective spread of 7.04 cents, or 17.6 basis points against a quoted 2.5. The ratio of the two numbers is the whole reason §11.4 distinguishes them. A quoted spread describes the first hundred shares; the slope of the staircase describes every share after them, and it is that slope, not the quote, that a large holder pays and that Kyle's lambda measures. The corollary is the one the tick paragraph draws: a fall in quoted spreads is not by itself evidence that trading got cheaper, because the cost can move from the quote, where it is measured, into the slope, where it is not.

**The tick, which binds all of this from below.** No quoted spread can be narrower than the minimum price increment, so wherever the tick binds, a spread measure reports the rule rather than the market. Decimalization in 2001 cut the US equity tick from a sixteenth of a dollar to a penny, and what followed had been forecast in advance and in detail: narrower quoted spreads, thinner displayed depth at each price, quotes revised so often that the displayed book flickers, and a weakened time priority for the public limit order, since a competitor can step ahead of it for a hundredth of a dollar. The predictable consequence is that liquidity supply migrates from displayed public orders toward fast intermediaries who can afford to requote continuously. Which means that a fall in measured spreads across this period is not by itself evidence that executing a large order became cheaper — the cost may simply have moved from the spread, where it is measured, to the impact of working an order against depth that is no longer displayed.

**Roll's estimator, in two lines.** Suppose the efficient price follows a random walk and the transaction price is the efficient price plus or minus half the spread, with the sign an independent coin flip — bid-ask bounce, exactly the structure Table 11.1 generated. Write the trade indicator as $$\pm 1$$ and $$s$$ for the spread. Then $$\Delta p\_t$$ contains $$(s/2)(\text{sign}\_t - \text{sign} \_{t-1})$$, and

$$
\mathrm{Cov}\big(\Delta p\_t, \Delta p\_{t-1}\big) = -\frac{s^2}{4} \quad \Longrightarrow \quad \hat s = 2\sqrt{-\mathrm{Cov}\big(\Delta p\_t, \Delta p\_{t-1}\big)}
$$

because the efficient-price innovations are serially uncorrelated and drop out, leaving only the cross-term $$-(s/2)^2 E\[\text{sign}\_{t-1}^2]$$. Roll (1984) is the reason this chapter can be done with free daily closing prices: a first-order autocovariance of $$-0.0225$$ in squared dollars implies a spread of thirty cents, which on a forty-dollar stock is seventy-five basis points. What the estimator misses is everything that makes the sign of the autocovariance positive — trends, price continuation, autocorrelated order flow — and in practice the sample autocovariance comes out positive for a substantial minority of stock-months, where the estimator is undefined. There is no clean fix — Problem 3 works through the choices, none of them innocent — and the honest use of Roll is as a cheap, noisy, comparable measure across many securities rather than a precise one for any single security.

**Amihud's ILLIQ.** Amihud (2002) proposed measuring price impact directly from daily data by asking how much price movement a dollar of trading generates:

$$
\mathrm{ILLIQ}\_i = \frac{1}{D}\sum \_{d=1}^{D} \frac{|r \_{i,d}|}{\text{dollar volume} \_{i,d}}
$$

It is a poor man's Kyle lambda, in return space rather than price space, using absolute returns in place of signed order flow because daily data cannot sign the flow. Its virtue is that it can be computed for every security in the world with a price and a volume, going back decades. Table 11.2 shows the range it spans.

**Table 11.2: The Amihud measure across three representative stocks**

|           | Mean absolute daily return | Mean daily dollar volume (millions) | $$\mathrm{ILLIQ}\times 10^{6}$$ | Implied price move per ten million dollars traded | Ratio to mega-cap |
| --------- | -------------------------- | ----------------------------------- | ------------------------------- | ------------------------------------------------- | ----------------- |
| Mega-cap  | 1.10%                      | 12,000                              | 0.92                            | 0.0009%                                           | 1                 |
| Mid-cap   | 1.70%                      | 120                                 | 141.7                           | 0.14%                                             | 155               |
| Small-cap | 2.60%                      | 3                                   | 8,667                           | 8.67%                                             | 9,455             |

*Source: author's calculation from the stylized inputs in the first two columns, which are illustrative of the orders of magnitude found across US equity size deciles rather than figures for any named security. Returns in decimal form, dollar volume in millions.*

Four orders of magnitude separate the top and the bottom of the cross-section, and the ordering is monotone in size. It is also the fact behind two verdicts in Chapter 6 — §6.2's, that the size premium sat disproportionately in the smallest microcaps, and §6.4's, that equal-weighted results carry a microcap tilt. The size effect and the liquidity effect are measured on nearly the same stocks, and any claim that small firms earn a premium has to reckon with the fact that trading them is expensive in a way that the returns of a paper portfolio never record. What ILLIQ misses is that it does not separate volatility from illiquidity — a stock whose price moves a lot for good fundamental reasons will look illiquid — and that dollar volume is itself endogenous to liquidity, so the measure is partly a scaled inverse of trading activity.

**Price impact regressions.** With trade-and-quote data, estimate $$\Lambda\_K$$ directly, as above. This is the most faithful measure and the least available: intraday data are licensed, large, and demanding. The literature's practical wisdom is that the cheap measures rank securities well and track conditions over time poorly — which matters, because the time series is what Section 11.5 needs.

***

## 11.5 Liquidity as a Priced Characteristic and a Holder Constraint

Everything so far has been about the cost of a trade. This section is about the price of a claim, and it is the reason the chapter exists.

**The premium, from the holder's arithmetic.** Suppose a claim costs $$s$$ to trade, round trip, and a holder expects to hold it for $$H$$ years before selling. The trading cost, amortized, is $$s/H$$ per year. A holder will not accept a lower gross return than she could get elsewhere; she needs the illiquid claim's expected return to exceed the liquid claim's by roughly the difference in amortized cost. Table 11.3 does the arithmetic for two claims and three holders.

