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An Analysis of Brokers’ Trading, With Applications to Order Flow Internalization
and Off-exchange Block Sales
Sugato Chakravarty
Purdue University
West Lafayette, Indiana, 47906
765-494-6427
Fax: 765-494-0869
EMAIL: SUGATO@PURDUE.EDU
Asani Sarkar
Federal Reserve Bank of New York
33 Liberty Street
New York, New York, 10045
212-720-8943
Fax: 212-720-1582
EMAIL: ASANI.SARKAR@NY.FRB.ORG
Current version: April 1998
Previous version: May 1997
The views expressed in the paper are those of the authors’ and do not represent those of the Federal Reserve
Bank of New York or the Federal Reserve System. The usual caveat applies.
A General Model of Brokers’ Trading, With Applications to Order Flow
Internalization and Off-exchange Block Sales
JEL Classification number: G12
Abstract
We study a wide variety of issues related to brokers’ trading. In our model, multiple informed
traders and noise traders trade through multiple brokers. Brokers may trade with their customers
in the same transaction (simultaneous dual trading) or trade after their customers in a separate
transaction (consecutive dual trading). We find that, with simultaneous dual trading, equilibrium
exists only if the number of informed traders exceeds the number of brokers; and, further, relative
to consecutive dual trading, brokers’ profits and uninformed losses are higher and informed
profits and market depth are lower. We endogenize the number of brokers and informed traders
and show that informed and noise traders choose one (all) broker(s) in the simultaneous
(consecutive) dual trading market. If the market entry cost is low, more informed traders enter
the simultaneous dual trading market and market depth may be higher relative to the consecutive
dual trading market. If the market entry cost is high, consecutive dual trading is clearly preferred.
We study order flow internalization by broker-dealers, and show that, in the free entry
equilibrium, internalization hurts retail customers and market quality; and that it is likely to be
more prevalent in thin markets with few informed traders. Finally, we examine off-exchange
block sales and find that, compared to exchange transactions, they are more liquid but less
informative.
Personal trading by brokers is pervasive throughout U.S. securities and futures markets,
and in financial markets around the world (Grossman (1989)). Yet, academics and policy makers
are unable to conclude if brokers’ trading is beneficial for customers and markets. For example,
the U.S. Congress passed the Futures Trading Practice Act in 1992, which directed the
Commodity Futures Trading Corporation (CFTC) to pass regulations prohibiting dual trading1/ in
high volume contracts. However, although dual trading is currently banned on the Chicago
Mercantile Exchange (CME) futures markets, it is practiced freely on other futures exchanges.2/
In contrast, regulators have not sought to restrict personal trading by brokers in the U.S.
equity markets, where the ratio of trading profits to brokers’ revenue increased from 0.18 to 0.26
in 1992.3/ Yet, even in equity markets, serious concerns remain regarding certain aspects of
brokers’ trading. A proposed New York Stock Exchange (NYSE) rule on front running aimed to
increase restrictions on proprietary trading by member firms.4/ Large institutional customers have
long been concerned that brokers may use knowledge of their orders to trade for their own
accounts.5/ Finally, The Securities and Exchange Commission (SEC) concludes that the
internalization of order flow, which occurs when a broker-dealer executes customer orders as a
1/
Dual trading in futures markets is the practice whereby futures floor traders execute trades for their own and customers’
accounts on the same day.
2/
The regulations allow affected exchanges to petition for relief based on 1) an acceptable audit trail, or ability to track a floor
traders’ activities, or 2) a threat to the hedging utility and price discovery function of futures markets, should the practice
of dual trading be prohibited. All affected exchanges have petitioned for relief. So far, only the Comex division (Comex)
of the New York Mercantile Exchange and the CME’s S&P 500 futures contract have received unconditional exemptions
from the dual trading ban. Seven other CME contracts and 13 contracts at the Chicago Board of Trade (CBOT) have
received conditional exemptions, allowing dual trading in low volume months only. See CFTC press releases, May 7 1997
and November 7 1997.
3/
The Securities Industry Factbook 1995. Figures relate to broker-dealers that are members of the New York Stock
Exchange (NYSE) and doing a public business, who account for the bulk of industry revenues.
4/
The Investment Dealer’s Digest, July 29, 1996. Specifically, NYSE has proposed widening its existing Rule 92 to allow
it to consider all member firms’ trades when looking for evidence of frontrunning---irrespective of whether they occurred
on the NYSE floor or not.
5/
See “Money Machine,” Business Week, June 10, 1991, pages 81-84.
1
market maker, “raise significant agency-principal concerns,” a feeling shared by other market
participants.6/
The lack of a consensus on brokers’ trading at the policy level is also reflected in the
academic literature. Empirical studies do not agree on the effect of dual trading on market
liquidity. Depending on the market studied, dual trading may increase, decrease or have no effect
on liquidity (Chang and Locke (1996), Chang, Locke and Mann (1994), Fishman and Longstaff
(1992), Smith and Whaley (1994)). There is no consensus in the theoretical literature, either.
Whereas Grossman (1989) argues in favor of a positive effect of dual trading on liquidity, Roell
(1990) and Sarkar (1995) conclude that dual trading may reduce liquidity. In Fishman and
Longstaff (1992), the effect of dual trading on liquidity is ambiguous. The literature also
disagrees on the effect of dual trading on customers’ welfare.7/
To provide clearer guidance to policy makers, we attempt to provide, in this paper, a fuller
examination of brokers’ trading than is available in the literature, by modeling differences in the
way brokers trade in different markets. Our approach draws on Grossman’s (1989) distinction
between simultaneous dual trading (where a broker trades for himself and a customer in the
same transaction) and consecutive dual trading (where a broker trades for customers as an agent
and for himself at other times, but not in the same transaction). The constraint on timing differs
by markets. Simultaneous dual trading occurs in securities markets, currency and interest rate
6/
The SEC comment is regarding preferencing, which is one way for broker-dealers to internalize customer order flow. More
generally, internalizing occurs when a broker-dealer directs order flow to an affiliated specialist. See “Report on the practice
of preferencing,” the SEC, April, 1997. For reaction of market participants, see the comments of Robert Murphy, president
of RPM Specialist Corp., as reported in Securities Week, March 11, 1996. Battalio, Greene and Jennings (1997) estimate
that 7 per cent of all orders, in a set of actively-traded NYSE securities, were internalized on the Cincinnati Stock Exchange
during September 1994.
7/
In Roell (1990), uninformed traders whose trades are observed (not observed) by the dual trader have higher (lower) profits
with dual trading. In Fishman and Longstaff (1992) and Sarkar (1995), the results depend upon whether brokers pass on
their entire trading profits to customers in the form of lower commissions (which is assumed to be independent of the order
size). Dual trading is generally bad for customers without the reduction in commissions, and good for customers with the
reduction.
2
swap markets, and the fixed income market, but not in futures markets. Consecutive dual trading
occurs in futures markets, and in all markets with simultaneous dual trading (Grossman (1989)).
While our model initially deals with dual trading, we later apply it to examine order flow
internalization by brokers. We further extend the model by reinterpreting brokers as “follow-on
traders”--i.e., those who trade based on the order flow of preceding traders. We use the extended
model to study off-exchange sales of large blocks of stocks, which are motivated by the desire of
large block sellers to hide their trades from “follow-on traders” (see below and section VI for a
fuller description).
In our model, based on Kyle (1985), multiple informed traders and noise traders trade by
dividing their orders equally among multiple dual traders. Later, we allow informed and noise
traders to choose the number of brokers, and endogenize the number of informed traders and
brokers entering the market. Informed traders observe private signals about the payoff of a single
risky asset and choose a quantity to buy or sell through brokers. The brokers, on receiving orders
from informed traders and noise traders, may: dual trade consecutively, by executing customers’
orders in the first period as agents and trading on their personal accounts in the second period as
principals; or dual trade simultaneously, by acting as agents for customers and trading for
themselves as principals in the same transaction.
If brokers dual trade simultaneously, each broker submits the net order (his personal
trades plus his customers’ trades) to the market maker for execution, and the game ends in a
single period. If brokers trade consecutively, trading occurs over two periods. In period one,
each broker submits the sum of informed and period one noise trades to the market maker for
execution. In period two, each broker submits his personal trade and period two noise trades to
the market maker. The market maker batches brokers’ trades and period two noise trades and
3
executes them at a single price. In all cases, the market maker prices the asset to earn zero
expected profits, conditional on the history of order flows realized so far.
Compared to the no-dual-trading benchmark, informed traders’ period one trading is
unaffected with consecutive dual trading, while noise traders’ losses per period are lower. Also,
price informativeness and marker depth are higher with consecutive dual trading. Thus,
consecutive dual trading is unambiguously beneficial for customers and the market, compared to
the no-dual-trading benchmark.
Simultaneous dual trading is generally bad for customers and the market relative to the
no-dual-trading benchmark. The reason is that brokers mimic informed trades by trading with
insiders in the same direction, causing insiders to trade less and at a higher price (in absolute
value), and lowering informed profits. In fact, we show that if the number of brokers exceeds the
number of informed traders, the simultaneous dual trading equilibrium does not exist.8/
Uninformed losses are unaffected but, since brokers offset a portion of noise trades, market depth
is reduced.
The only beneficiaries of simultaneous dual trading markets are brokers, whose expected
trading profits are higher than with consecutive dual trading. However, brokers are chosen by
customers, and so our results imply that markets with simultaneous dual trading may not be
viable---a prediction clearly at odds with institutional reality. To resolve this puzzle, we allow
informed and noise traders to choose the number of brokers and, further, we endogenize the
number of brokers and informed traders in the market.
