how your counterparty matters: using transaction networks ... · 1in studies that include portfolio...

Electronic copy available at: http://ssrn.com/abstract=1597738

How Your Counterparty Matters: Using Transaction Networks to

Explain Returns in CCP Marketplaces∗

Ethan Cohen-Cole† Andrei Kirilenko‡ Eleonora Patacchini§

December 1, 2011

Abstract

We study the profitability of traders in two fully electronic, highly liquid markets, the Dow

and S&P 500 e-mini futures markets. We document and seek to explain the fact that traders

that transact with each other in this market have highly correlated returns. While traditional

least squares regressions explain less than 1% of the variation in trader-level returns, using the

network pattern of trades, our regressions explain more than 70% of the variation in returns.

Our approach includes a simple representation of how much a shock is amplified by the network

and how widely it is transmitted. It provides a possible short-hand for understanding the

consequences of a fat-finger trade, a withdrawing of liquidity, or other market shock. In the S&P

500 and DOW futures markets, we find that shocks can be amplified more than 50 times their

original size and spread far across the network. We interpret the link between network patterns

and returns as reflecting differences in trading strategies. In the absence of direct knowledge of

traders’particular strategies, the network pattern of trades captures the relationships between

behavior in the market and returns. We exploit these methods to conduct a policy experiment

on the impact of trading limits.

Keywords: Financial interconnections, contagion, spatial autoregressive models, networkcentrality, trading limits.

JEL Classification: G10, C21

∗We are very grateful to Ana Babus, Lauren Cohen, Ernst Eberlein, Rod Garratt, Thomas Gehrig, Dilip Madan,

Todd Prono, Uday Rajan, Julio Rotemberg, Jose Scheinkman, Yves Zenou as well as conference participants at Centre

for Financial Analysis and Policy (CFAP) conference on Financial Interconnections and the FRAIS conference on

Information, Liquidity and Trust in Incomplete Financial Markets for constructive comments. Nicholas Sere and

Kyoung-sun Bae provided research assistance. All errors are our own.†Corresponding author. Robert H Smith School of Business; 4420 Van Munching Hall, University of Maryland,

College Park, MD 20742. Email: [email protected]; tel.: +1 (301) 541-7227.‡Commodity Futures Trading Commission; 1155 21st Street, N.W. Washington, DC 20581. Email: akir-

[email protected].§University of Rome La Sapienza, EIEF and CEPR. Email: [email protected]

Electronic copy available at: http://ssrn.com/abstract=1597738

1 Introduction

We study the profitability of traders in two fully electronic, centrally counterparty (CCP), highly

liquid markets, the Dow and S&P 500 e-mini futures markets. In these markets, as any other,

traders earn returns by buying and selling assets. Also akin to many other markets, there is a wide

variation in trader returns. We document and seek to explain the fact that traders that transact

with each other in this market have highly correlated returns. This pattern is remarkable for at

least two reason. First, in the markets that we study, because of an automatic matching algorithm,

agents cannot effectively choose their partners so the correlation in returns cannot be due to social

interactions or interpersonal connections. Second, the observed correlation in returns between

traders cannot be attributed to the observed characteristics of traders. The central question of this

paper is whether and to what extent using information on the interconnection between brokers’

trades helps us to understand the trading outcomes. By doing so, we have some insight into the role

of market structure in determining liquidity provision and in understanding shock amplification.

The presence of a computer match-maker linking buyers and sellers allows us to identify, in

a causal sense, the role of the network structure of contacts in shaping individual returns. In

particular, to understand the relationship between the returns of connected traders, we use the

structural properties of the entire network, i.e. all the actual (direct and indirect) connections

present in the market. This helps us both to explain the correlation between linked traders, and

also to understand the degree to which minor changes in the actions or outcomes of a single entity

can amplify into a system-wide effect. Approaches for the empirical estimation of network influences

are widely varied in the financial literature. Some use instrumental variables (Leary and Roberts,

2010), some use summary statistics of network characteristics (Ahern and Harford, 2010, Cohen et

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al. 2008, 2010, Hoberg and Phillips, 2010, 2011, Hochberg et al., 2007, Lin et al. 2011, Faulkender

and Yang, 2011), others use the tools of random networks (Allen and Gale, 2000, Freixas et al,

2000, Brunnermeier and Pedersen, 2009, Amini et al. 2011, Gai et al, 2011). Unlike these, our

approach capitalizes on the network structure itself to measure the properties of the system in a

recursive manner.

Our analysis is motivated by the following simple fact. We find that standard least squares

regression methods using observable characteristics of the markets and traders are able to explain

less than 1% of the variation in trader-level returns. When we expand the toolset to include

the network patterns of connections, the regressions are able to explain more than 70% of the

variation in returns. Indeed, observationally equivalent traders that sit in different networks or

in different network positions earn different returns. Our incremental approach makes this clear;

while some trader characteristics are correlated with returns, considering the spatial allocation of

traders into networks as an additional source of variation greatly improves the explanatory power

of the regressions.

Following these results, we dig deeper into our modelling approach and provide an estimation

of how much a shock can be amplified and how widely it can be transmitted as a function of the

network structure of traders. The link between structure and shock transmission provides some

guidance into how market structure can influence risk. In effect, our empirical approach provides

a short-hand for understanding the consequences of a fat-finger trade, a withdrawing of liquidity,

or other market shock. In the S&P 500 and DOW futures markets, we find that shocks can be

amplified as much as 50 times their original size and spread far across the network.

Why do networks matter for returns? They appear to have great significance, even in these

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two markets, in which a computer assigns trading partners by price and time priority alone. We

interpret the link between network patterns and returns as reflecting persistent differences in trading

strategies. Without direct knowledge of traders’particular strategies, the network pattern of trades

simply captures the relationships between behavior in the market and returns. While the link

between strategies and returns is unobservable, a set of trading strategies intermediated by a

computer can lead to a pattern of trades with correlated returns for connected traders.

Our study focuses on the analysis of the distribution of returns of traders in a single asset. Wide

literatures exist that discuss the investment performance of individuals across portfolios, the price

of individual or groups of assets, etc.1 Another literature exists on the profitability of financial

intermediaries, including specialists and trading desks.2 By studying an individual asset across all

traders, we can isolate the importance of financial interconnections. We contribute by suggesting

that the profitability of trading is influenced by the particular market role, as described by the

position in the network.

With the financial crisis and increasing concerns about financial integration and stability as a

leading example, a large number of theoretical papers have begun to exploit the network of mutual

exposures among institutions to explain financial contagion and spillovers. Allen and Babus (2009)

survey the growing literature. From an empirical point of view, there is little guidance in the

1 In studies that include portfolio management concerns, the heterogeneity in returns has been attributed to costs

differences (Anand, Irvine and Puckett, Venkataraman, 2009, Perold, 1988). Often the differences are found to be

explained by managerial ability in maintaining the persistence in returns over time. For mutual funds, Kacperczyk

and Seru (2007), Bollen and Busse (2005), and Busse and Irvine (2006)) show that mutual funds maintain relative

performance beyond expenses or momentum over multiple time periods.2Reiss and Werner (1998) suggest that interdealer trade occurs between the dealers with the most extreme inventory

imbalances. Sofianos (1995) disaggregates gross trading revenues into spread and positioning revenues and argues

that, on average, about one third of spread revenues go to offset positioning losses. Hasbrouck and Sofianos (1993) find

that specialists are capable of rapidly adjusting their positions toward time-varying targets, and the decomposition

of specialist trading profits by trading horizon shows that the principal source of these profits is short term.

