The order urgency will be updated by price movements and the order execution condition after intraday transactions. For example, if the stock price has increased by a significant level in one trading day, then the sell order urgency will increase and the buy order urgency will decrease for the next day. Another adjustment considers the order execution condition. Suppose that one trader has a task order size for a day ofSt, and the order executed that day isSexe. At the end of the day, if there are still a considerable number of orders that have not been fully executed ((St−Sexe)/St >
U T, whereU T is some threshold for unexecuted orders), this trader may continue to execute the order the next day, but with higher urgency. Suppose that the open price and close price for a trading day arePopenandPclose. At the beginning of the next day, the order urgency will be adjusted according to the following rules:
• ifPclose > Popen×(1+θ )(θis the threshold for urgency adjustment), traders will pick a new task. If it is a buy order, its urgency has a 50% probability to
Dark Pool Usage and Equity Market Volatility 25 minus 1 (50% chance of maintaining the original urgency); if it is a sell order, its urgency has a 50% probability to plus 1.
• ifPclose< Popen×(1−θ ), traders will pick up a new task. If it is a buy order, its urgency has a 50% probability to plus 1 (50% chance of maintaining the original urgency); if it is a sell order, its urgency has a 50% probability to minus 1.
• else if (St − Sexe)/St > U T, traders have an equal probability (50%) of continuing to execute the previous remaining order and increase its urgency with 1, or dropping it and picking a new one.
4 Experiment
To test how different extents of dark pool usage affect market volatility, T traders are divided into three groups, each distinguished from the others by different but fixed levels of dark pool usage, set asU1,U2, andU3, respectively. The probabilities of dark order submission of these three groups are denoted as [U1:U2:U3].U1is assigned as the benchmark probability of 0.U2andU3refer to low and high usage of the dark pool, respectively.
In addition to dark pool usage, there are two other factors that may have a significant influence on the order executed price in the dark pool, and thus reflect market volatility. The first is the dark pool cross-probability. A lower cross- probability indicates a relatively longer order execution delay in the dark pool.
In this case, there may be a significant difference between the midprice in the submission session and the execution price in the dark pool, especially when the market is volatile. The second factor is the proportion of market orders. Although an increase in market orders makes the market more volatile and implies a higher price improvement in dark trading, this increase also reflects a lack of liquidity in the main exchange, and implies the same situation in the dark pool. Thus, there is an increased execution risk for the dark orders. During the simulations, different values are assigned to these two parameters to examine their influence on market volatility.
Table3lists the parameter values used in the simulation.
These parameters fall into three categories. Some are effectively insensitive. For example, the experiments show that there is no significant influence on the results for 100–3000 agents. The experiments described in this paper consider a relatively small simulation time. The model parameters evaluated by this method includeT, D,S,F, andP0. Some of the parameters are based on the values used in the previous models, likeλ,θ, andU T [9,10], whereas others are taken from empirical studies, such ast ickandP Imax[16,19].
Before conducting the experiments, we first confirm that the model can account for the main characteristics of financial markets, which reflect empirical statistical moments of a market pricing series. In validating that markets are efficient, it is common practice to show there is no predictability in the price returns of assets. To demonstrate this, the autocorrelation of price returns should show that there is no
26 Y. Xiong et al.
Table 3 Parameters in simulation
Description Symbol Value
Number of traders T 100
Number of groups M 3
Trading days D 10
Intraday trading sessions S 100
Order split fractions F 10
Stock initial price P0 10
Tick size t ick 0.01
Dark pool cross-probability cp [0.1,1]
Order urgency distribution [a:b:c] [0.9-MO:0.1:MO]
Market order proportion MO [0.3,0.8]
Dark order submission probability [U1:U2:U3] [0:0.2:0.4]
Order placement depth λ 4
Max price impact P Imax 0.003
Urgency adjust threshold θ 0.05
Unexecuted order threshold U T 0.2
0 5 10 15 20 25 30 35 40 45 50
−0.2 0 0.2 0.4 0.6 0.8
Lag
Sample autocorrelation
Autocorrelation function of returns
Fig. 2 Autocorrelation of returns for dark pool trading model
correlation in the time series of returns. Figure2shows that the price movements generated by the model are in line with the empirical evidence in terms of an absence of autocorrelation.
