A Strategic Analysis of Portfolio Compression.

A Strategic Analysis of Portfolio Compression
Extended Abstract
Katherine Mayo
University of Michigan
kamayo@umich.eduMichael P. Wellman
University of Michigan
wellman@umich.edu
ABSTRACT
We analyze portfolio compression, the netting of cycles in a finan-
cial network, as a strategic decision made by firms within a debt
network. We define a network game in which firms have only lo-
cal information and ask what criteria the firms should consider
in their decision to compress. We propose a variety of heuristic
strategies and evaluate them using agent-based simulation and
empirical game-theoretic analysis. Our results show that some sim-
ple strategies based on local information perform better than the
unconditional strategies of always agreeing or disagreeing to a
compression and that when the decision is made strategically, the
price of anarchy is always high.
KEYWORDS
financial network; portfolio compression; game theory
ACM Reference Format:
Katherine Mayo and Michael P. Wellman. 2021. A Strategic Analysis of
Portfolio Compression: Extended Abstract. In Proc. of the 20th International
Conference on Autonomous Agents and Multiagent Systems (AAMAS 2021),
Online, May 3–7, 2021, IFAAMAS, 3 pages.
1 INTRODUCTION
The idea of portfolio compression is to simplify a web of financial
obligations by canceling a cycle of debt, leaving each party’s net
position the same. Consider Figure 1 where financial institution A
owes money to institution B, B owes a like amount to institution
C, and C owes the same amount to A and all debts have the same
interest rates and maturity dates. When such a cycle is identified
the institutions on the cycle must make the compression decision,
deciding to accept or reject the canceling of debts. Intuitively, can-
celing the three debts leaves all parties in the same net position,
while simplifying their balance sheets. If all are solvent and able to
pay the debts when due, the operation of compressing portfolios by
canceling the debts has no effect and the compression decision is of
little interest. With risk of default, however, portfolio compression
is not generally neutral and may affect stability in the network.
Prior analyses of compression have determined that preference
for compression among individuals in the network is connected
to characteristics of the network. In some cases, compression of a
cycle can benefit financial institutions because it limits financial
contagion by removing paths of default propagation. On the other
hand, portfolio compression can also limit the ability for parts of
the network to absorb losses from an insolvent institution, allowing
contagion to spread that may have been avoided [2, 4].
Proc. of the 20th International Conference on Autonomous Agents and Multiagent Systems
(AAMAS 2021), U. Endriss, A. Nowé, F. Dignum, A. Lomuscio (eds.), May 3–7, 2021, Online.
©2021 International Foundation for Autonomous Agents and Multiagent Systems
(www.ifaamas.org). All rights reserved.
Figure 1: A simple example of a debt cycle.
While these analyses offer us a look at what aspects of the net-
work play an important role in the effects of compression, they are
generally conducted with a global view of the network and where
details of the external shock are known. In practice, the compres-
sion decisions are made by institutions with a limited view of the
network and prior to resolution of uncertain financial events. Eval-
uation of compression decisions, therefore, necessitates a model
accounting for uncertainty and incomplete information on the state
of the network. Further, prediction of what these firms would de-
cide should also consider strategic effects, recognizing that optimal
policies generally depend on how the other firms make their deci-
sions.
To analyze the broader context of compression decisions, we
define a one-shot game on a financial network, with nodes repre-
senting financial institutions connected by directed edges represent-
ing debt obligations. Nodes make a strategic decision to accept or
decline a proposed compression using a strategy from a set of heuris-
tic strategies we propose, which may provide useful indications of
potential negative effects of compression to the voting node. To
evaluate these strategies, we randomly generate networks with debt
cycles, and employ agent-based simulation to determine compres-
sion decisions and outcomes at different insolvent node recovery
rates. Following the methodology of empirical game-theoretic anal-
ysis(EGTA) [ 3,5], we identify Nash equilibria among the heuristic
strategies.
Within this environment, we have found evidence that for an
individual node, the optimal policy is generally conditional on
features of the local network and the individual agent’s balance
sheet. When the compression decision is made strategically by the
nodes, we find that the price of anarchy is high for all recovery
rates.
2 COMPRESSION GAME
The game starts with a set of 𝑁={1,...,𝑛}nodes each represent-
ing a financial institution to which 𝑛agents are randomly assigned.
Each node is also given a random endowment of external assets,
𝑒𝑖∼𝑈[10,20].
Extended Abstract
AAMAS 2021, May 3-7, 2021, Online
1599
We model a liability as a directed edge 𝑙𝑖 𝑗of value𝑣representing
𝑣units owed by node 𝑁𝑖to node𝑁𝑗. The same edge represents an
internal asset for node𝑁𝑗. We refer to a node 𝑖’stotal liabilities,
𝐿𝑖as the sum of the values on its liability edges. And define the
sum of values on node 𝑖’s internal assets in addition to its external
asset holdings as the node’s total assets,𝐴𝑖.Insolvency can then be
formally defined for node 𝑖as the case where 𝐴𝑖−𝐿𝑖<0.
Liability edges are randomly added to the network, creating a set
𝐸of network edges, until the network contains at least one cycle
𝐶=(𝑁𝑐,𝐸𝑐), where𝑁𝑐⊆𝑁and𝐸𝑐⊆𝐸, of three or more edges.
Note that at this point, all nodes in the network remain solvent.
The game proceeds with a compression vote where agents on a
cycle independently cast their vote for the proposed compression
using their selected strategy from the strategy set. A compression
is performed only if all 𝑁𝑐nodes vote in favor of the compression.
We use𝜇=𝑚𝑖𝑛𝑖,𝑗∈𝑁𝑐,𝑖≠𝑗𝑙𝑖 𝑗to denote the smallest liability on the
cycle. Then a compression of cycle𝐶is defined by the revised set of
liabilities𝑙𝑐
𝑖 𝑗for edges on the cycle:
∀𝑙𝑖 𝑗∈𝐸𝑐𝑙𝑐
𝑖 𝑗=𝑙𝑖 𝑗−𝜇.
An example of a potential cycle and its subsequent compression
can be seen in Figure 2.
Figure 2: A simple debt cycle before and after compression
by two units.
