Noname manuscript No.
(will be inserted by the editor)Eﬀects of Dark Pools on Financial Markets’Eﬃciency and Price-Discovery Function:
An Investigation by Multi-Agent Simulations
Takanobu Mizuta · Shintaro Kosugi ·Takuya Kusumoto · Wataru Matsumoto ·Kiyoshi Izumi

Received: date / Accepted: date

Abstract In financial stock markets, dark pools, in which order books orquotes are not provided, are becoming widely used. However, increasing theuse of dark pools would raise regulatory concerns as it may ultimately affect the quality of the price-discovery function in the lit markets, which arenormal markets in which all order books are provided to investors. This may
destabilize a market and heighten financial systemic risk.
In this study, we investigated eﬀects of a dark pool on financial markets’
eﬃciency and the price-discovery function by using an artificial market model.
We found that the markets are made more eﬃcient by raising the share of the
trading value amount of the dark pool by a certain level. However, raising theshare above the level makes the market significantly ineﬃcient. This indicatesthat the dark pool has an optimal usage rate for market eﬃciency.
The smart order routing (SOR) is transmitting market orders to the darkpool, and this leads the depth of limit orders to become thicker. The thicker
limit orders absorb market orders, and thus a market price is still stable neara fundamental price. On the other hand, when too many waiting orders are
stored in the dark pool, the orders absorb market orders in the lit market by
SOR and prevent the market price converging to the fundamental price. This
causes the market price to stay very diﬀerent from the fundamental price andmakes the lit market ineﬃcient.
We also discuss mechanisms by which a dark pool makes a market eﬃcient
or ineﬃcient by using a simple equation model. The equations suggest that if
Takanobu Mizuta
SPARX Asset Management Co., Ltd., Tokyo, Japan
E-mail: mizutata@gmail.com
Takuya Kusumoto · Wataru Matsumoto
Nomura Securities Co., Ltd., Tokyo, Japan
Kiyoshi Izumi
School of Engineering, The University of Tokyo, Tokyo, Japan
CREST, JST, Tokyo, Japan

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Takanobu Mizuta et al.

the trading value amount is higher in dark pools than in lit markets, markets
become ineﬃcient. This suggests that when the usage rate of dark pools is low,dark pools rarely destroy the price-discovery function even though a large buysell imbalance occurs. On the other hand, when the usage rate of dark pools
is very high, dark pools very easily destroy the price-discovery function by avery slight buy-sell imbalance. We also compared results of the equations withthose of simulations and found similar tendencies.

1 Introduction
In financial stock markets, dark pools, in which order books or quotes are
not provided, are becoming widely used, especially by institutional investors[SEC(2010)]. In dark pools, investors can trade a large block of stock without market impacts because investors need not show their orders to anyone
else, and reducing market impacts by such investors may make markets more
eﬃcient [Johnson(2010)].
However, increasing the use of dark pools would raise regulatory concerns
as it may ultimately aﬀect the quality of the price-discovery function in thelit markets, which are normal markets in which all order books are provided
to investors. This may destabilize a market and heighten financial systemicrisk [EC(2010)], [Ye(2012)]. Therefore, for example in Europe, regulators are
discussing introducing a volume cap regulation for dark pools, specifically a
8% limit for the trading volume for each stock [Bowley(2014)].
Dark pools are very diﬃcult to discuss by only using results of empiricalstudies. Because so many factors cause price formation in actual markets, anempirical study cannot isolate the pure contribution of existing new types ofmarkets such as dark pools to price formation. Furthermore, empirical studies
cannot investigate situations that have never occurred before in real financial
markets.
We usually discuss whether new types of markets should be spread or noton the basis of their eﬀects on price formation. An artificial market, which isa kind of a multi-agent simulation, can isolate the pure contribution of these
new types of markets to the price formation and can treat situations that havenever occurred [LeBaron(2006)], [Chen et al(2012)], [Cristelli(2014)]. These arestrong points of the artificial market simulation study.
Many studies have investigated the eﬀects of several new regulations andeﬀects of new types of markets by using artificial market simulations, forexample, investigating eﬀects of short selling regulations [Yagi et al(2010)],transaction taxes [Westerhoﬀ(2008)], financial leverages [Thurner et al(2012)],
circuit breakers [Kobayashi and Hashimoto(2011)], price variations [Yeh andYang(2010)], [Mizuta et al(2015a)], tick sizes [Mizuta et al(2013)], speedupof exchange computer system [Mizuta et al(2015c)], and dark pools [Mo andYang(2013)], [Mizuta et al(2015b)].
Indeed, [Mo and Yang(2013)] investigated dark pools by using artificial
market simulations. However, they have not investigated situations that haveEﬀects of Dark Pools on Financial Markets’ Eﬃciency

3

never occurred before such as usage rates of dark pools that are much higher
than those at present because they also use real historical stock prices. [Mizuta
et al(2015b)] investigated whether dark pools reduce market impacts or not
by using artificial market simulations. However, they neither discuss market
eﬃciency nor markets’ price-discovery function, and their market selection
model was not realistic.
Therefore, in this study, we investigated eﬀects of a dark pool on financial
markets’ eﬃciency and price-discovery function by using an artificial market
model. This is a very important investigation into financial systemic risk because making a market ineﬃcient and losing the price-discovery function may
make the market unstable and increase financial systemic risk. In this study, we
additionally implemented smart order routing (SOR) to the model of [Mizuta
et al(2015b)] to treat actual market selection of investors. We discussed quantitatively how the spreading of dark pools beyond our experience could aﬀect
the price-discovery function and aimed to clarify the mechanism of dark poolsthat makes a market eﬃcient or ineﬃcient.