4

literature on how to estimate the propagation of financial distress.3 We contribute to this strand

of the financial connections literature by providing an empirical approach able to measure carefully

the pathways of spillovers in a market with a single asset. By peroviding details on the spread of

risk and the sources of profitability at this level of disaggregation will help with an understanding

of systemic risk and with the development of policy.

The remainder of this paper is organized as follows. Section 2 discusses data and institutional

features of the markets that we study. Section 3 discusses the empirics of trader-level returns.

This includes a standard least-squares estimation as well as a network estimation approach. We

also show how we can improve upon our baseline network regression model and elaborate on our

estimation results to understand the diffusion properties across the entire system following a shock.

We continue in section 4 to further highlight the role of network position to better understand

markets and trader profitability. Section 5 discusses the causal nature of our empirical results and

presents some additional robustness checks. Section 6 extends the work to implement a policy

experiment on the impact of trading limits. We conclude in section 7.

2 Data and Institutional Features

2.1 The CME and Futures Markets

Our data of interest are the actual trades completed on the Chicago Mercantile Exchange (CME)

for two contracts, the S&P 500 and Dow futures. The trades we observe are the result of orders

placed by traders that have been matched by a trading algorithm implemented by the CME. Using

the audit trail from the two markets, we uniquely identify two trading accounts for each transaction:

3See for example, Boyson, Stahel and Stulz (2008) on externalities in hedge fund sector, Adrian and Brunnermeier

(2009) and Danielsson, Shin and Zigrand (2009) on the argument that risk management must be based on more than

individual institutions due to connections between them.

5

one for the broker who booked a buy and the opposite for the broker who booked a sale. For these

two markets, First In, First Out (FIFO) is used. FIFO uses price and time as the only criteria for

filling an order: all orders at the same price level are filled according to time priority.

Each financial transaction has two parties, a direction (buy or sell), a transaction ID number, a

time stamp, a quantity, and a price. We have transaction-level data for all regular transactions that

took place in August of 2008 for the September 2008 E-mini S&P 500 futures and the Dow futures

contracts. The transactions take place during August 2008 during the time when the markets for

stocks underlying the indices are open. Both markets are highly liquid, fully electronic, and have

cash-settled contracts traded on the CME GLOBEX trading platform.

Because these two markets are characterized by the use of price and time priority alone in

determining trading partners, the only phenomenon that generates networks is the pattern of

trading strategies that links traders with each other. Particular patterns of trading will lead to

different probabilities of being at the center or periphery of the network, and to distinct chances

of trading with different types of counterparties. While for each period, we do not observe the

limit order book itself, we know that transactions occurred because market orders or limit orders

were matched with existing orders in the limit order book. We can then trace the pattern of order

execution—a trading network. Figure 1 illustrates this pattern.

[insert figure 1 here]

Empirically, we thus define a trading network as a set of traders engaged in conducting financial

transactions within a period of time; the presence of a link is simply a reflection of the ex-post

realization of a cleared trade.

The choice of the period of time within which a network is defined is important as it contains

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valuable information on the resulting network structure. Indeed, with more time, more transactions

are formed and more participants can form accurate beliefs about the valuation of a given asset.

Our approach is to define the network as a given number of transactions among traders that

are either directly or indirectly linked. Then, throughout the remainder of the paper, we will use

a range of network densities in order to ensure that our results are robust to this choice. More

specifically, we designate a network as a sequence of consecutive transactions. What we will call

‘sparse’networks are defined as containing 250 transactions, ‘moderately dense’networks contain

500 transactions and ‘dense’networks contain 1000 transactions.

While one could imagine alternate approaches,4 our evidence supports the above choice, i.e.

a definition of networks as a given number of transactions. Indeed, our results on the existence

of network effects are strongly robust when we vary the number of transactions. As well, the

fact that we find our chosen network definition has enormous empirical salience suggests that we

have chosen a reasonable concept for the network. In addition, there is no reason to believe that

an incorrect choice of network timing would lead to the spurious finding of a strong relationship

between networks and returns. Indeed, the opposite is true; a randomly defined network will show

no evidence of network effects by construction.

The networks that we define are distinct from one another over time. This occurs both because

agents may not be active in each time period and because their transactions are matched by the

trading algorithm in each time period.

4For example, an alternative would be to define the network based on some period of time or number of transactions

beginning at the point of a market shock, such as a significant price change.

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2.2 Returns and Descriptive Statistics

Each broker in this market earns a return. For example, buying a contract for a price of $1 and

selling it for $1.10 yields a profit of $0.10 and a return of 10%. Because some positions do not

clear during a given network time period, we report realized returns when positions clear during a

network time period. When they do not clear, we report the mark-to-market returns for the trader

in question.

Our S&P 500 futures dataset consists of over 7,224,824 transactions that took place among

more than 31,585 trading accounts. The DOW futures dataset consists of 1,163,274 transactions

between approximately 7,335 trading accounts. We show in Table 1 some simple statistics of the

data for each of the two markets that we analyze. For each definition of networks, we compute

returns for each trader, volumes for each trader as well as the variance of returns across traders

over the course of a trading day. Returns are shown as absolute levels of holding at end of time

period based on an initial investment of $1; thus, a return of 1 indicates that the trader broke even

during the time period. Average returns vary from a loss of 4 basis points to a gain of 4 basis

points. Of course, individual level results vary more widely.

Of note is that the average return across trading accounts is below 1, suggesting that traders

with high volume, on average, earn higher returns. We report the returns unweighted by volume; the

weighted average return across traders is, by construction in futures markets, equal to 1. Also note

that the standard deviation of returns and volume is increasing in the density of the networks. As

the number of transactions increase, the variance does so as well. Notice that the mean transaction

volume declines as the density of the network (i.e. the number of transactions) increases. This

pattern reflects the skewness in the data. There are large numbers of transactions of low volume

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and small negative returns, and a relatively smaller number of observations with higher volumes

and positive returns.

[insert table 1 here]

2.3 Why would individual returns be correlated?

The observed network of realized trades is a potential tool to describe the (unobserved) strategic

interactions at work in the market.

Consider a group of traders. These traders enter each day with a set of trading strategies. These

strategies can either be formal or informal, automated or manual. Indeed, the market contains

some of each of these. Among these formal strategies, for example, are algorithmic traders. These

computerized high-frequency traders composed approximately one-third of volume (Kirilenko et al,

2011) on any given day. The strategy of any given trader will depend on the anticipated strategies

of other traders as well as the observed actions during the day. As successful strategies become

known, followers emerge and copy the strategy. As long as traders either use correlated strategies

or condition their strategies on like information, their behaviors may be correlated in equilibrium

and thus as well in the observed data.

Of course, these correlated bidding patterns lead to similarity in returns. Because the matching

algorithm used by the CME is blind to identities of the traders, traders with correlated strategies

will trade with each other as well as with others. As they do so, and form links with one another,

correlation in trading strategies leads to a connection between strategies and network position.

Many traders will acknowledge that sitting between two traders with fundamental liquidity needs

can be profitable.

Note that futures markets are zero-sum markets in aggregate. Thus, while each transaction

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could potentially yield a profit for both parties, some portion of the network must absorb equal

losses for each gain. We illustrate how two traders could both profit from a transaction in figure 2.


Figure 2 shows the presence of two large traders (denoted ‘A’& ‘D’) that have fundamental

liquidity demands, one positive and one negative. Each of these participate in the futures market

by placing large one-sided orders either to buy or sell contracts.

A separate set of traders, denoted ‘B’, implements rapid offers to buy and sell. The objective

of such traders is to provide the liquidity needed by the large traders with fundamentals demands.

