Hybrid Markets, Tick Size and Investor Welfare 1

Evgenia Portniaguina

Michael F. Price College of Business

University of Oklahoma

Dan Bernhardt

Department of Economics,

University of Illinois

Eric Hughson

Leeds School of Business

University of Colorado

Draft: August 16, 2004

1The first author is grateful to the University of Utah Graduate School for financial support. The secondauthor acknowledges financial support from NSF grant SES-0317700. The third author is grateful to theGuiney Research Foundation for financial support. We thank Shmuel Baruch and seminar participants atthe New York Stock Exchange and at the University of Oklahoma for valuable comments and suggestions.The usual disclaimer applies.

Abstract

This paper shows how the tick size affects equilibrium outcomes in a hybrid stock marketsuch as the NYSE that features both a specialist and a limit order book. Reducing the tick sizefacilitates the specialist’s ability to step ahead of the limit order book, resulting in a reductionin the cumulative depth of the limit order book at prices above the minimum tick. If marketdemand is price-sensitive, and there are costs of limit order submission, the limit order book canbe destroyed by tick sizes that are either too small or too large. We show that an intermediate ticksize maximizes a market trader’s welfare on a hybrid market: excessively large ticks discourageparasitic undercutting by the specialist, but prices are bad, while if the price tick is too small,limit order depth again falls because of the parasitic undercutting by the specialist. In contrast,the specialist’s profits rise as the tick size is reduced as long as the tick is not too small.

Introduction

On January 29, 2001, the NYSE completed its shift to decimalization. The SEC mandated the

shift, saying that not only would it be easier for investors to understand trading, but it would

make stock prices “more competitive.”

Today, it seems clear that the opposite has, in fact, occurred. Following decimalization, there

was a massive 66% decline in the cumulative depth in the limit order book (Research Division of

the NYSE). Reflecting that reduction in depth, Bollen and Busse (2003) find that trading costs for

actively managed mutual funds increased by a remarkable 1.367 percent of fund assets. Likely re-

sponding to that reduction in depth, Chakravarty, Panchapagesan and Wood (2003) find that insti-

tutional traders re-allocated order flow toward electronic networks; and Ananth Madhavan, in pri-

vate discussion, indicated that institutions have broken their orders down into far smaller compo-

nents, reducing average share size by more than 50%, despite the associated fixed costs of doing so.

Another indication that decimalization has raised trader costs is a Charles Schwab’s report of a 22%

increase in cancellations or changes of limit orders in the five days following the NYSE’s completion

of decimalization (AP Feb. 12, 2001): traders have to monitor their orders more carefully. Per-

haps most surprisingly, a careful analysis by Chakravarty, Wood and Van Ness (2003) reveals that

decimalization significantly reduced not only trading volume, but even the total number of trades.1

An understanding of the market design is crucial for unraveling why the move to decimalization

seems to have backfired. The NYSE is a hybrid market in which a market order can be crossed

against both a limit order book and a specialist/floor broker. On the NYSE, limit orders are

submitted before a market order is realized, and accordingly have priority at the same price over

the specialist or competing floor brokers. Given the incoming market order and the limit order

book, the specialist or a floor broker can choose whether to undercut with a slightly better price

any portion of the book that they desire and take the remainder of the trade. The penny tick

dramatically reduced the cost of stepping ahead of limit orders, providing specialists and floor

brokers a significant advantage at the expense of other traders. The consequences for limit orders

is summarized by this complaint about the impact of decimalization by institutional traders that

“their efforts to buy large blocks of stock on the market are being blocked by specialists who ‘step1Chou and Lee (2003) also find that volume per trade decreased significantly after decimalization.

1

in’ at the last minute and bid a penny higher to buy stocks that institutional investors would have

gotten otherwise” (AP (February 17, 2001) report).

As long as submitting limit orders is either directly costly or indirectly costly because limit

orders can become stale due to information arrival—and a specialist can selectively step in front

of limit orders—then to offset the reduced likelihood of execution, the optimal response of limit

traders may be to submit fewer orders and set prices further from the quote mid-point. Harris

(1996) argues that a larger minimum price variation (tick) makes it less profitable for front-runners

to take trades away from large traders in markets that enforce time priority. Consistent with this

argument, he finds that order display increases with tick size.

Rock (1990) was the first to model a hybrid market structure. Seppi (1997) is the first to

analyze formally the effect of tick size on a hybrid market such as the NYSE. Seppi assumes

competitive limit order traders, price-insensitive market demand, and a monopolistic specialist.

The specialist decides which portion of the book to undercut, and limit order traders break even

conditional on being executed—the (exogenous) cost of order submission equals their (positive)

expected trading profits.

The contribution of this paper is to explore how the hybrid market design of the NYSE in-

teracts with the tick size to affect limit order depth, specialist profits and investor welfare. To

do this, we integrate rational, price-sensitive market traders into the model. If market orders

are not endogenized, then as Seppi finds, a smaller tick necessarily raises specialist profits. Both

the direct effect—it is less costly for the specialist to undercut a given tick—and the indirect

effect—cumulative depth in the limit order book falls, reducing the competition that the special-

ist faces—make this almost immediate. But, both when decimalization was first announced and

when it was implemented, the price of a seat on the NYSE fell, suggesting that the market did not

believe that decimalization would lead to greater specialist profit. For specialist profit not to rise,

it must be that there is an endogenous reduction in the size and volume of market orders. When

market order traders have price-sensitive demands, this is exactly what happens—they respond

to the reduced depth by submitting smaller orders, as Chakravarty, Wood and Van Ness find.

We find that in equilibrium, as in Seppi (1997), at every tick size save the smallest, the cu-

mulative depth of the limit order book falls as tick size is reduced, because the specialist finds

2

undercutting more attractive. In turn, the endogenous reduction in market demand reinforces this

direct effect on depth, as the reduced market demand further reduces the value of submitting a

limit order. Indeed, as in Glosten and Milgrom (1985), when market demand is too price-sensitive,

the limit book can become empty. We show that fixing the price-sensitivity of market orders, re-

ducing tick size reduces depth; and then show that fixing the tick size, increasing price sensitivity

reduces depth. When market demand is sufficiently price-sensitive, markets feature a non-empty

limit order book only when the tick size is sufficiently large. For smaller tick sizes, the increased

ability of the specialist to undercut the limit book makes it impossible for limit order traders to

break even at any price. The equilibrium limit order book is empty, and consequently, there are

wide quoted spreads. Parlour and Seppi (2001) recognize that a competing limit order market can

cause similar problems for a hybrid market.