We find that, with consecutive dual trading, informed and noise traders choose all the
brokers available in the market whereas, with simultaneous dual trading, they choose a single
8/
Our result is consistent with the institutional fact that, in futures markets, where typically many dual traders are present in
each contract, simultaneous dual trading is not permitted.
4
broker. These results reflect the facts that, with consecutive (simultaneous) dual trading,
uninformed losses (informed profits) decrease with the number of brokers.
In the free entry equilibrium, traders pay a fixed fee to enter the market. Consistent with
intuition, the number of entering brokers and informed traders is positively related to the asset
volatility and the size of noise trades, and negatively related to the entry fee. In the simultaneous
dual trading market, only one broker enters the market, anticipating correctly that customers will
choose just one broker. In the consecutive dual trading market, multiple brokers may enter the
market, but less in number than the number of informed traders.
If the entry fee is high, not enough informed traders may enter the simultaneous dual
trading market to make it viable. However, if the fee is low, more informed traders enter the
simultaneous dual trading market than the consecutive dual trading market. Surprisingly, if
enough informed traders enter the simultaneous dual trading market, market depth may be lower
compared to consecutive dual trading. Thus, with sufficiently low entry costs, and optimal choice
of the number of brokers by informed and noise traders, the simultaneous dual trading market
becomes viable.
In the simultaneous dual trading model, brokers act as market makers to noise traders by
offsetting a portion of their order flow in the same transaction. Thus, the model is appropriate for
studying order flow internalization, which primarily affects uninformed retail order flow (Easley,
Kiefer and O’Hara (1996), Battalio, Greene and Jennings (1997), and the SEC, 1997). We
assume that a subset of brokers are involved exclusively with internalizing the order flow of noise
traders, while the remaining brokers are involved exclusively with mimicking informed traders.
With internalization, broker’s commissions are lower, as predicted by the SEC and the NASD
5
research staff,9/ and, consistent with Battalio (1997), the price, net of commissions, may be lower
with internalization.
In the free entry equilibrium, however, internalization reduces market depth and price
informativeness, and increases uninformed losses. Further, the number of internalizing brokers is
negatively related to the market depth and the number of entering informed traders. Thus, our
model predicts that internalization is likely to be more prevalent in thin markets with few informed
traders. If thin markets have high spreads, this result supports advocates of purchased order flow
who argue that it primarily affects NYSE stocks with large spreads (Easley, Kiefer and O’Hara,
1996). However, contradicting these advocates, our results show that market quality is affected
adversely.
Next, we consider off-exchange block sales---i.e., the sale of very large blocks of stocks
by big investors or institutional holders to broker-dealers away from U.S. exchange floors. In a
typical off-exchange sale, a dealer offers to buy the block of stock at a discount to the closing
price on the exchange, acting as a principal. The recent popularity of such sales is due to the fact
that, if the sales were made on an exchange in piecemeal fashion (as predicted by Kyle (1985)),
the first sale may alert other traders, such as short sellers, who may then drive the price down.10/
Proponents say off-exchange sales provide liquidity for block trading, while opponents claim that
it reduces transparency (since the trades occur away from the floor of the exchange).
9/
The SEC suggests this possibility in its “Market 2000” report. The NASD research staff argues that many brokers who
use preferencing are discount brokers (see “The introduction of Nacquess into the Nasdaq stock market: Intent and
expectation,” June 1996).
10/
For example, Carl Icahn, who sold off-exchange 19.9 million shares in RJR Nabisco Holding Corp. to Goldman, Sachs
& Co, is quoted as saying that, with a piecemeal sale on the exchange floor, "word would have spread, giving short-sellers
a chance to drive the price down at his expense." See "More big stocks are handled 'off-Board'," the Wall Street Journal,
Wednesday, April 9, 1997, page C1.
6
Since the off-exchange block sale does not involve brokers’ trading, we use the model of
Holden and Subrahmanyam (1992)---where insiders sell shares piecemeal over time--- to analyze
the off-exchange transaction. We use our consecutive dual trading model to describe the
exchange transaction, reinterpreting brokers as follow-on traders (such as short-sellers). When
insiders sell a second time on the exchange floor, we assume they have to sell at the period two
price determined by follow-on traders. Our results show that, compared to the exchange market,
the off-exchange transaction may be more liquid but the off-exchange price is less informative.
Thus, while off-exchange sales are good for insiders, they are bad for the exchange (since the
exchange price would be more informative if insiders had traded on the floor).
The existing literature on dual trading does not consider multiple informed traders and
multiple brokers in the same model, nor do they endogenize the number of traders. Fishman and
Longstaff (1992) study consecutive dual trading in a model with a single broker and a fixed order
size. Roell (1990) and Chakravarty (1994) have multiple dual traders but a single informed
trader. Sarkar (1995) studies simultaneous dual trading with multiple informed traders and a
single broker. Our results imply that the coexistence of consecutive and simultaneous dual trading
markets cannot be explained by models where traders do not optimally choose the number of
brokers, and where market entry is fixed.
The idea of “follow-on” trading goes all the way back to Keynes (1937), who viewed
traders’ orders as being determined primarily by the observed or expected trades of other market
participants. Scharfstein and Stein (1990) and Shiller (1990, 1995) also suggest that traders track
each other. Welch (1992) shows that cascades are possible in a model where traders observe
whether prior traders bought or sold and then decide to buy or sell only one unit. In Chakrabarti
and Roll (1997), traders observe the trades and expenditures of prior large (but not small) traders
7
and, consequently, make a noisy inference of the large traders’ information. Their simulation
exercise shows that return volatility is lower and price efficiency is higher when traders observe
other traders.
Regarding internalization, Battalio and Holden (1996) show that, if brokers can distinguish
between informed and uninformed orders (as in our model), they can profit from internalizing
uninformed orders. However, unlike our model, they do not focus on the effect of internalization
on market quality. Dutta and Madhavan (1997) find that a collusive equilibrium is easier to
sustain with preferencing arrangements. In contrast to the theoretical results, the empirical
studies of Battalio (1997), Battalio, Greene and Jennings (1997) and Lightfoot, Martin, Peterson,
and Sirri (1997) and the experimental study of Bloomfield and O’Hara (1996) find no adverse
effect on market quality. As Battalio, Greene and Jennings (1997) state, their result may imply
that broker-dealers are not systematically “skimming” uninformed order flows or, alternatively,
that internalizing brokers have a cost advantage in executing orders. Macey and O’Hara (1997)
survey the literature on preferencing and internalization.
The remainder of the paper is organized as follows. Section I describes a trading model
with multiple customers and many brokers, when the number of informed traders and brokers is
fixed. Sections II and III solve the consecutive and simultaneous dual trading models. Section IV
endogenizes the number of informed traders and brokers. Section V analyzes order flow
internalization, while section VI studies off-exchange stock sales. Section VII concludes. All
proofs are in the appendix.
8
I.
A Model of Trading With Multiple Customers and Multiple Brokers.
We consider an asset market structured along the lines of Kyle (1985). There is a single
risky asset with random value v, drawn from a normal distribution with mean 0 and variance *v.
There are n informed traders, each of whom receives a signal about the true asset value and
submits market orders. For an informed trader i, i=1,..,n, the signal is si = v + ei, where ei is
drawn from a normal distribution with mean 0 and variance *e. A continuum of noise traders also
submit aggregate market orders u, where u is normally distributed with mean zero and variance
*u.
All random variables are independent of one another.
All customers, informed and uninformed, must trade through brokers. There are m
brokers in the market, who submit customer orders to the market maker. m and n are common
knowledge. We assume that orders are split equally among the m brokers. By observing
informed orders, brokers can infer the informed traders' signals. By observing orders of noise
traders, they are aware of the size of uninformed trades. Consequently, brokers have an incentive
to trade based on their customers' orders. However, brokers are not allowed to trade ahead of
(i.e., front run) their customers.
Brokers may trade in two possible ways. They may execute their customers' orders first,
and trade for their own accounts second in a separate transaction---i.e., engage in consecutive
dual trading. Alternatively, they may trade with their customers in the same transaction--i.e.,
engage in simultaneous dual trading. The sequence of events is as follows: in stage one,
informed trader i, i=1,..,n observes si and chooses a trading quantity xi. In stage two, broker j,
j=1,..,m trade consecutively or simultaneously, and places orders of an amount zj. In subsequent
sections, we describe in more detail how simultaneous and consecutive dual trading differ.
Finally, all trades (including brokers' personal trades) are batched and submitted to a market
9
maker, who sets a price that earns him zero expected profits conditional on the history of net
order flows realized.
Initially, the number of informed traders and brokers is fixed. Later, we allow informed
traders to choose the number of brokers to allocate their orders to, and study the free entry
equilibrium where informed traders and brokers decide whether to enter the market, depending on
a market entry cost and their expected profits upon entering.
II.
Consecutive Dual Trading.
In this section, we solve for the equilibrium in a market with consecutive dual trading. We
assume that brokers do not trade with their customers in the same transaction--i.e., simultaneous
dual trading is not allowed, as in futures markets.
A. The Consecutive Dual Trading Model
Trading occurs in two periods. In period one, brokers receive market orders from n
informed traders and the noise traders, which they then submit to the market maker. In period
two, brokers trade for themselves, along with period two noise traders. Each period, a market
marker observes the history of net order flow realized so far and sets a price such to earn zero
expected profits, conditional on the order flow history.
The sequence of events is as follows: in period one, informed trader i, i=1,..,n observes si
and chooses xi,d, knowing that his order will be executed in the first period. Accordingly,
informed trader i, i=1,..,n, chooses xi,d to maximize conditional expected profits E[(v-p1)xi,d si],
where the period one price is p1 = 1y1, the period one net order flow is y1 = xd+u1, the aggregate
informed trade is xd =*ixi,d and u1 is the period one noise trade.