**Table 11.3: Amortized trading cost, in basis points per year**

| Expected holding period | Liquid claim (spread 0.4%) | Illiquid claim (spread 2.0%) | Difference |
| ----------------------- | -------------------------- | ---------------------------- | ---------- |
| 3 months                | 160                        | 800                          | 640        |
| 2 years                 | 20                         | 100                          | 80         |
| 10 years                | 4                          | 20                           | 16         |

*Source: author's calculation, cost equal to the round-trip spread divided by the holding period in years. The two spread levels and the three holding periods are stylized, chosen to span the range found across US claim classes and holder types rather than measured for any named security or investor.*

Read down the last column: the compensation an investor requires for holding the illiquid claim depends on how long she intends to hold it, and varies by a factor of forty across these three holders. Amihud and Mendelson (1986) drew the consequence: in equilibrium, illiquid claims are held by long-horizon investors, because they are the ones for whom the cost is small — a **clientele effect**. And because the marginal holder of a high-spread claim has a longer horizon than the marginal holder of a low-spread claim, the equilibrium liquidity premium is **concave** in the spread. Doubling the spread does not double the premium; it partly reassigns the claim to a more patient owner. That is a pricing result derived entirely from who holds the claim and for how long. Chapter 18 §18.6 is the same argument at the extreme, where the claim cannot be sold at all and the clientele is endowments and pension plans.

**The evidence, in two forms.** Amihud (2002) established both. In the cross-section, stocks with high ILLIQ earn higher subsequent returns, controlling for size, beta, and the usual characteristics — the level of illiquidity is priced. In the time series, an unexpected *rise* in market illiquidity lowers contemporaneous stock prices, and a high level of expected market illiquidity forecasts high subsequent excess returns. The second finding is the more interesting: liquidity is not only a cost to be amortized, but a risk factor.

**Pástor-Stambaugh: the risk, not the level.** Pástor and Stambaugh (2003) made that precise and gave the literature its standard construction. They build a monthly measure of aggregate liquidity from the tendency of individual stocks' returns to reverse following volume — the signature of order flow that had to be paid for — average it across stocks, and extract the innovations. That series is a state variable describing the condition of the market as a whole. Then they ask a Chapter 6 question about it: does exposure to it earn a premium? Sort stocks on their **liquidity beta**, the loading on innovations in aggregate liquidity, and the high-beta portfolio outperforms the low-beta portfolio by an economically large margin — on the order of seven and a half percentage points a year in their 1966-1999 sample, after adjusting for market, size, value, and momentum exposure.

The economics is the same as everywhere else in this book. A stock whose price falls hard when market liquidity dries up is a bad claim to own, because it does badly in exactly the state where you may need to sell it and where everything else you own is also doing badly. It must therefore be cheap. In the language of Chapter 3, aggregate liquidity is a state variable that enters $$m$$; in the language of Chapter 6, its innovation is a factor $$f\_k$$ with a positive price of risk. Two claims can have identical average trading costs and different liquidity betas, and the second characteristic is priced separately from the first.

![Figure 11.4: Liquidity through time](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-5464b47890355cda4ec782813ff66d16536bae31%2Ffig_11_04_liquidity_through_time.png?alt=media)

**Figure 11.4: Liquidity through time.** Pástor and Stambaugh's innovation in aggregate market liquidity, monthly from 1962, on the series Pástor posts and updates. The six worst months are numbered, and the roll is worth reading before anything else: October 1987, September 1998, November 1973, September 2008, April 2000, May 1970. Nobody told the series about any of them. It is estimated from the tendency of individual stocks' returns to reverse following volume — the signature of order flow that had to be paid for to be absorbed — and the months it finds unaided are the months in which the market's capacity to intermediate failed. That is the strongest available evidence that the construction is measuring what it claims to measure, and it is why the series can serve as the state variable §11.5 prices claims against. Two cautions the figure also carries. The ordering among the extremes is not a ranking of severity, because the measure is an estimate and its standard error in a month of extreme volatility is large. And March 2020 comes tenth rather than second, which is a statement about what a monthly reversal-based measure can see — the March episode was concentrated in a fortnight, in Treasuries and corporate bonds more than in equities, and in the funding markets Chapter 19 is about rather than on the equity tape this series reads. *Source: Luboš Pástor's posted update of the Pástor and Stambaugh (2003) aggregate liquidity series; NBER recession dates.*

**Flight to liquidity.** The clearest evidence for a liquidity state variable is that it moves violently and that everything reprices when it does. Two episodes already in this book are instances. Chapter 9 §9.4 records that off-the-run Treasuries traded at large discounts to their nearly identical on-the-run twins in March 2020 — two claims with almost the same cash flows and very different prices, the difference being entirely which one could be sold quickly. Chapter 19's opening episode narrates the same days from the dealer's side, where the constraint sat. The same spread blew out in the autumn of 1998, during the Long-Term Capital Management unwind, which makes the pattern a regularity: in stress, investors do not merely move from risky claims to safe ones, they move from claims they can exit to claims they can exit faster, and they pay for the privilege. A liquidity premium that was twenty basis points in June becomes two hundred in October, on the same securities.

**Liquidity spirals.** Why does the premium move so much? Chapter 16 §16.5 gives the canonical statement, and this chapter supplies the microstructure application. Brunnermeier and Pedersen's mechanism links **market liquidity** — the cost of trading, the object of this chapter — to **funding liquidity**, the ease with which a levered holder can finance a position. A market maker's ability to hold inventory depends on how much financing her positions attract, and her financing terms depend on the volatility and liquidity of what she holds. A price shock raises volatility, which raises haircuts, which forces levered liquidity suppliers to reduce inventory, which means they widen quotes and pull depth, which raises volatility again. The spiral runs entirely through balance sheets: no information about the claim's payoff need change at any point.

That mechanism is what makes liquidity a *holder constraint* rather than only a priced characteristic. In Kyle's model, depth is a statement about how much private information exists. In a spiral, depth is a statement about how much unencumbered capital the liquidity suppliers have this morning. Both are real, they are not the same object, and the empirical measures of Section 11.4 do not distinguish them. When a lambda estimate rises, the market maker may have concluded that order flow has become more informative, or she may simply have run out of room. Section 11.6 argues that both were happening on May 6, 2010, and that they are hard to tell apart precisely because the market maker's optimal response to each is the same: quote wider, or stop.