The endogenous reduction in market demand suggests that the specialist may prefer a large

tick size. However, we show that as long as the tick size is not so small that the limit book

becomes empty, the specialist prefers a smaller tick because undercutting is so much easier with

a smaller tick. This result is consistent with the evidence in Coughenour and Harris (2004) that

specialist profits and participation rates increased after the decimalization for stocks in which

public order precedence used to be especially costly (small stocks and actively traded stocks with

tight spreads). For large stocks, they find specialist profits to decline. The reduction in profits for

the larger stocks may have resulted from the splitting of large orders into small components. In

fact, in our model the specialist’s share would fall to zero if all orders were sufficiently small.

Because limit order traders break even, average market trader losses rise as tick size falls. It is

complicated to determine how the price paid by each individual market order is affected when tick

size falls—some pay less and others pay more. The reason is that although the specialist pursues

a more aggressive undercutting strategy, limit depth also declines, so that a greater proportion

of the market order is filled by the specialist, who charges a high clean-up price. We derive

numerically how the tick size maximizing the utility of a particular market order trader varies

with the investor’s willingness to pay, and hence, how it varies across the order sizes that agents

trade. Investors who submit small orders benefit from smaller tick sizes; but aggregating across

all investors, the tick size that maximizes the utility of market traders (limit traders make zero)

3

on a hybrid market is strictly positive. Interpreting this result in the context of decimalization,

the only beneficiaries were sufficiently small retail traders, and to the extent that mutual funds

aggregate small investor trades, the move to decimalization may have hurt even small investors.

The contrast between the optimal tick size in a hybrid market and that in a pure limit order

market is sharp: in a pure limit order market, all market order traders prefer a tick size of zero.

However, given a particular positive tick size, over a wide range of tick sizes, traders who submit

relatively smaller orders prefer a hybrid market over one featuring an open limit order book.

The paper is organized as follows. We next present and analyze the model. We characterize

equilibrium outcomes in section 2. We illustrate the effect of tick size on market equilibrium in

section 3. In section 4 we analyze the effects of tick size on welfare, and show via a numerical

example that the optimal tick size on a hybrid market is positive—as opposed to an optimal tick

size of zero on a pure limit order market.

In section 5, we allow the value of the asset to move after limit orders are submitted. As a

result, a limit order can become stale, for example, a limit sell order may be priced below the

asset’s value. The specialist profits by taking the opposite side of such an order before the market

order arrives, thereby inflicting losses on the limit order trader. When limit orders can become

stale, limit order submission costs arise endogenously. Section 6 concludes.

1 The model

Our model builds on Seppi (1997). There is an asset that pays out v per share. This value is

common knowledge. Agents can submit market orders to a hybrid limit order/specialist market.

The limit market is made by a continuum of agents with preferences c1+qv, where c1 is consumption

of money, q is the number of shares of the asset held. Market orders can also be handled by a

specialist who shares the preferences c1 + qv.

We focus on the market buy orders and limit sell orders so that we consider market order

submitters with a relative preference for the asset, with preferences c1 + qβv, where β > 1.2 We

assume that β is distributed according to G(·) with density g(·). After Proposition 2, for ease of2The analysis of market sell orders is analogous.

4

exposition we let β be uniformly distributed on [βmin, βmax]. Fixing βmin, as we increase βmax, the

preference for the asset relative to cash increases, making market order demand less price sensitive.

That is, the greater is β, the less price sensitive are market orders. We normalize investors’ initial

endowments to zero. There is a distribution F (·) over the maximum number of N , that a liquidity

trader can buy, where N and β are independently distributed.

The liquidity trader can buy claims to the asset at prices that are positive integer multiples of

a tick, d > 0. At a given price, liquidity providers who submit limit orders have priority over the

specialist. For liquidity providers other than the specialist, there is a small per-share cost c > 0

of submitting a limit order, which is less than the tick size, i.e., c < d. Later, we endogenize this

assumption. The specialist can trade costlessly. Finally, as in Seppi (1997), there is a trading

crowd with a reservation price r that can absorb arbitrarily large orders. Alternatively, one could

assume that there is no trading crowd, but that the specialist never quotes a price greater than

X% above v. This closely mirrors the price continuity requirement on the NYSE that prevents

price-gouging. This alternative assumption leaves the qualitative results unaffected. Finally, in

Section 5, when we endogenize limit order submission costs by allowing the asset value to move

so that limit orders can become stale, the need to exogenously specify a trading crowd vanishes.

The market timing is as follows:

1. Liquidity providers submit limit orders. We denote the depth at price pj by sj .

2. A liquidity trader is selected and submits her market order.

3. The specialist offers a price, deciding which limit orders, if any, to undercut.

4. Trades are consummated and payoffs are realized.

To simplify the analysis, we initially assume that the common asset valuation, v, and the crowd’s

reservation price, r are multiples of the tick size, d. In our analysis of equilibrium, we discuss how

results are altered slightly if we relax this assumption. Later, we provide a welfare analysis that

relaxes this assumption altogether.

5

2 Equilibrium

Let p(·) be the price schedule faced by the liquidity trader. Given β, N , the market trader will

buy M shares, M = min(N,Y ), where Y is the greatest number such that β ≥ p(Y )v . Liquidity

providers other than the specialist submit limit orders so that the marginal limit trader at each

price earns zero expected profits.

Integrating over possible types (β, N), we compute the expected profits to each limit order.

The zero-profit limit order at pj solves Pr(executed)× (pj − v) = c. Finally, the specialist chooses

a clean-up price to maximize profits: he undercuts the limit book at the price that maximizes

trading profits. We assume the specialist undercuts the limit book if indifferent.

The specialist’s expected profit from undercutting price pj is:

E[πj−1] = (pj−1 − v)(M −j−1∑

i=1

si).

The specialist’s expected profit from not undercutting is:

E[πj ] = (pj − v)(M −j∑

i=1

si).

He therefore undercuts when

E[πj−1] = (pj−1 − v)(M −j−1∑

i=1

si) ≥ E[πj ] = (pj − v)(M −j∑

i=1

si). (1)

Given that the common valuation v is on the grid, equation (1) simplifies: the specialist undercuts

price pj if and only if

M ≤j−1∑

i=1

si + jsj . (2)

Let tj be the maximum market order size such that the specialist undercuts price pj :

tj =j−1∑

i=1

si + jsj . (3)

In turn, this implies that a liquidity trader faces cut-off price pj if and only if

tj < M ≤ tj+1. (4)

Substituting equation (3) into (4) we see that inequality (4) holds if and only if

sj+1 >j − 1j + 1

sj . (5)

6

If condition (5) is violated, then it cannot be an “equilibrium”. To see this, observe that if

condition (5) is violated, then it is never optimal for the specialist to quote pj ; he does better to

quote pj+1. That is, the specialist’s “clean-up” price jumps from pj−1 to pj+1. But, then execution

probabilities at pj and pj+1 are the same because limit orders at pj are executed only when those

at pj+1 are executed, and the only time that limit orders at pj are undercut by the specialist is

when those at pj+1 are also undercut. Because per-share revenues differ at the two prices, marginal

limit order submitters cannot be indifferent between them, and hence it cannot be an equilibrium.