The characteristic of volatility clustering is seen in the squared price returns for securities that have a slowly decaying autocorrelation in variance. In our model, the autocorrelation function of squared returns displays a slow decaying pattern, as shown in Fig.3.
The autocorrelation values are listed in Table4.
In addition, it has been widely observed that the empirical distribution of financial returns has a fat tail. The distribution of returns is shown in Fig.4. The kurtosis of this distribution is greater than 3, indicating the existence of a fat tail.
Dark Pool Usage and Equity Market Volatility 27
0 5 10 15 20 25 30 35 40 45 50
−0.2 0 0.2 0.4 0.6 0.8
Lag
Sample autocorrelation
Autocorrelation function of squared returns
Fig. 3 Autocorrelation of squared returns for dark pool trading model Table 4 Values of autocorrelation returns and squared returns
Returns 1 0.0282 −0.0253 0.0217 0.0628 0.0007 −0.0591 0.0323 0.0078 . . . Squared returns 1 0.1578 0.1481 0.1808 0.162 0.1216 0.1426 0.1401 0.0836 . . .
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
0 100 200 300 400 500 600 700 800
Sample return distribution
Count
Price return(%)
Kurtosis = 6.4
Fig. 4 Return distribution for dark pool trading model
In each simulation, there areT S =D×S =1000 trading sessions. Assuming the stock price at sessiontisPt, the return at sessiontisRt, which is calculated as:
Rt =Ln(Pt/Pt−1) (2)
28 Y. Xiong et al.
The daily volatility (Vol) of the market is calculated as:
Vol= S
⎛
⎝ 1 T S−1
n i=1
Rt2− 1 T S(T S−1)
n
i=1
Rt
2⎞
⎠ (3)
Assuming that the trading volume at sessiontisVt, the volume weighted average price (VWAP) of the market is calculated as:
VWAP(market)= T S
t=1(Pt×Vt) T S
t=1Vt
(4) In the same way, we can calculate the VWAP of traderk, denoted as VWAP(k).
The price slippage of traderk(P S(k)) is calculated as:
P S(k)=
(VWAP(k)−VWAP(market))/VWAP(market), sell order (VWAP(market)−VWAP(k))/VWAP(market), buy order (5) Let group(A) refer to the group ofntraders (k1,k2,. . . , kn) in which the dark pool submission probability isA.V (ki)refers to the trading volume of traderki. The VWAP slippage of group A (VWAPS(A)) is then calculated as:
VWAPS(A)= n
i=1P S(ki)×V (ki) n
i=1V (ki) (6)
The variance of executed prices (with respect to VWAP) of group A (VEP(A)) is calculated as the weighted square of the price slippage:
VEP(A)= n
i=1P S(ki)2×V (ki) n
i=1V (ki) (7)
Inference
Price volatility measures the extent of price changes with respect to an average price.
For different types of traders, the average prices set here are the same (VWAP of the market). Higher variance in the executed price of a group indicates that the execution of orders in this group tends to make the stock price deviate more from the previous price, thus making the market more volatile [17]. Take VEP(0) as the benchmark. The impact of group(A) on market volatility with respect to the benchmark (RVEP(A)) is calculated as:
RVEP(A)= VEP(A)
VEP(0) (8)
Dark Pool Usage and Equity Market Volatility 29 As the market volatility is the combination of the price volatilities of these three groups, higher price volatility in one group will increase the price volatility of the whole market. This leads to:
1. If RVEP(A) >1, usage of the dark pool makes the market more volatile;
2. If RVEP(A) <1, usage of the dark pool makes the market less volatile.
The price slippage is a derived measurement based on price volatility. The volatility level is mainly determined by the liquidity situation of the market, and the price slippage shows how different traders contribute to this volatility level.
The advantage in using price slippage is its ability to compare differences in the order execution situations of traders (with different dark pool usages) within one simulation, rather than measuring volatility differences among different simulations.
Because the order-balance, order-quantity, and order-urgency in different simula- tions may slightly affect market volatility levels, it is more accurate to compare the price slippages of different dark pool users within one simulation.
5 Result
During each simulation,cpwas assigned a value from the set{0.1,0.2, . . . ,0.9,1.0}
andMO took a value from{0.3,0.4, . . . ,0.7,0.8}. The simulation was repeated 50 times for each setting, giving a total of 3000 simulations. For each simulation, the market volatility, dark pool usage, and the variance of executed prices of each group were recorded. The analysis of variance (ANOVA) was used to test whether different dark pool usage, dark pool cross-probability, and market order proportion had a significant effect on the values of VEP and RVEP.
