A Specialist's Quoted Depth as a Strategic Choice Variable*
Kenneth A. Kavajecz
Department of Finance
The Wharton School of the University of Pennsylvania
2300 Steinberg Hall-Dietrich Hall
Philadelphia, PA 19104-6367
Ph: (215) 898-7543
Email: [email protected]
May 1998
*I gratefully acknowledge the comments of Matthew Clayton, Michael Fishman, Simon Gervais,
Lawrence Gorman, Kathleen Hagerty, Bjorn Jorgensen, Robert McDonald, Elizabeth Odders-White,
Maureen O’Hara, Mitchell Petersen and Robert Porter as well as Jim Shapiro (NYSE). All remaining
errors are of course my own.
A Specialist's Quoted Depth as a Strategic Choice Variable
Abstract
This paper presents a parsimonious model of a specialist choosing prices and depths jointly in
order to maximize profits. The model delivers many of the empirical results concerning posted spreads
and depths found in the literature. Specifically, the model predicts that (1) spreads will widen and depths
will fall in response to an increase in the amount of adverse selection, (2) spreads will widen and depths
will remain unchanged in the wake of an increase in price uncertainty, and (3) prices will shift upward
and depth will be shifted from the ask side to the bid side of the market in the event that either
expectations become more optimistic or the percentage of desired sales by liquidity traders increases.
Furthermore, the constrained models show that prices and depths are used as substitutes. In particular, a
narrow bid-ask spread induces small depth quotes whereas large depth quotes induce a wide bid-ask
spread.
I.
Introduction
At the most basic level, the study of how assets are traded in financial markets must include an
analysis of both the price and quantity components of an asset’s posted supply and demand schedules.
These schedules typically take the form of bid and ask prices as well as volume quotations or depths.
There has been a considerable amount of research on the price component of these posted supply and
demand schedules. In particular, that research describes how specialists determine their prices taking
into consideration such items as inventory costs and adverse selection costs; examples include: Kyle
(1985) and Glosten and Milgrom (1985).
Surprisingly, little work has been done on the quantity component of a specialist’s posted price
schedule. Strictly speaking, a specialist’s depths specify a willingness to purchase (sell) shares at the bid
(ask) for any amount at or below the bid (ask) depth. These depth quotes are not, however, the familiar
depth parameter discussed in the Kyle (1985) paper, rather they are quantities that specialists post in real
time that announce the number of shares available at the posted price. Observation of New York Stock
Exchange (NYSE) specialists’ quotes reveal that fifty percent of the quote revisions implemented by
specialists entail revision of only the quoted depth. The question is why are specialists so active in
changing their depth quotes?1
This paper presents closed form solutions for both unconstrained and constrained versions of a
specialist's problem of choosing prices and depths jointly to maximize profits in a market with adverse
selection. The unconstrained version allows the specialist to control both prices and depths whereas the
constrained versions allow the specialist to control either prices or depths. The results of the
unconstrained model demonstrate that a specialist has a unique optimal price/depth quotation.
An obvious explanation is that specialists are doing nothing more than reflecting the depth in place on the
limit order book. However, Kavajecz (1998) reveals that this is not the case. Between fifty and seventy-five
percent of the time specialists on the NYSE are adding liquidity to the market beyond that supplied by the limit
order book.
Furthermore, in addition to price, quoted depth is a fundamental tool used to address many of a
specialist's concerns. The comparative static results show how both prices and depths respond to changes
in: (1) the degree of adverse selection risk, (2) the distribution of asset payoffs, (3) the prior probabilities
of future prices and (4) the distribution of liquidity trades. The results from the constrained models show
that prices and depths are used as substitutes. In particular, depths that are constrained to be unusually
high induce wide spreads and spreads that are constrained to be small induce small depth quotations.
These results have particular relevance for discussions related to tick size and minimum depth
requirements.
The contribution of this work is fourfold. First, the model improves upon earlier theoretical
work by maintaining the actual form of the specialist's price schedule posted on the NYSE. In so doing,
the model enables the choice of depth to be independent of price, thereby allowing depth to be stated
explicitly rather than determined implicitly through the price choice. This aspect of the model design is
compelling both theoretically and practically. From a theoretical standpoint, while much of the microstructure theory has in mind a specialist price schedule that is continuous and complete, what we observe
in practice is a single price and a single quantity for each side of the market. Understanding the behavior
of the actual price schedule posted on the NYSE is a necessary first step toward establishing the link
between the actual schedule and its theoretical counterpart. Furthermore, from a practical standpoint, the
posted prices and quantities are the only current market information that market participants have when
deciding on a trading strategy; therefore, it is crucial to understand how the price schedules are
determined if we are to better understand the trading behavior that they influence.
Second, work by Lee, Mucklow and Ready (1993) shows that the quantity component of a price
schedule, in addition to the price component, is essential in determining the overall liquidity of a market.
Consequently, understanding how specialists strategically set depths is crucial. This model provides a
framework to begin analyzing quoted depth and highlights its effect on overall liquidity.
Third, the ability to set the depth parameter provides another tool for the specialist, a tool that
can be used to aid the specialist to mitigate adverse selection risk. For example, the approach of a
significant information event might leave the specialist open to a barrage of informed traders. Under
these circumstances, the specialist might collapse her quotes to one round lot or have the quotes merely
reflect the interest in the limit order book as in Rock (1990) and Kavajecz (1998). This way she is able
to minimize the profit potential of the informed traders at her expense. Although not addressed in this
model, a specialist could also use depths to manage inventory or promote price discovery. A specialist
might manage her inventory by increasing the bid (ask) depth when her position is lower (higher) than
the desired level. Moreover, the specialist may 'experiment', in the Leach and Madhavan (1993) sense,
with the depths in order to induce informed traders to reveal their information.