Seppi (1997) (Prop. 2) shows that when market order flow is price insensitive, then in equi-

librium, sj+1 is always large enough relative to sj that (5) holds. We will show that (5) holds (i)

independent of tick size only if market orders are sufficiently price-insensitive, and (ii) for more

price-sensitive market demand, only if the tick size is sufficiently large.

Given the strategies of the liquidity trader and the specialist, a limit order at price pj is

executed if (i) this price is not too high for the liquidity trader and (ii) the maximum number

of claims, N , exceeds the corresponding threshold, tj . Because at every price exceeding p1, the

threshold exceeds the cumulative depth, and because the specialist maximizes profits, either all

limit orders at a particular price are executed, or none are. The sole exception is when the asset

value is not on the grid so that p1−v is less than a full tick. Then, the specialist cannot profitably

undercut p1.3 The probability of execution at pj is

Pr(execution) = Pr(N > tj)Pr

(β >

pj

v

),

and the zero-profit/indifference condition for the marginal limit order at pj is

(1− F (tj))(

1−G

(pj

v

))=

c

pj − v=

c

jd. (6)

Using (6), we solve recursively for the limit book at each price. At p1,

(1− F (s1))(

1−G

(p1

v

))=

c

d,

3The zero-profit condition at p1 is not altered by this assumption, because for the marginal limit order trader,

the probability of execution is still the probability that the market order is enough to fill the depth at p1. If either

p1 is not affordable, or the liquidity shock is too small, the marginal limit order at p1 is not executed, so we compute

the depth at p1 in the same way regardless of whether the specialist finds it profitable to undercut p1 or not. At

prices above p1, all limit traders are marginal since either all limit orders at a particular price are executed, or none

are. At p1, unlike at higher prices, if v is not on the grid, not all limit orders are marginal.

7

and

s1 =

H

(c

d(1−G(p1v

))

): c

d(1−G(p1v

))≤ 1

0 : otherwise(7)

At the next price, the threshold is t2 = s1 + 2s2. Unless this threshold is exceeded, a limit order

at price p2 is not executed. Substituting the solution for s1 into the corresponding zero-profit

condition, we can solve for s2; and continuing we can solve for the entire book.

The following three propositions characterize the equilibrium and the effect of tick size and

price sensitivity of market demand on the equilibrium, provided that equilibrium exists. In Section

3, we show that under certain conditions, the equilibrium limit order book is empty.

Proposition 1 If the limit order book is non-empty, the equilibrium is unique and can be found

using Seppi’s (1997) solution procedure.

Proposition 2 Provided that the equilibrium limit order book is non-empty, the cumulative depth

Qj =∑j−1

i=1 si at or below any price pj on grids P such that pj ∈ P decreases as tick size decreases,

and the specialist’s expected profit increases as tick size decreases, regardless of the price sensitivity

of market demand.

Proposition 3 If the equilibrium limit order book is non-empty, increasing price sensitivity by

reducing βmax reduces depths at every price, provided that the probability density, f(·) = F ′(·) over

the maximum number of claims, N , does not increase “too” steeply in N , i.e., f(tj) is not too

large relative to f(tj−1) so that

−(j − 1)f(tj−1){(βmax − pj)2(βmin − pj) + (βmin − pj)(2d(βmax − pj) + d2)

}

+jf(tj){(βmax − pj)2(βmin − pj) + d(βmax − pj)2

}> 0.

If β and N are uniformly distributed, the above condition holds whenever the book is non-empty.

The restriction on the distribution over the maximum number of claims that a liquidity agent

would purchase is reasonable. In fact, the most reasonable parameterization of f(·) is that it is

nonincreasing, i.e., larger orders are less likely.

8

3 The effect of tick size on market liquidity

So far, we have characterized equilibrium outcomes when the book is non-empty. We next illustrate

numerically how both liquidity and volume decline both with tick size and with a rise in price

sensitivity. In particular, we highlight the interaction between tick size and price sensitivity. In the

next subsection, we illustrate equilibrium construction when market demand is price-insensitive.

We then show that liquidity declines with both increases in price sensitivity and with decreases in

tick size. Last, we show how, holding price sensitivity constant, decreasing tick size can lead to

an empty limit order book in equilibrium.

3.1 Example: Price-insensitive market demand

When demand is price-insensitive, (β = ∞), the framework corresponds to Seppi (1997). Let the

maximum number of claims be uniformly distributed between 0 and 10. Finally, let the asset

value, v be 1; the cost of order submission, c, be 14 ; the tick size, d, be 1

2 ; the crowd provide infinite

depth at price r = 3.

Consider the limit order at price p1, a tick above v. The specialist undercuts if and only if the

market order, M , is at or below the depth s1. The zero-profit condition for the marginal limit

order at p1 is therefore

Pr[executed] =c

d=

12,

so that s1 = 5. At the next price, p2, the zero-profit condition is

Pr[executed] =c

2d=

14,

and the specialist does not undercut as long as

(N − s1 − s2)(p2 − v) > (N − s1)(p1 − v).

Given the distribution of N and depth, s1, we solve for depth s2 = 54 . Next, at p3, three tick

sizes above v, we have Pr[executed] = 16 , and s3 = 25

36 . The specialist undercuts p1 if the order

size is less than s1: t1 = s1 = 5. The specialist undercuts p2 if the order size exceeds t1 but is

less than t2 = s1 + 2s2 = 712 . The specialist undercuts p3 if the order size exceeds t2 but is less

9

than t3 = s1 + s2 + 3s3 = 813 . If the market order exceeds 81

3 , the specialist maximizes profits by

undercutting the trading crowd by one tick in order to take the residual market order.

3.2 Example continued: Price-sensitive market demand

We begin by supposing that β is uniformly distributed on [1, 5]. We let v = 1, so that the asset

value lies on the price grid. We will demonstrate two results. First, as βmax declines, so that mar-

ket order flow becomes more price-sensitive on average, the limit order book thins out, and can

even vanish, because for sufficiently price-sensitive demand limit orders cannot earn zero expected

profits at any price. This is akin to the market breakdown found in Glosten and Milgrom (1985).

Second, fixing price sensitivity, as tick size falls, the limit order book can also vanish. We now

show that given a distribution for β, if market trade is sufficiently price-sensitive there exists a

critical tick size below which the equilibrium limit order book is empty.

At each price pj = v + jd, the equilibrium probability of execution times the execution profits

per share must equal the per-share cost of trading:

c = pr(execution)(v + jd− v),

so that

pr(execution) =c

jd,

where j is the number of ticks the price is above v. Here, the equilibrium probability of execution

at p1 = 32 is c

d = 12 . Similarly, at p2 = 2, it is 1

4 , and at p3 = 52 , it is 1

6 .

