Download Should Dark Pools Improve Upon Visible Quotes

Should Dark Pools Improve Upon Visible Quotes?
The Impact of Trade-at Rules∗
Michael Brolley†
Wilfrid Laurier University
June 8, 2016
Abstract
Dark pools—trading systems that do not publicly display orders—fill orders at a price
better than the prevailing displayed quote, but do not guarantee execution. This
improvement is known as the “trade-at rule”. I develop a model to examine the impact
of dark pool trade-at rules. In my model, investors, who trade on private information
or liquidity needs, can elect to trade on a visible market, or a dark market where limit
orders are hidden. A competitive liquidity provider participates in both markets. The
dark market accepts market orders from investors, and if a limit order is available,
fills the order at a price better than the displayed quote by a percentage of the bidask spread (the trade-at rule). The impact of dark trading on measures of market
quality and social welfare depends on the trade-at rule, and its impact on the relative
fill rates of visible orders to dark orders. A dark market with a large (but not too
large) trade-at rule improves market quality and welfare; a small trade-at rule, however,
impacts market quality and social welfare negatively. Price efficiency declines whenever
investors use the dark market. For a trade-at rule at midpoint or larger, the liquidity
provider does not post limit orders to the dark market.
∗
I thank Shmuel Baruch, Vincent Bogousslavsky, David Cimon, Hans Degryse, Rick Harris, Laurence
Lescourret, Katya Malinova, Angelo Melino, Jordi Mondria, Andreas Park, Peter Norman Sørensen, James
Upson, and Haoxiang Zhu for helpful comments, as well as participants of the 2015 NFA Meeting, 2014 EFA
Meeting, 2014 FMA Meeting, 2013 CEA Meeting, 9th CIREQ PhD Students’ Conference, and seminars at
Copenhagen Business School, the University of Toronto, and Wilfrid Laurier University. All errors are my
own. Financial support from the Social Sciences and Humanities Research Council is gratefully acknowledged.
†
Email: [email protected]; web: http://www.mikerostructure.com.
In recent years, concern has arisen over the impact of dark trading on equity markets.
While dark markets are not new, this concern comes as they begin to match the electronic
organization of visible markets, and gain a significant share of global trading activity. The
CFA Institute estimates that dark markets in the U.S. generate approximately one third of
all volume.1 Because dark markets benefit from visible market transparency, there is concern
from regulators that these benefits come at the expense of market quality (see Securities and
Exchange Commission (2010)).
Equity trading primarily conducts through two types of markets: visible limit order
markets, and dark markets. In visible markets, investors submit limit orders at pre-specified
(limit) prices; a trade occurs when a subsequent investor submits a market order. Limit
orders are visible to all market participants. In a dark market, liquidity is hidden, and
orders sent to the dark market are filled only if sufficient liquidity is present. To entice
investors to submit orders to trade against their hidden liquidity, orders sent to a dark
market usually fill at a better price than the prevailing quote offered by the visible market.
In many dark markets, orders receive only marginal improvement on displayed quotes.
Marketplaces generally give first-fill priority to visible orders over dark orders at the same
price, which provides an incentive for liquidity providers to display their trading intentions. However, any price improvement moves the dark order to the front of the queue,
according to price-visibility-time priority. To adequately incentivize liquidity providers to
display their orders, some countries now require dark liquidity to provide “meaningful price
improvement”—the so-called “trade-at rule”. For example, Canada and Australia require
dark liquidity to provide a price improvement of at least one trading increment (i.e., one
cent in most major markets).2 For securities with bid-ask spreads of one or two trading
increments, orders fill at the midquote.3 By imposing a minimum price improvement rule,
1
CFA Institute (2012): “Dark Pools, Internalization, and Equity Market Quality”.
The immediate impact of the trade-at rule in Canada was a 50% drop in dark trading volume. Rosenblatt
Securities Inc. (26-02-2013)
3
See Investment Industry Regulatory Organization of Canada (2011) and Australian Securities and Investments Commission (2012).
2
1
regulators suggest that dark markets may have a role in financial markets, but only if liquidity providers pay a premium for hiding their trading intentions. But how do trade-at rules
impact market quality and investor welfare? Is more improvement always better?
To analyze an investor’s optimal order placement decision, I model an investor’s order
placement strategy with three components: their valuation of the asset, the price of the order,
and its probability of execution. The choice of order type focuses on the trade-off between
price and execution risk. In a market where a competitive liquidity provider ensures the
limit order book is full, market orders face lower execution risk than visible limit orders or
dark orders. As compensation, visible limit orders and dark orders offer better prices. The
trade-off between dark orders and visible limit orders, however, is more complex, as both
orders face execution uncertainty. To compensate for execution risk, dark orders receive
price improvement on the prevailing quote, while visible limit orders permit investors to set
their own quote. Hence, a dark market’s trade-at rule plays a key role in how dark orders
compete with visible limit orders. In this paper, I analyze how trade-at rules impact overall
market quality and investor welfare. I then discuss my results in the context of trade-at rule
requirements by regulators.
I propose a dynamic trading model where investors trade for either informational or
liquidity reasons. They may trade at a visible market using either a limit order or market
order, or send an order to the dark market. Both the visible and dark markets are monitored
by a competitive, uninformed liquidity provider that acts as a market maker. The liquidity
provider possesses a monitoring advantage towards interpreting and reacting to market data,
an advantage they use to ensure that their limit orders are priced competitively. Moreover,
the liquidity provider ensures that the visible limit order book always contains limit orders
on either side of the book. Consequently, investors who submit market orders are guaranteed
execution at the available quote. Limit orders, however, are subject to execution risk, as
they may trade only if the subsequent investor submits the appropriate market order.
In the dark market, orders fill at a price derived from the visible market: the prevailing
2
quote, improved by a percentage of the bid-ask spread.4 Because the liquidity provider
cannot set the price of their dark orders directly, they instead choose the probability with
which they post orders to the dark market to ensure that they earn zero expected profits in
equilibrium. As a result, investor dark orders fill with a probability less than one.
I analyze the impact of introducing a dark market alongside a visible limit order market,
by first solving the case where all investors are indifferent between visible limit orders and
dark orders. I assume that when indifferent between visible and dark orders, investors use
visible orders. In this way, this setting serves as the “visible market only” benchmark. I find
that there exists a trade-at rule where investors are indifferent to visible limit orders and dark
orders, which occurs when trade-at rule equates price of a dark order and the price impact
of a visible limit order. I refer to this as the “benchmark” trade-at rule. In equilibrium,
investors use dark orders when the trade-at rule is above or below the benchmark level. I refer
to these trade-at rules as a “large trade-at rule” and a “small trade-at rule”, respectively.
I find that a dark market with a large trade-at rule improves market liquidity. By offering
a “discount” trading opportunity, investors who would otherwise be pushed out of the market
by the costs of visible orders can participate. The trade-off, however, is higher execution risk
of dark orders relative to visible limit orders. Because of this, only low-valuation visible limit
order investors migrate to the dark. This leads to an increase in the relative attractiveness
of visible market orders, and investors who submit limit orders migrate to visible market
orders. The result is an increase in volume both on the exchange, and overall. Quoted
spreads also narrow. However, if the trade-at rule is at the midquote (or better), then no
liquidity is provided to the dark market, and investors remain with the visible market.
Conversely, a dark market with a small trade-at rule incentivizes the liquidity provider
to ensure an investor’s dark order has lower execution risk than a visible limit order. Then,
because dark orders have lower execution risk than visible limit orders, the dark market
attracts investors away from both visible order types: the visible market order submitters
4
For instance, prior to minimum price improvement regulation in Canada, the dark pools MatchNow and
Alpha IntraSpread used trade-at rules of 20% and 10%, respectively.
3
with the lowest valuations, and the visible limit order submitters with the highest valuations.
Hence, investors who submit visible market orders have higher average valuations, which,
because some investors are informed, implies that visible market order submitters have a
larger price impact, on average. This leads to an increase in the trading costs of visible
market orders, thus negatively impacting visible market liquidity. The result is an increase
in quoted spreads, and a reduction in visible market and total volume.
By modelling both informed investors, and uninformed investors with private values, I can
study price efficiency and social welfare, respectively. I study price efficiency by measuring
the difference between the change in the fundamental value of the security and the price
impact of an investor’s order placement. I measure welfare in the sense of allocative efficiency,
similar to Bessembinder, Hao, and Lemmon (2012): the expected private valuation realized
by trade counterparties, discounted by the probability of a trade. I find that price efficiency
declines regardless of the dark market’s trade-at rule. Social welfare increases with a large
trade-at rule, but falls with a small trade-at rule.
My model suggests that the impact of dark market trade-at rules on equity markets is
dichotomous, depending crucially on the relative fill rates of dark and visible orders. My
results suggest that there may be a role for minimum trade-at rule requirements, but that a
midpoint trade-at rule is not welfare-maximizing. By measuring the fill rates of dark orders
and visible orders, my model predicts that setting a trade-at rule such that dark orders have
lower fill rates than visible orders would be welfare-improving, in comparison to the current
minimum trade-at rule regulation in markets such as Canada and Australia.
Several theoretical papers examine the impact of crossing networks that cross at the
prevailing visible market quote (Ye (2011)) or midquote of the spread (Buti, Rindi, and
Werner (2014); Degryse, van Achter, and Wuyts (2009); Hendershott and Mendelson (2000),
and; Zhu (2014)). The literature on non-midpoint crossing–executing trades at a discount
smaller than midpoint of the visible spread–in dark pools is relatively new. I contribute to
the dark pool literature by investigating how the pricing mechanism (the “trade-at rule”)
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affects visible market quality and price efficiency.
My work is most closely related to Buti, Rindi, and Werner (2014), who study the impact
of dark trading when uninformed large (institutional) and small (retail) investors choose
between a visible limit order book, and a dark pool as a crossing network that trades at the
midquote. Buti, Rindi, and Werner (2014) find that exchange orders migrate to the dark
pool, but that there is volume creation overall. Complementary, I study how (potentially)
informed investors choose between a visible limit order book, and a dark pool that improves
upon the displayed quote according to a trade-at rule. Similarly to Buti, Rindi, and Werner
(2014), I find that orders migrate to the dark pool, but that additional volume is created only
when the trade-at rule demands sufficient improvements over the prevailing visible quotes.
If the trade-at-rule is too loose, volume declines.
Zhu (2014) and Ye (2011) consider the impact of dark pools on market quality and
price discovery, when orders are crossed at the midquote and prevailing quote, respectively.
Zhu (2014) finds that access to a midpoint-crossing dark pool improves price discovery, but
worsens visible market liquidity, because informed traders concentrate their orders on the
visible market. Ye (2011), however, shows that access to a dark pool that crosses at the
quote harms price discovery. My model predicts that a dark market negatively impacts price
efficiency, regardless of the trade-at rule, because orders that migrate to the dark contribute
to price discovery only when a trade occurs.
Menkveld, Yueshen, and Zhu (2016) develop a “pecking order hypothesis” that uninformed investors choose a trading venue (whether a lit or dark venue), they prefer low-cost
(low immediacy) venues, but move down to higher-cost (higher immediacy) venues as their
immediacy needs dictate. They find support for their hypothesis empirically. I predict a
similar cost-immediacy trade-off, where investors with more extreme trading needs choose
order types with greater immediacy than investors with more moderate trading needs.
My work is also related to a wider theoretical literature on dark trading. Studies by
Boulatov and George (2013), Buti and Rindi (2013) and Moinas (2011) examine the impact
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of dark trading within a limit order market, where visible orders may interact with hidden
orders at the exchange. In this paper, I study impact of a dark pool that may divert orders
away from the visible market.
My predictions may also explain some of the seemingly contradictory results in the empirical literature.5 Using Australian data, Comerton-Forde and Putniņs̆ (2016) find that
dark trades are informative, but that orders migrating to the dark are less informed than
those that remain on the exchange. Moreover, this order migration worsens liquidity. In
the context of my model, their results are consistent with a dark pool that sets a small
trade-at rule. Comerton-Forde and Putniņs̆ (2016) state that a large majority of dark trades
execute at the bid-offer quotes, where liquidity providers trade against incoming investor
and institutional orders. In a study of U.S. crossing networks, Nimalendran and Ray (2014)
also find that dark trades have an informational impact on visible market prices. Foley and
Putniņs̆ (2016) study the impact of a minimum trade-at rule, using the recent minimum
price improvement restriction in Canada that requires all dark orders to improve upon the
current visible market prices by one tick. They examine their data in the context of one-sided
versus two-sided dark trading, and find that a relative increase of two-sided dark trading
is beneficial for market quality. Two parallel studies, Comerton-Forde, Malinova, and Park
(2015) and Anderson, Devani, and Zhang (2015), produce conflicting results to Foley and
Putniņs̆ (2016). Both studies find weak or no evidence for a change in overall market quality. However, in-line with the prediction of this paper, Comerton-Forde, Malinova, and Park
(2015) do find evidence that a midpoint trade-at rule leads to a sharp decline in two-sided
liquidity in all dark pools.