10
In period two, brokers choose their personal trading quantity after observing the n-vector
of informed trades {x1,d,..,xn,d}, u1 and p1. Thus, broker j, j=1,..,m, chooses zj to maximize
conditional expected profits E[(v-p2)zj {x1,d/m ,..,xn,d/m}, u1/m, p1], where the conditioning is based
on each broker observing his portion of the informed and uninformed orders received, plus the
period one price.
The m brokers submit their personal trades to the market maker, who sets p2 = 2y2+µ 2y1
where y2 = *jzj + u2, *jzj is the aggregate trade of all brokers and u2 is the noise trade in period
two. Finally, the liquidation value v is publicly observed and both informed traders as well as
brokers realize their respective profits (if any).
Define t=*v/*s, where t is the unconditional precision of si, i=1,..,n. Note that 0t1.
Further, define Q = 1+t(n-1), where (Q-1)si represents informed trader i's conjecture (conditional
on si) of the remaining (n-1) informed traders' signals. Since informed traders have different
information realizations, they also have different conjectures about the information of other
informed traders. t also measures the correlation between insider signals. For example, if t=1
(perfect information), informed signals are perfectly correlated, Q=n, and informed trader i
conjectures that other informed traders know (n-1)si--i.e., the informed trader believes other
informed traders have the same information as he.
Proposition 1 below solves for the unique linear equilibrium in this market.
Proposition 1: In the consecutive dual trading case, there is a unique linear equilibrium for
t>0. In period one, informed trader i, i=1,..,n, trades xi,d=Adsi, and the price is p1=1y1. In
period two, each broker j, j=1,..,m trades z = B1xd + B2u1, where xd = ixi,d
and the price is p2 = 2y2 + µ2y1, where:
11
t(u
Ad
(1)
n(v
nt(v
1
(2)
(u(1Q)
1
B1
(3)
✓mQ(1Q)
r
Q
B2
(4)
✓m(1Q)
Vnt(
r
m
1 m
✓Q(1Q)(u
2
v
µ2
(5)
nt(v
(6)
(1Q) (u
Wd is the expected trading revenue per broker, as given by:
Wd
✓nt(u(v
1
(7)
✓Q(1Q) m(1m)
The period one solution is identical to Kyle's (1985) single period equilibrium, extended to
include multiple informed traders and noisy signals.11/ Thus, informed trades are not affected by
dual trading in period two since informed traders trade only in period one.12/ In period two, dual
traders piggyback on period one informed trades (B1>0) and offset noise trades (B2<0).
Corollary 1: (1) B1/ n < 0, and B1/ t < 0 for n>1.
(2) B1/ m < 0,
B2 / m < 0.
B2 / n >0 and
B2 / t > 0 for n>1.
(mB1)/ m > 0, (m B2 )/ m >0.
11/
See, for example, lemma one in Admati and Pfleiderer (1988).
12/
The result is similar to the consecutive dual trading equilibrium in Fishman and Longstaff (1992), where the first trade
is not affected by dual trading.
12
The extent of piggybacking B1 decreases in the number of informed traders n and the
information precision t. Competition between informed traders leads insiders to use their
information less, making piggybacking less valuable. An increase in t increases the correlation
between insiders' signals, reducing (for n>1) the value of observing multiple informed orders.
Since the marginal value of observing noise trades is higher when the marginal value of observing
informed trades is lower, B2 , the extent to which noise trades are offset, increases in n and t.
Competition between brokers reduces the extent to which each broker exploits customer trades.
However, brokers in the aggregate exploit customer trades more.
B. The Impact of Consecutive Dual Trading on Traders' Profits and the Market
Since the period one equilibrium is identical to one without dual trading, we can compare
it with the period two equilibrium to obtain the impact of consecutive dual trading.
Corollary 2. (1) µ2 = 1 and 2 < 1.
(2) Price informativeness is higher with consecutive dual trading.
(3) Uninformed losses per period are lower while informed profits are
unaffected with consecutive dual trading.
µ 2 = 1 because covariance(y1,y2)=0: to the market maker, the period one order flow is not
informative about the period two order flow. The reason is that dual traders, by piggybacking on
informed trades, induce a positive serial correlation between the order flows. But, by offsetting
noise trades, dual traders also induce a negative serial correlation between the order flows. These
two effects are exactly offsetting. Also, 2 < 1 implies that the market maker's adverse selection
costs due to brokers' trading in period two is less than the adverse selection cost from insider
trading in period one.
13
Since the covariance between the order flows of the two periods is zero, information
revealed by the period two price is simply the information revealed by the period one price plus
the additional information revealed by brokers' trading.
Informed trades and, therefore, informed profits, are not affected by dual trading. Since
µ 2=1, it follows that the period two price p2 = p1 + 2y2: the dual trader trades at worse prices
than the informed trader. Thus, dual trading profits are lower than informed profits in the first
period and, consequently, uninformed losses to dual trading are lower than losses to informed
traders.
We now solve for the simultaneous dual trading equilibrium.
III.
Simultaneous Dual Trading.
A. The Simultaneous Dual Trading Model
Simultaneous dual trading is modeled in a single period Kyle (1985) framework. The
notations are the same as in section II. All variables and parameters related to simultaneous dual
trading are denoted either with superscript s or subscript s.
A group of n informed traders receive signals si about the unknown value v, and choose
quantities xi,s knowing that his order will be executed along with the orders of brokers and noise
traders in the same transaction. Accordingly, informed trader i, i=1,..,n, chooses xi,s to maximize
conditional expected profits E[(v-ps)xi,s si], where the price is ps = sys, the net order flow is ys =
xs+mz+u, the aggregate informed trade is xs=*ixi,s , z is the amount each broker trades and u is
the noise trade.
Upon receiving the orders of their informed and uninformed customers, brokers choose
their personal trading quantity after observing the n-vector of informed trades {x1,s,..,xn,s} and u.
14
Thus, broker j, j=1,..,m, chooses zj,s to maximize expected profits E[(v-ps)zj,s {x1,s/m,..,xn,s/m},
u/m], where the conditioning is based on each broker observing his portion of the informed and
uninformed orders.
The m brokers submit their customer trades and personal trades to the market maker, who
sets the price that earns him zero expected profits conditional on the net order flow realized.
Finally, the liquidation value v is publicly observed and both informed traders as well as brokers
realize their respective profits (if any).
Proposition 2 below solves for the unique linear equilibrium in this market.
Proposition 2: In the simultaneous dual trading case, there is a unique linear equilibrium
for t>0 and Q>m. Informed trader i, i=1,..,n, trades xi,s=Assi, broker j, j=1,..,m trades z =
B1,sxs+B2,su, where xs=*ixi,s, and the price is ps = sys, where:
As
s
1 (Q m) t(u
1m Q
n(v
_J
_j-
(1m)
nt(v
_j
_j
(u(1Q)
1
(1m)
B2,s
1
(Q m)
B1,s
(8)
(9)
(10)
(11)
Ws, the expected trading revenue per broker, satisfies:
Ws
_f
nt(u(v
Q(1m)
(12)
15
Equilibrium exists if Q>m. Since nQ, existence implies n>m: the number of informed
traders must exceed the number of brokers. The intuition behind this result is as follows.
Suppose an insider buys. Dual traders also buy in the same transaction, piggybacking on the
insider trade (i.e. B1>0), and increasing the price paid by the insider for his purchase. From
corollary three below, the order of an individual insider is exploited less as the number of insiders
increases, and is exploited more as the number of brokers increases. If the number of brokers is
too large relative to the number of insiders, the adverse price effect makes it too costly for
insiders to trade.
Corollary 3: (1) B1,s/ n < 0, and B1,s/ t < 0 for n>1.
(2) B1,s/ m > 0,
B2,s / m < 0.
B2,s / n = 0 and
B2,s / t = 0.
(mB1,s)/ m > 0, (m B2,s )/ m >0.
Unlike the consecutive dual trading case, the extent to which noise trades are offset
depends inversely on the number of brokers but is independent of the number of informed traders
n. In the consecutive dual trading model, knowledge of period one noise trades is relevant to
brokers because it influences the way the market maker prices the asset in period two. The
market maker's pricing problem, in turn, depends on the extent of private information. This
interaction between information and noise trades is absent in the case of simultaneous dual
trading.
B. The Impact of Simultaneous Dual Trading on Traders' Profits and the Market
In this section, we compare the simultaneous dual trading equilibrium with one where dual
trading is banned (i.e., the period one solution for the consecutive dual trading equilibrium).
Corollary 4. Compared to the no-dual-trading benchmark,
(1) Market depth is lower with dual trading.
16
(2) Price informativeness is the same with or without dual trading.
(3) Uninformed losses are the same and informed profits are lower with simultaneous dual
trading.
Market depth is lower with dual trading because brokers reduce the net order flow of
noise traders, increasing the market maker's adverse selection costs. Price informativeness
depends positively on the variance of the net order flow and inversely on market depth. Dual
trading reduces depth and decreases the variance of the net order flow in the same proportion.
Hence, price informativeness is invariant to dual trading.
Informed profits are lower with simultaneous dual trading because the dual traders'
piggybacking forces insiders to reduce the size of their orders (for the same level of information).
However, dual trading profits make up the reduction in informed profits exactly, and,
consequently, uninformed losses to dual traders are the same.
C.
Comparison Between Simultaneous And Consecutive Dual Trading
By combining results from propositions one and two, and corollaries two and four, we can
compare market parameters for the two kinds of dual trading, holding the number of informed
traders and brokers fixed.