Now, the probability of execution at p1 is

pr(execution at p1) = pr(N > s1) ∗ pr(β > v + d).

This implies thatc

d=

Nmax − s1

Nmax −Nmin× βmax − (v + d)

βmax − βmin.

Solving for s1 yields

s1 =307

.

In equilibrium then, the specialist undercuts to v if M ≤ s1.

10

Next, we solve for s2, the depth at p2 = 2. At s2, the probability of being executed is the

probability that M is such that the specialist would prefer not to undercut to p1. That is

Pr[executed] = Pr[(M − s1 − s2)(2d) > (M − s1)d]

= Pr[M > s1 + 2s2] ≡ Pr[N > s1 + 2s2]Pr[β > p2].

This implies thatc

2d=

Nmax − s1 − 2s2

Nmax −Nmin× βmax − p2

βmax − βmin.

Solving yields

s2 =2521

.

Now, we find t2, the minimum order so that limit orders at p2 are executed:

t2 = s1 + 2s2 =203

.

Next, we solve for s3, the limit depth at p3 = 52 .

c

3d=

Nmax − s1 − s2 − 3s3

Nmax −Nmin× βmax − p3

βmax − βmin.

This equation is valid only when the specialist, if he undercuts, undercuts to p2 and not to p1.

One must verify that this is indeed the specialist’s optimal strategy.

If it is the optimal strategy, then

s3 =1321

.

The cutoff at the highest price is then

t3 = s1 + s2 + 3s3 =223

.

Now, we must determine whether the specialist, facing this limit order schedule, has an incentive

to undercut to p1 instead of p2. For such undercutting not to be optimal, it must be that

s3 >13s2.

In this case, s3 − 13s2 = 2

9 > 0. Table 1 presents the results of this example.

For β uniformly distributed between 1 and 4, we obtain the results summarized in Table 2.

11

Still, s3 − 13s2 = 5

36 > 0. Observe that as market demand becomes more price-sensitive, depth

falls at every price in the limit book. This follows immediately from Proposition 3.

Now, suppose that β is uniformly distributed between 1 and 72 . Were we to follow the same

solution procedure, assuming that s3 > 13s2, so that for some M the specialist prefers to undercut

to p2 rather than to p1, we would obtain: s1 = 154 , s2 = 25

24 , s3 = 2572 and t1 = 15

4 , t2 = 356 , t3 = 35

6 .

Now, s3 = 13s2, violating our assumption.

What happens? There is no equilibrium in which the specialist trades at p2. We now show

that there is also no equilibrium where the specialist undercuts to p1. In fact, the limit order book

must be empty! Essentially when β is uniformly distributed on [1, 72 ], market order demand is too

price elastic.

Suppose the specialist were to undercut to p1. Then, the probability that orders are executed

at p2 and p3 must be equal, because the only time when orders at p2 execute is when p3 is not

undercut. Therefore, marginal limit orders cannot earn zero profits at both p2 and p3 unless both

s2 and s3 are zero–the limit book is empty above p1. Solving the limit order zero profit condition

at p1 where the book above is empty implies that the probability of execution at p1 must equal

the probability that not undercutting p1 yields specialist profits that exceed zero (which is what

undercutting to v yields). That is,

12

=Nmax − s1

Nmax −Nmin× βmax − p3

βmax − βmin.

The only way this equation can hold is if s1 < 0, which cannot be. The limit order book is

therefore empty at p1. Hence, the entire limit order book is empty.

It is important to emphasize the fact that the entire limit order book is empty follows because

v is on the price grid. If v is not on the price grid, then the book is empty at every price exceeding

p1, but at p1, the limit order book fills up until the expected profit of the marginal limit order is

zero. Consequently, limit orders at p1 are profitable on average.

Next, we illustrate the effects on depth of changing the tick size. First, we show that when

tick size shrinks, cumulative depths decline. Second, we show that as tick size shrinks, eventually,

this can lead once again to an empty limit order book.

To illustrate, let tick size in the example fall to 13 . Let β be uniformly distributed between 1

12

and 5. The solution is found in Table 3.

The cumulative depth at p = 2 is then s1 + s2 + s3 = 4.66. In contrast, when the tick size was12 , the cumulative depth was s1 + s2 = 5.48.

Now we show that when β is uniformly distributed between 1 and 4, with a tick of 13 , the limit

book is empty. Because the limit book had depth at every price, ceteris paribus, when the tick

was 12 , we will have shown that reducing the tick size alone can lead to an empty limit order book.

We begin by constructing an equilibrium candidate. We obtain: s1 = 2516 , s2 = 405

224 , s3 = 215224 ,

s4 = 257448 , s5 = 771

2240 and t1 = 2516 , t2 = 105

28 , t3 = 254 , t4 = t5 = 53

8 . But now, s5 = 35s4, violating the

assumption that the specialist will undercut p5 to p4. This implies that the book is empty at both

p4 and p5. Continuing using the same argument, one can show that the entire book must be empty.

3.3 A more general case

Our example shows that (1), when market demand is sufficiently price sensitive, the equilibrium

limit order book is empty and (2), for a given distribution for β, decreasing the tick size causes

the limit book to thin out and for small enough tick sizes, can eventually lead to an empty limit

order book. We now extend these results to a general probability distribution over the maximum

number of claims N .

Proposition 4 Suppose that market demand is sufficiently price-sensitive in the sense that βmax <

2rv − 1. Then there exists a critical tick size, d∗ = 1

3 (2(r − v)− (βmax − 1)v), such that for all

d < d∗, the limit book is empty.

4 Welfare

4.1 Hybrid market

The main issue surrounding the Common Cents legislation of the 1990’s was how changing tick

size affected the welfare of various types of traders. Our environment allows us to analyze the

welfare of liquidity traders (liquidity providers who make up the limit order book essentially earn

zero). We now show that different classes of liquidity traders are differentially affected by changes

13

in tick size: traders who submit small orders prefer small tick sizes, and those who want to trade

in volume prefer larger tick sizes.

In our environment, market order size is not exogenous, and in particular, it is affected by

tick size. This means that different traders who happened to trade M when the tick size was d

buy many different amounts when the tick size is d′, so that it is not simple to determine what

happens to the welfare of a cohort who trades M in a particular regime. What we do is consider

the welfare of an individual market order trader whose type is given by the pair, {β, N}. Because

each trader begins with a zero endowment, the trader’s welfare is given by his expected utility,

c1 + qβv.

We next show numerically that the tick size maximizing the welfare of the average market

order trader is positive, and that traders with different {β,N} disagree about the optimal tick

size that maximizes their expected utilities. Consider our example from the previous section, with

β ∼ U [1, 5], N ∼ U [0, 10], the true value, v = 1 + ε, the crowd reservation value, r = 3, the limit

order submission cost, c = 133 . We calculate the expected utilities for representative {β, N}, for

five different tick sizes, 12 , 1

4 , 18 , 1

16 , and 132 . We assume that the specialist does not undercut the

first price in the limit order book when indifferent. This is realistic, because the probability that

the true value, v is exactly on the price grid is a probability zero event. The results are given in

Table 4.