Because the large traders may not appear on the market at precisely the same time, the liquidity

providers can extract profits from the large traders by being willing to transact when needed.

The combination of the liquidity traders’actions can generate a diamond-shaped network pattern

illustrated in figure 2, panel A. On one side, the liquidity traders buy as needed and on the other

they sell as needed. By being willing to buy and sell, the agents in the center can generate profits.

Of course, knowledge that agents can achieve these profits leads to a new set of trading strategies.

Figure 2 shows the emergence of additional agents, denoted ‘C’. Effectively, the second set of agents

hopes to intermediate between one large trader with fundamental demand and the initial set of

liquidity traders. Now, if one evaluates the correlation in returns over a given period of time of

these traders, she will observe that the profits of the liquidity traders are inversely correlated with

those at the ends of the diamond. As the large traders lose money the liquidity traders earn (or

vice-versa). However, our new entrants ‘C’, over time, yield returns that are positively correlated

with the other liquidity traders.

An example is presented in Panel B. The table shows the returns of each set of traders in a

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hypothetical case. The outcome in case 1 is that the returns of A and D are negatively correlated

with the returns of B. That is, the market will act like a shock absorber. As new shocks hit the

system, the reaction is to ameliorate the impact.

In case 2, the returns of A and D continue to be negatively correlated with B and C; however, B

and C show positively correlated returns. It is straightforward to see that as the number of traders

in the center of diamond increases, a market change that impacts one of the traders will have a

similar impact on the others.

To see how two traders could both profit from an interaction, consider A, B, and C. A purchases

a contract from C for $2 at time t and buys one from B for $1 at time t + 1. At time t + 2, C

purchases a contract from B for $1.25. The final transaction yielded C a profit of $0.75 and B

a profit of $0.25. Of course, A has lost the full dollar in the process. The trade between A and

C allows them to share the $1 gain. Repeated interactions of this type will generate a positive

correlation in returns.

These examples help understand how returns can be correlated across trading strategies, but

also importantly help illustrate how shocks can be propagated. That is, they are a representation

of the pathways of the transmission of risk in the system.

3 Empirics of Trader-Level Returns

3.1 Traditional regression models

Assume that there are N traders divided in k = 1, ...,K networks, each with nk members, i =

1, ..., nk,

K∑k=1

nk = N . In this section, we ignore the network structure of the connections. We

simply use the network as the relevant "market" (or trading period) for each trader. As explained

before, we consider networks as sequences of trading of 250, 500 and 1000 trades. Parsing trading

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activity in this way allows one to avoid variations in returns that may occur solely due to the ebbs

and flows of trading. We thus run least squares regressions including simple statistics of a number

of relevant characteristics to capture aggregate variations of market conditions.

For each day or half day of activity (indexed by t), we evaluate the role of a range of control

variables on returns. Our specification is

ri,κ,t = β0 + β1buyvolumei,k,t + β2sellvolumei,k,t + β3variancei,t + ... (1)

+β4volumet−1 + β5laggedreturnsi,t−1 + υi,κ,t

for i = 1, ..., nκ; κ = 1, ...,K.

where buyvolume is the number of contracts purchased in the relevant trading period for each

trader i (defined by network k each trader belong to) and similarly sellvolume is the number of

contracts sold in the trading period. Recall that returns are calculated as mark-to-market value

of contracts at the end of the day (or half-day). As a result, buyvolume does not always equal

sellvolume during a particular trading time period. V ariance is the variance of a trader’s return

during the full (or half) trading day, volume is the net trading (buy-sell) aggregate volume in the

prior time period, and laggedreturns is the trader’s mean return in the first half of the trading

day. This final variable only appears in specifications estimated on data from the second half of the

day and captures potential persistence in returns. In all specifications, we also include a control

for trades late in the day.

Table 2 collects the estimation results. Note first that the vast majority of coeffi cients have

economically insignificant magnitudes. With the exception of half-day returns, other variables have

essentially no impact on returns. Some variables show statistical significance in places, though the

large number of observations makes this unsurprising. Second, we highlight that the adjusted R

12

squared statistics of these regressions are very small. The largest is .003. This reflects the fact that

individual level returns are very, very diffi cult to explain. Indeed, there is little theory that would

point towards an ability to do so without linking the traders to particular investment strategies or

variations in access to information.


3.2 Network regression models

Having seen that the OLS results of the above specifications do a relatively poor job of characterizing

the distribution of returns, we look to augment our specification to capture the information content

of the network connections.

Effectively, we will augment specification 1 with an regressor capturing the returns of connected

agents. For example, to consider the influence on i of only a single other agent (j), specification 1

would read

ri,κ = α0 +

M∑m=1

βmxmi,κ + γrj,k + υi,κ, (2)

where xm denote the set of m variables considered above and rj denotes the returns of the trading

partner. So an estimated coeffi cient γ greater than zero indicates that returns for trader j are

positively correlated with returns for trader i. Extended to a simple network of three agents (i, j, s)

t t ti j s

the equation expands to

ri,κ = α0 +

M∑m=1

βmxmi,κ + γ1rj,k,d + γ2rs,k,2d + υi,κ, (3)

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where the subscript d and 2d indicate agents j and s at one node and two nodes distant from i,

respectively. The coeffi cient γ1 captures correlation in returns between directly connected traders,

whereas γ2 the correlation between agents further away in the network structure. These multiple

steps are important; they are similar in spirit to multiple lags in a time series regression. The set

xm now also includes additional regressors for the characteristics of every other agent. Thus, as the

number of agents increases and the network expands, we can continue to add additional regressors

to the right hand side of this specification for each agent and each degree of separation from agent

i. Eventually, we will add n− 1 regressors for each degree of separation, leading to a very complex

specification that takes into account each type of influence of every agent on every other.

To include every other agent and every degree of separation, and to simplify notation, we can

introduce a matrix that keeps track of the links between agents. This is a N−square adjacency

matrix G = {gij} whose generic element gij would be 1 if i is connected to j (i.e. interacts with

j) and 0 otherwise. Here gij = 1 if trader i and j have concluded a transaction during a period of

time, and gij = 0, otherwise. This matrix represent the interaction scheme of the traders in the

market. The G matrix associated with the simple network in the picture above is:

G =

i j s

i 0 1 0

j 1 0 1

s 0 1 0

indicating that i trades with j, s with j, and j with i and s.

Then, we can collapse the above specification with all traders at every level of interaction into

the following simplified one:

ri,κ = α0 +

M∑m=1

βmxmi,κ + θ1

gi.,k

nκ∑j=1

gij,κrj,κ + υi,κ, for i = 1, ..., nκ ; κ = 1, ...,K. (4)

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where ri,κ is the idiosyncratic return of trader i in the network k, gi.,k =∑nκ

j=1 gij,κ is the number

of direct links of i, 1gi.,k

∑nκj=1 gij,κrj,κ is the average returns of trading partners, υi,k is a random

error term, xmi,κ is a set ofM control variables at the individual and/or network level. This model is

the so-called spatial lag model or spatial autoregressive model in the spatial econometric literature

(see, e.g. Anselin 1988) and can be estimated using standard software via Maximum Likelihood.

The object of estimation is now θ.

Table 3 shows the estimation results of this model specification vis-a-vis the previous traditional

one. We report the identical regressions as in Table 2 and the augmented regressions based on

the network approach. It is striking to see the substantive improvements in fit coming from the

additional regressor. Indeed, as we inspect the adjusted R squared coeffi cients, we note that they

are considerably higher than the OLS specification. The additional information in trading partner

returns is systematically important in predicting my own returns. The R squared coeffi cients now

range from .05 to .37 for the S&P and .04 to .18 for the Dow. In one case, these regressions explain

more than 1/3 of the variation in trader returns.