Corollary 5. Suppose m and n are fixed. Then:
(1) If Qm, only the consecutive dual trading equilibrium is viable.
(2) If Q>m, then both types of dual trading equilibrium are viable. Relative to
simultaneous dual trading, uninformed losses and brokers’ profits are lower, while
informed profits, market depth and price informativeness are higher with consecutive dual
trading.
17
Corollary five implies that only brokers prefer the simultaneous dual trading market.
However, since brokers are chosen by customers, it is puzzling that markets with simultaneous
dual trading should continue to exist. To resolve this puzzle, in the next section, we allow
informed traders to choose brokers, which may be one way for informed traders to protect
themselves from excessive piggybacking by brokers. We also allow endogenous market entry by
traders.
IV. Free Entry by Informed Traders and Brokers
In this section, we study the two forms of dual trading, given that informed and
uninformed traders optimally choose the number of brokers to give their orders to, and given that
there is free entry into the asset market. The decision-making sequence of agents is as follows: 1)
Informed traders and brokers simultaneously decide whether to enter the market; (2) Informed
and noise traders choose the number of brokers to give the order to; (3) Informed and noise
traders divide their orders equally among the chosen brokers. From then on, the game continues
as before.
Informed traders choose the number of brokers to maximize expected profits.
Uninformed noise traders choose the number of brokers to minimize their expected losses to
informed traders and brokers. The following proposition describes traders’ choice of the number
of brokers to trade with.
Proposition 3. (1) When brokers trade consecutively, informed and noise traders choose all
the brokers available.
18
(2) Suppose there are at least two informed traders, so that the simultaneous dual trading
equilibrium exists. Then, with simultaneous dual trading, informed and noise traders give
orders to only one broker.
With consecutive dual trading, informed traders' profits are independent of m but noise
trader losses are decreasing in m. Thus, noise traders choose all available brokers while informed
traders are indifferent to the choice of m. When brokers trade simultaneously, the situation is
reversed: informed profits are decreasing in m whereas uninformed losses are independent of m.
Thus, informed traders choose one broker and noise traders are indifferent to the choice of m.
For concreteness, we assume that informed and noise traders choose the same number of brokers
in each case.
Next, consider free entry by informed traders and brokers. Let ki be the cost of entering
market i, i=d (consecutive dual trading),s (simultaneous dual trading). To obtain analytic
solutions, we assume t=1. The free entry equilibrium satisfies two conditions. Traders enter a
market until their expected profits, net of the entry cost, are zero. And the cost is low enough so
that entry is profitable for the minimum number of traders necessary to sustain equilibrium.
In the consecutive dual trading equilibrium, the equilibrium number of informed traders,
nd, entering the market is independent of m, since informed profits are independent of m. But
broker profits are dependent on nd, and so the equilibrium number of brokers entering the market,
md, depends on nd. In the free entry equilibrium, we show below that md< nd. Thus, costs must be
low enough to sustain at least one broker and two informed traders.
In the simultaneous dual trading equilibrium, proposition three implies that only one
broker enters the market and, proposition two implies that at least two informed traders must
19
enter the market, or the market will not be viable. Again, costs must be low enough to sustain at
least one broker and two informed traders.
Proposition four solves for the free entry equilibrium in the two markets.
Proposition 4. (1) In the market with consecutive dual trading, nd informed traders and md
brokers enter the market, where md< nd, and md and nd are given by:
_j
nd(1
_j
(1
n d)
md
)
(u(v
kd
(u(v
_j
md
_j
1
nd
____,/
kd
1
(13)
(14)
At least two informed traders and at least one broker enter the market if:
kd
(u(v
_j
(15)
_j
3 2
(2) In the market with simultaneous dual trading, ns informed traders and one broker
enter the market, where ns is given by:
1)
(ns
n s)
ns(1
(u(v
_j
_j
(16)
2ks
At least two informed traders and one broker enter the market if:
ks <
(u(v
_j
(17)
_j
12 2
Proposition four is intuitive: the number of informed traders and brokers entering a market
is inversely related to the cost of entering the market, and positively related to the volatility of the
asset value and the size of noise trades. In the consecutive dual trading market, a broker expects
20
to make less trading profits than an informed trader since he trades second. Thus, the equilibrium
number of brokers is less than the equilibrium number of informed traders in this market.
An important observation is that the participation constraint (17) is more restrictive than
the inequality (15), implying that the simultaneous dual trading equilibrium is viable only at lower
market entry cost, relative to the consecutive dual trading equilibrium. In the following
proposition, we compare the two types of dual trading markets given free entry by brokers and
informed traders and the optimal choice of the number of brokers by informed traders.
Proposition 5. In the free entry equilibrium of proposition four,
1) Suppose kc = ks = k. If k is high (low), then nc is greater (less) than ns.
2) Market depth is higher (lower) with simultaneous dual trading if ns is high (low) relative
to nc and mc.
The proposition says that, if entry costs are low, the simultaneous dual trading market may
have more informed traders and lower market depth in a free entry equilibrium. The reason is that
informed traders protect themselves from excessive piggybacking by choosing only one broker,
thus maximizing their expected profits in the by simultaneous dual trading market (see corollary
three). Consequently, if entry costs are low enough, many informed traders enter the
simultaneous dual trading market and market depth is higher. Thus, the proposition provides
some intuition as to why we see markets with simultaneous dual trading exist in the real world.
V. Internalization of order flow by broker-dealers
Internalization is the direction of order flow by a broker-dealer to an affiliated specialist or
order flow executed by that broker-dealer as market maker. Broker-dealers can internalize order
flow in several ways. For example, large broker-dealer firms, particularly NYSE member firms,
21
purchase specialist units on regional exchanges and direct small retail customer orders to them.13/
In off-exchange internalizations, NYSE firms execute orders of their retail customers against their
own account, with the transaction taking place in the so-called third market or over-the-counter
market. Such transactions, also called 19c-3 trading,14/ has become a major source of profits for
broker-dealers.
We use our simultaneous dual trading model to analyze order flow internalization. Since
internalizing brokers typically handle order flows of small retail investors, we assume that there
are m1 internalizing brokers who handle all of the noise trades, and m2 piggybacking brokers who
handle all the informed traders, with m1+m2 = m. As before, the market maker sees the pooled
order, and, further, cannot distinguish between internalizing brokers and informed-order brokers.
Let z1 be the trade of an internalizing broker and let z2 be the trade of informed-order
brokers. From (10) and (11):
u
(1m1)
z1
z2
(18)
xs
(Q
(19)
m2)
We compare uninformed losses, brokers' commissions and the market quality between the
internalization model and a model with no brokers' trading. For simplicity, we assume t=1. The
following proposition shows the effect of order flow internalization on noise trader losses and the
market, for both the fixed entry and free entry equilibrium solutions.
13/
See “Report on the practice of preferencing,” the SEC, April, 1997.
14/
The name refers to Rule 19c-3 allowing NYSE stocks listed after April 26, 1979 to be traded off-exchange. Brokerdealers were rumored to have earned over $500 million in 1994 from 19c-3 trading. See “In-House trades can be costly
for small investors,” the Wall Street Journal, December 20, 1994, page C1.
22
Proposition 6. (1) Suppose market entry is fixed. With internalization of order flow,
market depth is lower while uninformed losses and price informativeness is unchanged.
The competitive broker’s commission is lower with order flow internalization.
(2) Suppose market entry is free. Then, with internalization of order flow, (i)
one piggybacking broker, no informed traders and mo internalizing brokers enter the
market, where mo varies inversely with no, and mo < no. (ii) Relative to a market with no
order flow internalization, market depth, price informativeness and the number of
informed traders are lower, while uninformed losses are higher.
With fixed entry, the results on market depth, price informativeness and uninformed losses
follow from corollary four. Brokers make positive profits, reducing the competitive commission.
The expected profits of internalizing brokers are inversely related to the market depth. As more
informed traders enter the market, market depth increases, and so internalizing brokers’ expected
profits are lower. Thus, in the free entry equilibrium, there is less internalization when there are
more informed traders. Further, the number of entering informed traders is lower with order flow
internalization, since internalizing brokers reduce market depth by offsetting the uninformed order
flow. Thus, compared to the market without internalization, price informativeness is lower and
uninformed losses are higher.
Based on proposition six and the discussion above, we state the following corollary.
Corollary 6. The extent of order flow internalization is inversely related to market depth
and the number of informed traders.
23
VI. Off-exchange stock sales by broker-dealers.
In this section, we re-interpret the brokers in our model as follow-on traders--i.e., traders
whose trading is derived from observing the order flow of preceding traders. In the context of
this broader definition, we study off-exchange sales of large blocks of stocks.
In a typical off-exchange sale, the market maker offers to buy a block of stock at a
discount to the closing price on the exchange, acting as a principal. Thus, an off-exchange
transaction involves no dual trading or follow-on trading. The appropriate model for the offexchange transaction is Holden and Subrahmanyam (1992), since insiders optimally dribble a large
trade over time. We simplify by assuming two periods of trading and letting the time between
two auctions in the Holden and Subrahmanyam (1992) model ( tn in their notation) equal one.
Thus, n insiders commit to sell their block of stock off-exchange over two periods. In deciding
how much stock to sell, insiders maximize their total expected profits in the current and future
period. The market is structured along the lines of Kyle's (1985) sequential auctions model.
We use our consecutive dual trading model to describe the insiders' alternative of selling
on the exchange floor. In period one, insiders sell stock on an exchange. m follow-on traders
(equivalent to the consecutive dual traders) observe the stock sale in period one, and engage in
"follow-on" trading in period two. When insiders sell a second time, we assume they have to sell
at the price p2 determined by follow-on traders in period two.