Table 4 reveals that regardless of β, traders who prefer to trade small quantities, N ≤ 5, prefer

the smallest tick size of 132 . Traders with N = 6 are indifferent between tick sizes of 1

32 and 116 .

Traders with larger liquidity shocks, (N = 7, 8, 9) and traders with N = 10 and who are price

sensitive, so that β ≤ 1.75 prefer a tick of 116 . Finally, traders with N = 10, but who are less price

sensitive prefer a still larger tick of 18 .

This is not surprising. Traders who submit small orders trade only at the lowest price in

the book, and thus benefit from a small tick size. Further, if such traders trade small quantities

because they are price-sensitive, a larger tick will preclude them from trading. Traders who submit

larger orders, either because N is large or because β is large, benefit more when depth in the book

at higher prices is greater, which is the case when the tick size is larger. The table illustrates this

tradeoff, and shows that for large orders, adequate depth is more important than receiving a very

14

Table 1: Equilibrium with β ∼ U [1, 5]

Prices Limit Depths Specialist’s Price

p0 = 1 0 M ≤ 307

p1 = 32

307

307 < M ≤ 20

3

p2 = 2 2521

203 < M ≤ 22

3

p3 = 52

1321

223 < M

Table 2: Equilibrium with β ∼ U [1, 4]

Prices Limit Depths Specialist’s Price

p0 = 1 0 M ≤ 4

p1 = 32 4 4 < M ≤ 25

4

p2 = 2 98

254 < M ≤ 20

3

p3 = 52

3772

203 < M

Table 3: Equilibrium with β ∼ U [1, 5] with tick size 13

Prices Limit Depths Specialist’s Price

p0 = 1 0 M ≤ 2011

p1 = 43

2011

2011 < M ≤ 11

2

p2 = 53

8144

112 < M ≤ 20

3

p3 = 2 397396

203 < M ≤ 115

16

p4 = 73

40016336

11516 < M ≤ 52

7

p5 = 83

3157173920

527 < M

15

Table 4: Optimal Tick Size as a Function of Trader Type {N, β},

β/N 1 2 3 4 5 6 7 8 9 10

1.25 116

1.50 116

1.75 116

116

2.00 or 18

2.25 132

18

2.50 18

2.75 18

3.00 132

116

18

3.25 18

3.50 18

3.75 18

4.00 18

4.25 18

4.50 18

4.75 18

5.00 18

Notes: Optimal tick size preferred by each trader type, {N, β}over the choice set { 12, 1

4, 1

8, 1

16, 1

32}.

Traders with N ≤ 5 prefer a tick of 132

regardless of β. Traders with N = 6 are indifferent between tick

sizes 132

and 116

. Traders with N = 7 − 9 prefer a tick of 116

, regardless of β. Finally, for traders with

N = 10, those with small β ≤ 1.75 prefer a tick of 116

; the remainder prefer a larger tick of 18.

16

good price on a small portion of one’s order.

Next, we integrate over trader types in our example and consider the effect of altering tick size

on aggregate volume, average utility, specialist participation, and specialist profits. Our measure

of aggregate volume is simply the average trade size. Specialist participation is measured as the

fraction of the order flow that the specialist takes. Specialist profit is also a per-trade average.

The results are given in Table 5.

First note that specialist participation rises sharply as tick size falls, because undercutting is

so much easier with a smaller tick. When the tick is 18 , specialist participation is still only 3.6%,

but at a tick of 132 , the specialist takes nearly half of all order flow. This is consistent with the

empirical analysis of Coughenour and Harris (2004).

Next, observe that specialist profits also rise sharply as the tick size falls, again because the

specialist undercuts more aggressively. Surprisingly, in practice, reducing the tick size has not led

to a large rise of the price of a seat on the NYSE. Recent reductions in trading volume over time

on the NYSE are clearly too small to explain why seat prices have not risen. Perhaps the true

explanation is that the specialist’s increased ability to undercut has led traders to split their orders

into ever-smaller components, reducing the specialist’s advantage. This is also consistent with the

finding in Coughenour and Harris (2004) that for large stocks, specialist profits and participation

has fallen with the switch to decimalization. Indeed, in our model, were all orders smaller than

the aggregate depth at p1, when p1 is not on the grid, the specialist’s share would fall to zero.

Notice that reducing tick size first increases, and then decreases aggregate volume. For traders

who wish to submit large orders, a smaller tick reduces trading volume, on average. To see why,

observe that from (6), at any fixed price pj , the threshold market order tj is independent of tick

size. Consider a trader with highest affordable price p′. If the tick is d, the next available price

is p′ + d = p′′. The corresponding threshold amounts are t′ and t′′. Suppose N is large enough

to exceed t′′. Under the tick size d, she can submit an order as large as t′′ and still trade at her

highest affordable price p′ since the specialist will undercut p′′ unless the market order exceeds

t′′. Consider now what happens if the tick size is reduced to d2 . The threshold t∗ at the new

intermediate price p∗ satisfies the condition t′ < t∗ < t′′, since (i) t′ and t′′ do not change with

tick size, and (ii) thresholds are monotonically increasing in price in equilibrium. Now, to ensure

17

that the specialist still trades at p′, the trader must not submit an order larger than the new

intermediate threshold, which is smaller than t′′. Hence, trading volume falls with tick size. What

happens if N is not sufficient to exceed t′′? In that case, as tick size falls, the market order will

either stay the same size if N ≤ t∗, or decrease if N > t∗. Finally, what happens if the highest

affordable price is p∗? In that case, reducing tick size does not affect trading volume because the

same threshold t′′ must not be exceeded either for p′ (under tick size d) or for p∗ (under tick sized2).

For traders who wish to submit small orders, it is clear that reducing tick size can increase

order size. These two countervailing effects produce the hump-shaped relationship observed in the

table.

Last, note that average trader utility is maximized at a tick of 116 . However, recall from Table

4 that different trader types prefer different tick sizes.

4.2 Pure limit market

Here, we consider the welfare of the market order trader in a pure limit market. Welfare analysis

for the pure limit market is analogous to that in a hybrid market, except there is no specialist to

consider. The strategy of the market order trader is still to trade as many claims as possible until

the highest affordable price is reached, or until the demand for shares is exhausted, whichever

happens first. The marginal limit order traders at each price still make zero (although the infra-

marginal limit traders expect profits), so the equilibrium depths at every price in the limit order

book are still determined from the zero-profit condition. The cumulative depths in the pure limit

market are then equal to the previously defined threshold amounts tj at the corresponding prices

in the hybrid market. This is because in the pure limit market, the probability of executing a

marginal limit order is the probability that the market order exceeds the cumulative depth at that

price; and in the hybrid market, the probability of execution is the probability that the market

order exceeds the threshold amount at the same price.