The means of VEP(0), VEP(0.2), and VEP(0.4)over the 3000 simulations were compared by one-way ANOVA to see whether different dark pool usages led to different values of VEP. The results in Table5indicate that VEP changes with the dark pool usage.
In addition, two-way ANOVA was applied to test the effects of dark pool cross- probability and market order proportion on the mean of RVEP(0.2)and RVEP(0.4).
Table6presents the analysis results for RVEP(0.2).
Table 5 One-way ANOVA to analyze the effect of dark pool usage on VEP ANOVA table
Source of variation SS df MS F P-value Fcrit
Between groups 12,534 2 6267 9.62 <0.01 3.0
Within groups 5,862,478 8997 652
Total 5,875,012 8999
SS: sum of the squared errors; df: degree of freedom, MS: mean squared error
30 Y. Xiong et al.
Table 6 Two-way ANOVA to analyze the effect of dark pool cross-probability and market order proportion on RVEP(0.2)
ANOVA table
Source of variation SS df MS F P-value Fcrit
Cross-probability 0.51 9 0.057 0.78 0.63 1.88
Market order 5.48 5 1.10 15.2 <0.01 2.22
Interaction 3.20 45 0.07 0.99 0.50 1.37
Within 211 2940 0.07
Total 221 2999
SS: sum of the squared errors; df: degree of freedom; MS: mean squared error Table 7 Linear regression
(1) (Objective variable:
RVEP. Explanation variables:
dark pool usage, cross-probability, market order proportion, volatility)
Regression statistics
Multiple R 0.92
R square 0.84
Adjusted R square 0.84 Standard error 0.02
Observation 6000
Table 8 Linear regression (2) (Objective variable: RVEP. Explanation variables: dark pool usage, cross-probability, market order proportion, volatility. SE: standard error)
Coefficients SE tStat P-value Lower 95% Upper 95%
Intercept 0.73 0.003 247 0 0.72 0.73
DP usage −0.24 0.003 −68 <0.01 −0.24 −0.23
Cross-prob. 0.001 0.0012 0.88 0.38 −0.001 0.003
Market order 0.3 0.006 50 <0.01 0.29 0.32
Volatility 1.33 0.12 11 <0.01 1.10 1.57
Table6illustrates that the value of RVEP is affected by the proportion of market orders but is not affected by the cross-probability of the dark pool.
Next, we analyzed how RVEP changes according to these factors. Taking RVEP as the objective variable, we conducted a linear regression to observe the effect of dark pool usage, cross-probability, market order proportion, and volatility on RVEP.
Among these factors, market volatility is a statistic. RVEP and the market volatility were calculated after each simulation, and the analysis investigates the relationship between the two. The following tables present the linear regression results.
Tables7and8 indicate that the market volatility and market order proportion are important and have a positive relationship with RVEP. Moreover, the dark pool usage exhibits a negative relationship with RVEP, which suggests that higher usage of the dark pool leads to smaller RVEP values and tends to stabilize the market. The dark pool cross-probability does not have a significant effect on RVEP.
Figures5,6,7plot RVEP(0.2)and RVEP(0.4)according to different dark pool cross-probabilities, market order proportions, and market volatilities, respectively.
Dark Pool Usage and Equity Market Volatility 31
0.14 0.3 0.46 0.64 0.8 0.96
0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96
RVEP(0.2) RVEP(0.4)
RVEP(0.2) & RVEP(0.4) wrt Cross probability
Ratio
Dark cross probability Fig. 5 RVEP with respect to dark pool cross-probability
0.3 0.4 0.5 0.6 0.7 0.8
0.7 0.75 0.8 0.85 0.9 0.95 1
RVEP(0.2) RVEP(0.4)
Ratio
RVEP(0.2) & RVEP(0.4) wrt Market order proportion
Market order proportion Fig. 6 RVEP with respect to market order proportion
In Fig.5, the results are ordered by the cross-probability of the 3000 simulations and the RVEP is shown as a moving average (over intervals of 500). The red solid line and blue dashed line are the RVEP values given by low and high usage of the dark pool, respectively. Both RVEP(0.2)and RVEP(0.4)are less than 1 for all cross- probabilities, which indicates that using the dark pool decreases market volatility.