> **Box 11.1 — Market design, compressed**
>
> Where trading happens has changed more in twenty years than in the previous hundred, and four features of the current US equity market are worth naming.
>
> **Fragmentation.** A given stock trades on a dozen-odd exchanges and on a larger number of off-exchange venues simultaneously, rather than at one specialist's post. Regulation NMS, adopted in 2005, is what makes this coherent: its order protection rule requires that a trade not execute at a price inferior to a better quote displayed at another venue, which stitches the fragments into one notional book at the cost of a great deal of message traffic and a permanent race to be first.
>
> **Dark pools.** Venues that do not display quotes and execute at or inside the public midpoint, used mainly by institutions trying to move size without revealing it — a direct response to the Kyle problem, since an order that never appears in the flow has no impact. Off-exchange execution is a large fraction of US share volume. The externality is that the public book, which is where the price everyone references is formed, is left with a residual flow that is more informed than the whole.
>
> **High-frequency market makers.** The firms that replaced the specialists quote continuously and hold inventory for seconds. May 2010 left an ambiguous verdict on them, since they both absorbed the first wave of selling and were among those who stopped; Section 11.6 takes it up.
>
> **Payment for order flow.** Retail brokers route customer orders to wholesalers who pay for them, and the wholesalers can profitably execute those orders inside the public spread for exactly the Glosten-Milgrom reason: retail orders are, on average, uninformed, so a market maker who can buy the right to trade only against them faces a lower $$n$$ and can quote a tighter spread than one who must quote to everybody.
>
> The serious treatment of all of this — limit order book dynamics, venue competition, the empirical industrial organization of exchanges — is Foucault, Pagano and Röell's, and it is the depth reference in the readings. This chapter takes from market design only what the pricing argument needs.

***

## 11.6 Who Holds Liquidity, and What Their Constraints Do to Its Price

Part III's mandated question, asked here, is unusually concrete: not who holds this claim in general, but who is holding it right now, in the seconds between a seller's arrival and a buyer's — who supplies liquidity, in other words, and when they stop.

**Four suppliers, in historical order.** The **specialist** on the New York Stock Exchange floor held a monopoly franchise in an assigned set of stocks and, in exchange, an affirmative obligation to maintain a fair and orderly market — to bid when nobody else would. The franchise paid for the obligation, and both are gone. **Dealers** in quote-driven markets, from the Nasdaq of the 1990s to the corporate bond market today, hold inventory and quote two-sided prices in competition, with no obligation to do so. **High-frequency trading firms** are the current marginal supplier in listed equities and futures: they quote continuously, hold inventory for seconds rather than days, and finish most days close to flat. And **patient institutions** supply liquidity passively without thinking of themselves as doing it, every time an index fund rebalancing over a week works a limit order and lets others trade against it — a channel nearly invisible in the market-making literature and enormous in aggregate.

**What each requires to stay.** Three inputs, and the models above name two of them. First, **capital**: any supplier who buys from a seller must finance and bear the position until she can sell it, and how much she can hold is set by her equity, her haircuts, and her risk limits. This is Chapter 19 §19.5's argument at trade frequency — the marginal value of a dollar of intermediary capital is what prices the claim, and the spread and depth she quotes are that shadow value made observable. Second, **hedging capacity**: a dealer who buys a corporate bond can lay off the interest-rate risk in Treasuries and much of the credit risk in an index swap, retaining only the idiosyncratic residual, so her willingness to make a price depends on whether those hedges are cheap. When the hedge stops working, the same position consumes far more risk budget and quotes widen with no change in the bond. Third, **protection from adverse selection**: the Glosten-Milgrom spread is only viable if the supplier can charge it. Every institution that segments order flow — payment for order flow, dark pools with participant restrictions, dealer relationships in over-the-counter markets, the size at which a quote is firm — exists to lower the $$n$$ that a given quote faces, which is why arrangements that look anticompetitive can coexist with historically narrow spreads.

**The withdrawal mechanics.** Liquidity supply stops in three ways, and they usually happen together. **Inventory limits bind**: a market maker who has absorbed one-directional flow for an hour has a position, and every additional unit of the same sign is worth less to her than the last; her quotes shade away from the flow, then stop. **Adverse selection spikes**: a sustained imbalance is itself evidence, and a supplier who cannot distinguish an uninformed algorithm from an informed one must assume the worst, which in the model of Section 11.2 means raising her estimate of $$n$$ and widening. The two are observationally identical from outside — both produce wider quotes and less depth — and they call for opposite policy responses. And **funding tightens**: haircuts rise, financing is pulled, and the position becomes unholdable at any price the supplier is willing to quote. Chapter 16 §16.5's spiral is the general statement; Chapter 19 §19.5 prices it.

**May 6, 2010, with the tools.** The sell program that started at 2:32 p.m. was, in this chapter's language, a very large liquidity trade: an uninformed order executed by an algorithm indifferent to price. Kyle's model says such an order should be absorbed at a cost proportional to $$\Lambda\_K$$, and the market's initial response was exactly that: HFT market makers took the other side and accumulated inventory. Then three things went wrong at once. The algorithm's participation rule made its selling *increase* with volume, so the market makers' own recycling of inventory among themselves — the hot-potato trading the staff report documents — called forth more selling, a feedback loop no model in this chapter contains. Inventory limits bound, and suppliers who had been buying became sellers, flipping the sign of their contribution to depth. And the adverse-selection question became unanswerable: a persistent one-way flow of that size, on a day of European sovereign stress, could not be distinguished in real time from someone who knew something. The correct Glosten-Milgrom response to an unidentifiable spike in $$n$$ is to widen without limit, and a quote widened without limit is a stub quote at one cent or one hundred thousand dollars. The trades that printed there were the model's prediction, executed literally.