To determine whether off-exchange trading benefits insiders, we compare market depth
and price informativeness in period two for the two cases. We assume t=1 (perfect information)
in the consecutive dual trading model and. Since analytic solutions are not possible, we use
numerical illustrations for our results. The following proposition (illustrated in figure one)
compares the market quality in period two for off-exchange and exchange trading by insiders.
24
Proposition 7. Price informativeness is lower but market depth higher with off-exchange
trading.
Off-exchange trading provides more liquidity to insiders since "follow-on" trading drives
up period two prices, making it costly for insiders when they trade a second time on the exchange.
However, greater liquidity is obtained at the expense of the price efficiency of the exchange
market.
VII. Conclusion.
In this article, we study a wide variety of issues related to brokers’ trading. Multiple
informed traders and noise traders trade through multiple brokers, who either trade in the same
transaction as their customers (simultaneous dual trading) or in a separate transaction
(consecutive dual trading).
While the consecutive dual trading equilibrium always exists, the simultaneous dual
trading equilibrium fails when the number of brokers is greater than the number of informed
traders. The reason is that brokers trade with informed traders in the same direction, thus
worsening informed traders’ terms of trade. This effect is magnified with many brokers, leading
informed traders to stop trading. When both equilibria exist, informed profits and market depth
are lower, while uninformed losses and brokers’ profits are higher with simultaneous dual trading,
relative to consecutive dual trading. Thus, only brokers prefer simultaneous dual trading.
We allow informed and noise traders to choose the number of brokers, and endogenize the
number of brokers and informed traders in the market. In the simultaneous dual trading market,
informed and noise traders choose only one broker whereas, with consecutive dual trading,
informed and noise traders choose all available brokers. If the market entry cost is low, more
25
informed traders enter the simultaneous dual trading market, and market depth may be lower,
relative to the consecutive dual trading market. Thus, the adverse effects of simultaneous dual
trading on customers and the market are mitigated in the free entry equilibrium. If the market
entry costs are high, consecutive dual trading is better.
In the simultaneous dual trading model, we allow some brokers to internalize the
uninformed order flow, by selling to noise traders as dealers, out of inventory. In the free entry
equilibrium, we find that although internalization results in lower brokers’ fees, market depth and
price informativeness are lower, and uninformed losses are higher. In addition, the number of
internalizing brokers is inversely related to market depth and the number of informed traders.
We reinterpret brokers as “follow-on traders”, who trade based on the order flows of
preceding traders, and apply our consecutive dual trading model to explain large block sales
conducted off the exchange floor to avoid “follow-on” trading. We find that such off-exchange
transactions may provide additional liquidity for large block sales, but at the expense of reducing
the informativeness of the exchange market.
26
References
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Review of Financial Studies, 1, 3-40.
Battalio, Robert and Craig W. Holden (1996), Why doesn’t decimal trading eliminate payment
for order flow and internalization? Working Paper, Indiana University.
Robert Battalio (1997), Third market broker-dealers: Cost competitors or cream skinners?
Journal of Finance, 52, 341-352.
Battalio, Robert, Greene, Jason and Robert Jennings (1997), Do competing specialists and
preferencing dealers affect market quality? Review of Financial Studies, 10, 969-993.
Bloomfield, R. and M. O’Hara (1996), Does order preferencing matter? Working Paper, Johnson Graduate
School of Management, Cornell University.
Chakrabarti, Rajesh and Richard Roll (1997), Learning from others, reacting and market quality,
Working Paper, University of California, Los Angeles.
Chakravarty, S. (1994), Should actively traded futures contracts come under the dual trading
ban?, Journal of Futures Markets, 14, 661-684.
Chang, E. C. and P. R. Locke (1996), The performance and market impact of dual trading:
CME Rule 552, Journal of Financial Intermediation, 5, 23-48.
Chang, E. C., Locke, P. R. and S. C. Mann (1994), The effect of CME Rule 552 on dual
traders, Journal of Futures Markets, 14(4), 493-510.
Dutta, P. and A. Madhavan (1997), Competition and collusion in dealer markets. Journal of Finance, 52,
245-276.
Easley, David, Kiefer, Nicholas M. and Maureen O’Hara (1996), Cream-skimming or profit-sharing? The
curious role of purchased order flow. Journal of Finance, 51, 3, 811-833.
27
Fishman, M. and F.A. Longstaff (1992), Dual trading in futures markets, Journal of Finance,
47, 643-672.
Grossman, S.J. (1989), An economic analysis of dual trading, working paper, University of
Pennsylvania.
Holden, C.W., and A. Subrahmanyam (1992), Long-lived private information and imperfect
competition, Journal of Finance, 47, 247-270.
John Maynard Keynes (1937), The general theory of employment, interest and money, (New
York: Harcourt, Brace).
Kyle, A.S. (1985), Continuous auctions and insider trading, Econometrica, 53, 1315-1335.
Lightfoot, Lois E., Martin, Peter G., Peterson, Mark A., and Erik Sirri (1997), Preferencing and
market quality on U.S. equity exchanges, Working Paper, the Securities and Exchange
Commission.
Macey, Jonathan R. and Maureen O’Hara (1997), The law and economics of best execution,
forthcoming, Journal of Financial Intermediation.
Roell, A. (1990), Dual-capacity trading and the quality of the market, Journal of Financial
Intermediation, 1, 105-124.
Sarkar, A. (1995), Dual trading: winners, losers and market impact, Journal of Financial
Intermediation, 4, 77-93.
Scharfstein, David S. and Jeremy C. Stein (1990), Herd behavior and investment, American
Economic Review, June 1990, 80, 3, 465-479.
Shiller, Robert J. (1990), Market volatility and investor behavior, AEA Papers and Proceedings,
May 1990, 80, 2, 58-62.
28
Shiller, Robert J. (1995), Conversation, information and herd behavior, AEA Papers and
Proceedings, May 1995, 85, 2, 181-185.
Smith, T. and R. E. Whaley (1994), Assessing the cost of regulation: The case of dual trading,
Journal of Law and Economics, 37(1), 329-36.
Spiegel, M. and A. Subrahmanyam (1992), Informed speculation and hedging in a
noncompetitive securities market, Review of Financial Studies, 5(2), 307-329.
Subrahmanyam, A. (1991), Risk aversion, market liquidity and price efficiency, Review of
Financial Studies, 4, 417-441.
29
Table 1
Trading Volume and Price Impact Around Boesky's Trading in Carnation Stock
We estimate the following regression:
STKPR/SIZEt = a0 + a1STDPRt + a2BKVOLt + a3NONBKVOLt + a4PREIt + a5CURIt +a6POSTIt
where STKPR is the hourly average of Carnation stock trading price, SIZE is the total hourly volume divided by the number
of (hourly) trades, STDPR is the hourly standard deviation of Carnation's stock price, BKVOL is the hourly Boesky volume,
NONBKVOL is the difference between the hourly Boesky volume and the hourly total volume, PREI equals 1 in the hour
immediately before the hour Boesky trades on a day and zero otherwise, CURI equals 1 in the hour when Boesky trades and
zero otherwise and POSTI equals 1 in the hour immediately following the Boesky trading hour and 0 otherwise.
The p-values of the coefficient estimates are in parentheses below the coefficients. The number of observations are 336. The
sample period is June 1, 1984 through August 31, 1984.
Independent variables
Dependent variable: STKPR/ SIZE
Intercept
12.0050
(0.0001)
STDPR
-5.2618
(0.1103)
BKVOL
0.000082
(0.9895)
NONBKVOL
-0.010043
(0.0001)
PREI
-1.3124
(0.4233)
CURI
-6.8576
(0.0074)
POSTI
-3.1477
(0.0436)
R-Squared
0.22
30
An important aspect of market competition, analysed in our next result, is the number of traders--in
our case, the relative number of insiders and brokers on the market.
Corollary 4: (1) µ2 is independent of m. µ2 is decreasing (increasing) in n if n is greater (less) than (2t)/t.
(2) 2 is decreasing in m. 2 is decreasing (increasing) in n if n is greater (less) than [(2-t)(1-t)]1/2/t.
(3) The informativeness of period two prices is increasing in n and m.
(4) Suppose the fixed cost of brokerage per trade is small. Then net informed profits and net
uninformed losses are decreasing in m and n.
2 is decreasing in m because of increased competition between brokers. The effect of increasing the
number of informed traders on 2 and µ 2 is non-monotonic, echoing earlier results by Subrahmanyam (1991)
in the context of risk-averse informed traders. Here, the "switch" value of n depends on information
precision. If information is perfect (t=1), market depth increases with n, as in Holden and Subrahmanyam
(1992). For very imprecise information (t close to 0), market depth decreases with n. The reason is as
follows. As information becomes more precise, each insider's conjecture about the information of other
insiders becomes more highly correlated, increasing the variability of aggregate insider trades and the market
depth.15/
An increase in the number of insiders and brokers increases the informativeness of period two prices
because increased competition makes insiders and brokers use more private information in the aggreagate.
Increased competition also drives down informed and brokers' profits, reducing commissions and uninformed
losses.
15/
If t=1, the variance of aggregated insider trades is proportional to the square of n. If t=0, the variance is proportional to
n.