In addition, the value of RVEP(0.4)is consistently lower than that of RVEP(0.2), suggesting that higher usage of the dark pool makes the market more stable. Both
32 Y. Xiong et al.
0.62 0.80 1.04 1.44 2.10 2.69
0.7 0.75 0.8 0.85 0.9 0.95 1
RVEP(0.2) RVEP(0.4)
Ratio
RVEP(0.2) & RVEP(0.4) wrt Volatility
Volatility (%)
Fig. 7 RVEP with respect to market volatility
curves fluctuate within small ranges, indicating that the dark pool cross-probability does not affect market volatility. These fluctuations are caused by different market order proportions.
The market tends to be stable when there is a balance between buy orders and sell orders. In this case, the price will move up and down around the midprice. Such order-balance guarantees the executions in the dark pool, and these executions at the midprice of the exchange stabilize the price movements. However, the market tends to be more volatile when there exists an extreme order-imbalance. When these imbalanced orders are submitted to the exchange, they cause rapid price changes and larger spreads. However, if such an imbalance occurs in the dark pool, only a few executions are made until a new balance is formed between buy orders and sell orders. In this sense, orders submitted to the dark pool tend to inhibit the trading volume when the price is changing rapidly, but enhance the trading volume when the price is relatively stable.
According to the above analysis, increased usage of the midpoint dark pool leads to a less volatile market. Although different midpoint dark pools have different crossing mechanisms, the results indicate that the crossing time may not have a significant influence on market volatility.
Figure 6 shows an upward trends in RVEP with an increase in market order proportion. Moreover, the difference between RVEP(0.2)and RVEP(0.4)decreases as the market order proportion rises. This indicates that when the proportion of market orders is small, higher usage of the dark pool decreases market volatility (corresponding to lower RVEP values and a larger difference between RVEP(0.2) and RVEP(0.4)). For larger numbers of market orders, higher usage of the dark pool has less of an effect on decreasing the market volatility (corresponding to higher RVEP values and a smaller difference between RVEP(0.2)and RVEP(0.4)).
Dark Pool Usage and Equity Market Volatility 33 Similar to Figs.6,7reveals the relationship between RVEP and market volatility.
This graph shows that when market volatility is low, higher usage of the dark pool has a definite suppressing effect. However, when market volatility is high, higher usage of the dark pool causes a less noticeable suppression.
When market volatility is low, there is a balance between buy orders and sell orders, and executions in the midpoint dark pool tend to make the market stable.
When market volatility is high, only a small number of executions will happen in the dark pool, so this volatility-decreasing effect is weakened. The difference between RVEP(0.2)and RVEP(0.4)decreases under high market volatility, possibly because the higher usage of the dark pool will not lead to many more executions when the market is very volatile and there is an extreme order-imbalance.
This volatility-decreasing effect of the dark pool is consistent with the findings of Buti et al. [1,2] and Mizuta et al. [10]. Buti et al. [1,2] stated that increased dark pool trading activity tends to be associated with lower spreads and lower return volatilities. Similarly, Mizuta et al. [10] found that as dark pool use increases, the markets become more stable. In contrast, Ray [15] reported that following dark pool transactions, the bid–ask spreads tend to widen and price impacts tend to increase.
This may be because the CN in his sample was relatively small: the average dark pool usage in Ray’s sample was 1.5%, whereas we considered a usage of 20%
(which is closer to the present market rate of 15%). In another model, Zhu [20]
suggested that a higher dark pool market share is associated with wider spreads and higher price impacts. This difference may be because Zhu’s conclusion is based on the assumption that all traders are fully rational. Our model, however, used zero-intelligence agents to investigate the relationship between dark pool usage and market volatility, and did not consider the impact of specific trading strategies.
6 Conclusion
We used an agent-based model to analyze the relationship between dark pool usage and market volatility. We focused on one typical type of dark pool in which customer orders are matched at the midpoint of the exchange’s bid and ask prices.
Simulation results showed that the use of this midpoint dark pool decreases market volatility. This volatility-suppressing effect becomes stronger when the usage of the dark pool is higher, when the proportion of market orders is lower, and when market volatility is lower. However, changes in the cross-probability of the dark pool were not found to have any effects.
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