We force insolvency in the network by a shock to external assets,
modeled as the failed realization of outside investments. Nodes on
a cycle are randomly assigned to invest their remaining external
assets in one of two investments with equal probability. All nodes
not on a cycle are randomly assigned to either invest their remain-
ing external assets in one of two investment or to not make an
investment, with equal probability. One of the investments will be
randomly chosen to fail and nodes assigned in it will not recoup
any of their investment.
We then make all nodes pay off their debts, using the greatest
clearing payment algorithm by Veraart [4]to calculate the payments
for each node the resolve the network. At the end of the game, the
payoff to the agent assigned to node 𝑁𝑖is the final equity 𝐴𝑖−𝐿𝑖.
2.1 Strategies
Agents may choose from among a set of heuristic strategies to make
the compression decision. Each strategy emphasizes a different
piece of information available to the node making the decision
including assets, liabilities, and cycle’s 𝜇-value. We do not claim that
our collection of strategies is exhaustive, but rather that they cover a
variety of strategically relevant features. The financial institutions
we are modeling may always accept a potential compression to
comply with regulation or out of a desire to simplify balance sheets,
so we include in our analysis the unconditional acceptance strategy
YES and its complement the unconditional reject strategy NO.3 ANALYSIS
We analyze the compression games by extensive simulation of se-
lected strategy profiles through a process known as EGTA. We
follow an iterative procedure for selecting profiles to simulate,
geared toward identifying symmetric mixed-strategy Nash equilib-
ria, using methods similar to those employed in prior EGTA studies
[1,6]. For some of our games instances, we also apply a method
called deviation-preserving reduction (DPR), which approximates a
many-player game by one with fewer players [7].
Through this approach, we obtain approximate Nash equilibria
for six𝑛=10player games, defined by varying settings of the re-
covery rate: 𝛼∈{0,0.1,0.3,0.5,0.7,1.0}. Games with 𝛼∈{0.3,0.7}
were reduced from 10-player to 4-player games using DPR.
To analyze the effects of a strategic compression decision on
the network, we use the price of anarchy using total equity as the
measure of social welfare. Total equity is defined as:
𝐸=Õ
𝑖[𝑒𝑖+Õ
𝑗𝑙𝑗𝑖−𝐿𝑖].
We run the compression game 10,000 times, comparing for each
network the outcome with a strategic compression decision made
with the Nash equilibria to the outcome under the optimal decision
for the network. Then we can define the price of anarchy, 𝑃, as:
𝑃=𝐸𝑠
𝑚𝑎𝑥{𝐸𝑐,𝐸𝑛𝑐}
where𝐸𝑠is the total equity in the network where the compression
decision comes from a strategic vote, 𝐸𝑐is the total equity in the
network where the cycle is compressed, and 𝐸𝑛𝑐is the total equity
in the network without compression.
4 PRELIMINARY RESULTS
We find that strategies employing simple, local network informa-
tion are preferred to the unconditional strategies by nodes making
the compression decision. The only exception is when 𝛼=0and
the pure Nash equilibrium is YES. The results of the price of an-
archy experiment shows that the average price of anarchy is high
and close to 1 for all 𝛼values. Thus, while self-interested decision
makers will not choose the social optimum, the cost to the network
of a strategic compression decision is not high.
5 CONCLUSION
Our study provides a first look at the strategic decision facing nodes
in a financial network presented with compression opportunities.
We focus on the implications for financial contagion, anticipating
how a compression may insulate or expose an institution to the
effects of an asset shock. Taking a perspective ex ante to the poten-
tial shock, we ask what types of strategies nodes with imperfect
information should use to make such a decision.
From our results, we can confirm that adopting a heuristic strat-
egy based on privately held information will often be the better
response for an individual node in the network. When we compare
the choice made by strategic choice to the optimal decision, we
see that the price of anarchy remains high. Therefore, allowing
compression to be a decision made strategically by nodes in the
network will not necessarily always negatively affect systemic risk.
Extended Abstract
AAMAS 2021, May 3-7, 2021, Online
1600
REFERENCES
[1]Ben-Alexander Cassell and Michael P. Wellman. 2013. EGTAOnline: An experiment
manager for simulation-based game studies. In Multi-Agent Based Simulation XIII.
Lecture Notes in Artificial Intelligence, Vol. 7838. Springer, 85–100.
[2]Steffen Schuldenzucker and Sven Seuken. 2020. Portfolio Compression in Financial
Networks: Incentives and Systemic Risk. (Aug. 2020). https://ssrn.com/abstract=
3135960 Working Paper.
[3]Karl Tuyls, Julien Perolat, Marc Lanctot, Edward Hughes, Richard Everett, Joel Z.
Leibo, Csaba Szepesvári, and Thore Graepel. 2020. Bounds and dynamics for
empirical game-theoretic analysis. Autonomous Agents and Multi-Agent Systems
34, 7 (2020).[4]Luitgard A. M. Veraart. 2019. When does portfolio compression reduce systemic
risk? (Jan. 2019). https://ssrn.com/abstract=3311176 Working Paper.
[5]Michael P. Wellman. 2016. Putting the agent in agent-based modeling. Autonomous
Agents and Multi-Agent Systems 30 (2016), 1175–1189.