31
7
7
9
9
11
11
13
-~
13
Market Depth for
off-Exchange
Transaction
Market Depth for
Exchange
Transaction
Price
Informativeness of
off-Exchange
Transaction
Price
Informativeness of
Exchange
Transaction
Lambda
I
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
1
3
5
7
m
9
I
11
13
-.L.O-----J------
1
2
3
4
5
6
m
7
8
9 10 11 12 13
Price Informativeness for n = 10
0.86
0.88
0.9
0.92
0.94
0.96
0.98
1
1.02
0
X
(Inverse of) Market Depth for n = 10
Figure 1: (Inverse of) Market Depth and Price Informativeness with off-Exchange Block Sales
5
~
~
~
(Inverse of) Market Depth for n = 2
0.3
3
5
r
r
X
0.25
0.2
0.15
0.1
0.05
0
1
m
l
Price Informativeness for n = 2
2.5
3
I
I
2
1.5
1
0.5
0
1
m
Price lnformolivenss
+
*
+I
*
Lambda
Price Informativeness
Appendix
Proof of Proposition 1:
Consecutive dual trading equilibrium.
In period one, the insider's problem is identical to the single period model in Kyle (1985), but
with multiple informed traders. The solution to this equilibrium is in Lemma 1 of Admati and
Pfleiderer (1988). This gives Ad and λ1 .
In period two, dual trader j observes p1 , xd / m and u1 / m and trades z j . The net order flow is
m
y2 =
∑
z j + u2 = mz + u2 . We will show that z j = z for each j since each z j is a function of
j =1
xd and u1 .
Using the law of iterated projections, it is easy to show that
x 1,d x 2 ,d
x n ,d u1
x 1,d x 2 ,d
x n ,d u1
Εv
,
,...,
, , p1 = Ε v
,
,...,
,
.
m m
m m
m m
m m
LM 1
RS - - - UV- OP LM 1
RS - - - UV-OP
W Q NT
W Q
NT
The jth dual trader’s problem is given by:
LMb
N
g
E v − p2 z j
Max
j
z
LMc
N
x1,d x 2,d
x n,d u1
,
,....,
, , p1
m m
m m
h
E v − λ2 z j − λ2 z − j − λ2u2 − p1 z j
or, Max
j
z
OP
Q
x1,d x 2,d
x n ,d u1
,
,....,
, , p1
m m
m m
OP
Q
(1.1)
where z − j = ∑ z k . From above, the first and second order conditions are:
LM
N
Foc: E v
j≠ k
OP
Q
x1,d x 2 ,d
x n ,d
,
,.....,
− 2 λ2 z j − λ2 z − j − µ 2 Ad s − µ 2 u1 = 0
m m
m
(1.2)
n
where s =
∑s
i
i =1
Second Order Condition: − 2λ2 < 0, ⇒ λ2 > 0
(1.3)
30
A
L
x
x
x O F x I m nΣ
F x IF m I nΣ
= G JG J
,
,...,
E Mv
=G J
P
N m m m Q Hm K FG A IJ Var b sg Hm KHA K nQΣ
Hm K
1,d
2 ,d
d
n ,d
--
v
d
2
-
-
d
t
s
Q
v
-=d
d
s
(1.4)
Substituting (1.4) in (1.1) and solving for z = z j , for every j, we get
z=
Tµ
ts
µ2
⋅
−
u1
λ1Q 1 + Q λ2 1 + m λ2 1 + m
b
g b g b
where, T = λ b1 + Qg− µ Q
µ
1
g
(1.6)
2
Now,
y2 = mz + u2
ts
m 1
m 1
=- Tµ - − - - µ 2 u1
λ1Q 1 + Q
1 + m λ2
1 + m λ2
ts
=
Tµ Tλ − Tλµ 2 u1 + u2
λ1Q 1 + Q
b
b
g
g
where, Tλ =
(1.5)
+
m 1
1 + m λ2
(1.7)
(1.8)
y1 = Ad s + u1
ts
=
+ u1
λ1 1 + Q
b
u2
(1.9)
g
b g λ ntb1Σ+ Qg
ntΣ
= Cov b v , y g =
TT
λ Q b1 + Q g
= Covb y , y g
t TT
=
Var b sg− µ T Σ
Q[λ b1 + Qg ]
Σ 01 = Cov v , y1 =
v
(1.10)
1
Σ 02
v
2
(1.11)
µ λ
1
Σ 12
2
1
2
µ λ
=
=
t Tµ Tλ
[ b
Q λ21 1 + Q
ntΣ v
[λ1 b1 +
2 λ
2
2
1
2
Q
g]
u
(1.12)
nQΣ s − µ 2 TλΣ u
g]
2
2
Tµ Tλ − µ 2 TλΣ u
31
b g
Var b sg + Σ
b g]
nQΣ + Σ
b g]
+ Σ
b g]
= Var b y g
t Var b sg
=
dT T i + bT µ g Σ + Σ
[λ Qb1 + Qg]
t nQΣ
=
d T T i + [b T µ g + 1]Σ
[λ Qb1 + Qg]
Σ 11 = Var y1
t2
=
[λ1 1 +2 Q
t
=
[λ1 1 + Q
ntΣ v Q
=
[λ1 1 + Q
Σ 22
2
u
2
s
2
(1.13)
u
u
2
2
2
2
µ λ
2
µ λ
2
λ
u
2
(1.14)
u
1
2
2
s
2
λ
2
u
1
Next, we solve for µ 2 and λ2 . First, we show that, if Σ12 = 0, then λ1 = µ 2 . From (1.12),
ntΣ v Tµ
Σ12 = 0 ⇒
2 = µ 2Σ u
λ21 1+ Q
b
g
Substituting for λ1 , Σ12 = 0 ⇒ Tµ = µ 2
Substituting for Tµ from (1.6), Σ12 = 0 ⇒ λ1 = µ 2
Further, if λ1 = µ 2 , then Σ12 = 0 .
If Σ12 = 0, then λ2 = Σ 02 / Σ 22 . Substituting for Σ 02 and Σ 22 from (1.11) and (1.14), we get (5).
Finally, it is easy to check that, if λ1 , λ2 and µ 2 are as given in (2), (5) and (6), Σ12 = 0.
Alternatively, instead of assuming Σ12 = 0, we can solve for µ 2 and λ2 directly from the
projection formulas
Σ 02 Σ11 − Σ 01Σ 12
D
Σ 01Σ 22 − Σ 02 Σ12
µ2 =
D
2
where D = Σ11Σ 22 − Σ12 .
λ2 =
b g
The calculations are long and involved, but the same result obtains.
32
ts
1
Tµ
µ 2 u1
z=
−
Q λ b1 + Qg λ b1 + mg
λ 1+ m
Σ Qb1 + Qg L ts
ntΣ u O
=
−
M
P
m ntΣ
NM Qb1 + Qg Σ 1 + Q QP
F1 1 I
F 1 Q Iu
x − G
HG m Qb1 + QgKJ
H m b1 + QgKJ
1
2
2
~t_t
_f
_f
u
v
_f_t
v
∴ z=
1
(1.15)
u
_t
===----t _f
====----t _f
d
1
Therefore, comparing coefficients:
1
1
B1 =
m Q 1+ Q
Q
−1
B2 =
m 1+ Q
The jth broker profits:
====----t _f
b
===----t
_f
_t
b
g
(1.16)
g
b
b
c
g
b
Wd = E v − p2 z
= E v − λ2 y2 − µ 2 y1 z
= E v − λ2 mz + u2 − µ 2 xd + u1 z
[
g
g b
[
gh
[
Substituting for z from above, taking expectations and simplifying we obtain:
t
Wd =
~t
ntΣ u Σ v
b
Q 1+ Q
1
g ⋅ mb1 + mg
(1.17)
_f
Proof of Corollary 1:
(1) We use the results that ∂Q / ∂n > 0 and ∂Q / ∂t > 0 for n>1. The proof follows since
∂B2
1
∂B / ∂Q < 0 and
=
3 > 0.
∂ Q
1+ Q 2
_t
-II
1
b
g
(2) It is clear that B1 and B2 are decreasing in m from definition. Further,
mQ
m Q
are increasing in m.
and m B2 =
1+ Q
1+ Q
_f_tll
_f
_J
mB1 =
_J
Proof of Corollary 2:
(1) µ 2 = λ1 follows from (2) and (6).
33
1+ Q
_J
λ2 = λ1
Q
_f
J
m
b 1 + mg
Note that Q ≥ 1 and m ≥ 1 . For Q = 1 and m = 1 , λ2 =
1+ Q
_J
J
Further,
Q
is decreasing in Q and
b
λ1
_t
/_t
2
< λ1 .
g
m 1 + m is decreasing in m. Thus, λ2 < λ1 for Q > 1
and m > 1 .
c h
b g
(2) Define period one price informativeness as: PI 1 = Σ v − Var v p1 = 2λ1 Σ 01 − λ1 Σ 11 , where
b g
b g
Σ 01 = Cov v , y1 and Σ 11 = Var y1 . Thus, PI 1 =
2
ntΣ v
.
1+ Q
Similarly, period two price informativeness is defined as:
PI2 = Σ v − var v p1, p2
c
h
c
h
c h
PI2 > PI1 because var v p1, p2 < var v p1 .
(3) The expected informed profit for insider i is
tΣ u Σ v
E v − p1 x i ,d =
n 1+ Q
which is the same as with no dual trading.
g
[
_f
b
_f
]
b
g
~t_f
n tΣ u Σ v
Now, the aggregate insider profits Π I =
=
n 1+ Q
Additionally, aggregate broker profits are given by
_f
____,t
ntΣ u Σ v
g b1 + Q g .
b
(1)
ntΣ u Σ v
m
Π B = m Wd =
(2)
1+ Q Q 1+ m
1
Hence, from (1) and (2), the noise traders' per period losses are: U d = Π I + Π B .