In terms of the coeffi cient of interest, θ, the estimates for the S&P are between .02 and .27

depending on the day and the network type; similarly, they are between .02 and .10 for the Dow.

Most of these are estimated with a very high degree of precision. This suggests that increases in

trading partner returns could be important for individual outcomes.

In order to highlight the relative importance of the network effects vis-a-vis other possible

controls, such as those included in the least squares estimation table 2, we estimate model (4)

without controls. Table 4 shows the results. The coeffi cient estimates for the network effect, θ, as

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well as the values of the R squared are nearly the same as in table 3, suggesting that the improved

fit of the regression is due to the network approach. The principle difference that we observe is

that for some specifications, the t-stats are lower than they were in table 3. This reveals that while

the control variables do not explain much of the variation in returns, they absorb some estimation

noise, allowing the network effects to be estimated more precisely.


In the remainder of this section we will improve model (4) and elaborate upon this specification

to highlight the role of networks in propagating changes in returns across traders.

3.3 Weighted Networks

We extend the simple network model above to use the network equivalent of importance weights

in an OLS regression. While the baseline network model had strong results, we hypothesized that

the size of the transactions in the network would be a determinant of outcomes. Trading with large

counterparties would be different than trading with smaller ones. We will measure the importance

of traders by total trading value and replace the binary matrix G with a new matrix capturing

both the number of links and the importance of each link. Let the matrix W = GD, where G

is as defined above and D = {dij} is a matrix that weights the links within the network. The

scalar dij is a scaling factor, calculated as the total trading volume in the same trading period (the

network) of each i and j. Total trading volume is defined as the sum of all trades, both buys and

sells, made by trader i with all other traders. As a result, W= {wij} is now a weighted network

w. Figure 3 provides an illustration of the calculation of these weights. It shows a set of four

transactions amongst four traders, A,B,C and D. Each arrow is a single transaction, with the arrow

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pointing towards the buyer of a contract. In panel A, each value along an arrow shows the number

of contracts traded. The accompanying matrix is an unweighted network representation of the

transactions, i.e. the G from above. Each cell contains a ‘1’where two brokers have transacted and

0 otherwise. Panel B shows the same set of transactions, but having along each arrow a calculation

equal to our measure of importance: (total number of contracts bought or sold by buyer + total

number of contracts bought or sold by seller)/2. These values are the ones used for the weights

D = {dij} to get a weighted accompanying matrix W .


Results for this method are presented in table 5. We replace G with W in model (4) and run

the same regression again. We again follow the format of displaying results by the density of the

network.


All of the results are now stronger. First, note that the estimated correlation between trader

returns is now greater than 0.9 in the S&P and greater than 0.8 in the Dow. That is, the returns

a trader earns are very similar to those of her trading partners.

Second, across densities of network structure, we find estimates of θ that are large and always

statistically significant. Across each specification, the observed t-statistics increase; the estimation

is now much more precise than without the weights.

Third, these new specifications are able to explain a much larger fraction of the variation in the

trader-level returns. The adjusted R squared values are now uniformly above 70% in both markets.

Both the structure of the connections and their importance are important in understanding returns.

17

The last row of table 5 report values for the average multiplier, φ. Indeed, our network approach

(model 4) allows us to measure the aggregate amount won or lost by agents connected at any level

to a trader. Thus, if a trader wins $1, the multiplier measures how much traders in the network win

or lose. Because the coeffi cient θ measures the average correlation in returns across traders linked

by a single node in the network, θ2 measures the average correlation across two links, θ3 the average

across three, etc. Thus, a simple calculation allows us to measure the impact of a shock to any

given trader. Consider a shock of $1. On average, this will lead to a change in earnings of directly

connected agents of θ ∗1, agents two links away of θ2 ∗1, etc. One can see then, that for each dollar

won or lost by a trader, φ = 11−θ is the aggregate amount won or lost by agents connected at any

level to the trader. For example, an estimate of θ equal to 0.5 produces a multiplier of 2, suggesting

that for each dollar lost by a trader hit by an exogenous shock, the individuals connected to the

trader will lose an aggregate of 2 dollars. Because the market is a zero-sum one, if all agents are

marked-to-market at the time of the idiosyncratic loss, the two dollar losses of the trading partners

will be offset by two dollar of gains elsewhere in the network. Our measure is thus a calculation of

the degree of re-allocation of profits.

We report the value of φ below each specification. We can see that the multiplier is between

16 and 66 for the S&P and 5 and 21 for the Dow. These large numbers imply that these trading

networks have very high sensitivity to shocks. Small changes to one individuals rapidly spread and

magnify. As it will be explained in greater detail in the next section, these effects depends on both

the structure of the connections and on the strength of the interaction, as captured by θ.

The average multiplier can be interpreted as a compact statistic for a type of systemic risk of

a system. Indeed, the calculation of an average spillover following a shock defines the degree to

18

which idiosyncratic losses become systemic ones.

4 Networks, Centrality and Profitability

Now that we have established the strong empirical relationship link between structure of a network

and the returns of individual traders, we dig more deeply into the network structure to understand

to role of the individual position in the network in shaping individual returns. First, we discuss

one particular measure of network centrality, Bonacich Centrality (Bonacich 1987). This measure

is simply a characterizatio that emerges from our network model and helps to provide a simple

correspondence to returns. Second, we illustrate how this relationships between individual centrality

and returns leads to measurable distributional impacts in returns.

4.1 Centrality and returns

Different measures of individual centrality have been proposed in the network literature (Wasserman

and Faust, 1994). We consider here a particular measure, Bonacich centrality, which can be derived

from the estimation of model (4) and has a number of useful properties. First, it measures the

importance of each agent in a network taking into account not only the number of direct connections

but also their importance in terms of number of connections in a recursive manner, so that the

entire network structure is taken into account. Second, it is not parameter-free, but it depends

on the level of strategic interactions that stems form the network, as captured by our parameter

θ. Specifically, the Bonacich Centrality is a count of the number of all direct and indirect paths

starting at node i and ending at node j, where paths of length p are weighted by θp. Recall that

this was how we compressed the large number of variables into a simple specification, (4). More

paths from i to j imply a more central trader. A full description of the Bonacich measure, including

19

the connection with our model (4), is contained in Appendix A.

To illustrate the idea and its relevance to a trading network, consider the original Bonacich

(1987) example. Bonacich considers a network of individuals that communicate with each other.

The parameter θ measures the probability that a communication will be transmitted by any indi-

vidual to any of his contacts. θW is the expected number of these communications that are passed

on to direct contacts, θ2W 2 are the ones passed on to contacts two links away and θpW p is the

expected number of messages that reach agents at path-length p.

In the context of a trading network of mutual exposures, the magnitude of θ, combined with

the network structure W each trader is embedded in, thus reflects the degree to which a shock

is transmitted locally or to the structure as a whole. Small values of θ heavily weight the local

structure, while large values take into account the position of agents in the structure as a whole

(Bonacich, 1987). We include an illustrative figure 4 that shows the impact of a shock to a trader

in networks with different θ values. One can see that as θ becomes larger, the shock transmits more

widely across the network, i.e. it impacts traders much further away in the network.


To understand the link between network topology and returns, we can use our estimated θ to

calculate the Bonacich centrality for each trader in our networks (formula 7 in appendix A).5

With this distribution of positions in the network, we can look at the outcome differences across

traders of different centrality levels. We do so by looking at the impact of a one unit change in

centrality on returns. Table 7 reports both the standard deviation of the centrality measure as

5This calculation will generate a distribution of individual centralities depending on the strength of network

interactions and on the heterogeneity of network links (as captured by the estimate of θ and the matrix W in formula

(7), respectively).