2 Artificial Market Model
We built a simple artificial market model in which smart order routing (SOR)
was additionally implemented to the model of [Mizuta et al(2015b)], which
had been built on the basis of the model of [Chiarella and Iori(2002)].
The model of [Mizuta et al(2015b)] succeed at replicating high frequency
micro structures such as execution rates, cancel rates, one tick volatility, andso on, which were not replicated by the model of [Chiarella and Iori(2002)].
Their model [Chiarella and Iori(2002)] was very simple but replicated longterm statistical characteristics observed in real financial markets: fat-tail andvolatility-clustering.
The simplicity of the model is very important for this study. We explain
the basic concept for building our artificial market model in the Appendix.

2.1 Order Process
The model treats only one risk asset and non-risk asset (cash). The numberof agents is n. First, at time t = 1, agent 1 orders to buy or sell the risk
asset; after that at t = 2, agent 2 orders; at t = 3, 4, , , n, agents 3, 4, , , n orderrespectively. At t = n + 1, going back to the first, agent 1 orders, and at
t = n + 2, n + 3, , n + n, agents 2, 3, , , , n order respectively, and this cycle isrepeated. Note that time t passes even if no deals are done.
Agents always order only one share and can short-sell freely. The quantityof holding positions is not limited, so agents can take any shares for both longand short positions to infinity.

4

Takanobu Mizuta et al.

2.2 Lit Market and Dark Pool
The model has two markets: one lit market, which provides all order books toinvestors, and one dark pool, which provide no order books.
The lit market adopts a continuous double auction to determine a market
price of the risk asset. A continuous double auction is an auction mechanism
where multiple buyers and sellers compete to buy and sell some financial assets
in the market, and where transactions can occur at any time whenever an oﬀer
to buy and an oﬀer to sell match [Friedman(1993)], [TSE(2012)]. The minimum
unit of price change is δP . The buy order price is rounded oﬀ to the nearest
fraction, and the sell order price is rounded up to the nearest fraction. Whenan agent orders to buy (sell), if there is a lower-price sell order (a higher-pricebuy order) than the agent’s order, dealing is immediately done. We call this a“market order”. If there is not, the agent’s order remains in the order book.
We call this a “limit order”. The remaining order is canceled tc after the order
time.
There are many ways to determine trade prices in dark pools. In the model,
the dark pool adopts an average of the highest buy order price and the lowest
sell order price in the lit market as a trade price. This method is adopted
by many dark pools in real financial markets [Johnson(2010)]. Agents do not
specify an order price in the dark pool. When the agent orders one unit buy(sell) to the dark pool, trading is done immediately if the dark pool contains
opposite waiting sell (buy) orders. If there are no opposite orders, the orderremains and waits for opposite orders to come in. In the dark pool, therefore,
only either buy or sell orders remain. The same as in a lit market, the remaining
order is canceled tc after the order time.

2.3 Market Selection Model
We investigated two cases: one without and the other with smart order routing(SOR).
In the case without SOR, the agents order to the dark pool with probabilityd and to the lit market with probability 1 − d.
In the case with SOR, agents select a market in the following way. Whenthe order is a market order and opposite limit orders are waiting in the darkpool, the agents order to the dark pool. When the order is a limit order, or
when the order is a market order and opposite limit orders are not waiting in
the dark pool, the agents order to the dark pool with probability d, the sameas in the case without SOR.
For example, when the agent makes a market buy (sell) order, if limit sell(buy) orders are waiting in the dark pool, the order is always matched in thedark pool. If no waiting limit sell (buy) orders are waiting in the dark pool,the order goes to the dark pool with probability d. Also, when an agent makes
a limit order, the order goes to the dark pool with probability d.

Eﬀects of Dark Pools on Financial Markets’ Eﬃciency

5

Of course, investors prefer to make a market order in a dark pool than ina lit market if there are opposite waiting orders in the dark pool, because a
trade price of a market order in a dark pool is always improved by half of abid ask spread1 than that in a lit market.
Indeed in actual financial markets, an investor cannot see whether opposite
limit orders are waiting in dark pools or not. However, it is said that, in thecase of making a market order, at first the investor orders to a dark pool, andthen if the order is not matched, s/he cancels the order soon and orders to a lit
market as a market order. Therefore, this SOR model is not only reasonable
but also essential for a market selection model with a dark pool.
Because [Mizuta et al(2015b)] did not implement such a SOR in theirmodel, they could not discuss eﬀects of the SOR to market eﬃciency. An
important contribution of this study is to discuss these SOR eﬀects.

2.4 Agent ModelAn agent j determines an order price and buys or sells by the following process.
Agents use a combination of fundamental value and technical rules to form
expectations on a risk asset’s returns. An expected return of the agent j is
(
)
Pf
1
t
t
w1,j log t + w2,j rh,j+ uj ϵtj .
(1)
re,j
=
w1,j + w2,j + uj
Pwhere wi,j is the weight of term i of the agent j and is determined by randomvariables uniformly distributed in the interval (0, wi,max ) at the start of thesimulation independently for each agent. uj is the weight of the third term of
the agent j and is also determined by random variables uniformly distributedin the interval (0, umax ) at the start of the simulation independently for eachagent. Pf is a fundamental value that is constant. P t is a market price of therisk asset at time t2 . ϵtj is a noise determined by random variables of normalt
distribution with an average 0 and a variance σϵ . rh,j
is a historical price returntinside an agent’s time interval τj , and rh,j = log (P t /P t−τj ). τj is determinedby random variables uniformly distributed in the interval (1, τmax ) at the startof the simulation independently for each agent.
The first term of Eq. (1) represents a fundamental strategy: an agent expects a positive return when the market price is lower than the fundamental value, and vice versa. The second term of Eq. (1) represents a technicalstrategy: an agent expects a positive return when historical market return ispositive, and vice versa.
After the expected return has been determined, an expected price isttPe,j= P t exp (re,j
).
1

(2)

A bid ask spread is defined as the diﬀerence between the best bid price (the highest buy
limit order price) and best ask price (the lowest sell limit order price).
2 When the dealing is not done at t, P t remains at the last market price P t−1 , and at
t = 1, P t = Pf

6Takanobu Mizuta et al.

tAn order price Po,jis determined by random variables normally distributedtin an average Pe,j and a standard deviation Pσ , where Pσ is a constant.
Buy or sell is determined by a magnitude relationship between the expectedttprice Pe,jand the order price Po,j
, i.e.,
tt
When Pe,j > Po,j , the agent orders to buy one share,ttWhen Pe,j< Po,j, the agent orders to sell one share.