[6]Michael P. Wellman, Tae Hyung Kim, and Quang Duong. 2013. Analyzing Incen-
tives for Protocol Compliance in Complex Domains: A Case Study of Introduction-
Based Routing. In Twelfth Workshop on the Economics of Information Security
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[7]Bryce Wiedenbeck and Michael P. Wellman. 2012. Scaling simulation-based game
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Extended Abstract
AAMAS 2021, May 3-7, 2021, Online
1601


Stability Effects of Arbitrage in Exchange Traded Funds: An
Agent-Based Model
Megan Shearer
University of Michigan–Ann Arbor
Ann Arbor, Michigan, USA
shearerj@umich.eduDavid Byrd
Bowdoin College
Brunswick, Maine, USA
d.byrd@bowdoin.edu
Tucker Hybinette Balch
J.P. Morgan AI Research
New York, New York, USA
tucker.balch@jpmchase.comMichael P. Wellman
University of Michigan–Ann Arbor
Ann Arbor, Michigan, USA
wellman@umich.edu
ABSTRACT
An index-based exchange traded fund (ETF) with underlying se-
curities that trade on the same market creates potential opportu-
nities for arbitrage between price deviations in the ETF and the
corresponding index. We examine whether ETF arbitrage trans-
mits small volatility events, termed mini flash crashes, from one
of its underlying symbols to another. We address this question
in an agent-based, simulated market where agents can trade an
ETF and its two underlying symbols. We explore multiple market
configurations with active and inactive ETF arbitrageurs. Through
empirical game-theoretic analysis, we find that when arbitrageurs
actively trade, background traders’ surplus increases because of the
increased liquidity. Arbitrage helps the ETF more accurately track
the index. We also observe that when one symbol experiences a
mini flash crash, arbitrage transmits a price change in the opposite
direction to the other symbol. The size of the mini flash crash de-
pends more on the market configuration than the arbitrageurs, but
the recovery of the mini flash crash is faster when arbitrageurs are
present.
KEYWORDS
ETF arbitrage; mini flash crash; empirical game-theoretic analysis
ACM Reference Format:
Megan Shearer, David Byrd, Tucker Hybinette Balch, and Michael P. Well-
man. 2021. Stability Effects of Arbitrage in Exchange Traded Funds: An
Agent-Based Model. In 2nd ACM International Conference on AI in Finance
(ICAIF’21), November 3–5, 2021, Virtual Event, USA. ACM, New York, NY,
USA, 9 pages. https://doi.org/10.1145/3490354.3494414
1 INTRODUCTION
Anexchange traded fund (ETF) is a portfolio of securities that trades
on the stock market. The underlying securities in an ETF’s portfolio
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https://doi.org/10.1145/3490354.3494414can be any traded entity, such as stocks, bonds, or commodities.
ETFs have become a popular investment vehicle, as they provide
convenient access to portfolio trades, offering average investors
apparent liquidity [ 2]. Investing in an ETF, rather than each secu-
rity in its portfolio, requires fewer trades, and therefore offers the
opportunity for diversification with lower trading costs [18].
We focus on index-based ETFs , which are ETFs whose under-
lying securities are based on a market index . Some examples of
market indexes are the Standard and Poor’s (S&P) 500 and Dow
Jones Industrial Average (DJIA) which track market performance,
or the Volatility Index (VIX) which tracks market volatility. Though
designed to track the index, the actual price of the index-based ETF
is determined through trading it as a symbol on the stock market.
When the market is open, participants can simultaneously observe
the trading price of the ETF and calculate the index. Any disparity
composes a potential arbitrage opportunity between the ETF and
its underlying securities.
Arbitrage trading can help an ETF’s price track its corresponding
market index [ 2]. It may also introduce or reinforce other depen-
dencies. For example, arbitrage may tether the price volatility of the
ETF’s underlying symbols [ 4]. The rise in market-wide price volatil-
ity in recent years has raised concerns about economic growth
and investor trust [ 11,21]. Price co-movements in ETF portfolios
have led some to question the role ETFs play in amplifying volatil-
ity [4,28]. We study this issue, focusing specifically on mini flash
crashes , which are short volatility events where the price rapidly
changes, then quickly reverts [ 31]. ETF arbitrage may channel mini
flash crashes from one underlying symbol to others. ETF arbitrage
may also have other effects, for example on the distribution of
surplus among traders.
To explore these questions, we develop an agent-based model
(ABM), populated by trading agents implementing algorithms com-
monly employed in agent-based finance literature. Our market
consists of multiple stocks which compose the portfolio of an ETF.
We study our simulated market with active and inactive ETF arbi-
trageurs. We then induce a mini flash crash to one of the underlying
securities, by injecting a trader who submits a series of large mar-
ketable limit orders (a common pattern of behavior triggering such
mini flash crashes [ 6,17,31]). We then analyze the effect of arbitrage
activity on the market response.
Using agent-based simulation allows us to examine otherwise
identical market environments with and without the presence of
ICAIF’21, November 3–5, 2021, Virtual Event, USA Megan Shearer, David Byrd, Tucker Hybinette Balch, and Michael P. Wellman
ETF arbitrage. We explore the impact of mini flash crashes by com-
paring the two scenarios, with strategic equilibration of trading
agents under each setting. We implement two strategies that trade
on arbitrage opportunities between the ETF and its underlying
symbols. This allows us to observe whether trading strategies de-
pendent on an ETF and its portfolio directly impact their underlying
symbols’ price volatility.
We employ a simulated market model using a standard limit
order book and message system similar to the US stock exchange
NASDAQ [ 10]. Our market contains two symbols which compose
an ETF’s portfolio. The ETF also trades on the market. Our model is
populated with 27 background trading agents and four arbitrageurs.
A conservative and an aggressive arbitrageur trade only on the ETF,
and a conservative and an aggressive arbitrageur trade on both ETF
and underlying symbols. We also use one impact agent to submit a
series of large, marketable sell orders to create a mini flash crash.
Through empirical game-theoretic analysis, we determine the
optimal trading strategies for all agents when arbitrageurs are ac-
tive and inactive, in order to examine the impact of ETF arbitrage
on market welfare and volatility. Arbitrageurs are highly profitable
and background agents’ average surplus is impacted significantly
by the arbitrage activity. Background traders’ final payoffs increase
with active arbitrage because arbitrageurs increase liquidity by
submitting marketable orders. Active arbitrage also increases the
volatility of its underlying symbols around events like mini flash
crashes. When one underlying symbol experiences a mini flash
crash, the other symbol experiences a price change in the opposite
direction when arbitrageurs are active. With active arbitrage there
is a faster price reversion of the mini flash crash in the symbol
which originally experienced the event. The competitiveness of the
background traders, rather than the arbitrageurs, influences the
magnitude of the mini flash crash. The price of the index, ETF, and
symbol that experiences the mini flash crash are lower than the
price preceding the event. When arbitrageurs are active, the aver-
age price of the other underlying symbol is higher than the price
preceding the event. Overall, the study demonstrates the effect of
ETF arbitrage on welfare and how ETFs can spread volatility events
through their portfolios and contribute to market-wide volatility.