This model takes a first step toward understanding the use of this tool in a setting with adverse
selection. Acknowledging that specialists can potentially use depth strategically to address adverse
selection risk implies that investigations of specialist behavior utilizing only quoted prices provide an
incomplete picture. For example, research such as Glosten and Harris (1988), Huang and Stoll (1997),
and Madhavan, Richardson, Roomans (1997), that estimate the adverse selection component of the bidask spread completely ignore the impact that the choice of depth would have on the measurement of these
effects. A further implication of a specialist possessing a tool in addition to price is that it may provide a
way to identify various rationales for specialist behavior. If the joint response of price and depth can be
shown to be distinct, then those distinct responses can be used to differentiate between competing microstructure effects.
Lastly, the constrained models provide a framework to analyze the impact of various regulatory
or exchange mandated requirements placed on the specialist. For example, the model predicts that the
recent movements by markets to reduce the minimum tick size coupled with the requirement that
specialists reflect ‘quote-improving’ limit orders in their quotes should induce quoted depth to fall to
compensate for the diminished spread. In addition, the model predicts that requirements forcing
specialists to honor their posted prices for substantial quantities should induce large bid-ask spreads.
The remainder of the paper is organized as follows. Section II reviews earlier theoretical work.
Section III contains the model and comparative statics. Section IV contains the price constrained and
depth constrained models. Section VI concludes.
II.
Literature Review
In a related piece, Easley and O'Hara (1987) demonstrate that large trades may execute at worse
prices due to adverse selection costs. Their model is one of the first to allow variable quantities (small
and large) to be traded. The model has two risk neutral market makers along with informed and
uninformed traders. The competing market makers post 'no regret' prices for each quantity, while the
informed traders maximize profits conditional on those price schedules. The Glosten (1989) model
generalizes the problem of multiple quantities by solving for the complete optimal price schedule under
the assumptions of negative exponential utility and normally distributed random variables. Contrary to
Easley and O’Hara (1987), the market maker in the Glosten model maximizes profits by quoting a price
schedule that specifies the price per share as a function of the quantity traded, given the strategies of the
market participants. The results demonstrate that a monopolist specialist is able to provide a more liquid
market compared to a group of competitive market makers because the monopolist is able to average
profits across trades. While both models allow the specialist to quote prices for different trade sizes,
neither allow the specialist to control depth independently of price. Rather, in their models depth is
determined implicitly through the specialist’s choice of the slope of the price schedule; consequently,
there is no role for quoted depth as a choice variable.
Papers by Rock(1990) and more recently Seppi (1997) explore how a specialist competes with a
limit order book for the ability to provide liquidity to the market. The Rock (1990) model obligates the
specialist to post a complete price schedule accounting for the fact that the interest from competing limit
order traders at weakly better prices take precedence over the specialist’s interest. In contrast, the
specialist in the Seppi (1997) model chooses a ‘clean-up’ price given the state of the limit order book as
well as the size of market order demanding liquidity. Although the models are structured differently,
both models demonstrate the ‘second adverse selection problem’ inherent to limit order trading. This
problem stems from a specialist’s incentives to take the most profitable trades and leave the most
unprofitable ones for the limit order book. While the interaction between the specialist and the limit
order book is important to better understand the behavior a specialist price schedule, the structure of
these models limits any discussion of quoted depth. Like the Easley and O’Hara (1987) and Glosten
(1989) models, in choosing a price, the specialist implicitly defines the depth at that price. Therefore,
although multiple quantities are allowed, maximum quantities can not be controlled independently of
price.2
Other researchers have focused on modeling the specialist's choice of depth. Charoenwong and
Chung (1994) develop a model of a specialist's choice of depth in a continuous time setting; however,
they assume that the specialist's quoted prices are exogenous. In particular, they assume that the ask
(bid) price is simply the expected value plus (minus) some profit margin. Consequently, their focus is on
the choice of depth abstracting from any joint interaction with prices. Dupont (1995) improves upon
Charoenwong and Chung (1994) by modeling the specialist's joint choice of prices and depths. Dupont's
focus is how the relation between price and depth changes if the specialist is risk adverse rather than risk
neutral. His simulation results show that depths, in absolute value, are decreasing in the information
quality of the traders and, in addition, depths are more sensitive than the bid-ask spread to a change in the
quality of the informed trader's information.
The model presented here does not explicitly account for the presence of a limit order book. Under certain
assumptions, however, it can be interpreted as that portion of depth contributed by the specialist above that already
provided by the limit order book as in Kavajecz (1998).
III.
The Model
Consider a one-period model of trading a single risky asset. The payoff to the risky asset is the
random variable
. The terminal value of the security is θ with probability µ and θ with probability 11
2
µ where θ1<θ2. There is a continuum of traders who are differentiated by the amount of private
information they possess; λ of the trading population are perfectly informed about the payoff of the risky
asset (the informed) and 1-λ of the trading population possess no private information of the terminal
payoff of the risky asset (the uninformed), where 0 < λ < 1.
In addition to the traders there is a single risk-neutral specialist who makes a market in the risky
asset. Her objective is to post a price schedule in order to maximize profits; however, the price schedule
must have a special form.3 In particular, the specialist must post a bid (ask) price and a quantity up to
which she will honor the posted bid (ask) price, above which there will be no trade.4
The chronology of the trading process is as follows. The specialist posts the price schedule
consisting of a bid and ask price (denoted b and a, respectively where in equilibrium b<a) along with the
corresponding bid and ask depth quotes (denoted β and α, respectively). In doing so, the specialist takes
into consideration the likelihood of trading with both types of traders and the amount that they are likely
to trade. A trader is chosen at random from the population of market participants and decides to trade or
not to trade. If the trader decides to trade, he chooses a quantity, where the quantity is constrained to be
less than or equal to the relevant depth quote.