3 Simulation Results[Mizuta et al(2015b)] searched for adequate model parameters verified bystatistically existing stylized facts and market micro structures to investigate
eﬀects of dark pools on market stability. They found parameters to replicate
both long-term statistical characteristics and very short-term micro structuresof real financial markets3 . Specifically, we set, n = 1, 000, w1,max = 1, w2,max =10, umax = 1, τmax = 10, 000, σϵ = 0.06, Pσ = 30, tc = 20, 000, δP = 0.1, Pf =10, 000. We ran simulations to t = 10, 000, 000.

3.1 Market Eﬃciency
In this study, we compared several statistical values of simulation runs forvarious d = 0, 0.025, 0.05, 0.075, 0.1, 0.125, 0.15, 0.175, 0.2, 0.25, 0.3, 0.4, 0.5,
0.6, 0.7, 0.8, 0.9 under not only other parameters that were fixed but also thesame random number table. We simulated these runs 100 times, changing therandom number table each time, and we used averaged statistical values of
100 times runs.
We introduced “market ineﬃciency” Mie that directly measures market
eﬃciency,te
1 ∑|P t − Pf |
Mie =
,
(3)
te t=1
Pf
where || means absolute value. Mie is always greater than zero, and Mie = 0
means a market is perfectly eﬃcient. The larger the Mie , the less eﬃcient themarket4 .
Fig. 1 shows market ineﬃciency Mie for various shares of a trading value
amount of the dark pool (D) in the cases with and without SOR (Smart Order
Routing). Note that D is diﬀerent from the probability of ordering to the darkpool d. We defined D as D = VD /(VD + VL ), where VD is the trading value
amount in the dark pool and VL is the trading value amount in the lit market.
3

We explain the verification of the model in the Appendix.
This index is sometimes used in experimental financial studies of people, in which
this market ineﬃciency is sometimes called “RAD” (Relative Absolute Deviation) [Stöckl
et al(2010)]. Many indications for measuring market eﬃciency have been proposed [Verheyden et al(2013)]. A feature of Mie is that Mie is calculated by using a fundamental pricePf directly, which is never observed in empirical studies. We can also use Mie in simulationand experimental studies because we can exactly define a fundamental price.
4

Eﬀects of Dark Pools on Financial Markets’ Eﬃciency

7

Market Inefficiency with and without SORMarket Inefficiency (Mie)

0.35%
0.30%

0.25%
0.20%
0.15%
0.10%

with SOR
without SOR

0.05%
0.00%
0%

25%
50%
75%
100%
Share of Trading Value of Dark Pool (D)Fig. 1 Market ineﬃciency Mie for various shares of trading value amount of the dark pool(D).

d is almost the same as the share of numbers of all orders that include limit
and market orders. On the other hand, D counts only market orders.
In the case without SOR, Mie monotonically and gradually decreased withD as Fig. 1 shows. On the other hand, in the case with SOR, Mie decreased
more sharply in D ≲ 70%, and Mie increased significantly in D ≳ 70%. This
indicates that there is an optimal usage rate of the dark pool for the market
eﬃciency. Next, we discuss why Mie decreased in D ≲ 70% in section 3.2 andthen why Mie increased significantly in D ≳ 70% in section 3.3.

3.2 Becoming Eﬃcient: lower than optimal usage rate of Dark PoolFig. 2 shows execution rates in the lit market for various D in the cases withand without SOR. In the case without SOR, the execution rates were stable
and moderately decreased. On the other hand, in the case with SOR, theexecution rates more rapidly decreased. This indicates that the SOR reduces
the execution rates in the lit market.
Fig. 3 shows a mechanism for reducing the execution rates in the case with
SOR. As we mentioned in section 2.3, the agents order to the dark pool when
the order is a market order and opposite limit orders are waiting. In othercases, the agents order to the dark pool with probability d. Therefore, thenumber of market orders to the lit market is reduced by the number of market
orders sent to the dark pool by the SOR when opposite limit orders are waitingin the dark pool. Thus, the execution rates in the lit market with the SOR are
lower than those without the SOR.

8

Takanobu Mizuta et al.

Execution Rate with and without SOR
35%

Execution Rate (k)

30%
25%
20%
15%
10%

with SOR
without SOR

5%
0%

0%

25%
50%
75%
100%
Share of Trading Value of Dark Pool (D)Fig. 2 Execution rates in the lit market for various D.

Lit
MarketLimit
OrdersMarketOrdersSOR
Sent to

Sell

Price

84

101

176

100

Buy

99

2

98

77

Dark
Pool
Sell

Buy
175

Fig. 3 A mechanism for reducing the execution rates in the case with SOR.

This mechanism raises the limit orders relative to market orders in thelit market. Fig. 4 shows the depth of limit orders and bid ask spreads in thelit market for various D in the case with SOR. Here, depth of limit orders
is defined as an average number of buy limit orders from (1 − 0.001) × P t (amarket price) to P t per average number of executed trades within one day(20, 000 time steps), and a bid ask spread is defined as the diﬀerence between

Eﬀects of Dark Pools on Financial Markets’ Eﬃciency

9

0.20%

15%

0.15%

10%

0.10%

5%

0.05%

Bid Ask Spred

Depth of Limit orders

Depth of Limit orders and Bid Ask Spred20%

Depth of Limit orders
Bid Ask Spred

0%

0.00%
0%
25%
50%
75%
100%
Share of Trading Value of Dark Pool (D)Fig. 4 Depth of limit orders and bid ask spreads in the lit market for various D.