5
Several empirical studies on dark trading and market quality find mixed results: Degryse, de Jong, and
van Kervel (2015), Weaver (2014) and Foley, Malinova, and Park (2012) analyze Dutch, U.S. and Canadian
data, respectively, and conclude that dark trading negatively impacts market quality; Using U.S. data, Buti,
Rindi, and Werner (2011) find that liquidity improves. In a labratory experiment, Bloomfield, O’Hara, and
Saar (2015) conclude that dark trading has no impact on liquidity or price efficiency; Degryse, Tombeur, and
Wuyts (2014) examine both hidden orders on lit venues, and orders on dark venues, providing evidence that
dark and hidden orders are substitutes, and that the choice of opaque order type depends on the prevailing
market conditions.
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1
Model
I propose a sequential trading model (in the sense of Glosten and Milgrom (1985)) where
(potentially) informed, risk-neutral investors have access to a visible limit order book and
a dark market. A competitive, uninformed professional liquidity provider supplies liquidity
to both markets. The dark market prices its limit orders according to a “trade-at rule”
that improves upon the prevailing visible market quotes. The dark market in my model
reflects the structure of several dark pools, where liquidity is sourced from outside liquidity
providers (e.g., Alpha Intraspread, Getco, Sigma X, and Crossfinder). I frame the model from
the perspective of a buyer at period t; the description is analogous for sellers, by symmetry.
Security. There is a single risky security with an unknown fundamental value, V , that
follows a random walk. An innovation δt to the fundamental value occurs at each period t;
δt is uniformly distributed on [−1, 1]. Vt is given by,
Vt =
X
δτ
(1)
τ ≤t
Market Organization. The market consists of a visible limit order book, and a dark
market. The visible limit order book accepts market orders and limit orders from market
participants. The dark market sources its liquidity from a professional liquidity provider,
and accepts (market) orders from investors.6 Visible limit orders and liquidity posted to the
dark market live for one period, after which any unfilled orders are cancelled. At period t,
the best prices at which to buy and sell at the visible market are denoted as askt and bidt ,
respectively. Figure 5 in the Appendix illustrates the timing of the model.
A (market) order sent to the dark market fills at a pre-specified price. The dark market
sets the price based on an exogenous trade-at rule, λ ∈ (0, 1), measured as a fraction of
the difference between the best visible market prices (i.e., the quoted spread) askt − bidt
at period t. For example, a buyer-initiated dark trade fills at askt − λ × (askt − bidt ). By
6
I refer to liquidity provided to the dark market as ‘posted’ liquidity.
7
modelling the trade-at rule as an improvement on the visible spread, the model effectively
imposes price-visibility-time priority, a common practice in industry. The probability that
the order is filled in the dark, is then determined endogenously by the probability that the
professional liquidity provider posts orders to the dark market in the previous period. Prices
at the dark market are denoted by the superscript ‘Dark’.
Similar to Foucault (1999), I assume that the security trades throughout a “trading day”
where the trading process ends after period t with probability (1 − ρ) > 0, at which point
the payoff to the asset is realized. The history of all transactions, as well as quotes and
cancellations on the visible market, are observable by all market participants. I denote this
history up to (but not including) period t as Ht . Moreover, I denote the public expectation
of Vt at period t as vt . The structure of the model is common knowledge to all participants.
Investors. At each period t, a single investor randomly arrives at the market from a
continuum of risk-neutral investors. With probability µ ∈ (0, 1) the investor is informed,
and privately learns the period t innovation δt .7 Uninformed investors are endowed with
liquidity needs, yt , uniformly distributed on [−1, 1].8 Informed investors have no liquidity
needs. Upon arriving at the market in period t and only then, an investor may submit an
order for a single unit (round lot) of the security.9 I assume that, if indifferent between a
visible order or a dark order, the investor chooses the dark order, an assumption supported
by the empirical results of Menkveld, Yueshen, and Zhu (2016). An investor leaves the
market forever upon the execution or cancellation of their order.
Professional Liquidity Provider. There is a single, competitive professional liquidity
7
Numerical simulations show that the results are qualitatively robust to densities of the innovation δt ,
where the innovation density first-order stochastically dominates the density for the private values (i.e., as
the average valuations increase, so does the price impact of the order).
8
Assuming that some investors have liquidity needs is common practice in the literature on trading with
asymmetric information, to avoid the no-trade result of Milgrom and Stokey (1982).
9
As dark trading has been historically popular for executing large orders, trading in a single unit may
seem unnatural. The CFA Institute notes, however, that order and transaction sizes in today’s dark pools
are similar to those on visible markets (CFA, 2012). Data from FINRA from May 16, 2016 finds that for
the 25 non-block-trading alternative trading systems (ATS), the average trade size is 229 shares. Moreover,
Foley and Putniņs̆ (2016) find that after the implementation of a midpoint trade-at rule in Canada, the total
average (dollar) trade size remained low, at approximately $7100 CAD.
8
provider (as in Brolley and Malinova (2014)) that participates in both markets. The liquidity
provider is risk-neutral, uninformed, and has no liquidity needs. Within any period t, the
liquidity provider updates his orders in response to market information (i.e. changes to the
public history) before the arrival of the next investor.10 The liquidity provider maintains a
“full” visible limit order book at any period t by filling vacancies on either side of the book,
for buy and sell limit orders, respectively.
and askLP
with limit orders priced at bidLP
t
t
The professional liquidity provider also supplies liquidity to the dark market. Because
prices at the dark market are derived from the prevailing visible quotes and the trade-at
rule, the liquidity provider maintains competitiveness by choosing the intensity with which
he post orders to both sides of the dark market, such that his expected payoffs are zero.
I choose a single liquidity provider that is competitive within each market for modelling
and exposition simplicity, but the model keeps the essence of a market where many liquidity
providers compete. First, the competitiveness assumption for the visible market is similar to
a market where many uninformed liquidity providers compete (in the Bertrand sense) in limit
order pricing, the result of which drives profits to zero. With a competitive visible market,
a liquidity provider in the dark market could implement a competitive liquidity provision
strategy across markets, by setting a lower than competitive price in the visible market, and
earning a positive profit in the dark market. However, the reality of dark markets has led to
a proliferation of dark venues, and as such, it is not unreasonable to assume that competition
between liquidity providers across dark venues would drive profits to zero. Lastly, I assume a
single liquidity provider at the dark market to abstract from the liquidity provision intensity
coordination problem that would arise from competition within a single dark market.
Investor Payoffs. I denote period t market and limit buy orders in shorthand by
MBt and LBt , respectively. An investor’s dark buy order in period t is denoted DBt . The
superscript ‘inv’ denotes limit prices submitted by investors. An investor’s payoff to any order
type is the difference between their valuation (their private value yt , plus their assessment of
10
See Brolley and Malinova (2014) for a detailed justification of this assumption.
9
the security’s value) and the price paid, discounted by the execution probability. Investors
who abstain from trading receive a payoff of zero. The payoffs to each order type are:
MB
πt,inv
= yt + E[Vt | infot , Ht ] − askt
(2)
inv
inv
LB
πt,inv
= ρ · Pr(fill | infot , Ht , bidinv
t+1 ) × yt + E[Vt+1 | infot , Ht , fill at bidt+1 ] − bidt+1
DB
πt,inv
= Pr(dark fill | infot , Ht ) × yt + E[Vt | infot , Ht ] − askDark
t
(3)
(4)
where infot is the period t investor’s information about the innovation δt ; Pr(fill | infot , Ht , bidinv
t+1 )
and Pr(dark fill | infot , Ht ) are the respective probabilities that a visible limit order is filled,
and that a dark order is filled.
Professional Liquidity Provider Payoffs. In any period t, the professional liquidity
provider observes the period t investor’s action, and then may submits limit orders to the
visible market, or post orders to the dark market. At period t, a visible buy limit order at
price bidLP
t+1 earns the payoff,
LB
πt,LP
= ρ · Pr(fill | investor action at t, Ht , bidLP
t+1 )
LP
× yt + E[Vt+1 | investor action at t, Ht , fill at bidLP
t+1 ] − bidt+1
(5)
I denote dark liquidity posted at period t as DLt . The liquidity provider chooses Pr(DLt ) to
satisfy the following zero expected profit condition (the bid-side equation is similar).
Dark
askDark
t+1 = E[Vt+1 | investor action at t, Ht , fill at askt+1 ]
2
(6)
Equilibrium
I search for a symmetric, stationary perfect Bayesian equilibrium in threshold strategies in
which the bid and ask prices at the visible market in period t are competitive with respect
to available public information at period t. Proofs are in the Appendix.
10
2.1
Order Pricing Rules
In equilibrium, the professional liquidity provider posts competitive limit orders to the visible
market. I use ∗ to denote “in equilibrium”. I denote the equilibrium bid and ask prices by
bid∗t and ask∗t , respectively, which, by the competitive pricing assumption, are given by:
bid∗t = E[Vt | Ht , MS∗t ] = vt + E[δt | MS∗t ]
(7)
ask∗t = E[Vt | Ht , MB∗t ] = vt + E[δt | MB∗t ]
(8)
A limit order posted by an investor in period t is posted competitively if the period t
investor compensates the period t+1 investor for the fact that they may have been informed.
To compensate the period t+1 investor, a competitive limit price includes the expected price
impact, E[δt | LB∗t ] of the submitting investor. Hence, an investor’s limit order is posted at:
bid∗t+1 = E[Vt+1 | Ht , MS∗t+1 , LB∗t ] = vt + E[δt | LB∗t ] + E[δt+1 | MS∗t+1 ]
(9)
ask∗t+1 = E[Vt+1 | Ht , MB∗t+1 , LS∗t ] = vt + E[δt | LS∗t ] + E[δt+1 | MB∗t+1 ]
(10)
Prices in the dark market are derived from (a) the quote in the visible market, and, (b)
the trade-at rule, λ. Hence, the professional liquidity provider does not choose the price of
are given by,
and askDark
the dark orders they post, and their prices, bidDark
t
t
bidDark∗
= bid∗t + λ × (ask∗t − bid∗t )
t
(11)
askDark∗
= ask∗t − λ × (ask∗t − bid∗t )
t
(12)
Equations (11)-(12), by symmetry, simplify to (1+2λ)×bid∗t and (1−2λ)×ask∗t , respectively.
At the dark market, quotes are derived from the visible market, so the liquidity provider
chooses only to post or not post a limit order to the dark. However, the liquidity provider
must act competitively, choosing the intensity with which he post orders to the dark market
such that he breaks even on average. That is, the liquidity provider chooses this intensity
such that the price that he receives (pays) for a dark sell (buy) order at period t − 1 (based
11
on the trade-at rule, λ) must equal to the public value of the security at period t − 1, plus
the price impact of trading with an investor in the dark at period t. Hence, the price of
orders posted to the dark must satisfy:
bidDark∗
= E[Vt | Ht , DS∗t ] = vt + E[δt | DS∗t ]
t
(13)
askDark∗
= E[Vt | Ht , DB∗t ] = vt + E[δt | DB∗t ]
t
(14)
would have a post-trade price impact,
or bidDark∗
To see that a dark trade at askDark∗
t
t
note that the “ticker-tape” reveals that a trade has occurred in the dark. Because the price
at which the trade occurs is away from the midpoint (for all λ 6= 1/2), the liquidity provider
infers the direction of the trade, and uses this information to update their limit orders on
the visible market.11 In this way, trades in the dark have a post-trade price impact. In what
follows, I use the term ‘price impact’ to refer to post-trade price impact for visible market
trades and dark trades, and pre-trade price impact for visible quotes submitted by investors.
Lastly, all investors form a common prior vt from Ht ; because future innovations are
independent of past price history, expectations on δt are independent of Ht .
2.2
Investor Decision Rules
An investor will submit an order (to either market) only if, conditional on their information and on the submission of the order, their expected profits are non-negative. Moreover,
an investor who submits an order chooses the order type that maximizes their expected
profits. Similar to Brolley and Malinova (2014), I restrict my attention to equilibria where
investors submit visible limit orders that cannot be improved upon by the professional liquidity provider. That is, where bidinv
= bidLP
= bid∗t . Orders with non-competitive prices
t
t
yield the investor negative profits, or an execution probability of zero. In the Appendix,
I discuss the off-the-equilibrium-path beliefs about non-competitive limit orders such that
11
In a visible market with more than one (identical) liquidity provider, the model would require that all
liquidity providers to infer the direction of the dark trade from the ticker tape and update their quotes
simultaneously, to abstract from the sniping scenario detailed in Budish, Cramton, and Shim (2015).
12
market buy
πM (zt )
order filled
visible market
ρPr(MSt+1 )
limit buy
1 − ρPr(MSt+1 )
zt
Pr(DLt−1 )
dark market
1 − Pr(DLt−1 )
no trade
πL (zt | filled at t + 1)
πL (zt | not filled) = 0
πD (zt | filled at t)
πD (zt | not filled) = 0
πN T (zt ) = 0
leaves market
Figure 1: Investor Decision Tree: Given their valuation, zt , the period t investor chooses to submit
an order to the visible market, dark market, or abstain from trading. If the investor sends an order to the
visible market, the investor chooses either a market buy, or a limit buy (with the appropriate competitive
price). A market order is filled automatically in period t, while a limit order is filled in period t + 1 with
probability Pr(MSt+1 ). An order sent to the dark market, is filled with probability Pr(DLt−1 ).
competitive limit order pricing is supported in equilibrium.