In a pure limit order market, the optimal tick size is zero. Numerical results not reported here

confirm that this is the case. 4

4This conclusion would change in a dynamic environment where limit order traders can adjust their orders

18

The question that we address here is for a given tick size larger than zero, which types of

liquidity trader prefer the pure limit market, and which prefer the hybrid market? To answer

this question, we compute equilibrium limit order submission strategies and equilibrium utilities

for all types of trader over the five tick sizes detailed above in the absence of a specialist. Not

surprisingly, traders with smaller N prefer the hybrid market. However, as the tick size decreases,

more and more traders prefer the pure limit market to the hybrid market. But convergence to a

strict preference for pure market is extremely slow. Even at a tick size of 1256 (with smaller order

submission cost), traders with N < 7 still prefer the hybrid market.

In the case considered above, equilibrium in the hybrid market features active limit order

submission. Were it the case that market traders were sufficiently price-sensitive, for small tick

sizes, the limit order book is empty in the hybrid market. In that case, the pure limit market is

far superior for all traders except perhaps the specialist.

5 Endogenous limit order submission costs

Although it may seem that our results, as well as those in the previous literature, depend critically

on the tradeoff between fixed limit order submission costs and profits from limit order execution,

we now show that qualitative results are unaffected when we endogenize limit order submission

costs by allowing limit orders to become stale.

Until now, we ignored one of the advantages the specialist has over traders who submit limit

orders—he can react much more quickly to exogenous changes in asset value than can limit order

submitters. Put another way, the specialist may know much more about current asset value when

he trades, than do limit order traders. We model this information asymmetry by assuming that the

specialist knows the asset’s current value, but the limit order traders know only the distribution

of realized asset values, so that limit sell (buy) orders may become stale when asset value rises

(falls). In the case that limit orders become stale, we assume that the specialist cleans up the

limit order book by profitably buying from (selling to) the limit order book when the asset value

rises (falls).

intertemporally. In that case, an optimal tick size would be found by analyzing the tradeoff between monitoring

costs, and the possibility of being undercut by a new limit order.

19

To illustrate this point, we alter the model timing slightly so that limit orders can become

stale: the value of the underlying asset changes after the limit orders are submitted, but before

the market order size is realized. For simplicity, we consider a two point distribution for asset

value: the asset value is v with probability 1 − π, and v + ∆, ∆ > 0, with probability π. This

corresponds to the case where there is an informational announcement that is either good news or

bad news.

It is immediate that because ∆ is bounded, it is costless to submit limit buy orders at any

p ≤ v and to submit limit sell orders at any p ≥ v + ∆, because these limit orders cannot become

stale. This implies that there will be infinite depth, provided by a “crowd” of limit orders at the

first tick ≤ v, and at the first tick ≤ v + ∆. These limit orders earn (weakly) positive expected

profits; profits are strictly positive whenever either v or v + δ is not a feasible price.

It is also immediate that limit orders in the book cannot cross, and that limit sell (buy) orders

must exceed (be less than) the asset’s expected value. Sell limit orders at p < v + π∆ expect to

lose at least (1−π)∆ with probability π. If these limit orders always executed against an incoming

market order whenever the asset value were v, they would make slightly less than π(1−π)∆. The

same argument applies on the limit buy side.

The depth of the limit order book at price pj is found from the zero-profit condition,

π(v + ∆− pj) = (1− π)Pr(M > tj)(pj − v). (8)

The left-hand side of equation (8) is the expected loss of a limit sell order to the specialist at price

pj when the price moves against it times the probability the asset value rises. The right-hand side

is expected limit order profits when the asset value falls times the probability that it does so.

We begin by considering limit sell orders in the following motivating example. For simplicity,

let β = ∞, so that market demand is insensitive, and M = N . Let N ∼ U [0, 10], let π = 12 , and

consider the tick sizes d = 12 , and d = 1

4 . Let v = 1, and ∆ = 2.

Because ∆ is bounded, order submission costs are effectively zero at a price of 3 (they cannot

become stale), the book has equilibrium depth of at least 10 at price 3. Hence the “crowd” is

endogenized—it is replaced with limit orders at price 3.

Limit orders that do become stale (when the asset value is high) are assumed to be executed

20

with probability one at their limit prices. For example, an order at price 2 loses 1 per share when

stale. When the asset value is low, the limit orders do not become stale, and the specialist’s

optimal strategy is the same as in the basic model.

The resulting equilibria are given in Table 6.

As before, reducing tick size reduces cumulative depth at prices available at both tick sizes.

For example, at p = 52 , cumulative depth is 20

9 at tick size of 12 , and it is 4

5 + 4445 < 20

9 when the

tick size is 14 .

This example shows that both the crowd and the order submission cost can be endogenized.

Essentially, in this new setting, limit orders at each price face a different effective submission cost.

Limit sell orders at higher prices face lower costs than more competitive limit orders because they

lose less when stale. The example above shows that reducing tick size, ceteris paribus, reduces

cumulative depth at prices that are available on both grids. We now generalize formally these

results.

Proposition 5 Let the zero-profit condition be (8). The equilibrium is unique. Limit sell orders

are submitted at prices above the asset’s expected value, π∆ + v.

Proposition 6 In equilibrium, the cumulative depth Qj =∑j−1

i=1 si at any price pj > π∆ + v

on grids P such that pj ∈ P decreases as tick size decreases, and the specialist’s expected profit

increases as tick size decreases.

Next, we consider what happens when market demand is price-sensitive and β is uniformly

distributed between βmin and βmax. As before, when market demand is sufficiently price-sensitive,

the equilibrium limit order book is nonempty only when tick size is sufficiently large.

Proposition 7 Suppose that market demand is sufficiently price-sensitive in the sense that βmax ≤∆v +1. Then there exists a critical tick size, d∗ =

√12 (∆2 −∆v(βmax − 1)), such that for all d < d∗,

the limit book is empty.

21

Table 5: Effect of Tick Size on Specialist Participation, Profits , Average Trade Size, Average

Trader Utility

Tick Size Specialist Participation Specialist Profit Trade Size Trader Utility12 0.005779 0.029969 4.753267 8.30279514 0.012333 0.049906 5.077280 9.47720418 0.036066 0.085394 5.236166 10.05040116 0.117536 0.150817 5.232244 10.25624132 0.480349 0.277748 5.229295 10.20908

Notes: Specialist participation, measured as a fraction of total order flow, specialist profits, average

trade size, and average trader utility as a function of tick size. The maximum number of shares an agent

us willing to trade, N ∼ U [0, 10]; and β ∼ U [1, 5]. Utility is given by c1 + qβV , and agents have an

initial endowment of zero. Specialist profit is a per-transaction (not per-share) average integrating over

the uniform distributions for trader type, {N, β} Average trade size and average trader utility are also

averages integrating over trader type.