Absorb Market
OrdersMarketPricesMany Limit
OrdersFundamental
Many Limit Price
OrdersAbsorb Market
OrdersFig. 5 A mechanism for making the lit market eﬃcient in the case with SOR.

best bid price (the highest buy limit order price) and best ask price (the lowest
sell limit order price) per Pf (a fundamental price). Indeed, by increasing D,
the depth of limit orders becomes thicker, and this leads to narrowing down
the bid ask spread.
Fig. 5 shows a mechanism of how the thicker depth of limit orders makes
the lit market eﬃcient. First, P t is very near Pf . When the depth of limit
orders becomes thicker, even though agents make market orders departingfrom Pf , thicker limit orders absorb these market orders, and thus P t is still
stable near Pf .
We summarize why Mie decreased in D ≲ 70% as shown in Figures 3 and5. Execution rates in the lit market are reduced by more market orders being
sent to the dark pool by the SOR than limit orders increasing D. This leads

10

Takanobu Mizuta et al.

Waiting Orders in Dark Pool and Expected Returnby Market Price Ranges

0.3%

Waiting orders in Dark Pool（D=96%）
Waiting orders in Dark Pool（D=44%）
Expected Return（D=96%）
Expected Return（D=44%）

20
10

0.2%
0.1%

10130

10110

10090

10070

10050

10030

10010

-0.3%
9990

-30
9970

-0.2%

9950

-20

9930

-0.1%

9910

-10

9890

0.0%

9870

0

Expected Return

Number of Waiting Orders

30

Market Pricet
Fig. 6 Number of waiting limit orders in the dark pool and averaged re,j(expected return)tfor all agents by various P ranges.

MarketPricesMarket Sell Orders

Prevent toConverging

Stored Waiting Buy
Orders in Dark Pool
Fundamental Price

Fig. 7 A mechanism of how the lit market is made ineﬃcient.

to the depth of limit orders becoming thicker, and these thicker limit orders
absorb market orders. Thus, a market price is still stable near a fundamental
price.

3.3 Becoming Ineﬃcient: Too High usage rate of Dark Pool
We discuss why Mie increased significantly in D ≳ 70%. Fig. 6 shows a numbertof waiting limit orders in the dark pool and re,j(expected return) for all agentst
averaged by various P ranges in the case of D = 44% and 96% with SOR.

Eﬀects of Dark Pools on Financial Markets’ Eﬃciency

11

Table 1 Summarized Probabilities for all cases of Orders

buy (a)sell (1 − a)market order (k)limit order (1 − k)market order (k)limit order (1 − k)lit (1 − d)ak(1 − d)a(1 − k)(1 − d)(1 − a)k(1 − d)(1 − a)(1 − k)(1 − d)dark pool (d)
ad/2
ad/2
(1 − a)d/2(1 − a)d/2

A sign for the number of waiting orders in the dark pool indicates buy or sellorders: plus means buy orders, and minus means sell orders.
In the case of D = 44%, a few waiting orders are stored in the dark poolt
and averaged re,jare positive when P t is higher than Pf (= 10, 000) and vice5
versa . On the other hand, in the case of D = 96%, many waiting orders aret
stored in dark pools and averaged re,j
are the opposite to those in the case of
D = 44%.
Fig. 7 shows how the lit market is made ineﬃcient. When P t becomesmuch higher than Pf , many buy waiting orders are stored in the dark pool andtaveraged re,j
are negative, which means that agents make market sell orders.
These market sell orders could have led to P t reducing and converging to Pf .
However, many buy waiting orders stored in the dark pool absorb these market
sell orders and prevent P t converging to Pf . Therefore, P t is maintained at amuch higher price than Pf , and the lit market is made ineﬃcient. When P t
becomes much smaller than Pf , the opposite occurs. This is why Mie increased
significantly in D ≳ 70%.

4 Theoretical Discussion
4.1 Simple Equation ModelIn this section, we discuss mechanisms by which a dark pool makes a market
eﬃcient or ineﬃcient by using a simple equation model.
We define a, k, and d as the probabilities of an agent making a buy order,
making a market order in the lit market without SOR, and ordering to thedark pool the same as in previous sections, respectively. Obviously, 1 − a, 1 − kand 1 − d are the probabilities of an agent making a sell order, making alimit order in the lit market without SOR, and ordering to the lit market is,respectively.
In the simulation results, few limit orders in the dark pool were canceled.
This means that all limit orders in the dark pool were executed with market
orders and that all numbers of limit orders were the same as all numbers ofmarket orders in the dark pool. Therefore, we defined probabilities of an agent
making both limit and market orders in the dark pool as 1/2.
5 When there were not enough data in a P t range, we did not plot them in Fig. 6. In thecase of D = 44%, data exist in narrower P t ranges than those in the case D = 96% because
there are no trades outside these P t ranges.

12Takanobu Mizuta et al.

Table 2 Summarized Probabilities: Summation of buy and sell, a = 1/2
market order (k)limit order (1 − k)lit (1 − d)k(1 − d)(1 − k)(1 − d)dark pool (d)d/2d/2

Execution Rate from Equation and Simulations
35%

Execution Rate (k')

30%
25%

20%
15%

Equation Model

10%

Simulation (Difference fromwithout SOR to with SOR)

5%

0%
0%

25%
50%
75%
Share of Trading Value of Dark Pool (D)100%Fig. 8 k′ from Eqs.(9) and (10) and the diﬀerence between execution rates without andwith SOR of simulation results from Fig. 2 for various D.

These definitions derived that, for example, the probability of an agent
making a market buy order in the lit order is ak(1 − d). Table I summarizes
such probabilities for all cases of orders.
Note that from Table I, we can easily derive a share of the trading value
amount of the dark pool (D)6 as defined in the previous section as
D=

d/2
.
k(1 − d) + d/2(4)

In the following subsections, we investigate market eﬃciency by the simple
equation model from Table I and compare it with simulation results.