20

well as this impact for sparse, moderately dense and dense networks.6 We report absolute changes

in returns; because the benchmark return is 1, the numbers can also be interpreted as percentage

changes. They are changes in returns over a one-day time periods. We do not normalize to an

annual basis. It appears that high returns are associated with high degrees of centrality irrespective

of network complexity.


We note two patterns. One, the standard deviation of the centrality measure is nearly identical

across the three network types; indeed, it’s relatively similar across markets. Two, the impact of

a one unit change (approximately 1/3 of a standard deviation), is also relatively constant across

network densities.

We highlight this finding as it suggests, in part, that our network definition is effective. Even

though we construct our networks based on an ad-hoc choice of transactions, the impact of the

networks that we define remains consistently important throughout the measured time period.

While one could potentially improve upon the definition, the strength and consistency over time of

these findings suggests that we have captured a large portion of the network effect.

Taken as a whole, this evidence indicates that the number of direct and indirection connections

in the network, as weighted by θ, is a relevant factor that plays a role in explaining the cross-

sectional variation of returns.6Note that we do not put centrality on the right-hand-side of our regressions. The impacts are derived from a

simple transformation of the estimated THETA from Model (4).

21

4.2 Network structure and distributional effects

Our analysis so far shows to what extent network position (network centrality) of an individual

trader was important in explaining the level of individual returns. The more central a trader

emerges from the exogenous matching process, the higher his returns.

In the remainder of this section, we highlight the implication of differences in network structures

in terms of the distribution of outcomes in financial networks. That is, is there a difference in the

variance of returns for traders operating in different types of networks?

Recall first a few stylized facts. One, we find that network structure very well explains individual

level returns. Two, we find that the average multiplier, as measured by the ratio of an aggregate

impact to the level of an individual shock, is very high in the networks that we analyze. Three,

we find that an improvement in terms of centrality for an individual trader is associated with a

positive change in returns.

Given these three findings and the fact that futures markets are zero-sum, we can make two

claims. First, at the level of a network (250-1000 transactions), we should see that a change in

the distribution of the centrality measure should have no change on the mean return in a network.

That is, an arbitrary re-allocation of individuals around the network should change the distribution

of outcomes, but not the mean.7 Second, it thus follows that one should find differences in the

variance of returns. We find evidence of these two phenomena in our data.

Figure 5 displays the results. It relates the impact of network centrality to the variance of

returns in the network, and finds a positive relationship. It also shows that the aggregate mean

7We discussed above that the unweighted mean of returns at the network level may not always be one, given that

some traders earn large profits. Precisely, the re-allocation can impact to a small degree this unweighted return, but

cannot impact the weighted network returns—which must always be equal to one.

22

of returns remain roughly unchanged. As centrality becomes more important, the distribution of

returns widens. This is a logical implication; if being central leads to greater returns, in a zero-sum

market, this necessarily means that someone at the periphery must lose out, and the variance of

returns widens.


Technically, the relationship shows that the distribution of returns of the network with greater

sensitivity to centrality stochastically dominates (in a second order sense only) the distribution of

returns for a network with lower sensitivity to centrality.

5 Discussion and robustness checks

The validity of our analysis and its relevance for policy purposes hinges upon the correct identifi-

cation of the network effect, θ.

The core problem that emerges in estimating linear-in-means models of interactions is Manski

(1993)’s reflection problem. This arises from the fact that if agents interact in groups the expected

mean outcome is perfectly collinear with the mean background of the group: how can we distinguish

between trader i′s impact on j and j′s impact on i? Effectively, we need to find an instrument:

a variable that is correlated with the behavior of i but not of j. Cohen-Cole (2006) noted that

complex network structures can be exploited for identification. Bramoullé, Djebbari and Fortin

(2009) highlighted the same phenomenon and showed that in network contexts, one observes ‘in-

transitivities.’ These are connections that lead from i to j then to s, but not from s to j (see

picture). Thus, we can use the partial correlation in behavior between i and j as an instrument for

the influence of j on s.

23

That is, network effects are identified if we can find two agents in the economy that differ in the

average connectivity of their direct contacts. A formal proof is in Bramoullé, Djebbari and Fortin

(2009). As a result, the architecture of networks allows us to get an estimate of θ, while eluding

the “reflection problem.”Of course, a complex trading network such as the one we are concerned

with has a very rich structure of connections and identification essentially never fails.

Another traditional concern in the assessment of network effects in the social sciences is that net-

work structure can be endogenous for both network self-selection and unobserved common (group)

correlated effects. The first problem might originate from the possible sorting of agents. However,

given our definition of networks based on high-frequency data and a random matching algorithm,

we have no reason to believe that any selection effects exist in this context. Agents are assigned to

trading partners as we described above, based on time and price priority alone. Even if two traders

were to attempt to time a transaction as to ensure a match, the high volume of transactions on

these markets makes this nearly impossible to complete. As such, we have a strong claim that

individuals cannot choose their network partners and thus no selection effects should be present.

Network topology is exogenous here. The possible presence of unobserved correlated effects instead

arises from the fact that agents in the same group tend to behave similarly because they face a

common environment or common shocks. These are typically unobserved factors. For example,

traders with similar training, that sit in similar rooms or use trading screens that show similar

types of data, may be influenced in their trading patterns in ways that generate correlations in

returns. While we believe this to be very unlikely, we can control for these unobserved effects by

re-estimating our model after taking deviations in returns with respect to the group-specific means,

i.e. from the average returns of (direct) trading partners. That is, if agents in a given empirically

24

observed network have some similarity that leads them to earn higher returns as a group, we will

average out this group-level effect and look only for the presence of spillovers. Of course, our pri-

mary specification already largely nets out market level returns by virtue of the fact that aggregate

market levels returns are 1. In this case, we also control for group-level unobserved heterogeneity.

That said, there is little reason to believe that in an electronically matched market one would

observe any effect of this sort.

Results are in table 6 and indeed illustrate very small differences from those in table 5.


These results are useful also for another reason. The market that we are discussing is a zero-

sum one; benefits to a given individual are necessary reflected in losses to another. As a result,

complementarities in returns must necessarily be reflected in losses elsewhere in the network. By

estimating our results in deviations from average level returns for an individual’s own ‘network,’

we handle this issue. In deviations, complementarities will no longer be reflected elsewhere in the

network structure and we can consequently use our results to evaluate the impact of a shock to the

system. The particular context of analysis and our approach thus enables us to uncover a causal

relationship between network structure and profitability.

6 A Policy Experiment

One of the advantages of this approach is that it provides a mechanism via which policy makers

and regulators can understand the impacts of their choices on the risk in the system. As a leading

example, the August, 2010 passage of the Dodd—Frank Wall Street Reform and Consumer Protec-

tion Act (Dodd-Frank) included a call for the evaluation of position limits in futures markets. The

25

impact of such limits has been fiercely debated.

In this section, we construct a counterfactual study that explores the consequences of this

policy using our framework. Our exercise runs as follows. We set an arbitrary transaction limit

for a given period of time. Given the restriction, we re-estimate our model (4) assuming that any

traders that, in the data, transact a greater number than this amount, transacted only the fixed

maximum. Specifically, we restrict to C the number of contracts that can be purchased in 1/10 of

a trading day, thus setting artificial bounds on the weights of our matrix W . This does not change

the network structure other than the weights of the links. We consider C = {2, 3, 7, 10, 20, 30, 100}

for the Dow futures market and investigate the consequences of such limits in terms of the high

returns are associated with high degrees of centrality irrespective of network complexity. estimate,

θ̂. Figure 6 shows the results for each of the values of C.