Table 6: Equilibrium with insensitive liquidity trade with tick sizes 12 and 1

4

Prices LimitDepths Specialist′sPrice LimitDepths Specialist′sPrice

(d = 1/2) (d = 1/2) (d = 1/4) (d = 1/4)

p = 1 0 never 0 never

p = 54 na na 0 never

p = 32 0 never 0 never

p = 74 na na 0 never

p = 2 0 M ≤ 203 0 M ≤ 4

p = 94 na na 4

5 4 < M ≤ 203

p = 52

209

203 < M 44

45203 < M ≤ 60

7

p = 114 na na 428

441 M > 607

p = 3 ≥ 709 never ≥ 1066

147 never

22

6 Conclusion

This paper studies the effect of changing tick size on liquidity and on the welfare of market

participants in a hybrid market such as the NYSE. The general message supported by our results

is that decreasing tick size too much may have undesirable effects on both liquidity and welfare. In

the context of price-sensitive market demand, we demonstrate that cumulative depth in a hybrid

market decreases as tick size falls. We also show that for sufficiently price-sensitive market demand,

when tick size is too small, equilibrium features wide quoted spreads, very little trading activity,

and an empty limit order book. Market order sizes fall with tick size for all but the smallest orders.

Next, we demonstrate via a numerical example an intuitive result that the change in expected

utility of a market order trader is maximized in a hybrid market when tick size is positive. However,

different types of traders disagree on the optimal tick size: traders who submit small orders prefer

smaller tick sizes. Note however, that small investors who pool their trades in a mutual fund prefer

larger tick sizes, consistent with Bollen and Busse’s (2003) empirical evidence. Specialist profits

are maximized at a tick size of zero. Finally, we show that our results are not driven by fixed

limit order submission costs. We find qualitatively similar results where the cost of limit order

submission is driven by the possibility that limit orders can become stale.

References

Bollen, N.P.B., Busse, J., 2003. Common Cents? Tick Size, Trading Costs, and Mutual FundPerformance, Vanderbilt University and Emory University, unpublished.

Bollen, N.P.B., Whaley, R., 1998. Are teenies better? Journal of Portfolio Management 25,10-24.

Chakravarty, S., Wood, R., 2000. The effect of decimal trading on market liquidity. WorkingPaper, Purdue University, unpublished.

Chakravarty, S., Panchapagesan, V., and Wood, R., 2003. Institutional Trading Patterns andPrice Impact Around Decimalization. Working Paper, Purdue University, unpublished.

Chakravarty, S., Wood, R., and Van Ness R. 2004. Decimals And Liquidity: A Study Of TheNYSE. Journal of Financial Research, 27, 75-94.

Chou, R., W. Lee, 2003. Decimalization and Market Quality, National Central Universityand Ching-Yun Institute of Technology.

23

Coughenour, J., and L. Harris, 2004. Specialist Profits and the Minimum Price Increment.Working paper.

Glosten, L., Milgrom, P., 1985. Bid, ask and transaction prices in a specialist market withheterogeneously informed traders. Journal of Financial Economics 21, 71-100.

Goldstein, M., Kavajecz, K., 2000. Eighths, sixteenths, and market depth: changes in ticksize and liquidity provision on the NYSE. Journal of Financial Economics 56, 125-149.

Harris, L.E., 1996. Does a large minimum price variation encourage order exposure? WorkingPaper, Marshall School of Business, University of Southern California, unpublished.

Parlour, C., Seppi, D., 2001. Liquidity-based competition for order flow. Working Paper,Carnegie Mellon University, unpublished.

Rock, K., 1990. The specialist’s order book and price anomalies. Working Paper, HarvardUniversity, unpublished.

Seppi, D., 1997, Liquidity provision with limit orders and a strategic specialist. Review ofFinancial Studies 10, 103-150.

7 Appendix

Proof to Proposition 1: Seppi (1997) shows uniqueness when the 2nd term on LHS of (6) is

equal to 1. Here, for each distribution of β, and given the price grid, we get a unique value for the

2nd term. Hence, we still get a unique value for the 1st term on the LHS. Hence the solution is

unique even in the price-sensitive case, provided that the equilibrium exists.

Proof to Proposition 2: See Seppi (1997), Proposition 8. The validity of the proof does not

depend on the price sensitivity of the market order. It does, however, depend on the condition

that the threshold amount tj be independent of tick size and be monotonically increasing in price

pj . We show later that, with sifficiently price-sensitive market orders, the monotonicity condition

is violated for a small enough tick size.

Proof to Proposition 3: Express depth at price pj as

sj =1j(tj − tj−1 + (j − 2)sj−1).

It is straightforward to show that s1 increases with βmax. Therefore, as long as we can show that

the difference tj − tj−1 increases with βmax, we will have shown that the depth at every price in

the limit book increases with βmax.

24

To show this, differentiate tj and tj−1 with respect to βmax. Since dtjdβmax

> 0 for any j,

and threshold must be monotonically increasing in price for equilibrium to exist, all we need to

demonstrate is that dtjdβmax

− dtj−1

dβmax> 0. In the case of uniform distributions for both N and β,

this condition becomes

(βmin − pj + jd)(βmax − pj)2 − (j − 1)(βmin − pj)(2d(βmax − pj) + d2) > 0.

Given our assumption that jd = pj − v and βmin ≥ v, the above inequality is true.

In the case of general probability distribution for N , the condition becomes

−(j − 1)F ′(tj−1){(βmax − pj)2(βmin − pj) + (βmin − pj)(2d(βmax − pj) + d2)

}

+jF ′(tj){(βmax − pj)2(βmin − pj) + d(βmax − pj)2

}> 0.

Whether or not this inequality is true depends, in general, on the probability distribution over

the maximum number of claims, F (·). All of the terms in the expression above are positive except

the term jF ′(tj)(βmax − pj)2(βmin − pj) < 0. If the weight F ′(tj) on this term is sufficiently large

relative to the weight on the first two terms, F ′(tj−1) (which is equivalent to the density function

increasing too steeply in N), the condition above is not true. Therefore, the proof works under

the condition that the probability density is not too steeply increasing in N .