4.2 Becoming Eﬃcient: lower than optimal usage rate of Dark Pool
Here, let us discuss an execution rate in the lit market with SOR (k ′ ). For
simplicity, we fixed a = 1/2 and summed the cases with both a buy and a sellorder. Thus, Table I is changed to Table II.
The SOR reduces the probability of market orders to the lit market, k(1 −d). When k(1 − d) > d/2, which means the trading volume in the lit market
6 Note that in our model agents always order one unit. Therefore, D is exactly the sameas the probability of an agent making an market order in the dark pool per the probabilityof an agent making an market order in both the dark pool and the lit market.

Eﬀects of Dark Pools on Financial Markets’ Eﬃciency

13

Market Inefficiency
Diference with and without SORMarket Inefficiency (Mie)

0.50%

0.40%
0.30%
0.20%
0.10%

Difference from without SOR to with SOR
with SOR

0.00%
0%

25%
50%
75%
100%
Share of Trading Value of Dark Pool (D)Fig. 9 Market ineﬃciencies (Mie ) with SOR and diﬀerence between Mie without SOR and
Mie with SOR of simulation results from Fig. 1 for various D.

is greater than that in the dark pool (therefore, this is exactly the same as
D < 1/2), the number of market orders sent to the dark pool by SOR is limitedby the number of limit orders in the dark pool because existing market orders
to the lit market outnumber limit waiting orders in the dark pool. In short,
reduction of market orders to the lit market by SOR depends on the numberof limit orders to the dark pool. Therefore, we can assume the SOR reduces
the probability of market orders to the lit market by αd/2, where α is constantfrom zero to 1. Thus, the probability of market orders to the lit market becomesk(1 − d) − αd/2.
On the other hand, when k(1 − d) < d/2, the opposite occurs. The numberof market orders sent to the dark pool by SOR is limited by the number ofmarket orders to the lit market because existing limit orders in the dark pool
outnumber market orders to the lit market. Therefore, we can assume the SOR
reduces the probability of market orders to the lit market by αk(1 − d), so theprobability of market orders to the lit market becomes k(1 − d) − αk(1 − d).
Therefore, when k(1 − d) > d/2 (exactly the same as D < 1/2), by using
Eq.(4), k ′ becomes,
k ′ = {k(1 − d) − αd/2}/(1 − d)
(
)
D
= 1−α
k,
1−D

(5)
(6)

On the other hand, when k(1 − d) < d/2 (exactly the same as D > 1/2), by
using Eq.(4), k ′ becomes,
k ′ = {k(1 − d) − αk(1 − d)}/(1 − d)= k(1 − α).

(7)
(8)

14

Takanobu Mizuta et al.

Because these two equations should become exactly the same when k(1 −d) = d/2, we can derive α = 1/3, easily7 . Lastly,
(
)
1
1 D
k′ = 1 −k (When D < )
(9)
31−D
2
2
1
k′ = k(When D > ).
(10)
3
2
Fig. 8 shows k ′ from Eqs.(9) and (10) and the diﬀerence between execution
rates without and with SOR of simulation results from Fig. 2 for various D. We
adjusted zero points of them at D = 0 to make them exactly the same as theexecution rate of simulation results at D = 0. We used the diﬀerence between
execution rates without and with SOR because Eqs.(9) and (10) include theeﬀect only of SOR on reduction of the execution rates and we want to isolate
the pure eﬀect of SOR from simulation results.
k ′ from Eqs.(9) and (10) showed that behaviors of k ′ are very diﬀerent
between D < 1/2 and D > 1/2. When D < 1/2, the number of market orders
sent to the dark pool by SOR depends on the number of limit orders in thedark pool. This means that the more orders made in the dark pool (increasingD), the more orders sent to the dark pool by SOR. Thus, D increases, and k ′decreases. On the other hand, when D > 1/2, the number of market orders
sent to the dark pool by SOR depends on the number of market orders to
the lit market. This means that increasing D leads to fewer orders to the lit
market and fewer orders sent to the dark pool by SOR. Thus, even though D
increases, k ′ never decreases.
This indicates that, whether markets become eﬃcient or ineﬃcient, themagnitude relationship between the number of market orders to the lit market
and the dark pool is intrinsically important; in short, whether D > 1/2 or
D < 1/2. Therefore, this suggests the optimal usage rate of the dark pool forthe market eﬃciency is D = 1/2.
The simulation result of k ′ (the diﬀerence between execution rates without
and with SOR) was similar to that of Eqs.(9) and (10); when D < 1/2, D
increases, k ′ decreases, and when D > 1/2, k ′ is stable. However, decreasing
shapes are diﬀerent; Eqs.(9) and (10) draw a convex upward, the simulation
result draws a convex downward, and saturation levels are slightly diﬀerent.
We consider these diﬀerences are derived by a mechanism other than SOR and
are subjects for a future study.
Fig. 9 shows market ineﬃciencies (Mie ) with SOR and the diﬀerence between Mie without SOR and Mie with SOR for simulation results from Fig. 1
for various D. We adjusted the zero point of the diﬀerence at D = 0 to makeit exactly the same as Mie with SOR at D = 0. The diﬀerence in Mie showedthat when D > 1/2, D increases, Mie increases; this means that SOR could
7 We can interpret α = 1/3 as follows. There are three cases for waiting limit orders in
the dark pool: buy orders waiting, sell orders waiting, and no orders waiting. We assume
the three cases have the same probability of occurring. The market order is sent to the darkpool by SOR in the only one case of the three cases, i.e., when opposite orders are waitingin the dark pool. This derives α = 1/3.

Eﬀects of Dark Pools on Financial Markets’ Eﬃciency

15

Level of buy-sell Imbalance
Destroing a Price Discovery Function

100%
90%

a

80%
70%
60%
50%

0%

25%

50%

75%

100%

Share of Trading Value of Dark Pool (D)Fig. 10 The boundary of buy-sell order imbalance (a = (1 + D)/(1 + 3D)) at which theprice-discovery function is destroyed or not for various D.

not make the lit market eﬃcient when D > 1/2. This also suggests that a
trading volume in dark pools higher than that in lit markets makes markets
ineﬃcient.