This paper is organized as follows. We discuss prior work in
Section 2. In Section 3 we provide an overview of an ETF’s market
structure and the arbitrage opportunities this structure creates.
We then describe the market mechanism for our simulated ETF
environment in Section 4. Section 5 presents our findings on the
impacts of mini flash crashes in a market with active and inactive
ETF arbitrageurs. We conclude in Section 6.
2 RELATED WORK
Most prior work uses historical data and quantitative models to
analyze the impact of ETFs on the volatility of their underlying
symbols. Using high frequency data, Anatolyev et al . [1] found that
ETF arbitrage distorts reactions to market shocks in the US stock
market. Ben-David et al . [4] constructed a quantitative model and
used historical data to conclude that ETFs increase the volatility of
their underlying symbols. Da and Shive [13]also analyzed historical
data and found that ETFs contribute to price co-movement in the
symbols in their portfolios. In contrast, Madhavan and Morillo[23] found that underlying symbol price co-movements correlate
with macro-market movements, rather than the presence of the
ETF. Lastly, Lynch et al . [22] found that an implementable trading
strategy could generate profits using ETF arbitrage given realistic
portfolio price co-movements.
There are also previous studies using historical data to examine
what leads to ETF arbitrage opportunities. Box et al . [5] found that
a price shock or order imbalance in an ETF’s underlying portfolio
typically precedes ETF arbitrage opportunities. Marshall et al . [24]
found that spreads increase before ETF arbitrage opportunities.
Our agent-based market simulation builds on that of Byrd et al .
[10]. Simulated market models allow other traders to strategically
react given the current state of the market, which changes each
iteration. It also enables us to control factors of influence, and study
the effect of single factors (e.g., presence of ETF arbitrage).
Numerous other ABM studies have examined order and trade
activity around volatility events. Several have replicated aspects of
the Flash Crash on May, 6 2010 to determine which trading prac-
tices contributed to the price drop and recovery [ 27,34,35]. Some
other ABM studies [ 3,19,20] have explored abstract mini flash
crashes and price bubbles to provide insight into what might exac-
erbate these events. Paulin et al . [29] incorporated a dependency
network to study the micro and macro impacts of flash crashes
across multiple securities.
Two other prior ABM studies analyze ETF arbitrage. Mizuta [25]
examined the impact of trading costs and ETF arbitrage, finding
that lower trading costs and higher price volatility lead to more
arbitrage opportunities, more trading volume, and stronger corre-
lation between the index and ETF price. Torii et al . [32] model an
index fund and two underlying symbols, and examine the impacts
of ETF arbitrage on response to a downward volatility event in one
of the symbols.
The model of Torii et al . [32] is in fact quite similar to ours, but
with a few key differences. First their background traders employ
technical trading strategies (i.e., consider price trends), whereas
our agents are purely fundamental traders. Second, they generate
volatility events by directly manipulating the fundamental value.
We initiate a different type of volatility event, a mini flash crash,
through an agent who quickly submits multiple large orders. In a
mini flash crash, the price momentarily drifts from the asset’s true
value because of trading activity. Such events commonly occur in
the real stock market when a trader who wants to leave a large
position and starts to sell off all of their holdings [ 6,17,31]. These
two methods of creating volatility events exhibit entirely different
order-book dynamics, generating qualitatively different arbitrage
opportunities and outputs, and differences in response after the
shock. Third, the prior work does not model the primary market of
an ETF, which is also an essential element in defining the perfor-
mance of arbitrageurs. Despite these differences, both the study of
Torii et al . [32] and our own find that when one underlying symbol
experiences a downward price shock, the other underlying symbol
experiences a price shock in the opposite direction.
3 ETF MARKET STRUCTURE
A special feature of ETFs in our model is that the ETF security ac-
tually trades on two markets. The primary market allows purchase
Arbitrage in ETFs: An Agent-Based Model ICAIF’21, November 3–5, 2021, Virtual Event, USA
and sale of the ETF based on the prices of its underlying securities.
On the secondary market , the ETF trades directly like any other
stock symbol. The potential divergence between prices for the ETF
on these two markets is what gives rise to the arbitrage opportunity.
3.1 Primary Market
Access to the primary market is limited to select parties, commonly
referred to as authorized participants (APs) [ 26]. In our model, APs
are allowed to trade daily at the Net Asset Value (NAV), which is
calculated based on underlying security prices at the close of the
stock market. Once the primary market price (i.e., NAV) is deter-
mined, APs may submit orders (termed basket orders ) to exchange
between the ETF and underlying securities [15].
3.2 Secondary Market
The secondary market is the stock market. On the stock market, an
ETF trades like any other symbol [ 14], with each share representing
a fraction of the ETF’s portfolio. When a trader submits an order,
it either transacts with an order on the opposite side, or rests in
limit order book until either transacted or canceled. For the ETF
and any other security on the stock market, we define the bid price
andask price as the highest offers to buy and sell, respectively, on
the security’s limit order book.
An ETF’s secondary market lowers barriers to entry compared
to other portfolio funds [ 30], which is a principal reason for the
popularity of ETFs [ 18]. Participants in the primary market can
also trade in the secondary market.
3.3 Arbitrage between the Primary and
Secondary Markets
An ETF trades at the NAV and trading price on its primary and
secondary markets, respectively. Even though the NAV is calculated
only once per day, it can be estimated throughout the secondary
market’s trading day by calculating the market index. When the
bid price rises above the market index, a trader can profit by buying
the underlying symbols and selling the ETF. Similarly, when the
ask price falls below the market index, it is profitable to buy the
ETF and sell the underlying symbols. Such arbitrage is expected to
help the trading price closely track the index [2].
Whereas any secondary market participant can trade on price
deviations between the ETF and market index, participants with
access to both the primary and secondary market can implement
arbitrage more directly [ 30]. Such a participant can arbitrage in the
secondary market, then submit basket orders in the asset it is long
in for shares of the asset it shorted.
4 MARKET MECHANISM
4.1 ABIDES
We employ a discrete event market simulation constructed on the
ABIDES platform [ 10].1The platform provides a continuous dou-
ble auction market with securities priced in cents, a set of typical
background agents, and a kernel which drives the simulation with
nanosecond resolution while permitting sparse activity patterns to
be efficiently computed.