Since the informed trader has perfect information about the terminal value of the risky asset, he
0QVKEG VJCV VJG URGEKCNKUV CEVKPI CU C OQPQRQNKUV FQGU PQV RQUV HTGITGVHTGGI RTKEGU 4CVJGT UJG
OCZKOK\GU RTQHKVU VJKU UEGPCTKQ KU EQPUKUVGPV YKVJ )NQUVGPIU CTIWOGPV VJCV OQPQRQNKUV URGEKCNKUVU JCXG VJG CDKNKV[ VQ
CXGTCIG RTQHKVU CETQUU VKOG
+V KU YKFGN[ MPQYP VJCV VJG URGEKCNKUV KU QHVGP YKNNKPI VQ VTCPUCEV CV DGVVGT RTKEGU VJCP YJCV KU SWQVGF QT
YKNNKPI VQ JQPQT VJG SWQVGF RTKEGU HQT OQTG VJCP VJG COQWPV URGEKHKGF KP VJG FGRVJ 5KPEG VJGUG FGXKCVKQPU HTQO VJG
UVTKEV RTKEG UEJGFWNG CTG FKUETGVKQPCT[ VJG[ CTG TGOQXGF HTQO EQPUKFGTCVKQP VQ HQEWU CVVGPVKQP QP VJG LQKPV EJQKEG
QH RTKEGU CPF FGRVJU
will have an unbounded demand if the specialist has 'mispriced' the asset; however, the specialist's price
schedule precludes any trading of quantities greater than the posted bid and ask depths. In addition, the
fact that there is a continuum of informed traders means that each trader acts as if he will have at most
one chance to trade.5 Given these constraints, informed trader j will trade according to:
qj
i
if b > if a < otherwise
||
0
where θ* represents the true value of the risky asset. Note that the depths are quoted from the specialist's
perspective. Since the specialist buys (sells) when she transacts at the bid (ask), the bid depth is greater
than zero and the ask depth is less than zero, i.e. β
CPF
α The uninformed possess no private information; therefore, their motivation for trading cannot be
information driven, rather they can be thought of as a collection of traders all with various (exogenously
determined) motivations and propensities for trading. Uninformed trader j can be described by a pair of
numbers (ej, rj) which represent the quantity which he wishes to trade (his endowment) and his
reservation value, respectively. Positive (negative) values of ej imply that the trader wants to sell (buy)
some amount of the risky asset. In addition, high (low) values of rj imply a high (low) valuation of the
risky asset. Given the above assumptions, each individual uninformed trader places his order according
to the following strategy:
qj
u
min [ , ej ]
min [ ||, |ej|]
0
if ej>0 and b > r j
if ej<0 and a < r j
otherwise
The strategy suggests that an uninformed trader will buy (sell) if his reservation price is higher (lower)
A key aspect of the model is that the informed traders will always choose to trade the maximum depth.
than the ask (bid) and the amount that is traded is constrained to be no more than the ask (bid) volume
quote.6 The endowments of the uninformed traders have cumulative distribution function F(x) with
support [t1, t2] where t1 < 0 < t2. Similarly, the reservation values have a cumulative distribution function
G(x) with support [θ1, θ2]. Moreover, I assume that the endowments and reservation values are
independent. Conditional on a price schedule (b,β,a,α), the trades of the uninformed population come
from the following distribution:
q u(b,,a,) (,0)
||
(0,||)
0
wp
wp
wp
wp
wp
[1 F ()] G (b)
[ F () F (0)] G (b)
F () [1 G (a)]
[ F (0) F ()] [1 G (a)]
[1 [1 F (0)] G (b) F (0) (1 G (a)) ] ]
Taking the strategies of the informed and uninformed as given, the specialist maximizes her expected
profit using the following objective function:
E [ %(b, , a, ) ] µ [ (1 b)] (1 µ) [ (2 a)] µ (1) [ [1 F ()] G (b) (1 b) [ F () F (0) ] G (b) E [q|q(0,)] (1 b) F () [ 1 G (a)] (1 a) [ F (0) F ()] [ 1 G (a)] E [q|q(,0)] (1 a) ] (1µ) (1) [ [1 F ()] G (b) (2 b) [ F () F (0) ] G (b) E [q|q(0,)] (2 b) F () [ 1 G (a)] (2 a) [ F (0) F ()] [ 1 G (a)] E [q|q(,0)] (2 a) ]
The first two terms represent the expected loss from transacting with the informed traders and the
remaining terms represent the expected profit or loss due to transacting with the uninformed traders. The
0QVKEG VJCV VJKU HQTOWNCVKQP FQGU PQV UWHHGT HTQO VJG RTQDNGO CFFTGUUGF D[ 2CWN 6JG
WPKPHQTOGF VTCFGTU KP VJKU OQFGN CTG FKHHGTGPV HTQO VJG GZQIGPQWU NKSWKFKV[ VTCFGTU KP QVJGT OQFGNU DGECWUG VJG[
CTG WPYKNNKPI VQ VCMG CP KPHKPKVG GZRGEVGF NQUU VQ EQPFWEV VJGKT VTCPUCEVKQPU
objective function makes implicit assumptions about the relation between the specialist's choice
variables. These assumptions can be summarized by the following two restrictions and will be verified
later in equilibrium. The quoted depths are constrained to be less than or equal to the maximum liquidity
trade, i.e. t1 α < 0 < β t2, and the quoted prices are constrained to be between the two possible final
payoffs, i.e. θ1 < b < a < θ2. In order to proceed with solving the model, I assume that ej is distributed
uniformly from t1 to t2 and rj is distributed uniformly from θ1 to θ2, i.e., ej ~ U[t1, t2], and rj ~ U[θ1, θ2].