4.3 Becoming Ineﬃcient: Too High usage rate of Dark PoolIn this subsection, we explain how a too high usage rate of the dark pool makes
the lit market significantly ineﬃcient.
We discuss the case in which D is suﬃciently higher than 1/2, a large
enough number of waiting buy orders are stored in the dark pool, and themarket price is higher than the fundamental price. In this case, all market sellorders to the lit market are sent to the dark pool by SOR. Therefore, from
Table I, we can derive that the probability an agent makes a sell order in thedark pool is (1 − a)d + k(1 − a)(1 − d). Of course, the probability an agent
makes a buy order to the dark pool is ad. If there are more new buy orders to
the dark pool than new sell orders to the dark pool, that is
ad > (1 − a)d + k(1 − a)(1 − d),(11)waiting buy orders in the dark pool are not reduced and this prevents a market price from converting to the fundamental price. In short, Eq.(11) is thecondition in which the price-discovery function of the lit market is destroyed.
Using Eq.(4), we can simplify Eq.(11) to
a>

1+D
.
1 + 3D(12)

16

Takanobu Mizuta et al.

Remember that a is the probability of an agent making a buy order. In otherwords, a is the ratio of buy orders to all orders, and 1 − a is the ratio of sellorders to all orders. Thus, a means an imbalance between the numbers of buy
and sell orders, a = 50% means the numbers of buy and sell orders are perfectly
balanced, and a = 100% means no sell orders. Therefore, a = (1 + D)/(1 + 3D)
is the boundary of whether the price-discovery function is destroyed or not.
Note that this discussion is exactly the same in the opposite case of buy andsell.
Fig. 10 shows a = (1 + D)/(1 + 3D) for various D. The upper side of theline satisfies Eq.(12). When D = 20%, a = 75%; when more than 75% of allorders are buy orders, then less than 25% are sell orders and the number ofbuy orders is more than three times that of sell orders, so the price-discovery
function is destroyed. When buy orders are more than three times as numerous
as sell orders, we can say that a very large buy-sell imbalance has occurred andconsider this to be a rare case. This suggests that dark pools with D = 20%
rarely destroy the price-discovery function.
On the other hand, when D = 90%, a = 51% (which is a very slightbuy-sell imbalance), and this suggests that dark pools very easily destroy theprice-discovery function.

5 Conclusion and Future Studies
In this study, we investigated how a dark pool, in which no order books areprovided, aﬀects financial markets’ eﬃciency and price-discovery function byusing the artificial market model. This is a very important investigation into
financial systemic risk because making a market ineﬃcient and losing the pricediscovery function may make the market unstable and increase financial systemic risk. In this study, we additionally implemented a smart order routing(SOR) to the model of Mizuta et al. [Mizuta et al(2015b)] to treat actual market selection of investors. We discussed quantitatively how spreading of darkpools beyond our experience could aﬀect the price-discovery function. We also
aimed to clarify the mechanism of a dark pool that makes a market eﬃcient
or ineﬃcient.
We found that market ineﬃciently (Mie ) was decreased sharply by raising
the share of the trading value amount of the dark pool (D) in D ≲ 70%. On
the other hand, in D ≳ 70%, Mie increased significantly. This indicates that
there is an optimal usage rate of the dark pool for the market eﬃciency.
The reason Mie decreased in D ≲ 70% is that the execution rates in thelit market are reduced by more market orders being sent to the dark pool by
the SOR than limit orders increasing D. This leads the depth of limit orders
to become thicker, these thicker limit orders absorb market orders, and thus
the market price is still stable near the fundamental price (see also Figures 3
and 5).
The reason Mie increased significantly in D ≳ 70% is as follows. When amarket price (P t ) becomes much higher than the fundamental price (Pf ), many

Eﬀects of Dark Pools on Financial Markets’ Eﬃciency

17

waiting buy orders are stored in the dark pool and averaged estimated returnst(re,j) for all agents are negative, which means that agents make market sellorders. These market sell orders could have made P t converge to Pf , but many
waiting buy orders stored in the dark pool absorb these market sell orders andprevent P t converging to Pf . Therefore, P t maintains a much higher pricethan Pf , and the lit market is made ineﬃcient. When P t becomes much lower
than Pf , the opposite occurs (see also Fig. 7).
We also discussed mechanisms by which a dark pool makes a market eﬃcient or ineﬃcient by a simple equation model. The equations about an execution rate we derived indicate that whether D > 1/2 or D < 1/2 is intrinsicallyimportant to whether markets become eﬃcient or ineﬃcient. Therefore, thissuggests that the optimal usage rate of the dark pool for the market eﬃciency
is D = 1/2 and that a trading volume amount in dark pools higher thanthat in lit markets makes markets ineﬃcient. We also compared results of theequations with those of simulations and found similar tendencies.
We also derived an equation showing the boundary of a buy-sell imbalance
at which dark pools destroy the price-discovery function. We also discussed
that when the usage rate of dark pools is low, for example D = 20%, theequation suggests that dark pools rarely destroy the price-discovery function
even though a large buy-sell imbalance occurs. On the other hand, when theusage rate of dark pools is very high, for example D = 90%, this equationsuggests that dark pools very easily destroy the price-discovery function by avery slight buy-sell imbalance.
A future study is to investigate more details of the optimal usage rate ofdark pools for the market eﬃciency. Our results suggested the optimal usage
rate was around 50%−70%, which is much higher than about 8%, which is thecap level of dark pools that European regulators are discussing [Bowley(2014)].
However, we could not determine the precise level of the optimal usage rateof dark pools for the market eﬃciency.
Another future study is comparing the simulation results with empirical
data. Indeed, we cannot observe Mie of real markets by empirical data because
we cannot find fundamental prices in real markets. On the other hand, we canobserve an execution rate, depth of limit orders and a bid ask spread of eachstock in real lit markets from empirical data. In addition, we can estimate Dof each stock from some statistics. D are diﬀerent from one stock to another.
Therefore we can draw figures such as Fig. 2 and Fig. 4 from empirical dataplotting each stock having diﬀerent D, execution rates and so on. To compare
these figures with Fig. 2 and Fig. 4, we can compare the simulation resultswith empirical data.
We also observe buy-sell imbalances in real lit markets from empirical data.
Using them and Fig. 10, we can discuss how much D may destroy the pricediscovery function in real financial markets.
An artificial market can isolate the pure contribution of these new typesof markets to the price formation and can treat such markets on a usage rate
higher than we have ever experienced. However, outputs of the artificial market simulation study would not be accurate or credible forecasts of the actual