[insert figure 6]

The simulation has two policy interpretations. First, we see from the figure that tighter trading

limits leads to higher average multiplier values. Above we noted that these multipliers can be

interpreted as a measure of systemic risk; however, here some additional detail is warranted. In

our context, systemic risk is a measure of the size of the passthrough that occurs following an

idiosyncratic shock. This is conceptually distinct from increases in the frequency of shocks (which

we do not address). What we observe from this exercise is that the size of shock propagation

increases as trading limits become tighter. In the case explored here, a move from no position

limits to a strict one would increase the systemic risk multiplier in the system from approximately

13 to 16.

Second, we can also infer from the exercise that tighter limits distribute the impact of the shock

26

across a wider range of market participants. That is, while a shock in the constrained world may

be widely distributed, an equivalent shock in the unconstrained world to a large trader may pass

to only a small number of counterparties. This phenomenon arises because in our experiment we

do not simulate new links between traders, the mechanism by which systemic risk increases is to

decrease the centrality of the network; that is, it downplays the importance of the traders who had

previously exceeded the limit and been quite central.

Effectively, this highlights that the policy comes with a distinct tradeoff. One one hand, in

our simulation, it has the potential benefit of dispersing adverse shocks to a wider range of market

participants. On the other hand, the limits also appear to generate aggregate larger consequences

from each shock. The five dollar loss may now be magnified to 6 or 7. The trade off of the two

will determine the aggregate impact of the policy and its final impact will undoubtedly be market

specific.

7 Conclusions

Our analysis explained a conjectured but to date unproven feature of financial markets: returns

from trading are correlated with the position agents occupy in a trading network. Using our

network-based empirical strategy on two highly liquid financial markets, we are able to explain a

large portion of the individual level variation in returns. This finding has potentially large salience.

Most importantly, one of our results is that individual level shocks are greatly amplified and

spread in these markets: a one unit change in individual level returns can be amplified even 50 times.

This implies very rapid propagation of shocks and little ability to avoid contagion. Because these

results are a function of the network structure, these results point policymakers in the direction of

potential interventions. Notice that the rapid spread and amplification derives from the network

27

structure; adjust the structure and adjust the speed of spillovers. This points towards interventions

in the matching algorithm, potentially during times of anticipated crisis. For example, one could

alter or eliminate most of the network spillovers altogether by concentrating trading into hourly

auctions.

A long literature in sociology and economics would suggest that network patterns are important

in non-market interactions based on a variety of plausible mechanisms. These include social stigma,

information sharing, peer pressure, and more. The diffi culty in translation of the methodologies

developed in the social science to financial markets, particularly electronic ones, is that there is

little basis to believe that any of the mechanisms are at work. Orders are matched at random by

a computer based on time and price priority, leaving little room for social impact even if traders

had a motivation to do so. Thus, our conclusions are statements about the empirical importance of

the networks that emerge as a result of equilibrium order strategies. We find that these strategies

not only lead to networks of note, but that an empirical mapping of the networks to returns shows

important effects.

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8 Appendix A: Spatial autoregressive model and network central-ity

For ease of interpretation, let us write model (4) in matrix notation and derive the reduced form.

The following derivations are helpful to understand why model (4) captures recursively the network

effects at any degree of separation and the link with a particulat network centrality measure,

Bonacich centrality (Bonacich, 1987).

Model (4) can be written as:

r = θGr+βx+ ε, (5)

where r is a N × 1 vector of outcomes of N agents, x is a N × V matrix of V variables that may

influence agent behavior but are not related to networks, G is the row normalized version of the G

32

matrix which is used to represent average returns and ε is a N × 1 vector of error terms, which are

uncorrelated with the regressors.

Given a small-enough value of β ≥ 0, one can define the matrix

[I−θG]−1 =+∞∑p=0

θpGp (6)

The p − th power of the matrix G collects the total number of paths, both direct and indirect, in

the network starting at node i and ending at node j. The parameter θ is a decay factor that scales

down the relative weight of longer paths, i.e. paths of length p are weighted by θp. It turns out

that an exact strict upper bound for the scalar θ is given by the inverse of the largest eigenvalue of

G (Debreu and Herstein, 1953).

In a row-normalized matrix, such as the one used in model (4), the largest eigenvalue is 1. If |θ| <

1 expression (6) is well-defined, that is, the infinite sum converges. The condition |θ| < 1 capture

the idea that connections further away are less influential than direct contacts and guarantees that

the matrix [I−θG]−1 is able to capture all the effects that stems from a given network topology,

that is the cascades of effects stemming from direct and indirect connections.

If |θ| > 1, the process is explosive. In a financial network context, it is equivalent to a complete

financial collapse. While interesting in its own right, we do not analyse this case here. We focus on

how, even in the absence of a complete financial collapse, a small shock can cascade causing large,

measurable and quantifiable damage. Therefore we consider |θ| < 1.

If one solves for r in model (5), the result is a reduced form relationship:

r = [I−θG]−1 βx+ [I−θG]−1 ε

Definition 1 (Bonacich, 1987) Consider a network g with adjacency N−square matrix G and a

33

scalar θ such that M(g, θ) = [I−θG]−1 is well-defined and non-negative. The vector of centralities

of parameter θ in g is:

b(g, θ) = [I−θG]−1 · 1. (7)

The centrality of node i is thus bi(g, θ) =∑n

j=1mij(g, θ), and counts the total number of paths

in g starting from i. It is the sum of all loops mii(g, θ) starting from i and ending at i, and all

outer paths∑

j 6=imij(g, θ) that connect i to every other player j 6= i, that is:

bi(g, θ) = mii(g, θ) +∑j 6=i

mij(g, θ).

By definition, mii(g, θ) ≥ 1, and thus bi(g, θ) ≥ 1, with equality when θ = 0.

Therefore, once one has on hand an estimate of θ it is possible to derive the distribution of

Bonacich centralities for all the agents in the network. Observe that in the original Bonacich (1987)

paper, the centrality measure is presented for unweighted networks. However, all the techniques

apply to the weighted network case, i.e. G =W (see Newman, 2004, for a discussion).8

8We are grateful to Jose Scheinkman for calling our attention to this.

34

A B C D E F Order Strategies

Order Submissions

Order Book

A B C

D E F

Empirical Pattern (Network)

Matching Engine

Figure 1

Note: Each node in the section labeled ‘order strategies’ represents a single trader’s plans for trading. The ovals beneath each trader, next to the label ‘ordersubmissions’ represents actual placed order. Below this, we denote with a box the complete order book. This is the aggregation at each point in time of allthe orders submitted by traders. This order book is passed through the box beneath it, which we have labeled a ‘matching engine’. This computer matchesorders based on price and time priority. Finally, beneath the matching engine, we provide a sample representation of the network patterns that could emergefrom a set of 6 completed transactions.

35

A D

BB

BB

BB

BB

BB

A D

BB

BB

BB

BB

BB

CCCCCC

Panel A

Panel B

Note: Panel A shows two agents with fundamental liquidity needs, marked A and B, and a series of agents that have traded with them. Each edge is marked asan arrow, pointing from the seller to the buyer. Panel B shows the same configuration with the addition of a few additional agents. The example assumes thatthe market price is constant at 100.