2
1+ Q
m
⋅
< 1 . The last
1+ m
Q
_f
_J_f
b
Noise traders' losses without dual trading Π I > U d if: Π B < Π I or,
inequality holds since λ2 < λ1
g
_ f _f
J_ _
34
Proof of Proposition 2:
Simultaneous dual trading equilibrium
There are m dual traders and n informed traders. The net order flow ys is given by
m
ys = ∑ z j + xs + u
(2.1)
j =1
n
where xs = ∑ x i ,s .
i =1
The dual trader j’s problem:
LMb
N
g
E v − ps z j
Max
j
z
LMd
MN
1-- --OPQ
x 1,s x 2 ,s
x n ,s u
,
,...,
,
m m
m m
(2.2)
i
E v − λs z j − λs z − j − λs xs − λs u z j
or, Max
j
z
x 1,s x 2 ,s
x n,s u
,
,...,
,
m m
m m
where z − j = ∑ z k
OP
PQ
k≠ j
The first order condition gives us
x 1,s x 2 ,s
x n ,s u
E v
,
,...,
,
− 2λs z j − λs z − j − λs x s − λs u = 0
m m
m m
LM
N
1-- --OPQ
(2.3)
n
∑x
L
x
x
x
uO t
$
$
where E M v
= s ; and s =
,
,...,
,
A
N m m m m PQ Q
1, s
2,s
n ,s
i ,s
=
i =1
s
xs
.
As
Substituting z = z j , for each j, and solving for z, we obtain
LM
gN
OP
Q
xs
t
u
- − λs − z =λs 1 + m QAs
1+ m
b
(2.4)
Insider i’s problem:
d
i
Max
E [ v − λs mz − λs x i ,s − λs x − i ,s − λs u x i ,s Isi ]
i
,
s
x
(2.5)
where x − i ,s = ∑ x k ,s
k ≠i
or, Max
i ,s
x
LMF
MNGH
E v−
LM
N
xm
t
1 + m QAs
OP
Q
I
JK
λs u i ,s i
x Is
1+ m
s
- − λs − λs x i ,s − λs x − i ,s − -
35
OP
PQ
LM x Fts − x Lλ + m F t − λ QA IO
MN 1 + m GH QA JKPQ
MN GH
i ,s
Max
i ,s
i ,s
i
s
s
s
x
−
L m F t − λ QA IO I O
P
-N 1 + m GH QA JKPQ JK PQ
[ I]M
E x − i ,s s i λs +
s
s
s
From above, the first order condition gives us
n − 1 As tsi
λs
m
t
mt
2 x i ,s
=−
+ ts i
+
⋅
λs +
1 + m 1 + m QAs
1+ m
QAs
ts i
= - [Q 1 + m − λs As n − 1 Q − mt n − 1 ]
Q 1+ m
ts i
= - [Q + m − λs As n − 1 Q]
Q 1+ m
LM
N
s
OP b g LM
OP
Q
N
Q
b g b g b g
b g
b g
b g
(2.6)
The second order condition is:
mt
> 0.
QAs
λs +
Thus, given λs > 0 , As has to be positive to satisfy the second order condition.
Simplifying (2.6), we get
x i ,s =
~[ n
b
g
--------;-
Comparing coefficients,
As =
__
b gs ]b
g
m + Q 1 − λs As n − 1 1 -+ m QAs
tsi
⋅
Q 1+ m
2[tm + λs QAs ]
-
n
b gs
Ast m + Q 1 − λs As n − 1
(2.7)
(2.8)
2[tm + λsQAs ]
Simplifying the above,
b
g
b g
t Q− m
> 0 if Q > m and λs > 0
λs Q 1 + Q
Note that for t=1, Q=1 + t(n-1) = n. Thus, As > 0 iff n > m.
As =
(2.9)
t
x
xs
u
⋅ s −−z =- λsQ 1 + m As 1 + m 1 + m
b
LM
N
g
OP
Q
=
xs
t
u
−1 −
1 + m λs As Q
m+ 1
=
xs
u
−
, after substituting for As
Q− m m+ 1
Therefore, B1,S =
b
(2.10)
1
−1
and B2,S =
Q− m
1+ m
g
b
g
36
Now,
y s = xs + mz + u
xsm
1
+
u
Q − m 1+ m
= xs +
LQ O 1
=x M
N Q − m QP + 1 + m u
F Q I
HGQ − mKJnA Σ
λ =
Q
F 1I
b Q − mg b A g nQΣ + HG1 + mKJ Σ
F Q I tb Q − mg
HGQ − mKJ⋅λ Qb1 + Qg ⋅nΣ
=
t b Q − mg
Q
F 1I
b Q − mg ⋅λ Q b1 + Qg nQΣ + HG1 + mKJ Σ
ntΣ λ b1 + Qg
=
F 1I
HG1 + mKJ Σ λ b1 + Qg
(2.11)
s
s
s
2
2
v
2
2
s
s
u
v
s
2
2
2
2
2
2
2
2
s
(2.12)
u
s
v
s
2
ntQΣ v +
2
2
u
s
After further simplification, we obtain
b1 + mg ntΣ
λ =
Σ b1 + Qg
t b Q − mg
A =
λ Qb1 + Qg
t b Q − mg b1 + Qg Σ
⋅
=
Qb1 + Q g ntΣ b1 + mg
t b Q − mg Σ
=
b g nΣ > 0, for Q > m
_f
v
s
(2.13)
_f
u
s
s
_f
u
(2.14)
_f
_f
_f _
v
_f
u
Q 1+ m
v
Finally, the jth broker profits
37
W = E b v − p gz
LF λ Q x − λ u Iz O
= E v−
NMHG Q − m 1 + mKJ QP
λ Q b A g nQΣ
λΣ
nA Σ
=
−
⋅
+
Q− m Q− m
Q − mg
b
b1 + mg
ntΣ Σ
1
nA Σ
=
⋅
+
Q − m 1 + Q b1 + Qgb1 + mg
nΣ b Q − mg tΣ
ntΣ Σ
1
=
b Q − mg⋅ Qb1 + mg ⋅1 + Q + b1 + Qgb1 + mg
ntΣ Σ 1
b1 + mg ⋅Q
s
]
s
s
s
s
2
s
v
---s
s
s
s
s
u
2
u
v
_f
v
(2.15)
v
_J
=
[
----
_t
_t
u
u
v
_t
u
v
Proof of Corollary 3:
I I
Follows immediately from the definitions of B1,s and B2,s in (10) and (11) in the text, and from
∂Q / ∂n > 0 and ∂Q / ∂t > 0 for n>1.
Proof of Corollary 4:
(1) λs =
ntΣ v 1 + m
= λ1 1 + m
Σu 1 + Q
_f
_f
b
g
Thus, λs > λ1 . The above implies that market depth is lower with simultaneous dual trading compared to
the no-dual trading benchmark (i.e., the inverse of λ1 ).
(2)
b g
PI = λ Cov v , y
s
s
s
Q
= λs As nΣ v
Q− m
t Q− m
Q
=
nΣ v
Q− m
Q 1+ Q
ntΣ v
=
1+ Q
which is the same as when there is no dual trading.
(3)
b
g
g
b
tΣ u Σ v
1
tΣ u Σ v
Q− m
-_J~ - - _ J
Individual insider profit =
,
<
n b1 + Qg1 + m Q
n b1 + Q g
where the term to the right of the inequality is an insider's profit without dual trading. The inequality
FQ − mI
F1I
HG1 + mKJ< 1 and HG Q KJ< 1 .
_f
_f
follows since
38
Uninformed losses with simultaneous dual trading equals the sum of aggregate insider and broker profits,
or
_t
m ntΣ u Σ v
b1 + mgQ
+
b
g
_t _t
Q − m ntΣ u Σ v
=
Q 1+ Q 1+ m
b
gb
g
ntΣ u Σ v
1+ Q
Uninformed losses with no dual trading equals
~t
ntΣ u Σ v
b1+ Qg
The final result follows directly.
Proof of Proposition 3:
(1) Under consecutive dual trading, informed profits (Id) is Id =
_t
Noise trader losses U d = nId + mWd =
L
NM
ntΣ u Σ v
1+
1+ Q
tΣ u Σ v
_J
b
n 1+ Q
_f
g , which is independent of m.
O
QP
1+ Q
m
which is decreasing in m.
⋅
Q 1+ m
_J
_t _ f
(2) Informed profits with simultaneous dual trading
Is =
tΣ u Σ v
_J
b
nQ 1 + Q
_f
LQ − m O
g⋅NM 1 + m QP
Consider the term in the square bracket above
O b g
QP b g
L
NM
− 1+ Q
∂ Q− m
=
< 0.
2
∂m 1 + m
1+ m
ntΣ u Σ v
,
Noise trader losses U s =
1+ Q
which is independent of m.
_J
Proof of Proposition 4:
1) In the consecutive dual trading model, the profit of a broker is Wd =
__f
When t = 1, Wd =
1
Σv Σu
ntΣ v Σ u
_f
-_ tt
_f
b
Q 1+ Q
1
1
g ⋅ m ⋅1 + m
1
--_f
b1 + ng ⋅ m ⋅1 + m
39
The profits of an informed trader in the consecutive dual trading model, I d =
tΣ v Σ u
_J
b
n 1+ Q
_f
g
Σv Σ u
When t = 1, I d =
n 1+ n
The equilibrium number of informed traders, nd , satisfies I d = k d since I d is decreasing in n. So
_J
_f
b g
Σv Σ u
nd 1+ nd =
kd
Similarly, the equilibrium number of brokers, md , satisfies:
b
g
_J
_f
(4.1)
Σv Σu
1
m 1+ m =
⋅
d
d
kd
1 + nd
From (4.1) and (4.2), md < nd .
g
_J
b
(4.2)
_J
_f
Thus, for the equilibrium outcome to be non-null, md =1 and nd =2.
md =1 requires:
Σ Σ
kd ≤ v u
2 3
where the RHS of (4.3) is Wd evaluated at m = 1 and n = 2.