Investors choose their order based on available public information, plus: (a) their private
valuation yt (if uninformed), or; (b) their knowledge of δt , (if informed). Because these
valuations enter investor payoff functions identically, I can summarize investor decisions
in terms of an investor’s “valuation”, regardless of type. I denote the period t investor’s
valuation by, zt = yt + E[δt | infot ]. zt is symmetrically distributed on the interval [−1, 1].
Using this notation, I depict the decision tree diagrammatically in Figure 1.
Because innovations δt are time-independent, I can decompose an investor’s valuation
into two parts: the expected value of the security based on the public information, plus zt ,
yt + E[Vt | infot , Ht ] = E[Vt−1 | Ht ] + yt + E[δt | infot ] = vt + zt
(15)
In addition, when submitting a limit buy order to the visible market, an investor incurs an
adverse selection cost of trading with a potentially informed investor in period t + 1, equal to
E[δt+1 | Ht , LB∗t , MS∗t+1 ]. This cost is identical for informed and uninformed investors, because
13
information about period τ ≤ t innovations does not inform about future innovations. I
summarize all the costs of submitting an order as “transaction costs”, which includes: order
prices, and, if trading in the following period (with a visible limit order), the expected adverse
selection associated from being picked off.
Hence, the payoff expressions (2)-(4) can be reduced to three components: (a) the investor’s valuation, (b) the order’s transaction costs, and (c) the order’s execution probability.
The payoff of any order type, k, is then generally characterized by,
πtk = execution probability × (zt − transaction costs)
(16)
Thus, an investor chooses an order type where i) the investor’s valuation exceeds the the
order’s transaction costs, and ii) the order type also maximizes their profits.
2.3
Equilibrium Characterization
I now derive properties that, under the aforementioned order pricing and investor decision
rules, must hold in equilibrium. Moreover, I focus my attention to the cases where all order
types—visible market orders, visible limit orders, and dark orders—are used in equilibrium.
I use these properties to successively reduce the set of candidate equilibria.
Using the intuition that an investor’s valuation must exceed the transaction costs of their
submitted order, the following lemma provides that in any symmetric, stationary equilibrium,
I show that an investor will not trade in the opposite direction of their valuation (though
some investors with non-zero valuations may abstain from trading because transaction costs
are too high). Moreover, an order type will (weakly) move the mid-quote in the direction of
the investor’s valuation, by the price impact (the average informativeness) of the order.
Lemma 1 (Transaction Costs and Price Impact) In any symmetric, stationary equilibrium where investors use all order types, investors do not sell if zt ≥ 0 and do not buy if
zt ≤ 0. Moreover, the transaction costs of a buy (sell) order in period t are equal its price
impact, which is positive (negative).
14
I interpret the difference in mid-quote across time periods, vt+1 − vt , as an order’s “price
impact”; this impact can occur pre-trade or post-trade, depending on the order type. Hence,
an investor earns a positive payoff from an order type whenever their valuation exceeds the
price impact of their order.
Given that an investor’s valuation exceeds the price impact of their order type, the
investor’s expected profit from that order type is increasing in the order’s fill rate. Then,
investors with the most extreme valuations (those away from zero) have the most to gain
from increased execution probability, but consequently, are the most informed (on average),
and hence contribute to a higher price impact.
I conjecture that these incentives result in a threshold strategy where those with the
most extreme valuations opt for orders with the highest fill rate, but in turn, contribute the
highest price impact, such that investors with a valuation below some threshold prefer to
accept a lower execution probability, in exchange for lower price impact. I define a threshold
strategy below, and prove this conjecture in the following lemma.
Definition 1 (Threshold Strategy) Let I and J be two order types. A threshold strategy
is defined as a set of valuation thresholds z I and z J , such that for z I < z J :
(1) an investor with valuation zt > z J chooses order type J;
(2) an investor with valuation z I ≤ zt ≤ z J chooses order type I;
(3) an investor with valuation zt < z I chooses neither.
Lemma 2 (Threshold Strategy and Execution Probability) In any symmetric, stationary equilibrium where investors use all order types, investors use a threshold strategy, as
described in Definition 1. Moreover, the (absolute) valuation thresholds |z J | are increasing
with the order’s execution probability.
Lemma 2 produces a result similar to the prediction of Hollifield, Miller, and Sandås (2004),
that investors with more extreme valuations use order types with higher fill rates. They
test this prediction empirically, as a monotonicity restriction on investors’ order submission
15
strategies, and find support when the choices of buy and sell orders are considered separately.
Taken together, Lemmas 1 and 2 summarize the optimal order placement decision: the
higher an investor’s valuation, the larger the price impact that they are willing to absorb, in
exchange for better execution.
Corollary 1 (Execution Probability and Price Impact) In any symmetric, stationary
equilibrium where investors use all order types, if an order I has higher execution probability
than order J, then I must also have a greater (absolute) price impact than J.
The intuition of Corollary 1 is similar to the pecking order hypothesis of dark order types,
coined by Menkveld, Yueshen, and Zhu (2016). They generate a pecking-order result in a
model with symmetric information: investors with high private valuations submit market
orders against quotes on the visible market, where they are guaranteed execution, but contribute a high price impact; investors with lower private valuations opt to send orders to a
dark market to receive a price improvement, at the expense of higher execution uncertainty.
Here, I show that the result is preserved in a model with asymmetric information and visible limit orders: an investor weighs an order’s execution likelihood against its transaction
cost—the latter of which is equivalent to the order’s price impact.
2.4
Equilibrium Existence
In what follows, I drop all time subscripts, as I focus on stationary equilibria. Lemmas 1-2
reduce the set of candidate equilibria to threshold strategies that are increasing in order execution probability: investors with higher valuations choose order types with higher likelihood
of execution. It also follows from Lemma 2 that investors with the most extreme valuations
submit visible market orders, because visible market orders are guaranteed execution, by
virtue of the professional liquidity provider ensuring the visible limit order market is always
full. Investors with valuations closest to zero abstain from trading, which loosely fits the
narrative where, the closer an investor’s valuation is to the public prior, the less likely they
16
are to trade. An investor who abstains from trading is similar, in a sense, to choosing an
order type with a fill rate of zero.
The remaining candidate equilibria therefore characterize the behaviours of investors with
valuations not too close to zero, but not too far from zero. Lemma 2 reduces these candidates
to two possible cases: (a) the execution probability of a dark order is lower than that of a
visible limit order, implying that the equilibrium thresholds are 0 ≤ z D ≤ z L ≤ z M ≤ 1,
and; (b) the execution probability of a dark order is higher than that of a visible limit order,
implying that the equilibrium thresholds are 0 ≤ z L ≤ z D ≤ z M ≤ 1.
2.4.1
Benchmark Equilibrium
I examine the model from the perspective of introducing a dark market alongside a visible
limit order market. In the absence of a dark market, the model follows Brolley and Malinova
(2014) which has unique equilibrium where investors use both market order and limit orders,
when the innovations δ are drawn from a uniform distribution.
When a dark market is available to investors, the “visible market only” equilibrium
exists for all λ ∈ (0, 1), if the professional liquidity provider chooses not to supply liquidity
to the dark (Pr(DL∗ ) = Pr(DL∗ ) = 0). I summarize this result in the following theorem; in
discussions that follow, I refer to this as the “benchmark equilibrium”.
Theorem 1 (Benchmark Equilibrium) For any λ ∈ (0, 1), there exist values z M ∗ , z L∗ ,
z D∗ , and Pr(DL∗ ), such that 0 < z D∗ = z L∗ = zBL < z M ∗ = zBM < 1, Pr(DL∗ ) = 0, and limit
prices bid∗ and ask∗ satisfy (9)-(12), that constitute a symmetric, stationary equilibrium in
threshold strategies (as in Definition 1).
Figure 2 illustrates how an investor with valuation z behaves under Theorem 1.
Because I focus on equilibria where investors use the dark market, it is useful to determine for which trade-at rules—if any—that the benchmark equilibrium may be the unique
equilibrium (i.e., investors who send orders to the dark are of measure zero). By the equilib17
Market Sell
-1
Limit Sell
−z M
Limit Buy
Abstain
−z L
zL
0
Market Buy
zM
1
Valuation zt
Figure 2: Benchmark Equilibrium: In the benchmark equilibrium described by Theorem 1, an investor
with valuation zt ∈ [−1, 1] optimally chooses their order type (market order, limit order, abstain from trade)
based on the region their valuation falls into, as illustrated by the figure above.
rium dark order pricing equations (11)-(12), the range of prices for dark sell liquidity, given
trade-at rule λ, is bounded by (−ask∗ , ask∗ ).
Consider any trade-at rule λ ≥ λ = 1/2. These trade-at rules imply askDark∗ ∈ (−ask∗ , 0),
which prices sell orders posted to the dark at mid-quote or better (i.e., equal to or lower
than the public value v). But, by Lemma 1, an investor’s dark buy order has a positive
(post-trade) price impact, and hence, the liquidity provider would expect to earn a loss by
trading against it. Thus, the liquidity provider chooses Pr(DL∗ ) = 0, for all λ ∈ (λ, 1), and
the unique equilibrium is as in Theorem 1.
The set of trade-at rules that may yield an equilibrium with dark trading reduces to
(0, λ). Now consider the investor’s problem. If we introduce a dark market with a trade-at
rule λ̂ such that an order sent to the dark by an investor is identical in every payoff-relevant
aspect to submitting a visible limit order, investors have no incentive to submit limit orders
to the visible market by the assumption that investors choose dark orders when they are
indifferent between a visible order and a dark order. This scenario is equivalent to a tradeat rule λ̂ such that the transaction costs from sending an order to the dark market versus
submitting a visible limit order satisfy,
(1 − 2λ̂) × E[δt | MB∗t , z M ∗ = zBM ] = E[δt | LB∗t , z M ∗ = zBM , z L∗ = zBL ]
⇐⇒ λ̂ =
1 − zBL
2(1 + zBM )
(17)
(18)
When the transaction costs of a dark order and a visible limit order are equal (to the
18
transaction costs of a limit order in Theorem 1), the professional liquidity provider must
choose Pr(DL∗ ) = ρPr(MSB ), which is the unique dark liquidity provision intensity that
solves both the equilibrium indifference conditions, and the zero expected profit condition.
Hence, given the thresholds z M ∗ = z L∗ = zBM , and z D∗ = zBL , as well as Pr(DL∗ ) = ρPr(MSB ),
we have the following proposition.
Proposition 1 (Payoff-Equivalent Orders) Given λ = λ̂ that satisfies (18), the valuation thresholds z M ∗ = z L∗ = zBM , and z D∗ = zBL , and the intensity of dark liquidity provision,
Pr(DL∗ ) = ρPr(MSB ), constitute a symmetric, stationary equilibrium in threshold strategies
(as in Definition 1).
The equilibrium at λ = λ̂ plays a critical role in determining how dark trading impacts
the market. Because λ̂ is unique, positive, and less than λ, I can use it to partition the
remaining trade-at rules (0, λ) into two regions. Moreover, this partition yields the following
Lemma describing the possible equilibrium types for each region, λ ∈ (0, λ̂) and λ ∈ (λ̂, λ).
Lemma 3 (Trade-at Rules and Valuation Thresholds) In any equilibrium where all
order types are used, equilibrium thresholds must satisfy:
(i) λ ∈ (0, λ̂) ⇒ 0 ≤ z L∗ ≤ zBL ≤ z D∗ ≤ zBM ≤ z M ∗ ≤ 1
(ii) λ ∈ (λ̂, λ) ⇒ 0 ≤ z D∗ ≤ zBL ≤ z L∗ ≤ z M ∗ ≤ zBM ≤ 1
Lemma 3 illustrates that, when the trade-at rule is large enough, the only possible equilibrium with dark trading is one in which investors with valuations closer to zero (and thus,
with smaller contributions to price impact) trade in the dark, compared to those who use
visible limit orders. For a smaller trade-at rule, the investors who use dark orders contribute
more to the price impact of their order than those who use visible limit orders.
2.4.2
Dark Orders with a Small Trade-at Rule.
If the dark market has a trade-at rule λ ∈ (0, λ̂)—a “small trade-at rule”—then dark orders have higher transaction costs in equilibrium than visible limit orders, by Lemma 3.
19
For investors to submit dark orders in equilibrium, the professional liquidity provider must
compensate investors for these higher transaction costs by ensuring that the fill rate of an
investor’s dark order is greater than the fill rate of a visible limit order, Pr(DL∗ ) ≥ ρPr(MS∗ ).
Lemma 3 further dictates that equilibrium thresholds must satisfy 0 < z L∗ ≤ z D∗ ≤ z M ∗ ≤ 1.