Using the uniform distribution assumptions, the specialist's maximization problem can restated as
follows:
E [ %(b, , a, ) ] µ (1 b) (1µ) (2 a)
(1)
t2 t2 t1
t2 t1
t1
(1)
t2 t1
(1)
(1)
t2 t1
b 1
{µ (1b) (1µ) (2b)}
2 1
b 1
1
{µ (1b) (1µ) (2b)}
2 1 2
2 a
{µ (1a) (1µ) (2a)}
2 1
2 a 1
{µ (1a) (1µ) (2a)}
2 1 2
Remember that the depths take on the sign of the specialist's implied inventory change; therefore, the bid
depth is positive and the ask depth is negative, i.e., β > 0 and α < 0. Moreover, the objective function
takes into account the fact that setting depths greater than the support of the uninformed trader's
endowment is a strictly dominated strategy.
An equilibrium in this model is a set of four functions that describe the price schedule of the
specialist given the strategies of the traders and the implicit assumptions embedded in the objective
function.
Proposition 1: If µ satisfies the condition:
1
t2
t1
< µ <
t2
1
t1
1
t1
t2
then the unique equilibrium for the one-period model is:
bu 1
1
8
(3 Eµ [] 1) (1µ) (2 1) 1 4
4
t2
u 3 t2 1 t2 1 8
2
au 2
t2
1
µ
1µ
2
2
t1
1
µ
1µ
(t2 t1)
(t2 t1)
1
1
8
(3 Eµ [] 2) µ (2 1) 1 4
4
t1
u 3 t1 1 t1 1 8
1
1
1µ
(t2 t1)
µ
1µ
(t2 t1)
µ
Proof: See Appendix.
The restriction on µ is equivalent to requiring the specialist to keep both sides of the market open. The
left hand inequality covers the ask side of the market, au* < θ2 and αu* < 0, and the right hand side covers
the bid side of the market, bu* > θ1, βu* > 0. There are two ways of interpreting the restriction on the
primitives (µ, λ, t1, t2). The first interpretation is that the restriction requires sufficient uncertainty about
the final payoff of the risky asset, i.e., µ not to close too zero or one, so that the specialist cannot afford
to forgo the expected profit on one side of the market by shutting that side down. The second
interpretation argues that the restriction requires that there be enough uninformed traders in the trading
population as to make the specialist's position profitable, i.e., λ must be close to zero.
In order to better understand the optimal price and depth functions, it is useful to highlight the
special case where there are no informed traders. If λ = 0, the radicals disappear and the price schedule
reduces to bu* = 1/2(θ1 + Eµ[θ]), βu* = t2, au* = 1/2(Eµ[θ] + θ2), αu* = t1 where 0 < µ < 1. Consequently,
the prices are just the midpoints between the expected value and the respective terminal values and the
depths allow all traders to trade their desired quantities. A rich set of comparative static results can be
obtained from the posited equilibrium. A standing assumption for all the forthcoming Corollaries is that
the parameter restrictions placed on µ in Proposition 1 hold.
Corollary 1.1: As the percentage of informed traders (λ) increases, the bid-ask spread increases and
both the bid and ask depths decrease.
0 bu
0 au
0 u
0 u
< 0,
> 0 and
< 0,
> 0
0
0
0
0
As the adverse selection problem becomes more acute, the standard result of a widening bid-ask spread
obtains and in addition, both depths narrow. In other words, as the adverse selection problem worsens,
the specialist is unwilling to supply as much liquidity to the market, either in terms of prices or in terms
of quantities. This result is consistent with the theory work done by Dupont (1995). It is also consistent
with the empirical work done by Lee, Mucklow and Ready (1993) and Kavajecz (1998). Specifically,
Lee, Mucklow and Ready (1993) demonstrate that quoted spreads widen and quoted depths drop in
advance of quarterly earnings announcements. They report that the quoted spread widens an average of
1.4% the day preceding the announcement and 8.2% on the day of the announcement while the quoted
depths fall an average of 5.3% and 4.4% for the same time periods. Their work shows that both spreads
and depths are needed to infer changes in liquidity. Kavajecz (1998) furthers the analysis by showing
that specifically, depth contributed by the specialist falls surrounding information events such as
quarterly earnings announcements and Federal Open Market Committee interest rate movements.
Collectively, the empirical work illustrates that specialists and other liquidity providers actively manage
information asymmetry risk by adjusting both spreads and depths.
Corollary 1.2: Given a mean preserving spread of the distribution of θ, the bid-ask spread increases and
the depths remain unchanged.
0 bu
0 au
0 u
0 u
< 0,
> 0 and
0,
0
0 (2 1)
0 (2 1)
0 (2 1)
0 (2 1)
An increase in the spread of possible terminal values can be thought of as an increase in the amount of
uncertainty that the specialist faces. The specialist is less sure of the possible terminal value; put
differently, the cost of being wrong about the terminal value increases. Notice that changes in (θ2 - θ1)
widen the bid-ask spread but, unlike changes in adverse selection risk, have no effect on the depth
parameters. In this model, depths are used to mitigate the adverse selection problem but are not used to
address uncertainty about the terminal value. These results are broadly consistent with the findings of
Kavajecz and Odders-White (1997), who show that depth quotes are used more often to address adverse
selection concerns while prices are used more often to address uncertainty associated with an asset’s
payoff.
Corollary 1.3: As the probability that
= θ , (µ), increases, the bid and ask prices decrease and the bid
1
depth decreases while the ask depth increases.
0 bu
0 au
0 u
0 u
< 0,
< 0 and
< 0,
< 0
0µ
0µ
0µ
0µ
Changes in the prior probability measure merely shift prices toward the more likely terminal value and
redistribute depth toward the side of the market with the less likely terminal value. For instance, an
increase in µ causes both the bid and ask prices to fall in order to readjust to the new expected value of
the risky asset, the bid depth falls in order to curtail probable losses, and the ask depth increases to allow
larger trades with a higher probability of profit. Therefore, by adjusting the quoted prices and quoted
depths this way the specialist reveals what he believes the new expected value of the asset to be.