_J
(4.3)
_t
nd =2 requires:
Σ Σ
Σv Σ u
kd ≤ v u <
(4.4)
3 2
2 3
where the RHS of (4.4) is I d evaluated at n = 2. Hence, md =1 and nd =2 hold simultaneously if
(4.4) is satisfied.
_J_J
_f
_t
2) In the simultaneous dual trading model, only one broker enters the market. Since Q > m in
equilibrium, at least two informed traders enter the market.
The profits of a broker with simultaneous dual trading, Ws , is
____,t
ntΣ v Σ u
=
Q 1+ m
Ws =
b
g
Σ v Σu
_J
b
g, at t = 1. .
n m+ 1
_f
The profits of an informed trader with simultaneous dual trading
tΣ Σ
Σv Σu
Q− m
n− m
_J
Is =
b
v
u
n 1+ Q
_J
g⋅Qb1 + mg = n b1 + ng⋅nb1 + mg at t = 1.
_f
_f
Given that two informed traders are in the market, at least one broker enters the market if:
40
ks ≤
Σ v Σu
2 2
(4.5)
Given that one broker enters the market, at least two informed traders enter the market if:
ks ≤
Σ v Σu
12 2
(4.6)
Thus, if (4.6) is satisfied, the participation constraints of both brokers and the informed traders
are simultaneously satisfied.
The equilibrium number of informed traders, ns (given ms = 1 ) is:
b
g
2 ns ns 1 + ns
=
ns − 1
b
g
Σ v Σu
ks
(4.7)
Proof of Proposition 5:
1) Let k s ≥ k d . Suppose k s satisfies (4.6). Then both equilibrium outcomes, consecutive and
simultaneous, are non-null. From (4.1) and (4.7) in the proof of proposition 4:
b
b
g b
g
g
F--
n − 1 kd
ns 1 + ns
⋅
= s
<1
nd 1 + nd
2 ns k s
(5.1)
From (2) and (11) in the text,
- r-FFF
λs
1 + md
n
1 + nd
= 2 1 + md ⋅
⋅ s ⋅
λ2
1 + ns
md
1 + ns
> 1 since ns < nd
(5.2)
kd
and ns ≥ 2 , then ns > nd . Consider (5.2) again. Suppose md is large.
2
1 + md
Then approximately,
≅1 , and
md
(2) From (5.1), if k s <
b
b
g
gb
g
λs 2 1 + md 1 + nd
≅
λ2
1 + ns
< 1 if ns is large relative to md and nd .
λ
For example, if ns > 2nd , then s < 1 .
λ2
b
g
41
Proof of Proposition 6:
1) Let the subscript " o" refer to outcomes in the order flow internalization model. Analogous to
proposition 2, we can show that:
b Q − m g ⋅ tΣ
A =
Qb1 + m g nΣ
b g
b g
_f
_f -
2
o
1
u
(6.1)
v
1 + m1 ⋅ ntΣ v
(6.2)
λo =
Σu 1 + Q
In the market without internalization, m2 = m and m1 = 0 . Since the market depth without
internalization is:
ntΣ v
λ1 =
, λo > λ1 .
Σu 1 + Q
PI = λ cov v , y , where
o
o
o
n
Q
u1
i
yo = xo ⋅
+
x
,
o = Ao s , s = ∑ s .
Q − m2 1 + m1
i =1
_ f _f
b
g
b g
From (6.1) and (6.2), and t = 1,
PI o =
nΣ v
= PI1 (price informativeness without internalization).
1+ n
Uninformed losses with order flow internalization is:
b
g
b
g
b
g
U o = nΕ v − po xo + m2Ε v − po z2 + m1Ε v − po z1
]
[
]
[
where po = λo yo . From (6.1) and (6.2),
_t
U0 =
ntΣ uΣ v
b1 + Qgb1 + m g ⋅
1
Q − m2
+
Q
_t
=
ntΣ u Σ v
m2
b1 + Qgb1 + m g ⋅ Q
+
1
_t
ntΣ u Σ v
b1 + Qgb1 + m g ⋅ m
1
1
_t
ntΣ u Σ v
1+ Q
= uninformed losses without dual trading.
Consider the fixed entry equilibrium. Suppose each broker charges $c per trade, independent of
the order size. Also the fee is paid irrespective of whether the customer makes the trade. The
42
aggregate of noise trades is considered one trade, so there is one trade per internalizing broker.
Brokers have a cost of f per trade, where f > 0. Following Fishman and Longstaff (1992), each
broker chooses c so that the broker's expected revenues from trading plus the expected
commissions equal zero.
Assume:
tΣ u Σ v
Q − m2
n ⋅Q
b1 + Qgb1 + m g⋅
__J
c<
1
_J-
(6.3)
Then, informed trader's expected profits, net of commissions are positive. Let co be the
competitive commission with order flow internalization. m1 brokers internalize the order flow of
noise traders, and each obtains trading profits given by.
Wo =
t
ntΣ u Σ v
b1 + Qgb1 + m g
(6.4)
1
where Wo was derived in the proof of part one. By assumption, noise trades are considered one
trade, and so
co = f − Wo
(6.5)
Without order flow internalization, the competitive commission is cn where
cn = f
(6.6)
Since Wo > 0, co < cn . Further, ps + c0 < ps + cn .
2) (i)
From the proof of part one, informed profits are decreasing in m2 , so informed traders
choose m2 = 1. However, informed traders do not choose the internalizing brokers, and
uninformed losses are independent of m1 , so m1 is determined in equilibrium.
Equilibrium requires n > m2 , so at least two informed traders must enter.
Assume t = 1. For at least two informed traders to enter,
ko ≤
Σ v Σu
12 2
_f
(6.7)
_f
43
where ko is the entry cost in the market with order flow internalization.
For at least one piggybacking broker to enter (i.e., m2 = 1 ):
ko ≤
Σ v Σu
6 2
(6.8)
For at least one internalizing broker to enter (i.e., m1 = 1 ):
Σ Σ
ko ≤ v u
3 2
Hence, the constraint (6.7) is binding.
_J
_J-
(6.9)
Given entry by mo internalizing brokers and one piggybacking broker, the
number of informed traders, no , who enter is:
ko =
Σ u Σ v no − 1 1
⋅
⋅
no + 1 1 + mo
_ _ _f
_J
no no
(6.10)
Given entry by no informed traders, the number of internalizing brokers entering is given by:
_J
ko =
ΣuΣv ⋅
no
mo
⋅
1 + no
1 + mo
_f
(6.11)
From (6.10) and (6.11),
no − 1
m =
bn g
o
2
< no
o
And so the number of entering brokers varies negatively with the number of entering informed
traders for no > 1.
(ii) In the market with no internalization, m2 = 1 and m1 = 0 , and so the profits of an informed
trader is
Σ Σ n− 1
1
⋅ u v ⋅
.
1+ n
n
n
Assume the market entry cost is kw , and nw informed traders enter. Then
ΣuΣv
nw − 1
nw + 1
J
kw =
-_J
nw nw
(6.12)
Assume ko = kw = k . Comparing (6.10) and (6.12), no < nw .
44
Evaluate λw (market depth with no internalization) at n = nw and λo at n = no . Since
λw is simply λo evaluated at m1 = 0, we have:
Σ F 1 I n −1
λ =
G J
k H1 + n K n
b g
g
2
v
w
w
w
λo =
(6.13)
w
Σ v no − 1
k no no + 1
b
Therefore,
b
b
g
gb
b
b
(6.14)
g
g
nw + 1
nw nw + 1
λo
=
⋅
n
⋅
o − 1 ⋅
no + 1
no nw − 1
λw
g
> 1 since nw > no ≥ 2.
From the proof of part (i) of this proposition, informativeness of price
(6.16)
nw
⋅Σ v
1 + nw
PI w =
(6.15)
no
Σv
1 + no
PIo =
Hence, PIo − PI =
bn − n g
b1 + n gb1 + n g⋅Σ
o
w
o
V
< 0.
W
From the proof of part (i) of this proposition, uninformed losses in the two markets are:
no
ΣuΣ v
1 + no
Uw =
and
(6.17)
nw
1 + nw
ΣuΣ v
(6.18)
d n − n id n n − 1i
b1 + n gb1 + n g ⋅ Σ Σ
_ f _f _f
Uo =
_J _f
_J _f
_J
w
Uo − U w =
o
w o
u
o
v
w
>0
45
Proof of Proposition 7:
Consider the Holden-Subrahmanyam model (1992), specialized to two periods and ∆t n ≡1.
From propositions one and two of their paper, the informed trading intensity in period two, β 2 ,
b
g
the period two market depth, λ2 , and price informativeness Σ v − Σ 2 are:
and q1
Σ1 =
Σv
n 1 − 2q1 + 1
(7.1)
Σ2 =
Σ1
1+ n
(7.2)
λ2 =
nΣ 2
Σu 1 + n
(7.3)
β2 =
Σu
Σ2 1 + n
(7.4)
b
g
b g
b g
F 1I
∈ G0, J is the unique root satisfying the cubic equation:
H 2K
b g b gb g b12+ qng + b1 + 1ng
0 = 2n q1 − 1 + n q1 −
3
2
1
2
(7.5)
2
b
g
We numerically compare (7.3) with (4.5), (7.4) with (4.1), and Σ v − Σ 2 with
PI2 =
LM
OP
b g N b gQ
nΣ v
m
⋅ 1+
.
1+ n
n 1+ n
♥
46
@FedFRASER