The recovery is as informative as the crash. Prices came back in twenty minutes — resiliency, in Section 11.1's terms, and evidence that nothing about any company's cash flows had changed. What had changed, for those twenty minutes, was the capacity of the holders willing to stand in the middle. That is why the regulatory response — single-stock circuit breakers, then limit up-limit down bands, a ban on stub quotes, and clearer rules for breaking erroneous trades — is best read not as a fix but as an admission. The supply of immediacy comes from private firms with the right to withdraw it, and an orderly market is not a claim any of them has contracted to honor. The specialist's affirmative obligation was abolished because its monopoly franchise was worth more than the obligation cost. Nobody has replaced either half.

In Chapter 1 §1.2's terms, a widening spread is either news or a constraint binding, and from outside the two are indistinguishable — which is why none of §11.4's measures can tell you whether the market maker has learned something or has simply run out of room.

***

## Elsewhere in the Series

* **The full treatment of market microstructure** — Foucault, Pagano and Röell, *Market Liquidity: Theory, Evidence, and Policy* (Oxford University Press, 2013). Limit order book models, venue competition, the industrial organization of exchanges, and the empirical microstructure literature at book length. This chapter takes the two canonical models and the pricing consequence; that book is where a reader goes next, and it is the depth reference for Box 11.1.
* **Global equity market structure, cross-listing, and the international geography of trading** — *International Finance*, Chapter 14, which also owns ETF arbitrage mechanics and index-flow dynamics. Section 11.5's liquidity premium is priced here; where the trading happens is developed there.
* **The information economics this chapter implements** — this book, Chapter 7 §7.2. Grossman-Stiglitz is the demand-curve statement; Sections 11.2 and 11.3 are the price-setting statement, with the same two populations and the same noise.
* **On-the-run versus off-the-run Treasuries and the convenience yield** — Chapter 9 §9.4. **March 2020 from the dealer's side, and the intermediary discount factor** — Chapter 19, especially §19.5. **Fire sales, margin spirals, and the canonical constrained-capital statement** — Chapter 16 §16.5, which this chapter applies rather than restates. **The liquidity component of corporate credit spreads, and the swap-based decomposition that isolates it** — Chapter 10 §10.4, which cites §11.5's two results for the pricing of the level and the risk. **Illiquid claims and the horizon clientele at the extreme** — Chapter 18 §18.6. **Demand systems, in which this chapter's liquidity is one of the characteristics demand is written over** — Chapter 20 §20.2.

***

## Summary

1. **Liquidity has three dimensions and one caveat.** Tightness is the cost of a small round trip, depth the quantity tradable per unit of price movement, resiliency the speed of recovery after an uninformed shock. All three are conditional on urgency: what is actually bought is immediacy, and someone must sell it.
2. **The spread has three components, and only one survives perfect competition.** Order processing costs are small and shrinking; inventory costs reflect the supplier's balance sheet; adverse selection is what remains when costs are zero and capital is infinite, and it is what the two canonical models isolate.
3. **Glosten-Milgrom: the spread is a conditional expectation.** A competitive market maker quotes $$E\[x\mid\text{buy}]$$ and $$E\[x\mid\text{sell}]$$, which in the symmetric two-state case gives $$s = n(x\_H-x\_L)$$ — the probability of facing an informed trader times the size of what he knows. She loses to the informed and recovers it from the uninformed, netting to zero.
4. **Quotes are a martingale and prices discover.** The ask equals the post-trade expected value, so transaction prices bounce around a random walk; spreads narrow as the market maker's residual uncertainty falls (Table 11.1). This is Chapter 7 §7.2's economics with a price-setting firm in place of the auctioneer.
5. **Kyle: price impact is** $$\Lambda\_K = \sigma\_x/2\sigma\_z$$ **and depth is its reciprocal.** The informed trader scales his order to the noise, at $$\sigma\_z/\sigma\_x$$ shares per dollar of mispricing, and reveals exactly half his information whatever the parameters. Depth rises with uninformed volume and falls with the amount of private information available — in this model it has nothing to do with the market maker's capital.
6. **Kyle's lambda is the empirical workhorse and is not a structural constant.** Estimated as the slope of price changes on signed order flow, it measures the information content of trading; a lambda estimated in calm conditions understates the cost of trading in stressed ones by an order of magnitude.
7. **Four measurement families, in descending order of availability.** Quoted and effective spreads, with the effective spread splitting into a realized spread (the supplier's compensation) and a price impact (the adverse-selection component); Roll's estimator, $$\hat s = 2\sqrt{-\mathrm{Cov}(\Delta p\_t,\Delta p\_{t-1})}$$, which needs only daily closes and fails when the autocovariance is positive; Amihud's ILLIQ, a poor man's lambda spanning four orders of magnitude across the size cross-section (Table 11.2); and direct price impact regressions, which need licensed intraday data.
8. **Illiquidity is priced as a characteristic, through the holder's horizon.** Amortized cost $$s/H$$ makes the required premium depend on who owns the claim (Table 11.3), which produces a clientele — illiquid claims migrate to patient holders — and makes the equilibrium premium concave in the spread.
9. **Liquidity risk is priced separately from the liquidity level.** Pástor and Stambaugh's aggregate liquidity innovations are a state variable; sorting on liquidity beta produced a spread on the order of seven and a half percentage points a year in their sample, risk-adjusted. A claim that falls when the market becomes illiquid is a bad claim to own, so it must be cheap.
10. **Liquidity is supplied by holders who can stop, and the Flash Crash is what that means.** Specialists gave way to dealers, dealers to high-frequency firms, alongside patient institutions supplying passively; each requires capital, hedging capacity, and protection from adverse selection, and each withdraws when inventory limits bind, adverse selection spikes, or funding tightens. On May 6, 2010, all three happened at once, quotes widened to stubs, and prices recovered in twenty minutes because nothing about any claim's payoff had changed.