Consequently, this result suggests that the way in which specialist post their price schedule may reveal
information about the underlying asset; for example, asymmetries in the quoted depths may be associated
with changes in the expected value of the asset. Brown, Cohen and Polk (1997) and Kavajecz (1998)
supply supporting evidence for this result. Brown, Cohen and Polk (1997) explicitly test corollary 1.3
using a trading strategy that conditions the timing of trades on the asymmetry between the quoted bid
depth and quoted ask depth. They find that timing trades based on asymmetries in quoted depths
generates an average savings of sixteen basis points which adds an additional 2.5% to the return of the
stock per year. Kavajecz (1998) also finds that way in which a specialist positions his quotes reveals
information about the future value of the asset. In particular, changes in the quoted bid-ask spread
midpoint are shown to be predicted by whether the specialist contributes depth at the ask or the bid.
Corollary 1.4: Changes in the distribution of desired liquidity trades by increasing either t1 or t2, where
t1<0<t2, raise prices and increase the bid depth while decreasing the ask depth.
0 bu
0 au
0 u
0 u
> 0,
> 0 and
> 0,
> 0
0 t1
0 t1
0 t1
0 t1
0 bu
0 au
0 u
0 u
> 0,
> 0 and
> 0,
> 0
0 t2
0 t2
0 t2
0 t2
Although not shown, it is easy to verify that price reactions wash out if the change in the distribution of
endowments results in t1 = -t2 ; therefore, as long as the distribution of desired liquidity trades is
symmetric, the only adjustment a specialist makes is to the depth quotations.
A comparison of the comparative static results shows that the specialist has a unique price/depth
response for various changes in the trading environment. The consideration of either prices or depths
alone is insufficient to discriminate between these effects. For instance, notice that considering only
prices, it would be impossible to distinguish between a change in adverse selection risk (λ) and a change
in price uncertainty (θ2 - θ1) or between a change in the probability measure (µ) and an asymmetric
change in the distribution of liquidity trades (t1, t2). However, four distinct reactions can be established
from the model by appraising both prices and depths. First, a widening (narrowing) of the bid-ask spread
and a decrease (increase) in the depths is attributable to changes in the information asymmetry. Second,
a change in the bid-ask spread with no change in the depths is caused by a change in price uncertainty.
Third, a downward (upward) shift in the bid-ask spread accompanied by a reallocation of depth from the
bid (ask) to the ask (bid), is brought about by changes in the probability assessments for the risky asset.
Lastly, changes in the depths with no corresponding change in the bid-ask prices are generated by
symmetric changes in the distribution of liquidity trades. In addition, the comparative static results
suggest that the specialist may have 'preferred' applications for prices and depths. For example, a change
in the distribution of prices is handled by adjusting the bid and ask price alone, whereas a symmetric
change in the distribution of liquidity trades is handled by adjusting only the depths.
IV.
Extensions: Constrained Models
This section presents results describing the model under two separate constraints. In the first
model, the price is constrained to be a proportional spread around the expected value and the second
model constrains the depths to be a fraction of the maximum liquidity trade. These constraints are
investigated to assess the impact of various restrictions placed on specialists’ ability to set price and
depth quotes as well as to determine whether prices and depths are used as complements or substitutes.
Specifically, constraining the prices addresses how the specialist's depth quotes are affected by exchange
execution requirements and minimum tick size restrictions. Moreover, constraining the depths addresses
how specialist's prices are affected by minimum depth requirements.
IV.1
Constrained Prices
Consider the one period model described earlier, where the specialist must post a price schedule
for the risky asset that maximizes her profit given that λ of the trading population are perfectly informed
and 1-λ of the trading population are uninformed. Instead of allowing the specialist to change prices,
suppose that prices are exogenously set according to: a =(1+γ)[µ θ1+(1-µ)θ2] and b =(1-γ)[µ θ1+(1-µ)θ2].
The parameter γ is assumed to be an exogenous constant which forces a proportional spread of
2γ(µθ1+(1-µ)θ2) centered around the expected value of the risky asset. While a proportional bid-ask
spread is an oversimplification of reality, it is parameterized as proportional for tractability. The
assumption on the spread can be relaxed to allow for the bid and ask to be different proportions of the
asset’s expected value without any change to the intuition of the model. Substituting the above
constraints into the unconstrained model yields an objective function with only two choice variables, the
bid and ask depths. As in the unconstrained model the bid and ask depths are independent of each other
so the maximization problem reduces to two separate single-variable maximization problems.
Proposition 2: If γ satisfies the condition:
0 < < min
(1 µ) (2 1)
,
µ (2 1)
µ 1 (1 µ)2 µ 1 (1 µ)2
then the unique equilibrium for the price-constrained one-period model is:
c t2 1
c t1 1
(t2 t1) (2 1) µ
(µ 1 (1 µ)2)
(t2 t1) (2 1) (1 µ)
(µ 1 (1 µ)2)
Proof: See Appendix.
The condition in Proposition 2 ensures that both sides of the market remain open. A comparison of these
results with those of the unconstrained model are revealing. Like the unconstrained model, if there are
no informed traders (λ=0), the specialist allows all quantities to be traded, irrespective of the constrained
prices. Unlike the unconstrained model, a change in any of the primitives will change the depth quotes.
This suggests that the depth quotes can be used as a substitute for the bid and ask prices. Furthermore,
for a given change in one of the primitives, there is a larger reaction in the constrained depths than in the
unconstrained depths. A particularly pertinent result is that for spreads constrained to be unusually
narrow, the specialist will compensate by reducing posted depth all else equal.
Corollary 2.1: As the spread narrows, both the bid and ask depths decrease.