18

Takanobu Mizuta et al.

future. The artificial market simulations need to show possible mechanisms
aﬀecting price formation by many simulation runs and to gain new knowledge and intelligence; conversely, the artificial market simulations are limitedin that their outputs would not certainly but only possibly occur in actual financial markets. Therefore, for more detailed discussions, we should compare
the simulation results to results of studies using other methods, e.g. empiricalstudies.

Appendix
Basic Concept for Building ModelAn artificial market, which is a kind of a multi-agent simulation, can isolate the pure contribution of these system changes to the price formation andcan treat the changes that have never been employed [LeBaron(2006)], [Chenet al(2012)], [Cristelli(2014)]. These are the strong points of the artificial market simulation study.
However, outputs of the artificial market simulation study would not be
accurate or credible forecasts of the actual future. The artificial market simulation needs to show possible mechanisms aﬀecting the price formation by many
simulation runs, e.g. searching for parameters, purely comparing before/after
the changing, and so on. The possible mechanisms shown by these simulation
runs will give us new knowledge and intelligence about eﬀects of the changes toprice formation in actual financial markets. Other study methods, e.g. empirical studies, would not show such possible mechanisms. The artificial market
simulation studies also need to gain such new knowledge and intelligence.
Indeed, artificial markets should replicate macro phenomena existing generally for any asset and any time. Price variation, which is a kind of macro
phenomena, is not explicitly modeled in artificial markets. Only micro processes, agents (general investors), and price determination mechanisms (financial exchanges) are explicitly modeled in artificial markets. Macro phenomena
are emerging as the outcome interactions of micro processes. Therefore, thesimulation outputs should replicate general macro phenomena at least to showthat simulation models are probable in actual markets.
However, it is not a primary purpose for the artificial market to replicate
specific macro phenomena only for a specific asset or a specific period. An
unnecessary replication of macro phenomena leads to models that are overfitted and too complex. Such models would prevent our understanding anddiscovering mechanisms aﬀecting the price formation because of related factors
increasing.
Indeed, artificial market models that are too complex are often criticizedbecause they are very diﬃcult to evaluate [Chen et al(2012)]. A too complex
model not only would prevent our understanding mechanisms but also could
output arbitrary results by over-fitting too many parameters. The simpler theEﬀects of Dark Pools on Financial Markets’ Eﬃciency

19

Table 3 Statistics without Dark Pool
trading
standard
deviations

autocorrelationcoeﬃcient forsquare return

execution rate
cancel rate
number of trades / 1 day
for 1 tickfor 1 day (20000 ticks)
kurtosis
lag
1
2
3
4
5

32.3%
26.1%
6467
0.0512%0.562%
1.42
0.225
0.138
0.106
0.087
0.075

models, the more diﬃcult arbitrary results are to obtain, and the easier themodel is to evaluate.
Therefore, we built an artificial market model that is as simple as possible
and do not intentionally implement agents to cover all the investors who would
exist in actual financial markets.

Verification of the Model
In many previous artificial market studies, the models were verified to see
whether they could explain stylized facts such as a fat-tail, volatility-clustering,and so on [LeBaron(2006)], [Chen et al(2012)], [Cristelli(2014)]. A fat-tailmeans that the kurtosis of price returns is positive. Volatility-clustering meansthat the square returns have positive autocorrelation, and the autocorrelation slowly decays as its lag becomes longer. Many empirical studies, e.g.
[Sewell(2006)], have shown that both stylized facts (the fat-tail and volatilityclustering) exist statistically in almost all financial markets. Conversely, theyalso have shown that only the fat-tail and volatility-clustering are stably observed for any asset and in any period because financial markets are generally
unstable.
Indeed, the kurtosis of price returns and the autocorrelation of the square
returns are stably and significantly positive, but the magnitudes of these values
are unstable and very diﬀerent depending on asset and/or period. The kurtosis
of price returns and the autocorrelation of the square returns were observed
to have very broad magnitudes of about 1 ∼ 100 and about 0.01 ∼ 0.2,
respectively [Sewell(2006)].
For the above reasons, an artificial market model should replicate thesevalues as significantly positive and within a reasonable range as we mentioned.
It is not essential for the models to replicate specific values of stylized facts
because these stylized facts’ values are unstable in actual financial markets.
Table III lists statistics in which there is only one lit market. All statistics
are averages of 100 simulation runs, and all the following figures use the averageof 100 simulation runs. We define 20, 000 time steps as 1 day because the20

Takanobu Mizuta et al.

number of trades within 20, 000 time steps is almost the same as that in
actual markets per day. All statistics; execution rates, cancel rates8 , standard
deviations of returns for one tick and one day9 , kurtosis of price returns,
and the autocorrelation coeﬃcient for square returns10 are of course almost
the same as the results of [Mizuta et al(2015b)] because the models do not
diﬀer except for the market selection model, which has smart order routing(SOR). [Mizuta et al(2015b)] showed that this mode replicated very shortterm micro structure, execution rates, cancel rates, and standard deviationsof returns for one tick and replicated long-term statistical characteristics, fattail, and volatility clustering, observed in real financial markets. Therefore, themodel was verified to investigate the eﬀect of dark pools on market stability.