***

## Key Terms

* **Immediacy**: The service of trading now rather than waiting for a natural counterparty; what the bid-ask spread is the price of
* **Tightness, depth, resiliency**: The three dimensions of liquidity — the cost of a small round trip, the quantity absorbable per unit of price movement, and the speed of recovery from an uninformed shock
* **Bid-ask spread** $$s$$: The difference between the price at which a liquidity supplier will buy and the price at which she will sell; decomposes into order processing, inventory, and adverse-selection components
* **Adverse selection component**: The part of the spread that compensates the liquidity supplier for systematically losing to traders who know more than she does; the part that survives zero costs and infinite capital
* **Market maker**: A dealer who posts two-sided quotes and stands ready to trade at them, holding inventory in the interval between a seller's arrival and a buyer's
* **Stub quote**: A placeholder bid or offer posted far from the market to satisfy a nominal obligation to quote, with no intention of trading on it
* **Kyle's lambda** $$\Lambda\_K$$: The price impact coefficient, $$\Lambda\_K = \sigma\_x/(2\sigma\_z)$$ in the one-period model — the price change per unit of net order flow. Never written as a bare lambda in this book, which reserves that letter for the price of risk
* **Market depth** $$1/\Lambda\_K$$: The net order imbalance the market absorbs per unit of price movement
* **Net order flow** $$\mathrm{OF}$$: Signed buy volume minus sell volume over an interval; the informed trader's order plus the liquidity traders'
* **Batch auction**: A mechanism in which orders arriving over an interval are pooled and cleared at a single price, so that the market maker sees only the aggregate
* **Effective spread**: Twice the distance from the prevailing midpoint to the execution price; splits into the realized spread and the price impact
* **Realized spread**: The effective spread net of the post-trade revision in the midpoint; what the liquidity supplier actually keeps
* **Roll's estimator**: $$\hat s = 2\sqrt{-\mathrm{Cov}(\Delta p\_t, \Delta p\_{t-1})}$$; recovers the spread from bid-ask bounce in daily closing prices alone
* **Amihud's ILLIQ**: The average ratio of absolute daily return to daily dollar volume; a price impact measure computable from free daily data
* **Liquidity premium**: The higher expected return required on a claim that is costly to trade, scaled by the holder's expected holding period
* **Clientele effect (in liquidity)**: The equilibrium assignment of illiquid claims to long-horizon holders, which makes the liquidity premium concave in the spread
* **Liquidity beta**: A claim's loading on innovations in aggregate market liquidity; priced, per Pástor and Stambaugh
* **Flight to liquidity**: The reallocation, in stress, from claims that can be exited to claims that can be exited faster, at a price
* **Liquidity spiral**: The feedback between market liquidity and funding liquidity, in which falling prices raise haircuts, force liquidity suppliers to shed inventory, and reduce depth further

***

## Readings

### Required

* Kyle, A. S. (1985). "Continuous Auctions and Insider Trading." *Econometrica* 53(6): 1315-1335. *The origin of price impact as an equilibrium object. Read the single-period model in Section 2 for the three formulas of §11.3; the continuous-time version that gives the paper its title is the same economics with the informed trader smoothing his order over the day.*
* Glosten, L. R. and P. R. Milgrom (1985). "Bid, Ask and Transaction Prices in a Specialist Market with Heterogeneously Informed Traders." *Journal of Financial Economics* 14(1): 71-100. *The spread derived from adverse selection alone, with no inventory and no costs. The result that quotes are conditional expectations, and therefore that prices are a martingale in the market maker's information, is what makes the model tractable and what §11.2's table illustrates.*

### Recommended

* Foucault, T., M. Pagano and A. Röell (2013). *Market Liquidity: Theory, Evidence, and Policy*. Oxford University Press. *The depth reference for this chapter and the natural next book. Everything Box 11.1 compresses into a paragraph is developed there at length, with the limit order book models this chapter omits entirely.*
* Amihud, Y. (2002). "Illiquidity and Stock Returns: Cross-Section and Time-Series Effects." *Journal of Financial Markets* 5(1): 31-56. *Introduces the ILLIQ measure and establishes both halves of the pricing result — illiquid stocks earn more, and expected market illiquidity forecasts market returns. The measure's durability comes from its data requirements, which are nil.*
* Chordia, T., R. Roll and A. Subrahmanyam (2000). "Commonality in Liquidity." *Journal of Financial Economics* 56(1): 3-28. *Individual stocks' spreads and depths move together, which is the precondition for treating aggregate liquidity as a state variable at all. Without this result, Pástor and Stambaugh's factor would have nothing to load on.*
* Pástor, Ľ. and R. F. Stambaugh (2003). "Liquidity Risk and Expected Stock Returns." *Journal of Political Economy* 111(3): 642-685. *The move from level to risk: aggregate liquidity as a priced state variable rather than a cost. Read it alongside Chapter 6 as a factor-construction paper, and note how much of the result depends on the choice of liquidity proxy.*
* Easley, D., S. Hvidkjaer and M. O'Hara (2002). "Is Information Risk a Determinant of Asset Returns?" *Journal of Finance* 57(5): 2185-2221. *Takes §11.2's adverse-selection component, estimates it stock by stock as a probability of informed trading, and asks whether it is priced. A third candidate measure alongside ILLIQ and Pástor-Stambaugh, and the one closest to the theory in this chapter.*
* Roll, R. (1984). "A Simple Implicit Measure of the Effective Bid-Ask Spread in an Efficient Market." *Journal of Finance* 39(4): 1127-1139. *Two pages of algebra that made spread estimation possible for every security with a price history. The paper is candid about the positive-autocovariance problem, which later work has repeatedly rediscovered.*
* Hasbrouck, J. (2009). "Trading Costs and Returns for U.S. Equities: Estimating Effective Costs from Daily Data." *Journal of Finance* 64(3): 1445-1477. *The repair job on Roll's estimator: a Bayesian version that survives the positive-autocovariance problem of §11.4 and yields a long panel of effective costs. It then measures the level-of-cost result of §11.5 directly rather than through a proxy.*
* Amihud, Y. and H. Mendelson (1986). "Asset Pricing and the Bid-Ask Spread." *Journal of Financial Economics* 17(2): 223-249. *The horizon-clientele argument of §11.5 in its original form, including the concavity of the premium in the spread — the first paper to derive an asset pricing result from who holds a claim and for how long.*
* Mizrach, B. and C. J. Neely (2007). "The Microstructure of the U.S. Treasury Market." Federal Reserve Bank of St. Louis working paper. *The institutional survey for the market where §11.5's flight-to-liquidity result matters most — interdealer platforms, the on-the-run and off-the-run distinction, and the price impact of announcement flow. Read it with Chapter 9 §9.4 and Chapter 19's opening episode.*
* Harris, L. (1999). "Trading in Pennies: A Survey of the Issues." Working paper, University of Southern California, prepared for the New York Stock Exchange. *Written before decimalization, it forecasts what a one-cent tick would do: narrower quoted spreads, less displayed depth, flickering quotes, weakened time precedence, and liquidity supply shifting from public limit orders to fast dealers. A rare teachable artifact — microstructure theory used to predict, and then checkable against the record — and the source for the tick paragraph in §11.4.*
* Bloomfield, R. and M. O'Hara (1999). "Market Transparency: Who Wins and Who Loses?" *Review of Financial Studies* 12(1): 5-35. *Transparency is not Pareto-improving: dealers, informed traders, and uninformed traders are affected in different directions, and the aggregate verdict depends on which of them the designer is trying to protect. The honest framing of the venue-design debate Box 11.1 compresses.*
* Bloomfield, R., M. O'Hara and G. Saar (2005). "The Make or Take Decision in an Electronic Market: Evidence on the Evolution of Liquidity." *Journal of Financial Economics* 75(1): 165-199. *Supplying liquidity and consuming it are the same trader's choice, made order by order, which is the experimental counterpart of §11.6's question about who is standing in the middle. Read it against the make-take fee schedules that now price that choice explicitly.*
* Menkveld, A. J. (2013). "High Frequency Trading and the New Market Makers." *Journal of Financial Markets* 16(4): 712-740. *One high-frequency market maker's trading reconstructed in full, with its inventory, its holding periods, and its dependence on a new venue's fee schedule. The bridge from Glosten-Milgrom's abstract quoter to the population that actually quotes in §11.6.*