Figure 2

CCCC

Case 1 Case 2

Case1 Case2Action Returns Action Returns

A Sell100at99 0.99 Sell100at99 0.99B Buy100at99

Sell100at1011.02 Buy100at99

Sell100at1001.01

C Buy100at100Sell100at101

1.01

D Buy100at101 0.99 Buy100at101 0.99

36

Note: Panel A shows a set of transaction between 4 traders. Each arrow is a single transaction, with the arrow pointing towards the buyer of a contract. Eachvalue along an arrow shows the number of contracts traded. The accompanying matrix is an unweighted network representation of the transactions. Each cellcontains a ‘1’ where two brokers have transacted.Panel B shows the same set of transactions. Along each arrow is a calculation equal to the (total trades of buyer + total trades of seller)/2. These values arethen used as weights in the accompanying matrix.

Figure 3

D

A B C

102

3

4

D

A B C

(16+10)/2=13

(16+5)/2=10.5

(5+7)/2=6

(16+7)/2=11.5

Transactions

Bilateral Volume

11111111

A B C DA B

C D

5.115.10135.1165.106

13A B C D

A B

C D

Panel A

Panel B

37

0 5 10 15 20 25 30 35 40 45 500

0.02

0.04

0.06

0.08

0.1

0.12Impulse Response Diagrams for Various Estimated Theta

Trader distance from shock (# network links)

Ave

rage

Im

pact

Figure 4

Note: The figure shows the impact of a 1 unit shock to a network. Each line shows the impact of the shock for a different value of θ. The distance from shockshows the impact as one moves away from the origin of the shock to the remainder of the network, traversing along only trading relationships. For example, 2degrees from shock would indicate the impact of i on k, with j in between.

Impulse Response Diagram for Various Estimated Theta

θ = .7

θ = .6

θ = .5

38

Figure 5

Note: The top figure shows the relationship between a one-unit change in centrality and the mean of returns in the network. To calculate this, we take theaverage impact of a one-unit change across all traders in a given network and plot it against the mean of returns across traders in the same network.The bottom figure shows the relationship between a one-unit change in centrality and the variance of returns in the network. To calculate this, we take theaverage impact of a one-unit change across all traders in a given network and plot it against the variance of returns across traders in the same network.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

2.5x 10 -3

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

Mea

n of

Ret

urns

Varia

nce

of R

etur

ns

Impact of One Unit Change in Centrality

Impact of One Unit Change in Centrality

39

11

12

13

14

15

16

17

0 20 40 60 80 100 120

Aver

age

Syst

emic

Ris

k M

ultip

lier

Trading Limit

Impact of Trading Limits on Systemic RiskFigure 6

Note: Figure shows the results of a simulation in which traders face trading limits. Each simulation result is an estimate of the systemic risk multiplier. Thehorizontal axis, above, shows the maximum trading limit in the simulation. The vertical axis shows the average impact of the systemic risk multiplier. Limitson the horizontal axis indicate maximum trading volume during a pre-specified time period.

40

Mean Standard Deviation Min MaxPanel AS&P 500 e-mini futures

Sparse NetworksAverage Returns 0.98 0.01 0.97 1.05Volume 5.94 4.98 1.00 1215

Moderately Dense NetworksAverage Returns 0.96 0.02 0.96 1.09Volume 5.73 7.90 1.00 1,518

Dense NetworksAverage Returns 0.92 0.02 0.96 1.106Volume 5.32 12.68 1.00 2,060

Total number of # trading accounts 31,585

Panel BDOW futures Sparse NetworksAverage Returns 0.99 0.03 0.99 1.02Volume 6.39 1.42 1.00 150

Moderately Dense NetworksAverage Returns 0.98 0.05 0.98 1.03Volume 6.33 2.60 1.00 190

Dense NetworksAverage Returns 0.95 0.07 0.98 1.04Volume 5.91 4.86 1.00 341

Total number of # trading accounts 7,335

Note: As discussed in the text, sparse networks are defined as containing 250 transactions each, moderately dense networks as containing 500transactions each and dense networks as containing 1000 transactions each. The top half of the table includes statistics from the S&P 500 e-minifutures market. The bottom half includes statistics from the Dow futures market. The columns report the means, standard deviation, minimum andmaximum of each variable. Returns are defined as the gross return on an investment; thus a value of 1 indicates no change in value. Values greaterthan one are net gains and those less than one are net losses. For each density of network in each market, we report the average daily return as wellas the total daily volume at the trader level. Thus, we report the mean return across individual level traders, where for each trader, we havecalculated their own average return over the course of the trading day. Note that these trader-level returns are unweighted by volume. Because thefutures markets are zero-sum, volume weighted returns are zero by construction. Volumes statistics are average daily volumes at the level of thetrader. Standard deviations are measured as the variance over the returns at the trader level, again unweighted. Minimums and maximums are thesmallest and largest for a trader on any day.

Table 1: Summary Statistics

41

Table 2: Traditional estimation without network effects

full day 2nd half full day 2nd half full day 2nd half

S&P 500 e-mini futures

Sell Volume 8.63e-08*** 8.61e-08*** -3.12e-09 -3.44e-09 2.03e-08 2.02e-08(2.41e-08) (2.41e-08) (1.83e-08) (1.84e-08) (2.87e-08) (2.86e-08)

Buy Volume 6.47e-08** 6.46e-08** 1.85e-08 1.76e-08 4.01e-08 4.04e-08(3.14e-08) (3.14e-08) (1.97e-08) (1.98e-08) (2.71e-08) (2.72e-08)

Variance of returns 2.21e-09*** 2.20e-09*** 6.45e-09*** 6.47e-09*** 7.10e-09*** 7.07e-09***(2.75e-10) (2.74e-10) (1.92e-10) (1.92e-10) (3.43e-10) (3.41e-10)

Trailing total volume -3.99e-08* -3.99e-08* -2.84e-08* -2.88e-08* -3.95e-09 -3.76e-09(2.05e-08) (2.05e-08) (1.48e-08) (1.48e-08) (2.21e-08) (2.21e-08)

Half-day returns 0.495 0.174*** -0.0453(0.466) (0.0552) (0.0318)

Constant Yes Yes Yes Yes Yes YesLate Day Control Yes Yes Yes Yes Yes Yes

R-Squared 0.001 0.001 0.000 0.001 0.001 0.001

full day 2nd half full day 2nd half full day 2nd half

DOW futures

Sell Volume 1.39e-06* 1.57e-06** 1.61e-06* 1.63e-06* 1.30e-06 1.52e-06(7.58e-07) (7.79e-07) (9.53e-07) (9.68e-07) (9.92e-07) (1.03e-06)

Buy Volume -5.54e-07 -4.31e-07 -7.74e-07 -7.58e-07 -1.52e-06 -1.40e-06(8.13e-07) (8.33e-07) (9.60e-07) (9.69e-07) (9.93e-07) (1.01e-06)

Variance of returns 3.24e-07** 3.22e-07** 1.31e-06*** 1.31e-06*** 8.00e-07*** 7.99e-07***(1.32e-07) (1.32e-07) (1.25e-07) (1.25e-07) (7.73e-08) (7.72e-08)

Trailing total volume -6.18e-08 -6.15e-08 -1.56e-07 -1.58e-07 -8.58e-07 -8.41e-07(5.79e-07) (5.78e-07) (7.75e-07) (7.75e-07) (7.16e-07) (7.13e-07)

Half-day returns 0.540 0.0713 0.197(0.644) (0.490) (0.225)

Constant Yes Yes Yes Yes Yes YesLate Day Control Yes Yes Yes Yes Yes Yes

R-Squared 0.001 0.001 0.003 0.003 0.003 0.003

250 Trades per Time Period 500 Trades per Time Period 1000 Trades per Time Period

Panel B


Note: Panel A shows results from the S&P 500 futures market. Panel B shows results from the Dow futures market. Each of the two panels shows three sets of results from the estimation of aleast squares specification. The dependent variable in each case is the returns of an individual trader during a window of trading. The columns distinguish between different windows of time;250, 500 and 1000 trades for the market as a whole. For each market and each network density, we report the OLS estimation results over 21 days. T-statistics are reported below eachcoefficient estimate. Below, we report the adjusted R squared value from each specification. We include control variables for sell and buy volume during trading window, variance of returnsfor trader prior to trading period, volume in prior trading period, returns in the first half of the day, and an indicator for the final 2 periods of each day. We denote significance of coefficientsat the 10, 5 and 1% levels with ***, **, and *, respectively.