1The ABIDES source code is available at https://github.com/abides-sim/abides .For the ETF secondary market, we use the provided ABIDES ex-
change agent, which operates in a manner similar to NASDAQ. The
market is open from 09:30 to 16:00, lists any number of securities
for trade, and provides a distinct order book mechanism for each
security. The exchange accepts limit orders of any share volume,
and cancellation of same, and transacts (including partial execution)
those orders against a security’s limit order book with a typical
price-then-FIFO matching algorithm. The exchange responds to
requests for market hours, last trade prices, and market depth quote
requests, with depth one representing the current bid and ask.
4.2 Market Index
We define this model’s bid market index ,ιt,b, at time tas the sum
of the bid prices of a bundle of nstocks:
ιt,b=nÕ
i=1max
b(i)
t
. (1)
Where max
b(i)
t
is the bid price of underlying symbol iat time t.
We also define this model’s ask market index ,ιt,a, at time tas the
sum of the ask prices of a bundle of nstocks:
ιt,a=nÕ
i=1min
a(i)
t
. (2)
Where min
a(i)
t
is the ask price of underlying symbol iat time t.
Last, we define this model’s mid market index ,ιt,m, at time tas the
sum of the mid prices:
ιt,m=nÕ
i=1m(i)
t. (3)
Where m(i)
tis the midpoint between the best bid and ask prices,
more formally:
m(i)
t=1
2
max
b(i)
t
+min
a(i)
t
.
4.3 Primary Market in ABIDES
After the exchange agent stops accepting orders, the primary mar-
ket receives the close price ,p(i), of each underlying symbol iin the
ETF portfolio, defined as the price of i’s last trade on the secondary
market. The primary market uses these closing prices to calculate
the value of the index and uses this value as the NAV:
NAV =nÕ
i=1p(i). (4)
Then the primary market opens for basket orders. Every basket
order it receives is executed at the NAV, and the agent is notified of
its transaction.
4.4 Exogenous Price Time Series
We assume the existence of a mean-reverting exogenous funda-
mental value series for each underlying security, which represents
an immeasurable consensus of the “true” value at any point in
time. We model the fundamental as following a sparse discrete
Ornstein-Uhlenbeck (OU) process [ 33] augmented with periodic
“megashocks” of higher variance. Given a value at time 0 for the
ICAIF’21, November 3–5, 2021, Virtual Event, USA Megan Shearer, David Byrd, Tucker Hybinette Balch, and Michael P. Wellman
fundamental, a value can be obtained for any subsequent time tby
sampling from a normal distribution [12], with:
Eh
Q(i)
ti
=µ+
Q(i)
0−µ
e−γt(5)
Varh
Q(i)
ti
=σ2
2γ
1−e−2γt
,
whereµis the long-term mean fundamental value, Q(i)
0is the time 0
fundamental value of symbol i,γa mean reversion rate, and σ
a volatility value. This fundamental process allows us to obtain
Q(i)
tdirectly from Q(i)
0without the requirement to compute all
intermediate Q(i)
1. . .Q(i)
t−1. That is to say, it permits the simulation
to “skip time” whenever no agents will arrive at the market.
The OU process produces a time series with a single scale of
variance, which we configure to provide appropriate noise during
mean reversion. To obtain a price time series more similar to a stock
price over a longer window, we augment the OU process with the
application of stochastic “megashocks” [ 9] that arrive according to
a Poisson process and perturb the fundamental by a higher variance,
σ2s, bimodal normal distribution with mean zero, and positive and
negative modes substantially away from zero. This is intended to
represent exogenous events which can substantially alter value
perception and periodically provides much larger price moves from
which the OU process can revert.
An ETF’s fundamental is derived from the fundamental value of
its underlying symbols:
Q(ETF)
t=nÕ
i=1Q(i)
t.
4.5 Background Agents
We employ a population of background traders for our experiments
modeled after those by Brinkman [7]. Each agent is assigned a sin-
gle security of interest. They arrive according to a Poisson process
with rateλa, cancel any outstanding orders, and place an order
with equal probability to buy or sell. When arriving at the market,
an agent receives a noisy observation of the exogenous price time
series described above and uses this observation to update an indi-
vidual Bayesian belief concerning the current fundamental value of
the security. The agent then projects this belief forward to the end
of the market trading period to obtain an estimated final valuation
for the security.
When other agents trade, they supply relevant information about
the value of the asset. Our background agents use price information
from past transactions as if they are noisy observations of the
fundamental at the time of the transaction. When a transaction
occurs, the background agent will update its estimate of the final
fundamental, as if the current transaction price is E[Q(i)
t], and was
drawn from a Gaussian distribution around the fundamental.
An agent uses this final valuation and a private valuation to select
a limit price, employing an extended form of the Zero Intelligence
(ZI) strategy [ 16], with parameters 0≤Rmin≤Rmaxandη∈[0,1].
The ZI strategy draws R∼U[Rmin,Rmax], and sets its limit price
such that it would achieve surplus Rif transacted at that price
based on the current valuation estimate. If a market order wouldguarantee surplus at least ηR, the agent places that order instead of
its computed limit order.
4.6 Impact Agent
We introduce the impact agent to induce a mini flash crash. Begin-
ning at time τ, this agent submits a rapid series of nmarketable sell
orders of size q,δseconds apart. These orders consume the buy side
of the order book, causing a precipitous price drop. As subsequent
traders arrive and submit orders, the price typically recovers.
4.7 ETF Arbitrage Agent
We develop four strategies for arbitrage between the ETF and its
underlying symbols. Like the background agents, arbitrage agents
enter the market at Poisson arrival rate λa. On arrival at time t, a
conservative agent calculates the difference between the ETF bid
price, b(ETF)
tand the ask market index ιt,a:
∆t,1=max
b(ETF)
t
−ιt,a.
A conservative arbitrage agent must also find the difference be-
tween the bid market index, ιt,b, and the ETF ask price, a(ETF)
t:
∆t,3=ιt,b−min
a(ETF)
t
.