***

## Discussion Questions

1. **Is high-frequency liquidity real liquidity?** HFT market makers have narrowed quoted and effective spreads dramatically and hold inventory for seconds. Their critics say this is liquidity that is present when it is not needed and absent when it is. Make the strongest case on each side, using the chapter's vocabulary rather than adjectives: which of tightness, depth, and resiliency has improved, which has not, and what evidence from May 2010 and from ordinary trading days bears on the question? Then answer a harder version: if a supplier's inventory horizon is seconds, is she supplying immediacy or merely relaying it?
2. **Is liquidity a property of the asset or of the holder?** Chapter 16 §16.5 argues that what looks like an asset's illiquidity is often the binding constraint of whoever must hold it. Take a corporate bond that a dealer will quote in size on Monday and will not quote at all in a crisis. Has the bond's liquidity changed, or has the dealer's balance sheet? State what observable would distinguish the two, and say what follows for a risk model that assigns each security a liquidity score.
3. **Two lambdas, one number.** Section 11.5 notes that a rise in estimated price impact could mean order flow has become more informative or that liquidity suppliers have run out of capital, and that the market maker's optimal response to each is identical. Design an empirical strategy that separates them. What data would you need beyond trades and quotes, and which of the two interpretations does a regulator's response depend on?
4. **What is a market maker owed?** The NYSE specialist held a monopoly franchise and an affirmative obligation to make a market. Today's suppliers have neither. Is the modern arrangement better? Consider what a reinstated obligation would have to be paid for with, whether payment for order flow is already an implicit version of the old franchise, and whether an obligation that binds hardest exactly when it is most costly is enforceable at all.
5. **The measure and the thing.** Roll's estimator, Amihud's ILLIQ, effective spreads, and Kyle's lambda all claim to measure liquidity and disagree with each other, sometimes by a lot, in the time series. Is "liquidity" one thing that four instruments measure imperfectly, or four different things sharing a name? Which of this chapter's results survives if the answer is the second?
6. **Who loses when a market becomes transparent?** Displayed depth and prompt trade reporting are usually argued for as unambiguous improvements. Take the other side with the chapter's two models. Who is worse off when a dealer must report her trades promptly, and who when a large institution must display size — and what happens to the spread each liquidity supplier is willing to post once her position is visible to the people she will have to trade with next? Then say what follows for a venue that competes by offering *less* transparency than its rivals, and whether the existence of such venues is evidence of a market failure or of a demand being met.

***

## Problems

**Problem 1 — A Glosten-Milgrom quote, with an asymmetric prior.** A claim will be worth either 60 or 40. The market maker's prior probability on the high value is 0.6. A fraction $$n = 0.30$$ of arriving traders are informed and trade in the direction of their information; the rest buy or sell with probability one half each.

(a) Compute the pre-trade expected value of the claim. (b) Compute the posterior on the high value conditional on a buy, and hence the ask. (c) Compute the posterior conditional on a sell, and hence the bid. (d) Compute the spread, and the midpoint of the quotes. Explain why the quote midpoint does not equal the pre-trade expected value, and say which of the two a researcher should use as "the price" when constructing a return series. (e) A buy arrives and executes at the ask. Recompute the bid, the ask, and the spread. Verify that the new bid equals the old pre-trade expected value, and explain why that identity holds.

**Problem 2 — Kyle: impact, depth, and the informed trader's size.** A stock has a pre-trade price of eighty dollars. The standard deviation of its terminal value is eight dollars, and liquidity traders' net order flow has a standard deviation of four hundred thousand shares.

(a) Compute Kyle's lambda and market depth. Express depth both in shares per dollar of price movement and as the price impact in basis points of a fifty-thousand-share order. (b) The informed trader learns the claim is worth ninety-two dollars. Compute his order. (c) If liquidity traders happen to net to zero, what price results? Comment on where it sits relative to the pre-trade price and the true value. (d) Compute his expected profit conditional on that signal, and his unconditional expected profit averaged over signals. (e) A new venue doubles the volume of uninformed trading, so that $$\sigma\_z$$ becomes eight hundred thousand. Recompute lambda, depth, the informed trader's order, and his unconditional profit. Who gains from the new venue, who loses, and what happens to the informativeness of the price?