Thus, I look for equilibrium valuation thresholds where an investor with valuation: i) z M ∗
is indifferent to market orders and dark orders; ii) z D∗ is indifferent to dark orders and limit
orders, and; iii) z L∗ is indifferent to limit orders and abstaining from trade. The professional
liquidity provider must also choose Pr(DL∗ ) to solve their zero profit condition in (6). These
equilibrium indifference equations are respectively given by,
m
Zλ<
≡ z M − E[δ | MB∗ ] − Pr(DL) × z M − (1 − 2λ) × E[δ | MB∗ ]
λ̂
γ
≡ Pr(DL) × z D − (1 − 2λ) × E[δ | MB∗ ] − ρ · Pr(MS∗ ) × z D − E[δ | LB∗ ]
Zλ<
λ̂
(19)
(20)
l
Zλ<
≡ z L − E[δ | LB∗ ]
λ̂
(21)
d
Zλ<
≡ (1 − 2λ) × E[δ | MB∗ ] − E[δ | DB∗ ]
λ̂
(22)
Setting equations (19)-(22) equal to zero, I find that there exist z M ∗ , z L∗ and z D∗ such that
0 ≤ z L∗ ≤ z D∗ ≤ z M ∗ ≤ 1, and a dark order fill rate Pr(DL∗ ) ≥ ρPr(MS∗ ), that solve the
system (19)-(22). The result is the following existence theorem.
Theorem 2 (Small Trade-at Rule) There exist values z L∗ and z D∗ , where
0 ≤ z L∗ ≤ z D∗ ≤ z M ∗ ≤ 1, λ ∈ (0, λ̂), Pr(DL∗ ) > ρPr(MS∗ ), and limit prices bid∗ and ask∗
that satisfy (9)-(12), that constitute a symmetric, stationary equilibrium in threshold strategies (as in Definition 1), for all z M ∗ ∈ [zBM , 1].
Figure 3 illustrates how an investor with valuation z behaves in equilibrium, when choosing
between a visible market and a dark market with a small trade-at rule.
20
MS
-1
DS
−z M
LS
−z D
Abstain
−z L
0
LB
zL
DB
zD
MB
zM
1
Valuation z
Figure 3: Small Trade-at Rule: In an equilibrium where a dark market offers a trade-at rule λ ∈ (0, λ̂)
as described by Theorem 2, an investor with valuation z ∈ [−1, 1] optimally chooses their order type (market
order, limit order, dark order, abstaining from trade) based on the region their valuation falls into, as
illustrated by the figure above.
2.4.3
Dark Orders with a Large Trade-at Rule.
If the trade-at rule is equal to λ ∈ (λ̂, λ)—a “large trade-at rule”—then dark orders have
lower transaction costs in equilibrium than visible limit orders, from Lemma 3. When the
trade-at rule is large, dark orders provide substantial price improvement over posted prices,
making them attractive relative to visible limit orders in terms of transaction costs. To
ensure that the professional liquidity provider breaks even, however, he must disincentivize
moderate-valuation investors (who have, on average, larger contribution to price impact)
from sending orders to the dark, instead attracting only low-valuation investors. To do so,
the professional liquidity provider offers a lower fill rate for dark orders relative to limit
orders, Pr(DL∗ ) ≤ ρPr(MS∗ ).
Lemma 3 dictates that equilibrium thresholds must satisfy 0 ≤ z D∗ ≤ z L∗ ≤ z M ∗ ≤ 1.
Similar in method to a small trade-at rule, I look for valuation thresholds where, in equilibrium, an investor with valuation: i) z M is indifferent to market orders and limit orders; ii)
z L is indifferent to limit orders and dark orders, and; iii) z D is indifferent to dark orders and
abstaining from trade. The professional liquidity provider’s zero profit condition must also
21
MS
-1
LS
−z M
DS
−z L
Abstain
−z D
0
DB
zD
LB
zL
MB
zM
1
Valuation z
Figure 4: Large Trade-at Rule: In an equilibrium where a dark market offers a trade-at rule λ ∈ (λ̂, λ)
as described by Theorem 3, an investor with valuation zt ∈ [−1, 1] optimally chooses their order type
(market order, limit order, dark order, abstaining from trade) based on the region their valuation falls into,
as illustrated by the figure above.
hold. These equilibrium indifference equations are respectively given by,
m
Zλ>
≡ z M − E[δ | MB∗ ] − ρ · Pr(MS∗ ) × z M − E[δ | LB∗ ]
λ̂
γ
∗
∗
∗
L
L
≡
ρ
·
Pr(MS
)
×
z
−
E[δ
|
LB
]
−
Pr(DL)
×
z
−
(1
−
2λ)
×
E[δ
|
MB
]
Zλ>
λ̂
(23)
(24)
d
Zλ>
≡ z D − (1 − 2λ) × E[δ | MB∗ ]
λ̂
(25)
l
Zλ>
≡ (1 − 2λ) × E[δ | MB∗ ] − E[δ | DB∗ ]
λ̂
(26)
Similar to the small trade-at rule case, I find that there exist z M ∗ , z L∗ and z D∗ such that
0 ≤ z D∗ ≤ z L∗ ≤ z M ∗ ≤ 1, and a dark order fill rate Pr(DL∗ ) ≤ ρPr(MS∗ ), that solve the
system (23)-(26). The existence theorem follows.
Theorem 3 (Large Trade-at Rule) There exist unique values z M ∗ and z D∗ , where
0 ≤ z D∗ ≤ z L∗ ≤ z M ∗ ≤ 1, λ ∈ (λ̂, λ), Pr(DL∗ ) ≤ ρPr(MS∗ ), and limit prices bid∗ and ask∗
that satisfy (9)-(12), that constitute a symmetric, stationary equilibrium in threshold strategies (as in Definition 1), for all z L∗ ∈ (zBL , zBM ).
Figure 4 illustrates how an investor with valuation z behaves in equilibrium, when choosing
between a visible market and a dark market with a small trade-at rule.
22
3
The Impact of Trade-at Rules
In this section, I analyze the impact of dark trading on market quality and investor welfare.
I do so by comparing market quality and welfare measures when investors use a dark market
(with either a small or a large trade-at rule) to the levels under the benchmark equilibrium.
As before, the classification of trade-at rule level as large or small is relative to the benchmark
trade-at rule, λ̂. The following subsection focuses on aspects of market quality; subsections
3.2 and 3.3 discuss price efficiency and welfare, the results for which are numerical.
3.1
Market Quality Measures
Trading volume in the context of my model has two components: visible and dark market
volume. Visible market volume is measured by the probability that an investor submits either
a buy or sell market order to the visible market (these orders are always filled). Because
dark orders have some level of execution risk, dark market volume is therefore measured by
the probability that an investors submits an order to the dark market and that the order
will be filled. Total trading volume at period t is thus measured as,
Trading Volumet = 2 × (Pr(MBt ) + Pr(DLt−1 ) × Pr(DBt ))
(27)
where the factor of 2 accounts for sell orders. Overall market participation measures the
likelihood that an investor who arrives at the market in period t submits an order of some
type (i.e does not abstain from trading), which is measured by,
Market Participationt = 2 × (Pr(MBt ) + Pr(DBt ) + Pr(LBt ))
(28)
Lemma 3 allows us to compare the equilibrium threshold values z L∗ , z D∗ and z M ∗ , given λ, to
the benchmark equilibrium to make predictions on trading volume and market participation.
Proposition 2 (Visible Market Volume and Market Participation) Compared to the
visible market only equilibrium, introducing a dark market with a small trade-at rule, de-
23
creases volume on the visible market; a large trade-at rule increases visible market volume.
Market participation (weakly) increases for any trade-at rule.
Numerical Observation 1 (Total Volume) Compared to the visible market only equilibrium, introducing a dark market with a small trade-at rule reduces total volume; a large
trade-at rule increases total volume.
Figure 7 summarizes the findings of Proposition 2 and Numerical Observation 1 graphically.
I find that a dark venue with a large trade-at rule improves market quality by introducing
an attractive (i.e., affordable) order type for low valuation investors. Investors who would be
indifferent to visible limit orders and not trading without dark trading, now elect to trade
with dark orders, increasing the price impact of visible limit orders. As a result, visible limit
order submitters who would be indifferent to market and limit orders without a dark market,
now submit visible market orders with the availability of a dark market (See Figure 2). Thus,
there is net order and volume creation. A dark market with a small trade-at rule offers lower
execution risk versus visible limit orders. Dark orders then induce visible market orders
submitters with the lowest valuations, and visible limit orders submitters with the highest
valuations (in a ‘visible market only’ setting) to migrate to the dark market. The result is
volume migration from the visible market to the dark market. Because dark orders fill less
often than visible market orders, the result is a reduction in total volume.
My results on visible market volume and total volume provide some guidance for further
empirical research. In their 2010 Concept Release, the SEC takes as given that dark trading
siphons volume away from visible markets, as a premise to their concern about price efficiency loss from increased dark trading.12 Proposition 2 suggests, however, that the impact
of dark trading on volume is dependent on the role that dark orders play in the “costimmediacy pecking order”—to borrow from Menkveld, Yueshen, and Zhu (2016)—alluded
to by Lemma 2 and Corollary 1. My model predicts that, if dark orders offer a higher fill
rate than visible limit orders (i.e., with a small trade-at rule), then the dark market will lead
12
Securities and Exchange Commission (2010)
24
investors to migrate to the dark. However, if dark orders offer a lower fill rate than visible
limit orders (i.e., with a large trade-at rule), the model predicts increased visible market
volume, as limit order submitters migrate to visible market orders (as well as dark orders).
I also consider the impact of a dark market on visible market transaction costs, measured
in my model by the quoted (half) spread.13 The results are illustrated graphically in Figure 8.
Proposition 3 (Quoted Spread / Visible Market Order Price Impact) Compared to
the visible-market-only equilibrium, introducing a dark market with a small trade-at rule
widens the quoted spread (market order price impact increases). A large trade-at rule, narrows the quoted spread (market order price impact decreases).
Dark orders subject to a large trade-at rule act as a lower (transaction) cost alternative
to visible limit orders for low-valuation investors, who migrate to dark orders. Visible limit
orders become more informative on average, increasing their price impact, and making them
less attractive to those with the highest valuations who would submit limit orders in the
case with no dark market. These marginal limit order submitters now prefer market orders,
decreasing the average informativeness of visible market orders. The end result is a decline
in the price impact of market orders, and a narrowing of the bid-ask spread.
Conversely, a small trade-at rule dark order provides an alternative for marginal market
order submitters that has a higher execution probability; marginal market order submitters
then prefer dark orders, increasing the average informativeness of visible market orders,
thereby widening the quoted spread and increasing the price impact of visible market orders.
3.2
Price Efficiency
Price impact measures the impact of a trade on the public expectation of the security.
However, because of noise from uninformed investors, the change in the public expectation
13
In this framework, the bid-ask spread is synonymous with the price impact of a market order, as the
spread measures the adverse selection costs due to information of a trade on the visible market, which
occurs when a visible market order is placed. For better illustration, I use the quoted half-spread measure
(symmetry implies that the bid-ask spread is equal to twice the quoted half-spread)
25
does not coincide with the innovation to the true value of the security. Thus, one can think
of price efficiency in this framework as the ex-ante the mean squared difference between the
innovation’s value and the investor’s expected price impact (per period). More formally,
Price Efficiency = E[(δ − (1 − µ) × E[δ | uninformed] − µ × E[δ | informed])2 ]
(29)
That is, for each possible innovation draw, δt there is an expected price impact from
the incoming investor’s action, and this price impact may or may not be equal to the value
of the innovation (and in the case of uninformed investors, may not even be in the same
direction). I then measure price efficiency as the expected deviation from the value of the
innovation each period. Because, for any innovation value, an uninformed trader’s expected
price impact is zero, the term E[δ | uninformed] = 0. Then, we can re-write (29) as,
2
2
Z E[δ | LB∗ ]
E[δ | MB∗ ]
dδ + 2 ×
δ−µ×
dδ
Price Efficiency = 2 ×
δ−µ×
2
2
LB
MB
2
Z E[δ | DB∗ ]
∗
dδ
(30)
+2×
δ − µ · Pr(DL ) ×
2
DB
Z
where the factors of two account for symmetry. The following result is numerical. Figure 9
presents it graphically.14
Numerical Observation 2 (Price Efficiency) Compared to the visible market only equilibrium, price efficiency worsens with any trade-at rule where investors use the dark market.
As the general usage of dark orders increases, price efficiency falls because investors
migrate from visible orders, to an order that only impacts prices when it is filled. Moreover,
in both cases, the mass of investors that migrate to the dark market is larger than those
who would participate only in a market with dark market trading (they previously made no
contribution to price efficiency). The combined result is a reduction in price efficiency.
14
Figure 9 uses zero as the informationally efficient benchmark, and larger deviations from zero, correspond
with less informationally efficient prices.
26
3.3
Social Welfare
Uninformed investors earn gains from realizing their private valuation. Following the work of
Bessembinder, Hao, and Lemmon (2012), I use private valuations to define social welfare as
a measure of allocative efficiency. If a transaction occurs in period t, then the social welfare
gain is given by the private valuation of a buyer minus the private valuation of a seller.