A more aggressive agent finds the difference between the ETF
mid price, m(ETF)
tand the mid market index ιt,m:
∆t,2=m(ETF)
t−ιt,m.
An aggressive agent must also find the difference between the bid
market index, ιt,b, and the ETF ask price, a(ETF)
t:
∆t,3=ιt,b−min
a(ETF)
t
.
∆t,4=ιt,m−m(ETF)
t.
These agents submit only marketable orders, so their sales are at
the bid and their buys at the ask.
ETF Single Asset Arbitrageur. This single asset arbitrageur trades
exclusively on the ETF security. It decides when to trade based on
a threshold ε≥0:
(
∆t,1>εSell ETF,
∆t,3>εBuy ETF.
An alternative, less conservative version of this arbitrageur makes
decisions based on the midpoint prices ( ∆t,2and∆t,4) instead.
ETF Multiple Asset Arbitrageur. The multiple asset arbitrageur
trades both the ETF and its underlying symbols. Its trades are also
triggered by a threshold ε≥0:
(
∆t,1>εSell ETF and buy underlying symbols,
∆t,3>εBuy ETF and sell underlying symbols.
There is also a variant version using midpoints.
These agents also trade on the primary market. When the pri-
mary market opens, they receive the NAV from Equation 4, then
Arbitrage in ETFs: An Agent-Based Model ICAIF’21, November 3–5, 2021, Virtual Event, USA
decide if they should submit basket orders by comparing the NAV
and close price of the ETF, p(ETF). More formally:
(
NAV−p(ETF)>0ETF shares→underlying symbol shares,
p(ETF)−NAV >0Underlying symbol shares →ETF shares.
If they submit basket orders, then they can hopefully end the day
net zero. The market makers need to be the fastest agents in the
system in order to be profitable when trading on so many symbols.
5 EMPIRICAL GAME-THEORETIC ANALYSIS
We introduce a set of heuristic strategies for both ETF arbitrageurs
and the background agents. Using agent-based simulations and em-
pirical game-theoretic analysis, we find the combination of strate-
gies that agents utilize in equilibria. We determine the impact of
ETF arbitrage on market welfare volatility by analyzing the market
under equilibrium settings when arbitrageurs are active and inac-
tive, as well as when arbitrageurs are active but the background
agents have not recalibrated their trading strategies from when the
arbitrageurs were inactive.
5.1 Market Environment Settings
We examine a variety of market environments to analyze the robust-
ness of our results. Each environment contains one exchange agent,
one ETF primary market agent, two symbols, and an ETF whose
portfolio is composed of the two other symbols. The exchange agent
accepts orders between 12:30 and 13:30. The ETF primary market
accepts orders between 17:00 and 17:01. Following Wang et al . [37] ,
we consider three market environments vary by market shock σ2s
and observation noise σ2n. The first consists of low shock and high
observation noise ( LSHN ) withσ2s=5×104andσ2n=107. The
second consists of medium shock and medium observation noise
(MSMN ) withσ2s=5×105andσ2n=5×106. Lastly, the third holds
high shock and low observation noise ( HSLN ) withσ2s=5×106
andσ2n=106.
For each non-ETF symbol, we generate a fundamental (5), with
meanµ=105, reversion γ=1.67×10−13, and market shock
σ=0. The sparse fundamental experiences a series of megashocks
throughout the trading period, and these arrive according to a
Poisson distribution with λ=2.78×10−13. We draw the size of
these megashocks from a binomial normal distribution with means
µs,1=0andµs,2=103, and varying values of σ2
2.
This market is populated with 27 background agents, where each
agent is randomly assigned one symbol with equal probability to
trade for the duration of each market run. Table 1 specifies the
strategies of the background agents. The background agents arrive
to the market according to a Poisson distribution with λa=10−11.
These agents submit orders of size q=100, but can hold a maximum
number of units at any time qmax=103. When the background
agents consider past transactions to update their estimate of the
fundamental, they use a variance of σ2p=103. Lastly, the private
value variance is σ2
PV=5×106.
We create a mini flash crash in one underlying symbol with a
single impact agent. This impact agent is assigned an underlying
symbol to trade on with equal probability at the beginning of each
market run. It then submits n=5trades beginning at τ=13:00
with size q=100andδ=6seconds between each trade.We implement four ETF arbitrage strategies, with one conserva-
tive ETF one asset (SA-C) agent, one aggressive ETF single asset
(SA-A) agent, one conservative ETF multiple asset (MA-C) agent,
and one aggressive ETF multiple asset (MA-A) agent. When arbi-
trageurs actively trades, all arbitrageurs use a strategy with ε=103.
However, when the arbitrageurs are inactive, they only exercise a
strategy where ε=1012. Both arbitrageurs submit orders of size
q=100. All ETF arbitrageurs arrive to the market according to a
Poisson process with λa=5×10−3.
5.2 EGTA Process
Empirical game-theoretic analysis (EGTA) is a method to find equilib-
ria in games by a heuristic strategy space and simulated payoff data
[39]. We utilize EGTA to find equilibria in varying market settings.
EGTA iteratively searches for potential equilibria in subgames, and
incrementally adds strategies to confirm or refute these potential
equilibria by examining potential deviations. Previous studies uti-
lize EGTA to examine complex environments where applying a
standard analytic method is hard [8, 36–38].
We model our market as a role-symmetric game, where play-
ers are divided into five roles: background traders, ETF-SAs, and
ETF-MAs. We utilize deviation-preserving reduction (DPR), which
approximates a many-player game by aggregating a game with
fewer players [ 40], because a game grows exponentially in players
and strategies. DPR has shown to generate good approximations of
the full game in multiple settings.
Our game consists of 27 background traders, one ETF-SA, and
one ETF-MA, and reduces to three background traders and one of
each arbitrageur when using DPR. These quantities of traders en-
sure that the required aggregations from DPR come out as integers.
In this setting one background agent deviates to a new strategy
while the other 26 background agents are further reduced to two.