Disclaimer & Acknowledgments
Note that the opinions contained herein are solely those of the authors and do not necessarilyreflect those of SPARX Asset Management Co., Ltd. and Nomura Securities Co., Ltd. On
behalf of all authors, the corresponding author states that there is no conflict of interest.
The authors are grateful to Execution Service Department, Nomura Securities Co., Ltd. fortheir valuable comments from a practical financial perspective on this paper.

References[Bowley(2014)] Bowley A (2014) Agreement of mifid ii reforms, instinet incorporated.
URL http://instinet.com/docs/msr/2014/Agreement_of_MiFID_II_Reforms-Quick_Analysis.pdf[Chen et al(2012)] Chen SH, Chang CL, Du YR (2012) Agent-based economic models andeconometrics. Knowledge Engineering Review 27(2):187–219[Chiarella and Iori(2002)] Chiarella C, Iori G (2002) A simulation analysis of the microstructure of double auction markets. Quantitative Finance 2(5):346–353[EC(2010)] European Commission (2010) Public consultation review of the markets in financial instruments directive (mifid). Consultation Report 8[Cristelli(2014)] Cristelli M (2014) Complexity in Financial Markets, Modeling Psychological Behavior in Agent-Based Models and Order Book Models, Springer[Friedman(1993)] Friedman D (1993) The double auction market institution: A survey. The
Double Auction Market: Institutions, Theories, and Evidence pp 3–25
[Johnson(2010)] Johnson B (2010) Algorithmic Trading & DMA: An introduction to directaccess trading strategies. 4Myeloma Press[Kobayashi and Hashimoto(2011)] Kobayashi S, Hashimoto T (2011) Benefits and limits ofcircuit breaker: Institutional design using artificial futures market. Evolutionary andInstitutional Economics Review 7(2):355–372[LeBaron(2006)] LeBaron B (2006) Agent-based computational finance. Handbook of computational economics 2:1187–1233
8 The execution rate is the ratio of the number of trades to that of all orders. The cancel
rate is the ratio of the number of cancels to that of all orders + cancels.
9 In our model, though overnight returns do not exist, the standard deviations of returns
for one day correspond to intraday volatility in real financial markets.
10 We used returns for 10 time units’ intervals (corresponding to about 10 seconds) to
calculate the statistical values for the stylized facts. In this model, time passes by an agent
just ordering even if no dealing is done. Therefore, the returns for one tick (one time) include
many zero returns, and they will bias statistical values. This is the reason we use returns
for about 10 time units’ intervals.

Eﬀects of Dark Pools on Financial Markets’ Eﬃciency

21[Mizuta et al(2013)] Mizuta T, Hayakawa S, Izumi K, Yoshimura S (2013) Investigation ofrelationship between tick size and trading volume of markets using artificial market
simulations. In: JPX working paper, Japan Excgange Group, 2, URL http://www.jpx.
co.jp/english/corporate/research-study/working-paper/[Mizuta et al(2015a)] Mizuta T, Izumi K, Yagi I, Yoshimura S (2015a) Investigation of pricevariation limits, short selling regulation, and uptick rules and their optimal design byartificial market simulations. Electronics and Communications in Japan 98(7):13–21,
DOI 10.1002/ecj.11684, URL http://dx.doi.org/10.1002/ecj.11684[Mizuta et al(2015b)] Mizuta T, Kosugi S, Kusumoto T, Matsumoto W, Izumi K, Yagi
I, Yoshimura S (2015b) Eﬀects of price regulations and dark pools on financial market
stability: An investigation by multiagent simulations. Intelligent Systems in Accounting,
Finance and Management, DOI 10.1002/isaf.1374, URL http://dx.doi.org/10.1002/
isaf.1374[Mizuta et al(2015c)] Mizuta T, Noritake Y, Hayakawa S, Izumi K (2015c) Impacts ofspeedup of market system on price formations using artificial market simulations. In:
JPX working paper, Japan Excgange Group, 9, URL http://www.jpx.co.jp/english/
corporate/research-study/working-paper/[Mo and Yang(2013)] Mo SYK, Yang MPSY (2013) A study of dark pool trading using anagent-based model. In: Computational Intelligence for Financial Engineering Economics(CIFEr), 2013 IEEE Symposium Series on Computational Intelligence on, pp 19–26[SEC(2010)] Commodity Futures Trading Commission and Securities Exchange Commission (2010) Concept release on equity market structure. Federal Register 75(13):3594–
3614[Sewell(2006)] Sewell M (2006) Characterization of financial time series. http://finance.
martinsewell.com/stylized-facts/[Stöckl et al(2010)] Stöckl T, Huber J, Kirchler M (2010) Bubble measures in experimental
asset markets. Experimental Economics 13(3):284–298[Thurner et al(2012)] Thurner S, Farmer J, Geanakoplos J (2012) Leverage causes fat tails
and clustered volatility. Quantitative Finance 12(5):695–707[TSE(2012)] Tokyo Stock Exchange (2012) Guide to tse trading methodology.
URL
http://www.jpx.co.jp/learning/tour/books-brochures/detail/tvdivq0000005u66-att/trading_methodology.pdf[Verheyden et al(2013)] Verheyden T, De Moor L, Van den Bossche F (2013) A tale ofmarket eﬃciency. Review of Business and Economic Literature 58(2):140–158[Westerhoﬀ(2008)] Westerhoﬀ F (2008) The use of agent-based financial market models
to test the eﬀectiveness of regulatory policies. Jahrbucher Fur Nationalokonomie Und
Statistik 228(2):195[Yagi et al(2010)] Yagi I, Mizuta T, Izumi K (2010) A study on the eﬀectiveness of shortselling regulation using artificial markets. Evolutionary and Institutional Economics
Review 7(1):113–132[Ye(2012)] Ye M (2012) Price manipulation, price discovery and transaction costs in the
crossing network. Price Discovery and Transaction Costs in the Crossing Network(March 14, 2012)[Yeh and Yang(2010)] Yeh C, Yang C (2010) Examining the eﬀectiveness of price limits inan artificial stock market. Journal of Economic Dynamics and Control 34(10):2089–2108