A trade occurs when the period t investor submits either a market order to the visible
market, or an order to the dark market and the order is filled. I measure the total expected
gains from trade in period t, by the expected private valuation of the investor submitting the
market order and the expected welfare of the counter-party, discounted by the probability
of an order being filled. Thus, the total expected welfare in period t is given by,
Wt = 2 · Pr(MBt ) · (E[yt | Ht , MBt ] − Pr(LSt−1 ) · E[yt−1 | Ht−1 , MBt , LSt−1 ])
(31)
+ 2 · Pr(DBt ) · Pr(DLt−1 ) · E[yt | Ht , DBt ]
Expression (31) describes total expected welfare per period as the expected private value
realized in period t, conditional on a visible or dark trade, and discounted by the likelihood
of each trade type. The first term describes the (discounted) total expected welfare of a
visible market trade: the expected private valuation from the market order submitter, and
the expected private valuation of the limit order submitter (which is non-zero in expectation
for the investor, but zero for the professional liquidity provider). The second term describes
the (discounted) total expected welfare of a dark trade. The factors of two in expression (31)
account for the symmetry of buyer and seller trades.
Numerically, I find that introducing dark trading alongside the visible market has the
following effect on expected social welfare, Wt (see Figure 10).
Numerical Observation 3 (Social Welfare) Compared to the visible market only equilibrium, introducing a dark market with a small trade-at rule reduces social welfare. A large
trade-at rule improves social welfare.
27
A dark market with a large trade-at rule improves welfare by encouraging greater use
of visible market orders from investors who use visible limit orders (implying more frequent
trades). This occurs because the transaction costs of a limit order increases when the lowvaluation limit order submitters migrate to the dark market. Moreover, the dark market
attracts investors who would otherwise not trade in a purely visible market. Hence, private
valuations are realized more frequently per order (more visible market orders) and more
frequently per investor (higher market participation). This result suggests that when the
transaction costs of dark orders are low, an intermediated dark market is socially beneficial.
I find that not all dark markets are welfare-improving, however. A dark market with a
small trade-at rule leads to fewer visible market orders in equilibrium. Despite more limit
order submitters now migrating to the dark market (where execution risk is lower), the
likelihood that an investor’s private valuation is realized, declines. While more investors are
able to participate in the market because of the availability of dark trading, the decline in
visible market volume exceeds the increase in market participation, leading to the decline in
social welfare.
4
Policy Implications and Empirical Predictions
In Section 3, my model predicts that the impact of dark trading on visible market quality
and social welfare depends on the trade-at rule in the dark market. More importantly, a
minimum trade-at rule has the potential to improve welfare and visible market quality. A
minimum trade-at rule of λ = λ̂ would make it unprofitable for a dark market to attract
moderate valuation investors with a small trade-at rule (which implies low execution risk).
Instead, the dark market may attract low valuation investors, contributing positively to both
market quality and investor welfare.
The minimum trade-at rule discussion has important implications for equity markets
in Canada and Australia. On October 15th, 2012, the Investment Industry Regulatory
28
Organization of Canada (IIROC) implemented a trade-at rule that requires most dark orders
to provide a meaningful price improvement of one trading increment (or in the case of a one
tick spread, half the spread).15 Australia followed suit with a similar rule on May 26th, 2013.
Since many liquid securities operate at a spread of one or two ticks, the minimum trade-at
rule effectively implies dark trades must occur at midpoint. The security that I model is
reflective of this type of security, as investors trade in single units (i.e., never walk the book),
and the professional liquidity provider ensures that the book is always full.
My model predicts that a minimum trade-at rule requiring dark pools to execute orders
at midpoint would eliminate all market-making activities in dark markets. Empirical work
by Comerton-Forde, Malinova, and Park (2015) and Foley and Putniņs̆ (2016) support this
prediction, finding that two-sided liquidity provision sharply declined following the introduction of the minimum price improvement rule in Canada. Further, Foley and Putniņs̆ (2016)
find that nearly all remaining dark trades executed at the midpoint of the NBBO (national
best bid-offer). However, Foley and Putniņs̆ (2016) find that the impact on trading costs is
negative: quoted and effective spreads rose by 5% and 8%, respectively, and price impacts
also increase. In line with the predictions of my model, a reduction in two-sided dark trading
improves trading costs only in the case where dark order fill rates are greater than the fill
rates of visible limit orders placed by investors. Because the fill rate of a visible limit order is
equal to the probability of a market order arriving next period (adjusted for the probability
of the trading day ending), setting the trade-at rule to λ = λ will improve market quality
if ρPr(MS∗ ) ≤ Pr(DL∗ ). Then, using Pr(MS∗ ) (i.e., the trading rate) as an upper-bound for
the fill rate of visible limit orders, my model generates the following testable prediction.
Empirical Prediction 1 (Midpoint Trade-at Rule) If the trading rate at the visible
market is lower than the fill rate of dark orders, setting a trade-at rule λ ≥ λ will:
(1) reduce quoted spreads and lower price impact.
(2) increases visible market volume and improves price efficiency.
15
See Investment Industry Regulatory Organization of Canada (2011)
29
The reverse of Empirical Prediction 1 holds when ρPr(MS∗ ) ≥ Pr(DL∗ ), but because the
lower-bound (ρ = 0) provides no meaningful testable prediction, an empirical test requires an
estimate of ρ. However, a simpler solution presents when ρ can be set to one. This assumption
may be valid in the following cases: a) high-volume equities (i.e., a high probability of the
limit order filling or being cancelled before the trading day ends), and; b) where data is
truncated to avoid activity around the closing bell (i.e., limit order submitters have low
expectations of a market close relative to the resting time of their order).
A trade-at rule that is slightly less restrictive than a midpoint rule would lead to dark
liquidity provision that results in the dark market being beneficial to market quality and
welfare. To determine what this type of trade-at rule would be, my model suggests that we
need to understand where dark orders fit into the immediacy pecking order relative to visible
orders (in a similar sense to Menkveld, Yueshen, and Zhu (2016)). Dark orders can have a
positive impact on market quality and welfare if they fall to the bottom of the immediacy
pecking order (i.e., dark orders fill with a lower probability than visible limit orders). Given
an estimate of the fill rate of a visible limit order (i.e., a measure of visible market volume),
any minimum trade-at rule that ensures dark orders have a lower fill rate would (at least
weakly) improve market quality and welfare.
5
Concluding Remarks
In this paper, I study how trade-at rules impact the way dark pools affect market quality
in equity markets. I construct a model where both informed and uninformed investors can
access a visible limit order market, and a dark market where prices are governed by a tradeat rule. Investors can both demand and supply liquidity to the visible market. Due to
the presence of a professional liquidity provider that ensures competitive visible limit order
pricing, the limit price decision for visible limit orders is simplified.
I find that a trade-at rules contribute to the impact of dark trading on overall market
30
quality and welfare. Moreover, the qualitative contribution of the trade-at rule relies on
the relative fill rates of visible limit orders and dark orders submitted by investors. A dark
market with a trade-at rule that implies a large price improvement leads dark orders to have
a lower fill rate than visible limit orders has an improvement in market quality and social
welfare. The dark market trades off their discount pricing with higher execution risk. The
“large discount” dark market attracts investors from both the pool of investors that submit
visible limit orders, as well as investors who would otherwise abstain from trading. A small
trade-at rule incentivizes the professional liquidity provider to offer reduced execution risk,
relative to the dark market with a large trade-at rule. The execution risk of dark orders
facing a small trade-at rule is lower than the execution risk of visible limit orders, and this
leads to orders of investors with moderate valuations migrating to the dark market. Orders
migrate from both the pools of investors submitting visible market orders and visible limit
orders. Consequently, market quality and social welfare decline. I find that while adding
a dark market improves market participation in all cases, price efficiency declines, as fewer
investors submit order types that have guaranteed price impact (i.e., dark orders).
My results have implications for minimum trade-at rule regulation. My model predicts
that the impact of dark trading on visible market quality and social welfare depends on the
trade-at rule of the dark market, and as such, a minimum trade-at rule has the potential
to improve welfare and market quality. A minimum trade-at rule that equates the trading
rate for both displayed and non-displayed marketplaces would prevent dark markets from
attracting investors with moderate valuations, as the liquidity provider would not be willing
to provide liquidity with a high enough level of intensity (it would be unprofitable). However,
dark pools may attract low valuation investors by setting a large trade-at rule, and as such,
the dark market would contribute positively to both market quality and welfare.
The predictions of my model find support in the empirical literature, as Foley and Putniņs̆
(2016) provide evidence that a midpoint trade-at rule significantly reduces total trading
volume, and that this reduction in dark trading leads to an increase in trading costs. To
31
determine what trade-at rules would be beneficial, we must be able to assess the relative
immediacy pecking order between visible and dark orders (in the sense of Menkveld, Yueshen,
and Zhu (2016))—the assessment of which would allow regulators to discern a large from a
small trade at rule.
32
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35
A
Appendix
The appendix contains all proofs and figures not presented in the text.
A.1
Proofs: Lemmas
Proof (Lemma 1).
First, I show that for any buy order, the order’s transaction costs as in equation (16) are
equal to its price impact. Sell order types hold by symmetry. For a market buy order, we
take the investor’s payoff to a market buy order from (2), and substitute the ask∗t from (8).
πMB = vt + zt − askt = zt − E[δt | MB∗t ]
(32)
Hence, the transaction cost of a market buy order is equal to its price impact, E[δt | MB∗t ].
Now, consider the investor’s payoff to a limit buy order, given by (3). By substituting
bid∗t+1 from (9), I simplify the payoff function to:
πLB = ρPr(MS∗ )(E[Vt+1 | LBt , MS∗t+1 ] − bid∗t+1 ) = ρPr(MS∗ )(zt − E[δt | LB∗t ])
(33)
Thus, the transaction cost of a limit buy order is equal to its price impact, E[δt | LB∗t ].
An investor’s contribution to the price impact of a dark order follows from inputting
the simplification of askDark∗ that follows from (12) into the liquidity provider’s zero profit
condition (6). Then, I can rewrite the investor’s dark order payoff (4) as,
πDB = Pr(DL∗ )(vt + zt − vt − (1 − 2λ) × E[δt | MB∗t ]) = Pr(DL∗ )(zt − E[δt | DB∗t ])
(34)
which implies that the dark order (post-trade) price impact is equal to its transaction costs.
Now, I show that, in a stationary, symmetric equilibrium, it must be the case that any
investor who submits a buy order has zt ≥ 0, and symmetrically for sell orders. Time
subscripts are dropped, as I focus on stationary equilibria. Let γI denote the fill rate of
order type, I, and let pI denote order type I’s price impact. Further, let investor t have a
valuation equal to zt ≥ 0. For any order type I, there is a buy and a sell option, IB, and IS,
36
respectively. Then, for any investor zt , a buy order of type I is preferred to a sell order of
type I if and only if:
γIB × (zt − pIB) ≥ γIS × (zt − pIS )
(35)
In any symmetric equilibrium, γIB = γIS , and pIB = −pIS . Then, equation (35) becomes
zt ≥ 0, implying that no investor with zt ≥ 0 would prefer IS to IB.
It follows, then, that for any buy order type IB to be used in equilibrium, the price
impact pIB must be positive. To see this, suppose instead that pIB < 0. It must be then, by
symmetry, that pIS > 0. Now, because pIB describes the average informativeness of investors
who submit orders of type IB in equilibrium, it must be the case that some investor with
zt < 0 submits orders of type IB. But from the previous argument, any investor with zt < 0
must prefer IS to IB. A contradiction. Thus, in any equilibrium, buy orders that are used by
some investors must have a positive price impact, pIB > 0, and symmetrically for sell orders.
Lastly, I show that investors who use buy orders of type I do not prefer any other sell
order type J. Consider two order types, I and J. Again, symmetry allows us to consider
a buy order of type I, and a sell order of type J, with the reverse following analogously.
That investors with zt ≥ 0 will not use any sell order type, in equilibrium, follows from the
argument above. Suppose that an investor zt ≥ 0 prefers a sell order of type JS to a buy
order, IS. By the argument above, this investor must prefer order type JB to JS. Note,
finally, that we have only shown that investors with zt ≥ 0 do not submit sell orders in
equilibrium. Because pIB > 0, any investor with zt ∈ (0, pIB) will prefer to abstain from
trading, or prefer another buy order type J, with pJB < pIB .
The argument for investors not using buy orders if zj ≤ 0 follows by symmetry.
Proof (Lemma 2).
Let γI denote the fill rate of order type, I, and let pI denote order
type I’s price impact. In this proof, the buy and sell notation is dropped, as I focus on the
decisions between buy order types. The decisions for sell order types proceeds analogously.
I will show that in any symmetric, stationary equilibrium where all order types are used,
investors use a threshold strategy such that an investor with valuation zt ≥ z J > z K prefers
37
order type J to order type K. Moreover, if γJ ≥ γK , then z J ≥ z K for all pJ , pK .
Suppose there is an order type J such that, in equilibrium, γJ ≥ γK and pJ < pK .
Suppose order type K be used by some investor, t, in equilibrium (investor t earns non-zero
profits). Then, the profit of investor t from using order type K is,
t
πK
= γK × (zt − pK ) < γJ × (zt − pJ ) = πJt
(36)
But we supposed that investor t chooses order type K in equilibrium; a contradiction. Thus,
for an order to be used in equilibrium, there must be no order that has both, i) a higher fill
rate, and, ii) a lower price impact.