Proof to Proposition 4: The zero-profit condition for the marginal limit order at price pj is(

βmax − pj

v

βmax − βmin

)× (1− F (tj)) =

c

pj − v. (9)

We must check whether the threshold tj is monotonically increasing in price. Monotonicity is

important because if tj+1 ≤ tj , there does not exist a range of market orders for which the

specialist optimally trades at pj . For market orders below tj , the specialist prefers to trade at

pj−1 than at pj ; and for market orders above tj+1, the specialist’s profit is greater if he trades at

pj+1 than at pj . In equilibrium, the zero-profit condition must hold for the marginal limit order

at every price. However, if the specialist always skips price pj , the probability of execution at pj

is the same as the probability of execution at the price one tick above, since the only time that

the limit orders at pj get executed is when the limit orders at pj+1 are executed also, and the

only time the limit orders at pj get undercut is when the limit orders one tick above are undercut

25

also. Since the probabilities of execution at pj and at pj+1 cannot be positive and equal without

violating the zero-profit condition (due to the different per-share revenues at different prices), the

only way they can be equal is if they are both zero, which implies that the limit depths at those

prices must be zero as well.

From the zero-profit condition, tj+1 > tj as long as

(j + 1)(βmax − pj+1

v) > j(βmax − pj

v).

By plugging in pj+1 = pj + d, and pj = v + jd, we can rewrite the above as

d <(βmax − 1)v

2j + 1(10)

As long as (10) holds for j = jmax − 1 = r−vd − 2, it will also hold for any lower j. Hence,

monotonicity requires that

d > d∗ =13

(2(r − v)− (βmax − 1)v) (11)

As soon as d = d∗, monotonicity (10) is violated at the top of the limit book but not at the

lower prices yet. As the tick size keeps falling below this critical value, jmax will increase. With

tick size below d∗, we will have d > (βmax−1)v2(jmax−1)+1 , and at some point, we will have d = (βmax−1)v

2(jmax−2)+1 ,

violating monotonicity for the price one tick below; and so on.

For monotonicity to hold at every price for any positive tick size, we need the condition d∗ ≤ 0,

which implies

βmax ≥ 2r

v− 1.

Therefore, we have shown that, as long as β has a sufficiently wide support, threshold amount

of the market order is monotonically increasing in price for any tick size. However, if βmax < ∆v +1,

there exists a positive tick size, determined by (11), at which monotonicity condition is violated

for the maximum price (while still holding for every price below):

jmax(βmax − pmax

v) = (jmax − 1)(βmax − pjmax−1

v) (12)

If the tick size keeps falling below the critical value (11), monotonicity will eventually be

violated for lower prices as well. However, it turns out that the tick size does not have to fall any

26

lower than the critical value. As soon as the tick size equals d∗, the whole book unwinds from the

maximum price down. This is the next part of the proposition, which remains to be shown. The

following proves that, as soon as monotonicity condition is violated at the top of the book (and

hence, the book is empty at the top), the whole book becomes empty.

Consider any price pj , and assume that (1) the limit book above pj is empty but (2) the

depths below pj are still determined from (9) since the monotonicity condition still holds below

pj (that is, tj−1 > tj−2 > ... > t1). This will be the situation, for example, when the tick size

equals d∗, and the book is empty at the maximum price and one tick below the maximum price,

but monotonicity condition is not violated for the lower price yet. In this situation, we need to

reconsider the zero-profit condition for the marginal limit order trader at pj . Now, the probability

of execution at pj is the probability that the specialist prefers trading at the maximum price to

trading one tick below pj . The new zero-profit condition at pj becomes:{

βmax − pmax

v

βmax − βmin

}×

{1− F (tnew

j )}

=c

pj − v.

The difference between the new zero-profit condition and the old zero-profit condition is that

only the market order traders with β > pmax

v will submit orders above the new threshold.

Now it is straightforward to show that tnewj ≤ tj−1 if

j(βmax − pmax

v) ≤ (j − 1)(βmax − pj−1

v). (13)

If tick size equals its critical value d∗, which implies that (12) holds as an equality, it is straightfor-

ward to show that (13) is true for any j, j ≥ 2, j ≤ jmax. To see this, rewrite (13) by substituting

j = jmax − k:

(jmax − k)(βmax − pmax

v) ≤ (jmax − k − 1)(βmax − pjmax−k−1

v).

By plugging in pjmax−k−1 = pmax − (k + 1)d, we have

k2 + (2− jmax)k ≤ (jmax − 1)− βmax − pmax

vdv

= 0.

The last equality follows from (12). Since k2 +(2− jmax)k ≤ 0 for any k, k ≥ 0, k ≤ jmax− 2, (13)

is true for any j, j ≥ 2, j ≤ jmax.

Hence, as long as the limit book is empty above certain price, the monotonicity condition for

that price can no longer hold. Therefore, the book becomes empty at that price as well, and then

27

the monotonicity condition can no longer hold for the price below; and so on so that the whole

book is empty in equilibrium.

Proof to Proposition 5: From the zero-profit condition (8), Pr(N > tj) has a unique value at

each pj . Since tj is defined exactly in section 2.2.1, it follows that sj is also unique at each pj .

At any price pj , limit orders are not ex ante profitable unless (pj − v) ≥ π(v+∆−pj)(1−π) . Therefore,

there are no limit orders at or below π∆ + v. Limit orders are positive above that price.

Proof to Proposition 6: See Seppi (1997), Proposition 8. The validity of the proof depends

on the condition that the threshold amount tj at any fixed price pj > π∆ + v be independent

of tick size and be monotonically increasing in price pj . From the zero profit condition (8), it is

straightforward that tj satisfies both of these conditions.

Proof to Proposition 7: The zero-profit condition for the marginal limit order at price pj is(

βmax − pj

v

βmax − βmin

)× (1− F (tj)) =

π(v + ∆− pj)(pj − v)(1− π)

. (14)

We must check whether the threshold tj is monotonically increasing in price. From the zero-profit

condition, tj+1 > tj as long as

∆− (j + 1)d(j + 1)(βmax − pj+1

v )<

∆− jd

j(βmax − pj

v ).

By plugging in pj+1 = pj + d, and pj = v + jd, we can rewrite the above as

j(j + 1)d2 − (2j + 1)∆d + ∆v(βmax − 1) > 0 (15)

It is easy to verify that LHS of (15) is higher for j′ = j − 1 than for j, for any given value of

tick size. Therefore, as long as (15) holds for j = jmax − 1 = ∆d − 2, it will also hold for any lower

j. Hence, monotonicity requires that

d > d∗ =√

12

(∆2 −∆v(βmax − 1)) (16)

For monotonicity to hold at every price for any positive tick size, we need the condition d∗ ≤ 0,

which implies

βmax ≥ ∆v

+ 1.

28

Therefore, we have shown that, as long as β has a sufficiently wide support, threshold amount

of the market order is monotonically increasing in price for any tick size. However, if βmax < 2rv −1,

there exists a positive tick size, determined by (16), at which monotonicity condition is violated

for the maximum price.

As soon as the tick size equals d∗, the whole book unwinds from the maximum price down.

This conclusion follows from the same logic as that in Proposition 4. If the limit book is empty

at the maximum price and one tick below it, then the specialist will never optimally trade at the

price two ticks below the maximum price; and so on, so the whole limit book becomes empty.

29