0 c
0 c
> 0 and
< 0
0
0
Typically specialists have many requirements related to execution quality and stabilization that place
limits on the quotes that can be posted, e.g., price continuity rules, size of the spread and the obligation to
reflect ‘quote-improving’ limit orders in the posted quotes. All these requirements have the effect of
constraining the specialist’s ability to post her optimal bid and ask prices. Under these circumstances,
the model predicts a reduction in depth to compensate the specialist for having to transact at less
favorable prices. In this case, the depth quotes act as a substitute for the constrained bid and ask prices.
Quote revisions due to changes in adverse selection risk, the distribution of liquidity trades, etc., that
might otherwise have been adjusted using prices are now adjusted using depths.
This result has particular significance in light of the change that many exchanges have made to
smaller minimum tick sizes. Harris (1994) was one of the first to empirically explore the relation
between tick sizes, depths and volume. His analysis predicts that a smaller minimum tick size, which
implies a smaller minimum spread, would decrease displayed quotation size or depth. This finding is
consistent with the results of the price constrained model where we see that as the constrained spread
decreases around the expected value, the specialist compensates by reducing the number of shares he is
willing to trade. Goldstein and Kavajecz (1998) provides empirical evidence in support of this result.
Their investigation of the June 24, 1997 minimum tick size reduction on the NYSE shows a forty-eight
percent decrease in the quoted depth with a thirty-six percent decrease in the depth provided by
specialists and floor brokers.
IV.2
Constrained Depths
Again consider the one period model described earlier; however, now suppose that the specialist
has full control over her prices but the depth figures are fixed according to α = ρt1 and β = ρt2. The
parameter ρ is assumed to be an exogenous constant which forces the depths to be some fraction of the
maximum trade size desired by the uninformed traders. Like before, substituting the above constraints
into the unconstrained model yields an objective function with two independent choice variables, the bid
and ask prices.
Proposition 3: If 0 < ρ 1, then the unique equilibrium for the depth-constrained one-period model is:
bc ac 1
1 (µ 1 (1 µ) 2 1) 2
2 1
(t2 t1) (2 1) µ
1
1 (µ 1 (1 µ) 2 2) 2
2 1
(t2 t1) (2 1) (1 µ)
(1 1
') t
2 2
(1 1
') t
2 2
Proof: See Appendix.
As was demonstrated in the price constrained case, the depth-constrained model yields the same result as
the unconstrained model when λ = 0. Likewise, the bid and ask prices act as substitutes for the depths in
that they are affected by changes in any of the primitives, and their reactions are larger for a given change
in the primitives. An important implication of this model is the impact that depth requirements have on
the bid-ask spread.
Corollary 3.1: The bid-ask spread is increasing in the constrained depth.
0 bc
0 ac
< 0 and
> 0
0'
0'
There are two examples of this constrained model, the first is the London Stock Exchange (SEAQ) and
the second is the recent regulatory changes that have been implemented in the NASDAQ market.
Although both examples occur in dealer rather than specialist systems the intuition for the trade-off
between prices and depths still apply. The London Stock Market is relevant to this model because the
exchange requires that market makers honor their posted prices for quantities amounting to two or more
percent of average daily trading volume. The model predicts that this unusually large depth requirement
would have the effect of widening the bid-ask spread. These predictions’ are consistent with the results
of Reiss and Werner (1996). They show that SEAQ market makers post extremely wide bid-ask spreads,
in fact much wider than the NASDAQ market. Moreover, SEAQ market makers rarely change their
prices, opting instead to transact within the spread much of the time. This behavior is consistent with
market makers identifying the counter party to better understand whether the trade is motivated by
information or liquidity.
The recent regulatory changes on the NASDAQ market provide another example of the trade-off
between depth and price. Prior to the regulatory reforms, the minimum depth requirement on NASDAQ
was set at 1,000 shares. This requirement appeared to be binding as market makers rarely posted depths
greater than the minimum depth. If the equilibrium unconstrained depth was smaller than 1,000 shares,
the model predicts that the bid-ask spread would be increased to compensate for honoring prices at
greater than the desired depth. While there were many changes to the NASDAQ trading protocol, of
particular relevance for this study is the reduction in the minimum depth requirement from 1,000 shares
to 100 shares. Based upon the results from the depth constrained model, the minimum depth reduction
should result in a reduction in the quoted spread as well as the quoted depth. The preliminary results
compiled by the NASD corroborate these predictions in that, after the full phase-in of the new regulatory
changes in October 1997, quoted spreads declined by an average of thirty-two percent while quoted depth
declined by an average of twenty percent.
V.
Conclusion
The role of quoted depth in a specialist's posted price schedule has received little research
attention. Yet it is an important topic since approximately half of the updated price schedules posted by
specialists on the NYSE result in only changes to the quoted depths. The model provides a unique
analytic solution to a specialist's optimal choice of prices and depths, in so doing, the model demonstrates
that quoted depth is used strategically by the specialist. In addition, the model delivers a number of
results concerning quoted prices and depths that have been shown to hold empirically. For example, the
model predicts that (1) spreads will widen and depths will fall in response to an increase in the amount of
adverse selection, (2) spreads will widen and depths will remain unchanged in the wake of an increase in
price uncertainty, and (3) prices will shift upward and depth will be shifted from the ask side to the bid
side of the market in the event that either expectations about future prices becomes more optimistic or the
percentage of desired sales by liquidity traders increases. The constrained models show that constraining
one of these choice variables causes the specialist to compensate with the other, implying that prices and
depths are used as substitutes. In particular, a narrow bid-ask spread induces small depth quotes whereas
large depth quotes induce a wide bid-ask spread.
References
Brown, Gregory W., Randolph B. Cohen, and Christopher K. Polk. (1997). Does Quoted Depth Predict
Intraday Stock Returns? Theory and Evidence, University of Chicago, Working Paper.