For a specific strategy profile, we sample between 200 and 10,000
simulation runs to reduce sampling error from stochastic market
features.2
5.3 Impact on Market Welfare
We use EGTA to analyze the impact of ETF arbitrage on market
welfare across varying market environments. Figure 1 depicts the
surplus of each role when ETF arbitrageurs are active, when arbi-
trageurs are active but background agents have not adjusted to their
presence, and when arbitrageurs are inactive in each market setting.
The background agents are unadjusted to active arbitrageurs when
2Details of strategies employed in the Nash Equilibria identified will we presented in
an online appendix (link to be provided in final version).
Table 1: Strategies employed by the background traders,
where each agent chooses their desired surplus between Rmin
and Rmax, and have hyperparameter η.
Strategy ZI1 ZI2 ZI3 ZI4 ZI5 ZI6
Rmin 2000 2000 2500 3000 3000 3500
Rmax 2500 3000 3000 3500 4000 4000
η 1.0 0.8 1.0 1.0 0.8 1.0
ICAIF’21, November 3–5, 2021, Virtual Event, USA Megan Shearer, David Byrd, Tucker Hybinette Balch, and Michael P. Wellman
Figure 1: The average payoff over 200 simulation runs for
the background agents when arbitrageurs are active and in-
active in the three market environments.
they utilize the strategies optimal with inactive arbitrageurs in a
setting when those agents are actually active.
Background agents are better off when arbitrageurs are active.
When arbitrageurs are active they provide more marketable liquid-
ity, particularly during a mini flash crash where there are many
arbitrage opportunities. Thus, background traders have more op-
portunities to trade when arbitrageurs are active. These traders
submit less competitive orders when arbitrageurs are active. A less
competitive ZI strategy is when the agent selects a higher desired
surplus range. When arbitrageurs are active, they are willing to
trade with these high-margin orders, so the background traders’
profit increases. The background traders realize similar payoffs
when they do not adjust to arbitrage.
When the arbitrageurs are inactive, the surplus of the SA-Cs,
SA-As, MA-Cs, and MA-As is always zero. ETF-MAs have a higher
surplus than ETF-SAs because they have the opportunity to trade on
all symbols, and the ETF primary market where ETF-MAs receive
the difference in value between the ETF and index.
5.4 Impact on Symbol Price
We analyze ETF arbitrageurs’ impact on market volatility, when one
underlying symbol experiences a mini flash crash. To assess this
impact we examine and compare the price of the underlying sym-
bols in environments with active arbitrageurs, active arbitrageurs
and unadjusted background agents, and inactive arbitrageurs. An
environment with active arbitrageurs and unadjusted background
agents happens when background agents do not recalibrate their
strategies to consider arbitrage. In each figure we represent the
price as the midpoint price over 200 simulation runs.
The price of the ETF should track the market index. Figure 2
shows the midpoint price of the ETF and Figure 3 depicts the market
index of each environment. This allows us to see how arbitrage
impacts the ETF trading price and market index, and what trading
opportunities arbitrageurs have around the mini flash crash in
an underlying symbol. The ETF trading price sees a decrease in
price when arbitrageurs are active. The market index crashes a
large amount in every environment at the time of the mini flash
crash. In environments active and inactive arbitrage, the indexdrops because as the midpoint price of an underlying symbol drops,
the market index or the sum of prices s0ands1also drops. When
ETF arbitrageurs are present, as the market index falls below the
ETF price, the arbitrageurs sell the ETF and buy the underlying
symbols. As the arbitrageurs sell the ETF, the bid side of the order
book reduces and the price drops. When the arbitrageurs buy the
underlying symbols they absorb the ask side of the order book,
which leads to the market index recovering quicker and back to a
higher price when an ETF is present. Thus, arbitrage helps the ETF
track the index.
In all market environments, the impact agent causes a distinct
mini flash crash in the underlying symbol. This symbol experi-
ences similar trends to the market index, where more competitive
background agents creates a smaller spread and smaller price drop.
Figure 4 shows each environment in a reduced time frame around
the mini flash crash, and active arbitrageurs, inactive arbitrageurs,
and unadjusted background agents are depicted together. The mini
flash crash recovers faster with active arbitrage because the arbi-
trageurs submit marketable buy orders to the underlying symbol
and marketable sell orders to ETF, causing the price of the index
and underlying symbols to rise faster. In each environment, the
price of this symbol does not recover to the price before the mini
flash crash, though the price does revert to a higher level when
arbitrageurs are active.
Figure 5 depicts the average price of the underlying symbol
where the impact agent does not trade. When background traders
are more competitive, there is a small but distinct upward price
movement at the time of the mini flash crash in the other symbol.
This happens because the arbitrageurs buy the ETF’s underlying
symbols when the index is lower than the ETF trading price, causing
the price increase. In environments with arbitrage, the average price
is higher than the price preceding the mini flash crash. The increase
in price This implies that the presence of ETF arbitrageurs can
impact the trading price of a symbol because of trading activity
independent of the symbol itself.
6 CONCLUSION
We analyze an agent-based, simulated market model with a stock
market and ETF primary market. We explore varying market en-
vironments that contain an ETF and two symbols which compose
the ETF’s portfolio. The market is also populated with numerous
background agents, an impact agent which creates a mini flash
crash, and ETF arbitrage agents. To determine the impact of ETF
arbitrage, we examine equilibria in each environment under set-
tings when arbitrageurs are active and inactive. The arbitrageurs
are extremely profitable when they actively trade, and the surplus
of the background agents increases when they are conservative and
decreases when they are more competitive. We also find that this
type of arbitrage may transmit a volatility event, like a mini flash
crash, throughout its underlying symbols. The other underlying
symbol experiences a price change in the opposite direction of the
other underlying symbol at the time of the mini flash crash. The
magnitude of the mini flash crash in the original symbol is impacted
more by the competitiveness of the background traders than the
arbitrageurs.
Arbitrage in ETFs: An Agent-Based Model ICAIF’21, November 3–5, 2021, Virtual Event, USA
(a) LSHN, Active Arbitrage:
ETF
(b) MSMN, Active Arbitrage:
ETF
(c) HSLN, Active Arbitrage:
ETF
Figure 2: Average price time series over 200 simulation runs of the ETF. The ETF in a market is meant to track the market
index. The ETF is only present in three of the market environments.