**Problem 3 — Roll's estimator from a short price series.** The following fourteen daily closing prices are observed for a stock:

40.15, 40.15, 40.15, 39.85, 40.15, 40.15, 40.15, 39.85, 39.85, 40.15, 39.85, 39.85, 40.15, 39.85

(a) Compute the thirteen daily price changes. (b) Compute the first-order sample autocovariance of the price changes, using the twelve overlapping pairs. (c) Compute Roll's estimate of the spread, in dollars and as a percentage of a forty-dollar midpoint. (d) The stock's true quoted spread is thirty cents. Comment on the estimator's performance here, and state the two features of this constructed series that make it flatter than a real one. (e) The following month, the same stock's price changes have a first-order autocovariance of $$+0.004$$. What does Roll's estimator say? What would you report, and what would you do if this happened for a quarter of the stocks in your sample?

**Problem 4 — Amihud comparison.** Over a sixty-day window, stock A has a mean absolute daily return of 0.9 percent on mean daily dollar volume of eight billion; stock B has a mean absolute daily return of 3.1 percent on mean daily dollar volume of four million.

(a) Compute $$\mathrm{ILLIQ}$$ for each, scaled by ten to the sixth. (b) Compute the ratio, and express each stock's illiquidity as the percentage price movement caused by ten million dollars of trading. (c) A fund holding fifty million dollars of stock B wants to exit over five days. Using the Amihud measure as a linear impact estimate, what price movement does that imply? State two reasons the estimate is likely to be wrong, and say in which direction each cuts. (d) Stock B's return volatility is three times stock A's. How much of the ILLIQ ratio is attributable to volatility rather than to trading costs? What does this tell you about using ILLIQ to compare stocks in different industries?

**Problem 5 — The liquidity premium and the clientele.** Two claims have identical expected cash flows and risk. Claim L has a round-trip trading cost of 0.5 percent; claim I has a round-trip cost of 3 percent. Three investors have expected holding periods of six months, three years, and fifteen years.

(a) Compute each investor's annualized trading cost for each claim, in basis points. (b) For each investor, compute the extra gross expected return she requires on claim I to be indifferent between the two. (c) In equilibrium, who holds claim I, and what is the liquidity premium on it? Explain why it is not the largest of the three numbers in (b). (d) Claim I's trading cost doubles to 6 percent. By how much does the premium rise, if the same three investors are the entire market? Relate the answer to the concavity result in §11.5. (e) The fifteen-year investor is a pension plan that discovers it must meet an unexpected redemption in year two. Explain, using §11.6, why the loss it takes is larger than the 3 percent round-trip cost the table assumes.

**Problem 6 ★ — Kyle with** $$N$$ **informed traders.** Extend §11.3's model to $$N$$ identical informed traders, each observing $$x$$ and choosing a quantity simultaneously, with the market maker seeing only the total order flow.

(a) Write trader $$i$$'s expected profit taking $$\Lambda\_K$$ and the other $$N-1$$ traders' linear strategies as given, and derive his first-order condition. (b) Impose symmetry and solve for the common trading intensity in terms of $$\Lambda\_K$$ and $$N$$. (c) Impose the market maker's zero-profit condition and solve the fixed point. Show that $$\Lambda\_K = (\sigma\_x/\sigma\_z)\sqrt{N}/(N+1)$$ and verify that it reduces to §11.3's result at $$N=1$$. (d) Show that $$\mathrm{Var}(x \mid \mathrm{OF}) = \sigma\_x^2/(N+1)$$, and describe what happens to depth, to price informativeness, and to total informed profits as $$N$$ grows. (e) Interpret. If the informed traders in (a) are competing hedge funds, does adding one more of them make the market better or worse for a liquidity trader? Does it make it better or worse for the market maker? Relate your answer to Chapter 7's account of what analysts are hired to do.

***

## Selected Solutions

*Solutions to Problems 2 and 3 follow. Solutions to the remainder are in the instructor materials.*

**Problem 2.**

(a) $$\Lambda\_K = \sigma\_x/(2\sigma\_z) = 8/(2 \times 400{,}000) = 1.0\times 10^{-5}$$ dollars per share. Depth is $$1/\Lambda\_K$$, or **one hundred thousand shares per dollar** of price movement. A fifty-thousand-share order moves the price by $$50{,}000 \times 10^{-5} = 0.50$$ dollars, which on an eighty-dollar price is **62.5 basis points**.

(b) $$q = (x-p\_0)/(2\Lambda\_K) = 12/(2\times10^{-5}) = 600{,}000$$ **shares**. Equivalently, his intensity is $$\sigma\_z/\sigma\_x = 50{,}000$$ shares per dollar of mispricing, and the mispricing is twelve dollars.

(c) $$p = 80 + 10^{-5}\times 600{,}000$$, or **86**, exactly halfway between the pre-trade price and the true value of 92. The half-revelation property holds for any signal and any parameters: the market maker's residual variance is $$\sigma\_x^2/2 = 32$$, so her residual standard deviation is 5.66 against the prior's 8.

(d) Conditional on the signal, $$q(x-p) = 600{,}000 \times 6$$, or **3.6 million dollars**, which equals $$(x-p\_0)^2/(4\Lambda\_K)$$. Unconditionally, $$E\[(x-p\_0)^2]/(4\Lambda\_K) = \sigma\_x^2/(4\Lambda\_K) = \sigma\_x\sigma\_z/2$$, or **1.6 million dollars**.

(e) With $$\sigma\_z = 800{,}000$$: $$\Lambda\_K = 5.0\times10^{-6}$$, depth **two hundred thousand shares per dollar**, the informed order **1,200,000 shares**, and unconditional profit **3.2 million dollars**. Everything doubles except price informativeness, which does not move at all — the residual variance is $$\sigma\_x^2/2$$ regardless of $$\sigma\_z$$. The liquidity traders gain individually (each faces half the impact per share) and lose in aggregate to the informed trader, whose profit comes entirely from them and has doubled. The market maker is unaffected: she earns zero either way. This is Chapter 7 §7.2's noise comparative static in a different model — more uninformed trading finances more informed trading and buys no extra price accuracy.