Charoenwong, Charlie and Kee H. Chung. (1994). Determinants of Quoted Depths of NYSE and AMEX
Stocks: Theory and Evidence, University of Memphis Working Paper.
Dupont, Dominique Yves. (1995). Market Making, Prices and Quantity Limits, University of Chicago
Working Paper.
Easley, David and Maureen O'Hara. (1987). Price, Trade Size, and Information in Securities Markets,
Journal of Financial Economics 19, 69-90.
Glosten, Lawrence R. (1989). Insider Trading, Liquidity, and the Role of the Monopolist Specialist,
Journal of Business 62, 211-235.
Glosten, Lawrence R. and Lawrence E. Harris. (1988). Estimating the Components of the Bid/Ask
Spread, Journal of Financial Economics 21, 123-142.
Glosten, Lawrence R. and Paul R. Milgrom. (1985). Bid, Ask and Transaction Prices in a Specialist
Market with Heterogeneously Informed Traders, Journal of Financial Economics 14, 71-100.
Goldstein, Michael A. and Kenneth A. Kavajecz. (1998). Eighths, Sixteenths and Market Depth: Changes
in Tick Size and Liquidity Provision on the NYSE, The Wharton School, University of Pennsylvania,
Working Paper.
Harris, Lawrence E. (1994). Minimum Price Variations, Discrete Bid-Ask Spreads, and Quotation Sizes,
Review of Financial Studies 7, 149-178.
Huang, Roger D. and Hans R. Stoll. (1997). The Components of the Bid-Ask Spread: A General
Approach, Review of Financial Studies 10, 995-1034.
Kavajecz, Kenneth A. (1998). A Specialist's Quoted Depth and the Limit Order Book, Journal of
Finance, forthcoming.
Kavajecz, Kenneth A. and Elizabeth R. Odders-White. (1997). An Examination of Changes in
Specialists’ Posted Price Schedules, Rodney White Center for Financial Research Working Paper, The
Wharton School.
Kyle, Albert. (1985). Continuous Auctions and Insider Trading, Econometrica 53, 1315-1335.
Leach, Chris J. and Ananth N. Madhavan. (1993). Price Experimentation and Security Market Structure,
Review of Financial Studies 6, 375-404.
Lee, Charles M. C., Belinda Mucklow and Mark J. Ready. (1993). Spreads, Depths, and the Impact of
Earnings Information: An Intraday Analysis, Review of Financial Studies 6, 345-374.
Madhavan, Ananth., Matthew Richardson, and Mark Roomans, (1997). Why do Security Prices Change?
A Transaction-Level Analysis of NYSE Stocks, Review of Financial Studies 10, 1035-1064.
Implementation of the SEC Order Handling Rules: Preliminary Findings. NASD Economic Research.
April 1997.
Paul, Jonathan. (1994). Information Aggregation without Exogenous 'Liquidity' Trading, University of
Michigan Working Paper.
Reiss, Peter C. and Ingrid Werner. (1996). Transaction Costs in Dealer Markets: Evidence from the
London Stock Exchange, in “The Industrial Organization and Regulation of the Securities Industry”
(Andrew W. Lo, ed.), pp. 125-177. National Bureau of Economic Research, Chicago, Illinois.
Rock, Kevin. (1990). The Specialist's Order Book and Price Anomalies, Review of Financial Studies,
forthcoming.
Seppi, Duane J. (1997). Liquidity Provision with Limit Orders and a Strategic Specialist, Review of
Financial Studies 10, 103-150.
Appendix
Proof of Proposition 1
After defining some simplifying variables, the specialist's objective function for her one period
maximization problem can be expressed as follows. Notice that the bid and ask sides of the market are
completely separate and can be solved independently.
E [ %(b, , a, ) ] µ (1 b) M1 (b 1) (Eµ [] b) (t2 1 2
)
2
1
(1µ) (2 a) M1 (2 a) (Eµ [] a) ( 2 t1 )
2
Eµ [] µ 1 (1µ) 2
where
M1 1
(t2 t1) (2 1)
First and Second Order Conditions
The following are the first order conditions associated with an interior maximum:
0 E [%] µ M (t 1 2) [ (E [] b) (b ) (1) ] 0
1 2
µ
1
2
0b
0 E [%] µ ( b) M (b ) (E [] b) (t ) 0
1
1
1
µ
2
0
0 E [%] (1µ) M ( 1 2 t ) [ (1) (E [] a) ( a) (1) ] 0
1
1
µ
2
2
0a
0 E [%] (1µ) ( a) M ( a) (E [] a) ( t ) 0
2
1
2
µ
1
0
These simplified expressions can be derived from the first order conditions assuming the restrictions
listed below hold. I proceed assuming they do hold and verify later that in equilibrium the restrictions
are satisfied.
b 1
(E [] 1) 2 µ
t2 a 1
2 M1 (t2 )
2
µ
M1 (Eµ [] b)
1
(E [] 2) 2 µ
t1 µ
(2)
(1µ) 1
2 M1 ( t 1 )
2
(1µ) M1 (Eµ [] a)
given the following restrictions:
(1)
(3)
(4)
b 1 g 0
g0
2 a g 0
g0
(R1)
(R2)
(R3)
(R4)
The second order sufficient conditions to ensure a unique maximum are that the Hessian of second
derivatives be negative definite. In order for the Hessian to be negative definite, the naturally ordered
principle minors must alternate in sign. Because of the separability between the bid and ask sides of the
specialist's problem, it suffices to check each side of the market separately.