Thus, we must have that for three order types to be used in equilibrium, we must have
that γI ≥ γJ ≥ γK ≥ 0, it must be the case that pI ≥ pJ ≥ pK ≥ 0. By corollary,
γ I pI ≥ γ J pJ ≥ γ K pK ≥ 0
(37)
Recall that the profit function of order type I for some investor t, takes the form,
πI = γI × (zt − pI )
(38)
The profit function is linear in zt ; moreover, the order with the highest fill rate also has the
lowest intercept, −γI pI . The linearity of πI in zt implies that since γI ≥ γJ ≥ γK ≥ 0, if
πI crosses πJ at some zT , it remains above πJ for all zt > zT . Similarly for πK . Moreover,
because γpI ≥ γJ pJ ≥ γK pK , for order type I to be used in equilibrium, πI must cross both
πJ and πK . Thus, max {zJ , zK } ≤ zI ≤ 1.
Then, the relation in (37) implies that for I to be used in equilibrium, it must cross πJ
and πK at some zI < 1. Likewise, since γJ ≥ γK ≥ 0, for J to be used in equilibrium,
πJ must cross πK , before πI crosses πJ . Thus, zJ ≤ zI . Lastly, πK crosses the no-trade
threshold, πN T = 0 before πJ and πI , but at a positive zK = γK pK . Hence, zK > 0.
Proof (Lemma 3). I prove Lemma 3 in four steps: step 1 shows that, in an equilibrium
where investors use all order types, given that thresholds satisfy 0 ≤ z L∗ ≤ z D∗ ≤ z M ∗ ≤ 1,
then it must be that the thresholds also satisfy 0 ≤ z L∗ ≤ zBL ≤ z D∗ ≤ zBM ≤ z M ∗ ≤ 1.
38
Then, in step 2, I show that these valuation thresholds can only be an equilibrium for λ ≥ λ̂.
Steps 3 and 4 follow similarly, showing that for λ ≤ λ̂, equilibrium valuation thresholds must
satisfy 0 ≤ z D∗ ≤ zBL ≤ z L∗ ≤ zBM ≤ z M ∗ ≤ 1.
In what follows, I make the following simplifying notational substitution.
E[δ | zI ≤ δ ≤ zJ ] ≡ f (zI , zJ ) =
µ(zI + zJ )
2
(39)
From (39), it is immediate that f (zI , zJ ) is increasing in both zI and zJ .
Step 1. Using the simplification in (39), I rewrite the indifference conditions given by
equations (19)-(22) in the following way:
z M ∗ − f (z M ∗ , 1) − Pr(DL) × (z M ∗ − f (z D∗ , z M ∗ )) = 0
(40)
Pr(DL∗ ) × (z D∗ − f (z D∗ , z M ∗ )) − Pr(MS∗ ) × (z D∗ − f (z L∗ , z D∗ )) = 0
(41)
z L∗ − f (z L∗ , z D∗ ) = 0
(42)
Let z M ∗ < zBM and z D∗ < zBL . Then, Pr(DL∗ ) > Pr(MS∗ ) > Pr(MSB ), and f (z D∗ , z M ∗ ) <
f (zBL , zBM ). Moreover,
Pr(DL∗ ) × (z M ∗ − f (z D∗ , z M ∗ )) > Pr(MSB ) × (z M ∗ − f (zBL , zBM ))
(43)
This implies that any z M ∗ that solves (40) satisfies z M ∗ > zBM , ⇒⇐. Moreover, f (z L∗ , z D∗ ) ≤
f (zBL , zBM ) implies that to satisfy πM (z M ∗ ) ≥ πL (z M ∗ ), z M ∗ ≥ zBM .
Thus, z M ∗ ≥ zBM . Now, let z D∗ > zBM . If we look for an equilibrium where z M ∗ = zBM , we
can rearrange (40)-(42) to form the following equation:
zBM − f (zBM , 1) − Pr(MSB ) × (z D∗ − f (z L∗ , z D∗ ))
(zBM − f (z D∗ , zBM ))
=0
(z D∗ − f (z D∗ , zBM ))
(44)
By inspection, z M ∗ = z D∗ = zBM and z L∗ = zBL form an equilibrium (the benchmark equilibrium). Using this fact, I examine what properties an equilibrium must have if z M ∗ = zBM + ǫ,
39
where ǫ > 0. Inputting this into equation (40), we have that:
µ
zBM − f (zBM , 1) − Pr(DLB ) × (zBM − f (z D∗ , zBM )) + (1 − Pr(DLB )) 1 −
ǫ>0
2
(45)
Thus, in equilibrium, either i) z D∗ < zBM , or ii) Pr(DL∗ ) must be larger than in the case
where z M ∗ = zBM . If the answer is i), then we are done. Suppose z D∗ ≥ zBM , and instead,
that Pr(DL∗ ) = Pr(DLB ) + ∆, where ∆ > 0, such that (45) equals zero. Then, inputting z M ∗
and Pr(DL∗ ) into equation (41) yields expression h1 :
ǫ
ǫµ − Pr(MSB ) −
× (z D∗ − f (z L∗ , z D∗ ))
h1 = −(Pr(DLB ) + ∆) × z D∗ − f (z D∗ , zBM ) −
2
2
(46)
Equation (46) contains the expression (41) when the system is solved for z M ∗ = z D∗ = zBM
and z L∗ = zBL , and thus is equal to zero. Expression (46) simplifies to:
h1 = −Pr(DLB ) ×
µǫ
ǫµ ǫ +
× (z D∗ − f (z L∗ , z D∗ )) (47)
+ ∆ × z D∗ − f (z D∗ , zBM ) −
2
2
2
Because ǫ is bounded above by 1 − zBM , and z D∗ is bounded below by zBM (h1 is increasing in
z D∗ , for a fixed Pr(DL∗ )), evaluating the second term at these bounds shows that the term
is greater than zero. Then, we can sign h1 by comparing the first and third terms:
ǫ
µ(zBM + z L∗ )
µ
M
h1 > × zB −
− Pr(DLB ) ×
2
2
2
Recall that z L∗ =
M −f (z M ,z L )
Pr(MSB )(zB
B
B
.
M ,z M )
M
zB −f (zB
B
µz D∗
2−µ
(48)
from (42). We also require the benchmark value, Pr(DLB ) =
Using these two equations, expression (48) simplifies to:
µ(1 − zBM )
ǫ
M
h1 > × zB −
2
(2(1 − µ)
By rearranging expression (49), h1 > 0 ⇐⇒ zBM ≥
µ
,
2−µ
(49)
which is a necessary condition of
any equilibrium where market orders are used. Thus, if z M ∗ > zBM , then h1 > 0. This implies
that dark orders are preferred to limit orders if z D∗ = zBM , and thus any z D∗ that solves (41)
when z M ∗ > zBM , must be such that z D∗ < zBM . Hence, z M ∗ ≥ zBM ≥ z D∗ ≥ 0, in equilibrium.
40
Finally, because z D∗ ≤ zBM , by condition (42), z L∗ ≤ zBL .
Step 2. To prove that in any equilibrium with thresholds, 0 ≤ z L∗ ≤ z D∗ ≤ z M ∗ ≤ 1,
can only exist for λ < λ̂, note that the following must hold in equilibrium:
E[δ | DB∗ ] = (1 − 2λ) × E[δ | MB∗ ]
⇐⇒ λ =
1 − z D∗
1 − zBL
<
= λ̂
2(1 + z M ∗ )
2(1 + zBM )
(50)
(51)
which follows from the result in step 1 that z D∗ > zBL , and z M ∗ > zBM .
Step 3. Now I show that given that equilibrium thresholds satisfy 0 ≤ z D∗ ≤ z L∗ ≤
z M ∗ ≤ 1, in an equilibrium where all order types are used, equilibrium thresholds must also
satisfy 0 ≤ z D∗ ≤ zBL ≤ z L∗ ≤ zBM ≤ z M ∗ ≤ 1.
To show 0 ≤ z D∗ ≤ z L∗ ≤ z M ∗ ≤ 1 ⇒ 0 ≤ z D∗ ≤ zBL ≤ z L∗ ≤ z M ∗ ≤ zBM ≤ 1, the
equilibrium conditions must satisfy the following, via (23)-(26);
z M ∗ − f (z M ∗ , 1) − Pr(MS∗ ) × (z M ∗ − f (z L , z M ∗ )) = 0
(52)
Pr(MS∗ ) × (z L∗ − f (z L∗ , z M ∗ )) − Pr(DL∗ ) × (z L∗ − f (z D∗ , z L∗ )) = 0
(53)
z D∗ − f (z L∗ , z D∗ ) = 0
(54)
Let z M ∗ > zBM and z L∗ < zBL . Then it must be true that Pr(MS∗ ) < Pr(MSB ), and that
z L∗ >
µz M ∗
2−µ
>
M
µzB
2−µ
> zBL , a contradiction. Instead, suppose then that z L∗ > zBL . Then,
πL (z M ∗ ) < πL (zBM ), implying that the z M ∗ that solves (52) must be such that z M ∗ < zBM , a
contradiction. Hence, z M ∗ ≤ zBM .
Then, given z M ∗ ≤ zBM , let z L∗ < zBL . Because f (z L∗ , z M ∗ ) < f (zBL , zBM ), πL (z M ∗ ) >
πL (zBM ) ⇒ z M ∗ > zBM , a contradiction. Thus, z L∗ ≥ zBL . Finally, for any zBM ≥ z M ∗ ≥ z L∗ ≥
zBL , in equilibrium it must be that z D∗ ≤ zBL , which obtains from solving (54) for z D∗ :
z D∗ =
µz L∗
µzBM
≤
= zBL
2−µ
2−µ
(55)
Step 4. To prove that thresholds 0 ≤ z D∗ ≤ z L∗ ≤ z M ∗ ≤ 1 can only form an equilibrium
41
when λ > λ̂, rearrange E[δ | DB∗ ] = (1 − 2λ) × E[δ | MB∗ ] to isolate for λ:
λ=
1 + z M ∗ − z L∗ − z D∗
1 − z D∗
1 − zBL
>
>
= λ̂
2(1 + z M ∗ )
2(1 + z M ∗ )
2(1 + zBM )
(56)
which we arrive at by the fact in step 3 that z D∗ < zBL .
A.2
Proofs: Theorems and Propositions
This section contains the proofs of the existence theorems and propositions. I conduct the
existence proofs in similar steps. I select a single threshold to be the reference threshold,
showing that all other thresholds exist and are unique for all values of the reference threshold,
making use of the intermediate value theorem, Lemma (2), and the implicit function theorem.
Then, using the intermediate value theorem, I show that there exists a value of the reference
threshold that such that the equilibrium exists. As a remark on notation, I drop the time
subscripts in all proofs, because of the stationarity condition.
Proof (Theorem 1). For this proof, I proceed in 2 steps. In step 1, I conjecture that the
professional liquidity provider chooses Pr(DL∗ ) = 0, and show that the model reduces to the
(visible market only) equilibrium of Brolley and Malinova (2014). In step 2, I argue that no
agent has an incentive to deviate from the conjectured equilibrium behavior.
Step 1: Visible Market Only Equilibrium
Conjecture that Pr(DL∗ ) = 0. Then, Lemma 2 guarantees us that the thresholds of the
equilibrium must be of the form 0 ≤ z D∗ ≤ z L∗ ≤ z M ∗ ≤ 1. Because no limit orders are
supplied to the dark market, any investor that sends an order to the dark market earns
a payoff of zero. Moreover, the payoff to the liquidity provider is also zero (satisfying the
competitive liquidity provision assumption).
Hence, all investors are indifferent between submitting a dark order, and not trading.
In market with a visible market only (i.e., Brolley and Malinova (2014)), zBL marks the
threshold at which an investor of valuation zBL would be indifferent to submitting a visible
42
limit order, and not trading. Thus, in a model where dark trading is available, but is
equivalent to not trading, it must be the case that threshold z L∗ = zBL = z D∗ . Then, given
0 ≤ z D∗ = z L∗ = zBL ≤ z M ∗ ≤ 1, it must be the case that z M ∗ = zBM . Thus, in an equilibrium
dark trading yields a zero payoff, the model reduces to the “visible market only” equilibrium
of Brolley and Malinova (2014).
To prove the existence more explicitly, I summarize the above discussion into the two
benchmark equilibrium payoff conditions: i) investors with valuation zBM are indifferent between market orders and limit orders, and ii) investors with valuation zBL are indifferent
between limit orders and dark orders (or not trading). These two conditions are:
ZBl : zBL −
µzBM
=0
2−µ
(57)
ZBm : zBM − E[δ | MB∗ ] − ρPr(MS∗ ) × (zBM −
µzBM
)=0
2−µ
(58)
By inspection, condition (57) gives us that zBL ∈ (0, zBM ) exists and is unique for all zBM ∈ [0, 1].
I now show that ZBm must crosses zero from below at least once.