(a) LSHN, Active Arbitrage:
Index
(b) MSMN, Active Arbitrage:
Index
(c) HSLN, Active Arbitrage:
Index
Figure 3: Average time series over 200 simulation runs of the market index. This index is composed of two symbols. An
underlying symbol experiences a mini flash crash at 13:00.
(a) LSHN: s0
 (b) MSMN: s0
 (c) HSLN: s0
Figure 4: Average price time series over 200 simulation runs of underlying symbol, s0, that is randomly selected to experience
a mini flash crash at 13:00 through an impact agent submitting a series of large trades. This shows active arbitrageurs, inactive
arbitrageurs, and unadjusted background traders on the same plot.
Most real-world ETFs represent portfolios with many symbols.
Our study models an ETF with only two symbols, in order to fo-
cus the effect on a single relationship. We believe this captures
qualitative properties of realistic ETFs as well, but it could be that
further insights would be revealed by extending the model to cover
more symbols. This could be readily incorporated in our ABM, with
linear impact on computational cost of simulation, though perhapsimposing somewhat more complexity on the arbitrage strategy
and the analysis of results. An ETF with a larger portfolio could
cause arbitrage to have a larger impact on the ETF than the under-
lying symbols, so arbitrage may help the ETF track the index, but
introduce less volatility to the underlying symbols.
Another potential limitation is in the space of background trad-
ing strategies. We focus exclusively on background agents that
ICAIF’21, November 3–5, 2021, Virtual Event, USA Megan Shearer, David Byrd, Tucker Hybinette Balch, and Michael P. Wellman
(a) LSHN, Active Arbitrage: s1
 (b) MSMN, Active Arbitrage: s1
 (c) HSLN, Active Arbitrage: s1
Figure 5: Average price time series over 200 simulation runs of underlying symbol without a mini flash crash. An impact agent
does not trade on this symbol.
use a ZI strategy which consider previous transactions in their
estimate of an asset’s fundamental value. It could be beneficial to
utilize more trading strategies dependent on the order book and
price movement, such as market makers or trend followers, because
these agents might exacerbate mini flash crashes.
This paper provides insight into the impact of ETFs on market
welfare, market volatility, and stock valuation. Previous studies have
used historical data to examine associations between ETF activity
and price volatility in its underlying symbols. With agent-based
simulation we are able to examine causality through a direct A/B
test in market environments with and without active ETF arbitrage.
We find other agents are better off with arbitrage if they are more
conservative, but arbitrage reduces their profits if they trade more
competitively. An implication is that inclusion in an index ETF
may impact the pricing of a stock without any actual change in
the stock’s fundamental value, and solely due to trading activity
independent of the stock itself.
ACKNOWLEDGEMENTS
This material is based on research supported in part by the National
Science Foundation under Grant no. 1741026, and by two JPMorgan
AI Research Fellowships.
This paper was prepared for informational purposes in part
by the Artificial Intelligence Research group of JPMorgan Chase
& Co ˙and its affiliates (“JP Morgan”), and is not a product of the
Research Department of JP Morgan. JP Morgan makes no represen-
tation and warranty whatsoever and disclaims all liability, for the
completeness, accuracy or reliability of the information contained
herein. This document is not intended as investment research or
investment advice, or a recommendation, offer or solicitation for
the purchase or sale of any security, financial instrument, finan-
cial product or service, or to be used in any way for evaluating
the merits of participating in any transaction, and shall not consti-
tute a solicitation under any jurisdiction or to any person, if such
solicitation under such jurisdiction or to such person would be
unlawful.
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Megan Shearer
David Byrd
Tucker Hybinette Balch
Michael P Wellman


Special issue on autonomous agents modelling other agents: Guest editorial.

EditorialSpecial issue on autonomous agents modelling other agents: Guest editorial
===================================================================================

Abstract
--------

Much research in artificial intelligence is concerned with enabling autonomous agents to reason about various aspects of other agents (such as their beliefs, goals, plans, or decisions) and to utilise such reasoning for effective interaction. This special issue contains new technical contributions addressing open problems in autonomous agents modelling other agents, as well as research perspectives about current developments, challenges, and future directions.

References (11)
---------------

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There are more references available in the full text version of this article.Cited by (6)
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* ### [An Evolutionary Framework for Modelling Unknown Behaviours of Other Agents](https://doi.org/10.1109/TETCI.2022.3229308)

2023, IEEE Transactions on Emerging Topics in Computational Intelligence
* ### [Diversifying agent's behaviors in interactive decision models](https://doi.org/10.1002/int.23075)

2022, International Journal of Intelligent Systems
* ### [Behavior Reasoning for Opponent Agents in Multi-Agent Learning Systems](https://doi.org/10.1109/TETCI.2022.3147011)

2022, IEEE Transactions on Emerging Topics in Computational Intelligence
* ### [Deep reinforcement learning for multi-agent interaction](https://doi.org/10.3233/AIC-220116)

2022, AI Communications
* ### [Deep Reinforcement Learning for Multi-Agent Interaction](https://doi.org/10.48550/arXiv.2208.01769)

2022, arXiv
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2022, arXiv
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Artificial Intelligence, Volume 279, 2020, Article 103185Show abstractWe propose two models of random quantified boolean formulas and their natural random disjunctive logic program counterparts. The models extend the standard models of random *k*-CNF formulas and the Chen-Interian model of random 2QBFs. The first model controls the generation of programs and QSAT formulas by imposing a specific structure on rules and clauses, respectively. The second model is based on a family of QSAT formulas in a non-clausal form. We provide theoretical bounds for the phase transition region in our models, and show experimentally the presence of the easy-hard-easy pattern and its alignment with the location of the phase transition. We show that boolean formulas and logic programs from our models are significantly harder than those obtained from the standard *k*-CNF and Chen-Interian models, and that their combination yields formulas and programs that are “super-hard” to evaluate. We also provide evidence suggesting that formulas from one of our models are well suited for assessing solvers tuned to real-world instances. Finally, it is noteworthy that, to the best of our knowledge, our models and results on random disjunctive logic programs are the first of their kind.
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Stefano V
Albrechta
Peter Stone
Michael P
Wellman