02 E [%]
0b 2
02 E [%]
0b0
02 E [%]
0 2
02 E [%]
00b
M1 (t2 1 2) (2) < 0
02 E [%]
0a 2
2
0 E [ %]
0a0
02 E [%]
0 2
2
0 E [ %]
00a
M1 ( 1 2 t1 ) (2) < 0
2
µ M1 (t2 ) { (Eµ [] b) (b 1) (1)}
M1 (b 1) (Eµ [] b) (1) < 0
µ M1 (t2 ) { (Eµ [] b) (b 1) (1)}
2
(1µ) M1 ( t1) {(1) (Eµ [] a) (2 a) (1)}
M1 (2 a) (Eµ [] a) (1) < 0
(1µ) M1 ( t1) {(1) (Eµ [] a) (2 a) (1)}
It is easy to verify that the second derivatives with respect to b and β are negative. The final condition to
check is:
02 E [%] 02 E [%]
0b 2 0b0
det
> 0
02 E [%] 02 E [%]
0 0 b 0 2
@ 2 M12 (b 1) (Eµ [] b) (t2 1 2) [µ M1 (t2 ) (Eµ [] 1 2b) ]2 > 0
2
@ (1 µ) (2 1) (2t2 ) (b 1) t22 > 0
Substitution of the optimal b and β reduce the above equation to one of the conditions given in
Proposition 1, namely that µ < (1 - λ )/(1 - λ (t1/t2)). Verification of the second order conditions for the
ask side of the market is done in the analogous way.
Bid Side Solution
I proceed by solving for the bid price and the bid depth. The ask side is not shown due to the symmetry
of the problem, however it is available from the author upon request.
Substituting equation (2) into equation (1) yields:
µ M 1 t 2 1 t 2 2
µ
2M1 (Eµ [] b)
(Eµ [] 1 2b ) 0
After a few lines of algebra the above reduces to:
2b 2 (Eµ [] 1) b (Eµ []2 Eµ [] 1) µ
(Eµ [] 1)
t2 M1
0
The root that I am interested in is the negative root. To see why the positive root is not optimal, consider
the following case. Suppose λ = 0, then using the positive root, b = Eµ[θ]. Substituting b into the
specialist's profit function shows that the specialist yields zero profits; however, if b < Eµ[θ], the
specialist would make positive profits.
Switching over to the bid depth, I substitute equation (1) into equation (2) which yields:
µ M1 Eµ [] 1 (Eµ [] 1) 2
µ
2M1 (t2 1
)
2
(t2 ) 0
A bit of algebra can verify:
µ t2
1 2
3 t2 t22 M1 (Eµ [] 1)
2
2
0
Here, the positive root can be eliminated as a solution candidate. Since the first term (3/2)t2 is greater
than t2, it is outside the feasible range for β. The only way for β*< t2, is for the optimal solution to be the
negative root.
It is easy to verify that the condition on µ in Proposition 1 ensures the restrictions R1 through R4 by
substituting ( 1 - λ )/( 1 - λ(t1/t2) ) for µ into b* and β*.
QED.
Proof of Proposition 2
Imposing the constraints on the prices shown below and substituting them into the objective function
yields an optimization problem with the two choice variables β and α.
a ( 1 ) (µ 1 (1 µ) 2)
b ( 1 ) (µ 1 (1 µ) 2)
where 0 < < min
(1 µ) (2 1)
,
µ (2 1)
µ 1 (1 µ)2 µ 1 (1 µ)2
implies 1 < (1 ) Eµ [] < (1 ) Eµ [] < 2
E [ %(, ; ) ] µ (1 (1 ) Eµ []) M1 ((1 ) Eµ [] 1) (Eµ [] (1 ) Eµ []) (t2 1 2
)
2
(1µ) (2 (1 ) Eµ []) M1 (2 (1 ) Eµ []) (Eµ [] (1 ) Eµ []) ( 1 2 t1 )
2
First and Second Order Conditions
The following are the first order conditions associated with an interior maximum:
0 E [%] µ ( (1 ) E []) M ((1 ) E [] ) ( E []) (t ) 0
1
µ
1
µ
1
µ
2
0
2
0 E [%] M ((1 ) E [] ) ( E []) < 0
1
µ
1
µ
0 2
0 E [%] (1µ) ( (1 ) E []) M ( (1 ) E []) ( E []) ( t ) 0
2
µ
1
2
µ
µ
1
0
2
0 E [%] M ( (1 ) E []) ( E []) < 0
1
2
µ
µ
0 2
Constrained Depth Solutions
The first order conditions given above reduce to the following functions:
c t2 1
c t1 1
(t2 t1) (2 1) µ
(µ 1 (1 µ)2)
(t2 t1) (2 1) (1 µ)
(µ 1 (1 µ)2)
QED.
Proof of Proposition 3
Imposing the constraints on the depths shown below and substituting them into the objective function
yields an optimization problem with the two choice variables b and a.
' t1
' t2
where 0 < ' 1
E [ %(b, a; ') ] µ ' t2 (1 b) M1 (b 1) (Eµ [] b) (t2 ' 2
1 2 2
' t2 )
2
(1µ) ' t1 (2 a) M1 (2 a) (Eµ [] a) ( 1 '2 t12 t12 ')
2
First and Second Order Conditions
The following are the first order conditions associated with an interior maximum:
0 E [%] µ ' t M (t 2 ' 1 '2 t 2) (E [] 2 b ) 0
2
2
1 2
µ
1
2
0b
02 E [%] M (t 2 ' 1 '2 t 2) (2)) < 0
2
1 2
2
0b 2
0 E [%] (1µ) ' t M (E [] 2 b ) ( 1 '2 t 2 t 2 ') 0
1
1
1
1
µ
2
2
0a
02 E [%] M (2) ( 1 '2 t 2 t 2 ') < 0
1
1
1
2
0a 2
Constrained Depth Solutions
The first order conditions given above reduce to the following functions:
bc ac QED.
1
1 (µ 1 (1 µ) 2 1) 2
2 1
(t2 t1) (2 1) µ
1
1 (µ 1 (1 µ) 2 2) 2
2 1
(t2 t1) (2 1) (1 µ)
(1 1
') t
2 2
(1 1
') t
2 2