Panel A

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Table 3: Traditional versus network estimation results

low high low high low high low high low high low high


Variance explained by specification 0.001 0.001 0.05 0.09 0.000 0.001 0.09 0.17 0.001 0.001 0.19 0.37 (adjusted R-squared)

Network Effect Coefficient (θ) 0.02*** 0.04*** 0.05*** 0.08*** 0.10*** 0.17*** (t-statistic) 38.01 38.02 37.13 42.66 38.18 44.59

Constant Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes YesControl Variables Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes

low high low high low high low high low high low high

DOW futures

Variance explained by specification 0.001 0.001 0.04 0.08 0.003 0.003 0.07 0.09 0.003 0.003 0.14 0.18 (adjusted R-squared)

Network Effect Coefficient (θ) 0.02*** 0.04*** 0.04*** 0.05*** 0.08*** 0.10*** (t-statistic) 7.77 49.47 8.19 44.81 9.07 50.68

Constant Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes YesControl Variables Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes

Panel B

OLS OLS OLSw/ networks w/ networks w/ networks

Sparse Networks Moderately Dense Networks Dense Networks

Note: Panel A shows results from the S&P 500 futures market. Panel B shows results from the Dow futures market. The columns distinguish between different levels of network structure complexity. The table reports the adjusted R2 value from table 2's OLSspecification as well as the adjusted R2 from the estimation of model (1). We highlight the adjusted R2 for the network estimation method in bold. For each type network density and each market, we report the range of adjusted R2 results across 21 trading days. Weinclude the same controls as in Table 1 as well as a constant.

Panel A

OLS OLS OLSw/ networks


w/ networks w/ networks

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Table 4: Estimation with Network Effects

low high low high low high


Network Effect Coefficient (θ) 0.02 0.05*** 0.05*** 0.11*** 0.10*** 0.27***0.90 38.24 4.55 46.51 6.94 47.60

Constant Yes Yes Yes Yes Yes YesControl Variables No No No No No No

R-Squared 0.04 0.10 0.09 0.21 0.18 0.46


DOW futures

Network Effect Coefficient (θ) 0.02*** 0.02*** 0.04*** 0.05*** 0.08*** 0.10***7.75 50.09 8.05 11.38 9.06 50.00


R-Squared 0.04 0.04 0.07 0.08 0.14 0.17


Note: Panel A shows results from the S&P 500 futures market. Panel B shows results from the Dow futures market. The columns distinguish between different levels of network structurecomplexity. Each of the two panels shows three sets of results from the maximum likelihood estimation of model (4). For each type network density and each market, we report the range ofestimation results across 21 trading days. The first row shows the estimates of the parameter θ, the network effect coefficient, from the above specification. T-statistics are reported belowcoefficient estimates. We denote significance of coefficients at the 10, 5 and 1% levels with ***, **, and *, respectively.


Panel A


Panel B


44

Table 5: Estimation with Weighted Network Effects



Network Effect Coefficient (θ) 0.94*** 0.96*** 0.96*** 0.98*** 0.97*** 0.98*** t - statistic 1488.42 2594.92 619.46 619.46 516.89 669.17


R-Squared 0.74 0.77 0.73 0.77 0.73 0.77

Average Multiplier (φ) 16.12 26.99 25.61 45.45 37.01 66.46


DOW futures



R-Squared 0.71 0.80 0.71 0.79 0.71 0.78




Panel B


Note: This table extends the network model (4) to include weighted network to reflect the relative importance of traders in the system. Panel A shows results from the S&P 500 futures market.Panel B shows results from the Dow futures market. Each of the two panels shows three sets of results from the estimation of model (1). The columns distinguish between different levels ofnetwork structure complexity. This table uses a weighted matrix W defined as the element-by-element product of the adjancency matrix of realized trades and the sum of trading volume. For eachtype network density and each market, we report the range of maximum likelihood estimation results across 21 trading days. The first row shows the estimates of the parameter θ, the networkeffect coefficient, from the above specification. T-statistics are reported below coefficient estimates. Below, we include the adjusted R squared value from each specification and the averagesystemic risk multiplier. This multiplier is total network impact of a one unit shock to an individual. Averaging across the impact for all individuals in the network produces this number, which isequal to φ=1/(1-θ). We denote significance of coefficients at the 10, 5 and 1% levels with ***, **, and *, respectively.

Panel A


45

Table 6: Estimation with Weighted Network Effects and Network Fixed Effects




Constant Yes Yes Yes Yes Yes YesControl Variables No No No No No NoFixed Effects Yes Yes Yes Yes Yes Yes

R-Squared 0.74 0.77 0.73 0.77 0.73 0.77



DOW futures


Constant Yes Yes Yes Yes Yes YesControl Variables No No No No No NoFixed Effects Yes Yes Yes Yes Yes Yes

R-Squared 0.71 0.78 0.70 0.79 0.71 0.78




Note: This table extends the weighted network model to include network fixed effects. to include weighted network to reflect the relative importance of traders in the system. Panel A showsresults from the S&P 500 futures market. Panel B shows results from the Dow futures market. The difference between this table and table 5 is the construction of returns. The primary results inTable 2 used the individual level gross returns. Here we use the deviation in returns from the average return at the network level in each time period. The adjacency matrix of realized trades is asymmetric, non-directed matrix of 1's and 0's with 1's indicating the presence of a trade and 0 the absence. For each type of network density and each market, we report the range of maximumlikelihood estimation results across 21 trading days. The first row shows the estimates of the parameter θ, the network effect coefficient, from the above specification. T-statistics are reportedbelow coefficient estimates. Below, we include the adjusted R squared value from each specification and the average multiplier. This multiplier is total network impact of a one unit shock to anindividual. Averaging across the impact for all individuals in the network produces this number, which is equal to 1/(1-θ). We denote significance of coefficients at the 10, 5 and 1% levels with***, **, and *, respectively.


Panel A


Panel B

46

Table 7: Network topology and profitability



Impact of one unit change in Bonacich Centrality 0.06 0.70 0.07 0.78 0.10 0.76Stdeviation - Weighted Bonacich Centrality 3.41 4.30 3.42 4.32 3.40 4.33


DOW futures

Impact of one unit change in Bonacich Centrality 0.39 0.60 0.37 0.57 0.38 0.57Stdeviation - Weighted Bonacich Centrality 3.68 4.11 3.68 4.11 3.68 4.11


Note: Panel A shows results from the S&P 500 futures market. Panel B shows results from the Dow futures market. The columns distinguish between different levels of network structurecomplexity. The exercise in this table is to 1) report individual level variation in centrality and 2) evaluate the difference in returns for traders with different centrality. For each type of networkdensity and each market, we report the range of results across 21 trading days. Individual level Bonacich centralities are calculated using the formula: b(w,θ)=[I-θW]⁻¹⋅1, where the 1 signifies avector of 1's. We report the standard deviation of centrality as well as the change in returns for a trader that changes his centrality by one unit.


Panel A

Panel B

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