ZBm (0) = 0 − E[δ | MB∗ ] − ρPr(MS∗ ) × (0 − 0) < 0
(59)
ZBm (1) = 1 − E[δ | MB∗ ] − 0 × (1 − z L∗ ) > 0
(60)
Because Z m is continuous in zBM , the intermediate value theorem implies the existence of a
zBM ∈ [0, 1] that crosses zero from below for all zBL (zBM ). This holds for zBL by the implicit
function theorem, as ZBl is continuous for all zBM . Then, taking the first derivative of ZBm
with respect to zBM ,
µ
µ ρ
∂ZBm
∗
∗
M
= 1 − + × (zB − E[δ | LB ]) − ρPr(MS ) × 1 −
∂zBM
2 2
2−µ
2(1 − µ) + µ2 + 4ρ(1 − µ)zBM + 2(1 − µ) · (1 − ρ)
∂ZBm
>0
=
⇐⇒
∂zBM
2(2 − µ)
Therefore, a unique equilibrium exists in threshold strategies.
43
(61)
(62)
Step 2: Optimality of Investor and Liquidity Provider Strategies
Since the professional liquidity provider leaves the dark market empty, any order submitted
to the dark earns zero profits. Then, investors who submit visible limit orders or market
orders (z ∈ [z L∗ , 1]) do not have an incentive to deviate from their strategy. Moreover,
investors who abstain from trading, z ∈ [0, z D∗ ), also earn zero profits, and thus will not
deviate from their strategy.
An investor with z = z D∗ = zBL also has no incentive to deviate from the conjectured
strategy, as they are indifferent to visible limit orders, dark orders and not trading, earning
zero profits from any of the three actions. Then, because the group of investors that sends
orders to the dark is of measure zero, any limit orders sent to the dark have an execution
probability of zero. Hence, the professional liquidity provider does not benefit from changing
his liquidity provision strategy in the dark (he earns zero profit in either case).
The conjectured behavior above constitutes an equilibrium, as all investors behave consistently with Theorem 1, except for investors with z = z D∗ , who are of measure zero.
Proof (Proposition 1). Conjecture that an equilibrium exists where the strategy profile
of investors and the liquidity provider is given by,
(z D∗ , z L∗ , z M ∗ , Pr(DL∗ )) = (z D∗ ≥ zBL , zBL , zBM , ρPr(MSB ))
(63)
Where ’B’ denotes the value the threshold takes in the benchmark equilibrium. Further
conjecture that λ = λ̂, the parameter value of λ that solves:
(1 − 2λ̂) × E[δt | MB∗t , z M ∗ = zBM ] = E[δt | LB∗t , z M ∗ = zBM , z L∗ = zBL ]
(64)
To show that such a trade-at rule exists, rewrite equation (64) as the function h(λ),
h(λ) = (1 − 2λ̂) × E[δt | MB∗t , z M ∗ = zBM ] − E[δt | LB∗t , z M ∗ = zBM , z L∗ = zBL ]
= (1 − 2λ) ×
(1 + zBM )µ (zBM + zBL )µ
−
2
2
44
(65)
(66)
Now evaluate h(λ) at λ = {0, 1/2}. We then have,
(1 + zBM )µ (zBM + zBL )µ
−
>0
2
2
(z M + zBL )µ
h(1/2) = − B
<0
2
h(0) =
(67)
(68)
which implies, by the intermediate value theorem, that there exists a λ̂ such that (65) holds
given zBL and zBM . Moreover, hλ (λ) = −(1 + zBM )µ < 0 implies that λ̂ is unique.
Then, given λ̂ as in (65), the level of liquidity provision to the dark pool must be equal to
Pr(DL∗ ) ≤ ρPr(MS∗ ). To see this, note that in the benchmark equilibrium (no dark orders),
(z L∗ , z M ∗ , Pr(MS∗ )) = (zBL , zBM , Pr(MSB )). Without dark orders, Pr(MSB ) is the equilibrium
fill rate for limit orders with with transaction costs equal to E[δ | LB∗ ] =
M +z L )µ
(zB
B
.
2
Then, if
dark orders have identical transaction costs, as in equation (64), any Pr(DL∗ ) > ρPr(MSB )
means that all investors with zt ∈ [zBL , zBM ] strictly prefer dark orders, and investors with
zt = zBM + ǫ for some ǫ > 0 prefer dark orders to market orders. Thus, z M ∗ > zBM . A
contradiction.
I now show that Pr(DL∗ ) ≤ ρPr(MSB ) can sustain the equilibrium initially posited.
Because (1 − 2λ̂) × E[δ | MB∗ ] = E[δ | LB∗ ] and Pr(DL∗ ) ≤ ρPr(MS∗ ) implies that πL = πD for
any zt , and since investors choose visible orders over dark orders when payoffs are identical
(by assumption), no investor has an incentive to deviate from the benchmark equilibrium
strategy profile.
Moreover, as no investors submit dark orders on-the-equilibrium path, the liquidity
provider’s payoff for all dark limit orders is zero given the appropriate set of off-the-equilibrium
path beliefs. I assume that these beliefs are such that E[δ | DB∗ ] = (1 − 2λ̂) × E[δt |
MB∗t , z M ∗ = zBM ], which satisfies the liquidity provider’s zero profit condition. Thus, the liquidity provider has no incentive to deviate from the equilibrium liquidity provision strategy.
Hence, (z D∗ , z L∗ , z M ∗ , Pr(DL∗ )) = (z D∗ ≤ zBL , zBL , zBM , ρPr(MSB )) is an equilibrium strategy
profile for λ = λ̂.
To prove that no other equilibrium exists in which λ satisfies equation (64),
45
Lastly, 0 ≤ λ̂ < 1/2 if and only if,
1 − zBM
(1 − zBL )
1
≤
≤
M
4
2
2(1 + zB )
(69)
which, because zBL < zBM , holds trivially.
Proof (Theorem 2). Similar to the proof of Theorem 1, I proceed by showing the existence
and uniqueness of z L∗ , z D∗ and Pr(DL∗ ) for all z M , and then show that a unique z M ∗ exists
for all λ ∈ (0, λ̂). In doing so, I reference the functions from (19)-(22) (reproduced below),
γ
Zλ<
≡ z M − E[δ | MB∗ ] − Pr(DL) × z M − (1 − 2λ) × E[δ | MB∗ ]
λ̂
∗
∗
∗
m
D
D
Zλ<
≡
Pr(DL)
×
z
−
(1
−
2λ)
×
E[δ
|
MB
]
−
ρ
·
Pr(MS
)
×
z
−
E[δ
|
LB
]
λ̂
(70)
(71)
l
Zλ<
≡ z L − E[δ | LB∗ ]
λ̂
(72)
d
Zλ<
≡ (1 − 2λ) × E[δ | MB∗ ] − E[δ | DB∗ ]
λ̂
(73)
Preliminaries.
For an equilibrium to exist where thresholds satisfy 0 < z L < z D ≤ z M < 1, it must be
true that (1 − 2λ) × E[δ | MB∗ ] > E[δ | LB∗ ], by Lemma 2. Moreover, Lemma 3 gives
us the “only if”, as any equilibrium where 0 < z L < z D ≤ z M < 1 can only exist for
λ ∈ (0, λ̂). We can also apply Lemma 3 to restrict the ranges of the threshold values to
0 < z L ≤ zBL z D∗ ≤ zBM ≤ z M ≤ 1.
Step 1: Existence and Uniqueness of z L∗ (z M )
I now show that there exists a unique z L ∈ [0, z D ] that solves (73) for all z M ∈ [zBM , 1] and
z D ∈ [zBL , zBM ]. Simple rearranging yields, z L∗ =
µ·z D
.
2−µ
Thus, a unique z L∗ exists.
Step 2: Existence and Uniqueness of λ∗ (z M )
I now show that there exists a unique λ ∈ (0, λ̂) that solves (72) for all z D ∈ [zBL , zBM ] and
z M ∈ [zBM , 1]. I achieve the bounding on z D and z M through Lemma 3. Let ǫ > 0 be
46
arbitrarily close to zero. Simplifying (72),
µ
× ((1 − 2ǫ) · (1 + z M ) − z M − z D )
2
µ
≈ × ((1 + z M ) − z M − z D ) > 0
2 zBL + zBM
µ
M
M
D
λ
· (1 + z ) − z − z
Zλ>λ̂ (λ = λ̂) = ×
2
1 + zBM
λ
Zλ>
(λ = ǫ) =
λ̂
D
L
M
M
M D
µ −(z − zB ) − (z − zB ) − zB (z −
= ×
2
1 + zBM
where (75) is negative by the fact that πD > 0 only if z D∗ ≥
µz M
.
2−µ
(74)
µz M
)
2−µ
<0
(75)
Hence, λ∗ exists.
λ
To show that λ∗ is unique, differentiate Zλ>
with respect to λ:
λ̂
λ
∂Zλ>
λ̂
∂λ
= −µ · (1 + z M ) < 0
(76)
Thus, a unique λ ∈ (0, λ̂) exists for all z D ∈ [zBL , zBM ] and z M ∈ [zBM , 1].
Step 3: Existence and Uniqueness of Pr(DL∗ )(z M )
I show that there exists a unique Pr(DL∗ ) ∈ [ρPr(MS∗ ), 1] that solves (70) for all z M ∈ [zBM , 1].
γ
γ
First, evaluating Zλ<
at the lower bound Pr(DL∗ ) = ρPr(MS∗ ), we have that Zλ<
> 0.
λ̂
λ̂
γ
Zλ<
= z M − E[δ | MB∗ ] − ρPr(MS∗ ) × (z M − (1 − 2λ)E[δ | MB∗ ])
λ̂
µ(zBM + zBL )
∗
M
M
> z − E[δ | MB ] − ρPr(MSB ) × z −
2
M
µ(zB + zBL )
∗
M
M
=0
> zB − E[δ | MB ] − ρPr(MSB ) × zB −
2
(77)
(78)
(79)
Where the equality at (79) follows by the fact that this is an indifference condition for the
benchmark equilibrium, and hence, the expression is zero when evaluated at (zBL , zBM ).
Then, evaluating (70) at the upper bound, Pr(DL∗ ) = 1:
γ
Zλ<
λ̂
=z
M
µ(1 + z M )
µ(1 + z M )
M
−
<0
− 1 × z − (1 − 2λ) ×
2
2
47
(80)
Thus, by the intermediate value theorem, Pr(DL∗ ) exists. Now, substitute the condition that
(1 − 2λ) × E[δ | MB∗ ] =
µ(z D +z M )
2
γ
by Pr(DL∗ ), we have:
into (70). Then, differentiating Zλ<
λ̂
γ
∂Zλ<
λ̂
∂Pr(DL∗ )
=− z
µ(z D + z M )
−
2
M
<0
(81)
Thus, Pr(DL∗ ) is unique for all z M ∈ [zBM , 1] and z D ∈ [zBL , zBM ].
Step 4: Existence and Uniqueness of z D∗ (z M )
Lastly, I show that there exists a z D∗ ∈ [zBL , zBM ] for all z M ∈ [zBM , 1] that solves (71). First,
evaluate (71) at z D∗ = zBL , and substitute (1 − 2λ) × E[δ | MB∗ ] = E[δ | DB∗ ]:
d
Zλ<
(z D∗
λ̂
=
zBL )
µ(zBL + z M )
µ
L
= Pr(DL ) × zB −
− ρPr(MSB ) × 1 −
zL
2
2−µ B
∗
Using the fact zBM < z M , we have that zBL =
M
µzB
2−µ
≤
µz M
.
2−µ
(82)
Thus, expression (82) is negative.
Now, I evaluate z D∗ = zBM :
∗
d
D∗
M
Zλ<
(z
=
z
)
=
Pr(DL
)
×
zBM −
B
λ̂
M +z M )
µ(zB
2
− Pr(MS∗ ) × (1 −
µ
)z M
2−µ B
(83)
m
We know by step 2 of the proof of Lemma 3 that Zλ<
is zero when z M ∗ = z D∗ = zBM and
λ̂
m
z L∗ = zBL . Lemma 3 shows that when λ ∈ (0, λ̂), Zλ<
is zero for z M ∗ > zBM only if z D∗ < zBM .
λ̂
d
Then, because (83) is increasing in z M (from Lemma 3), it must be that Zλ<
(z D∗ zBM ) > 0.
λ̂
Thus, z D∗ exists, by the intermediate value theorem.
d
Finally, substitute Pr(DL∗ ) from step 3 into Zλ<
, and differentiate by z D∗ :
λ̂
d
∂Zλ<
λ̂
∂z D∗
=
µ(z M −E[δ|MB∗ ])
2(z M −E[δ|DB∗ ])
µ
× z D∗ − E[δ | DB∗ ] + Pr(DL∗ ) × (1 − µ/2) − Pr(MS∗ ) × (1 − 2−µ
)>0
(84)
Thus, z
D∗
is unique for all z
M
∈
[zBM , 1].
Proof (Theorem 3). Similar to the proof of Theorem 2, I proceed to show the existence
and uniqueness of z L∗ , z D∗ and Pr(DL∗ ) for all z M , and then show that unique z M ∗ exists,
48
for all λ ∈ (λ̂, λ). In doing so, I reference the functions from (23)-(26) (reproduced below),
m
Zλ>
≡ z M − E[δ | MB∗ ] − ρ · Pr(MS∗ ) × z M − E[δ | LB∗ ]
λ̂
γ
≡ Pr(DL) × z L − (1 − 2λ) × E[δ | MB∗ ] − ρ · Pr(MS∗ ) × z L − E[δ | LB∗ ]
Zλ>
λ̂
(85)
(86)
d
≡ z D − (1 − 2λ) × E[δ | MB∗ ]
Zλ>
λ̂
(87)
l
Zλ>
≡ (1 − 2λ) × E[δ | MB∗ ] − E[δ | DB∗ ]
λ̂
(88)
Preliminaries.