**Problem 3.**

(a) The thirteen changes are

0.00, 0.00, −0.30, +0.30, 0.00, 0.00, −0.30, 0.00, +0.30, −0.30, 0.00, +0.30, −0.30

(b) Pair each change with its predecessor, giving twelve pairs. The mean of the twelve lagged changes is exactly zero and the mean of the twelve current changes is −0.025, so the sample autocovariance is the mean of the cross-products, $$-0.0225$$, less the product of the means, which is zero:

$$
\mathrm{Cov}\big(\Delta p\_t, \Delta p\_{t-1}\big) = -0.0225
$$

(c) $$\hat s = 2\sqrt{0.0225}$$ is **0.30 dollars**, or **75 basis points** of a forty-dollar midpoint.

(d) The estimate is exact, which is the point of the constructed series and not a property of real data. Two features make it flat. The efficient price never moves, so all of the variation in closing prices is bounce; and the sequence of trade signs was chosen so that the sample autocovariance equals its population value, $$-s^2/4$$. In a real fourteen-day sample the estimate would be off by tens of percent in either direction, because both the efficient price and the sign sequence are random and fourteen observations identify neither. Roll's estimator is a large-sample instrument used, in practice, on samples of a month.

(e) A positive autocovariance makes $$-\mathrm{Cov}$$ negative and the square root undefined: the estimator has no value. Report it as missing, not as zero. A quarter of the sample coming out positive is normal in daily data for less actively traded stocks, and it is a real problem rather than a computational one, because the two obvious fixes bias the mean in opposite directions — setting the missing values to zero understates average illiquidity in exactly the stocks that are least liquid, while discarding them selects on the outcome. The defensible responses are to report the fraction of undefined estimates alongside every average, to use a longer estimation window, and to cross-check against a measure that is always defined, such as ILLIQ.

***

## Data Exercise: Two Liquidity Measures from Free Daily Data

The whole exercise runs on free daily open-high-low-close-volume data. Any free source with daily closes, adjusted closes, and share volume will serve; several are available through public APIs and through the sites of the exchanges themselves, and the exercise is written so that it does not depend on which one you use. Record the source and the retrieval date, and check whether the closing prices are split- and dividend-adjusted before differencing them, since an unadjusted split will produce a large spurious price change and wreck both estimators.

**Part A — Build the sample.**

1. Choose two US-listed common stocks: one mega-cap in a major index, and one small-cap with a market capitalization under a billion dollars and a share price above five dollars (below that, tick-size effects dominate the spread and the comparison stops being about liquidity). Download five years of daily prices and volumes for both.
2. Construct daily dollar volume as closing price times share volume. Report, for each stock, the mean and median daily dollar volume and the number of zero-volume days. Explain why zero-volume days must be dropped before computing either measure and what dropping them does to your estimates.

**Part B — Roll's estimator.**

3. For each stock, compute daily price changes and, month by month, the first-order sample autocovariance and the resulting Roll spread estimate. Use raw price changes rather than returns, so that the estimate is in dollars, then divide by the month's average price to express it in basis points.
4. Report the fraction of months in which the autocovariance is positive and the estimator is undefined, for each stock. Is the fraction higher for the small-cap or the mega-cap? Is that what §11.4 leads you to expect, and what does it do to a naive comparison of the two stocks' average estimated spreads?
5. Plot the two monthly series of estimated spreads on one panel, in basis points, with the undefined months shown as gaps rather than zeros. Mark any month in which the market as a whole fell more than five percent.
6. As a robustness check, implement the Corwin-Schultz high-low spread estimator, which uses daily highs and lows rather than closing-price autocovariances and is defined more often. Compare the two series. Where they disagree, which do you believe, and on what grounds?

**Part C — The Amihud measure.**

7. For each stock, compute daily $$|r\_d|/\text{dollar volume}\_d$$ with dollar volume in millions, and average within each month to get a monthly ILLIQ. Report the full-sample mean for each stock, scaled by ten to the sixth, and the ratio between them. Compare the ratio with Table 11.2's spread across size deciles.
8. Convert each stock's ILLIQ into the implied percentage price movement caused by ten million dollars of trading, and state plainly what that number would mean for a fund trying to build a position of fifty million in the small-cap.
9. Regress each stock's monthly ILLIQ on its monthly realized return volatility. How much of the variation in ILLIQ is volatility? Discuss what that does to the interpretation of the measure, referring to §11.4's caveat.

**Part D — Aggregate liquidity and its price.**

10. Build a crude aggregate illiquidity series: repeat step 7 for a cross-section of at least fifty stocks spanning the size distribution, and take the monthly cross-sectional mean (or, better, the median, and say why the choice matters). Plot it from your sample's start to the present and mark the episodes of market stress in the window.
11. Compute the innovations in your aggregate series — the residuals from an autoregression, since the level is highly persistent — and correlate them with the contemporaneous market return. Does illiquidity rise when the market falls? Report the correlation and interpret its sign against §11.5's flight-to-liquidity discussion.
12. State two reasons your series is not the Pástor-Stambaugh liquidity factor, and one thing you would need in order to test whether exposure to it is priced. (Appendix A's portfolio-sort and Fama-MacBeth machinery, as Chapter 6 uses it, is what such a test would need.)

**Part E ★ (if you have licensed data).** Using TAQ through WRDS, compute for the same two stocks the daily volume-weighted effective spread, the realized spread, and the five-minute price impact, and estimate Kyle's lambda from a regression of five-minute price changes on signed order flow with trades signed by the Lee-Ready algorithm. Compare each intraday measure with your daily-data proxies from Parts B and C, in the cross-section and in the time series. The literature's claim is that the cheap measures rank securities well and track conditions over time poorly; test it, and report which of your two proxies degrades more in the time series.