For an equilibrium to exist where threshold values satisfy 0 < z D < z L ≤ z M < 1, it must
be true that (1 − 2λ) × E[δ | MB∗ ] < E[δ | LB∗ ], by Lemma 2. As in the proof of Theorem
2, the “only if” follows from Lemma 3, as any equilibrium where 0 ≤ z D ≤ z L ≤ z M < 1
can only exist for λ ∈ (λ̂, λ). Lemma 3 restricts the ranges of the threshold values to
0 ≤ z D < z L ≤ z M ≤ zBM < 1.
Step 1: Existence and Uniqueness of Pr(DL∗ )(z L )
To show that there exists a unique Pr(DL∗ ) ∈ [0, Pr(MS∗ )] that solves (86) for all z L ∈ [zBL , zBM ]
and z M ∈ [z L , zBM ], rearrange (86):
Pr(DL∗ ) =
ρPr(MS∗ ) × (z L − E[δ | LB∗ ])
< ρPr(MS∗ )
z L − (1 − 2λ)E[δ | MB∗ ]
(89)
Thus, a unique Pr(DL∗ ) < ρPr(MS∗ ) exists, as the denominator of (89) is always positive,
as can be inferred from (94). Moreover, (1 − 2λ)E[δ | MB∗ ] < E[δ | LB∗ ] implies that the
inequality holds.
Step 2: Existence and Uniqueness of z D∗ (z L )
I now show that there exists a unique z D ∈ [0, z L ] that solves (87) for all z L ∈ [zBL , zBM ], and
z M ∈ [z L , zBM ]. Evaluate (87) at the end points z D = 0 and z D = zBL :
49
d
Zλ>
(z D = 0) = 0 − (1 − 2λ)E[δ | MB∗ ] < 0
λ̂
(90)
d
Zλ>
(z D = zBL ) = zBL − (1 − 2λ)E[δ | MB∗ ]
λ̂
To see that expression (91) is positive for all z L , consider λ = λ̂ =
(91)
L
1−zB
M),
2(1+zB
which maximizes
the negative term of the expression. Inputting λ̂ into (91), we have:
zBM + zBL µ
· × (1 + z M ∗ )
1 + zBM 2
µ(zBM + zBL )
L
> zB −
=0
2
d
Zλ>
(z D = zBL ) = zBL −
λ̂
(92)
(93)
Because z M ∗ < zBM by Lemma 3. Hence, z D∗ exists. Then, by inspection, for any (z M , λ)
pair, z D∗ is unique.
Step 3: Existence and Uniqueness of λ∗ (z L )
To show there exists a unique λ ∈ [λ̂, λ] that solves (88) for all z L ∈ [zBL , zBM ], and z M ∈
[z L , zBM ], I evaluate (88) at λ̂ and λ:.
zBM + zBL µ
µ(z L + z D∗ )
M∗
(1
+
z
)
−
>0
·
2
1 + zBM 2
µ(z D∗ + z L )
l
λ)
=
0
−
Zλ>
(λ
=
<0
λ̂
2
l
Zλ>
(λ = λ̂) =
λ̂
(94)
(95)
Hence, λ∗ exists. To show that it is unique, we know that z D∗ = (1 − 2λ)E[δ | MB∗ ].
Substituting this into (88), and then taking the derivative with respect to λ, we get:
l
∂Zλ>
λ̂
∂λ
= −2E[δ | MB∗ ](1 − µ2 ) < 0
Thus, λ∗ is unique for all z L ∈ [zBL , zBM ] and z M ∈ [z L , zBM ].
50
(96)
Step 4: Existence and Uniqueness of z M ∗ (z L )
To show that there exists a unique z M ∗ ∈ [z L , zBM ] that solves (70) for all z L ∈ [zBL , zBM ], I
m
begin by evaluating Zλ>
(z M ) at the endpoints.
λ̂
m
M
L
Zλ>
(z
=
z
)
=
zL −
λ̂
m
Zλ>
(z M = zBM ) = zBM
λ̂
µ(1+z L )
2
− ρPr(MS∗ )(z L − E[δ | LB∗ ])
µ(1 + zBM )
(1 − zBM )
µ(zBM + z L )
M
−
−ρ
× zB −
2
2
2
(97)
(98)
Equation (98) is positive by the following argument. We know that for z L = zBL , that
m
Zλ>
(z M = zBM ) is zero, as it satisfies the equilibrium indifference condition for z M ∗ in the
λ̂
equilibrium described by Theorem 1. Then, since z L > zBL by Lemma 3, the second (negative)
m
term is smaller, implying that Zλ>
(z M = zBM ) > 0.
λ̂
m
m
For Equation (97), if z L = zBL , then Zλ>
< 0 because Zλ>
(z M = zBM , z L = zBL ) = 0, and
λ̂
λ̂
∂Z m
λ>λ̂
∂z M
> 0. Then, coupled with the fact that (98) is positive, we have that there exists a
m
zˆM = z L such that Zλ>
= 0. Thus, for all z M < zˆM , equation (97) is negative. Therefore,
λ̂
m
z M ∗ < zˆM exists. Then, taking the first derivative of Zλ>
:
λ̂
m
∂Zλ>
λ̂
∂z M
µ ρ
µ
∗
∗
M
>0
= 1 − + × (z − E[δ | LB ]) − ρPr(MS ) × 1 −
2 2
2
(99)
which implies that z M ∗ < zˆM is unique.
M , I show that z M ∗ = z L is the unique value. Suppose that z M ∗ = z L . z D∗
For z M > zc
and λ∗ follow as in Steps 2 and 3. To show that a unique Pr(DL∗ ) exists, we can combine
equilibrium conditions (85) and (86), and substitute E[δ | DB∗ ] =
Z(z M = z L ) : z L −
µ(1+z L )
2
− Pr(DL∗ ) z L −
µz L
2−µ
µz L
2−µ
to achieve:
=0
(100)
M , z M ].
It is immediate that a unique Pr(DL∗ ) ∈ [0, ρPr(MS∗ )] exists to solve (100), ∀z L ∈ [zc
B
M , a unique equilibrium exists where z M ∗ = z L∗ . Combining these two
hence, if z M ∗ < zc
proofs, I conclude that z M ∗ exists and is unique for all z L ∈ [zBL , zBM ].
Proof (Proposition 2). The exchange volume is given by 2Pr(MB∗ ), which is a decreasing
51
function of z M ∗ . Thus, by Lemma 3, if λ ∈ (0, λ̂), then z M ∗ ≥ zBM , and 2Pr(MB∗ ) <
2Pr(MBB ). Further, if λ ∈ (λ̂, λ), then z M ∗ ≤ zBM , and 2Pr(MB∗ ) > 2Pr(MBB ).
Because market participation is defined as the probability of submitting any order, we
can write it as 1−Pr(NT∗ ). If λ ∈ (0, λ̂), Pr(NT∗ ) = z L∗ −(−z L∗ ) = 2z L∗ , and in equilibrium,
z L∗ < zBL . Thus, for any small trade-at rule, market participation increases. If λ ∈ (λ̂, λ),
Pr(NT∗ ) = z D∗ − (−z D∗ ) = 2z D∗ , and in equilibrium, z D∗ < zBL . Thus, for any large trade-at
rule, market participation increases.
Proof (Proposition 3).
The price impact measure is given by E[δ | MB∗ ], which is
an increasing function of z M ∗ only. Thus, by Lemma 3, if λ ∈ (0, λ̂), then z M ∗ ≥ zBM ,
and E[δ | MB∗ ] is always greater than in the benchmark case. Further, if λ ∈ (λ̂, λ), then
z M ∗ ≤ zBM , and E[δ | MB∗ ] is always lower than in the benchmark case.
A.3
Out-of-Equilibrium Limit Orders and Beliefs
The equilibrium concept I employ in this paper perfect Bayesian equilibrium. On-theequilibrium-path, investors submit limit orders with competitive limit prices. However,
I require an appropriate set of out-of-equilibrium beliefs to ensure that competitive limit
prices strategically dominate any off-equilibrium-path deviations in the limit price. Intuitively, any limit order that is posted at a price worse than the competitive equilibrium price
is strategically dominated by the competitive price, as the professional liquidity provider
reacts to the non-competitive order by undercutting it. For non-competitive limit orders
that undercut the competitive price (i.e., a price inside the competitive spread), however, it
is not immediate that the competitive price strategically dominates this set of prices.
Perfect Bayesian equilibrium prescribes that investors and the professional liquidity
provider update their beliefs by Bayes rule, whenever possible, but does not place any restrictions on the beliefs of market participants when they encounter an out-of-equilibrium
action. To support competitive prices in equilibrium, I assume (similar to Brolley and Malinova (2014)) that if a limit buy order is posted at a price different to the competitive
52
equilibrium bid price bid∗t+1 , then market participants hold the following beliefs regarding
this investor’s knowledge of the period t innovation δt .
c < bid∗ , then market participants assume
If a limit buy order is posted at a price bid
t+1
that this investor followed the equilibrium threshold strategy, but “made a mistake” when
pricing his orders. The professional liquidity provider then updates his expectation about δt
to the equilibrium value and posts a buy limit order at bid∗t+1 . The original investor’s limit
order then executes with zero probability.
c > bid∗ , then market participants believe
If a limit buy order is posted at a price bid
t+1
that this order stems from an investor with a sufficiently high valuation (e.g., zt = 1) and
c accordingly.The new posterior expectation
update their expectations about δt to E[δt | bid]
c The professional liquidity provider is then willing to post a
of Vt equals to pt−1 + E[δt | bid].
c
bid price bid∗∗
t+1 ≤ pt−1 + E[δt | bid] + E[δt+1 | MSt+1 ]. With the out-of-the-equilibrium belief
of δt = 1 and with the bid-ask spread< 1, a limit order with the new price bid∗∗
t+1 outbids
any limit buy order that yields investors positive expected profits.
The beliefs for an out-of-equilibrium sell order are symmetric. These out-of-equilibrium
beliefs ensure that no investor deviates from his equilibrium strategy. I emphasize that these
beliefs and actions do not materialize in equilibrium. Instead, they can be thought of as a
“threat” to ensure that investors do not deviate from their prescribed equilibrium strategies.
53
Figure 5: Entry and Order Submission Timeline
This figure illustrates the timing of events upon the arrival of an investor at an arbitrary period, t, until their departure from the market.
Value yt is the private valuation of the period t investor and δt is the innovation to the security’s fundamental value in period t.
Professional liquidity providers post limit
orders to empty side(s) of the visible book,
and limit orders to the dark market
with probability Pr(DLt−1 )
54
Period t investor
Period t + 1 investor
submits order (if any)
submits order (if any)
Period t + 1 dark market
Period t dark market
Period t investor
enters market,
learns yt and δt
t
trades are reported,
Period t − 1 investor leaves market
(if still present)
Period t − 1 limit orders either
Period t + 1 investor
trades are reported,
enters market,
Period t investor leaves market
learns yt+1 and δt+1
(if still present)
t+1
Period t limit orders either
trade against the period t market order
trade against the period t + 1 market order
or get cancelled
or get cancelled
Figure 6: Equilibrium Thresholds
The panels below depict equilibrium valuation thresholds z M , z L and z D as a function of the trade-at rule (λ < λ̂ on the left, λ > λ̂ on the
right). A vertical dashed line marks λ̂; the horizontal dashed line indicates the visible market only benchmark value. Parameter µ = 0.5.
Results for other values of µ are qualitatively similar.
55
Figure 7: Volume, Market Participation
The panels below depict volume (visible market and total), and market participation as a function of the trade-at rule (λ < λ̂ on the
left, λ > λ̂ on the right). A vertical dashed line marks λ̂; the horizontal dashed line indicates the visible market only benchmark value.
Parameter µ = 0.5. Results for other values of µ are qualitatively similar.
56
Figure 8: Quoted Half-Spread
The panels below depict the quoted half-spread (also price impact) as a function of the trade-at rule (λ < λ̂ on the left, λ > λ̂ on the
right). A vertical dashed line marks λ̂; the horizontal dashed line indicates the visible market only benchmark value. Parameter µ = 0.5.
Results for other values of µ are qualitatively similar.
57
Figure 9: Informational Efficiency
The panels below depict informational efficiency as a function of the trade-at rule (λ < λ̂ on the left, λ > λ̂ on the right). Higher
values than the benchmark are less efficient. A vertical dashed line marks λ̂; the horizontal dashed line indicates the visible market only
benchmark value. Parameter µ = 0.5. Results for other values of µ are qualitatively similar.
58
Figure 10: Total Expected Welfare
The panels below depict total expected welfare as a function of the trade-at rule (λ < λ̂ on the left, λ > λ̂ on the right). A vertical
dashed line marks λ̂; the horizontal dashed line indicates the visible market only benchmark value. Parameter µ = 0.5. Results for other
values of µ are qualitatively similar.
59