OPTIMAL EXECUTION WITH IDENTITY OPTIONALITY REN E CARMONA CLAIRE ZENG Department of Operations Research and Financial Engineering

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OPTIMAL EXECUTION WITH IDENTITY OPTIONALITY
REN´
E CARMONA*, CLAIRE ZENG*
*Department of Operations Research and Financial Engineering
Princeton University
Princeton, NJ 08544
Abstract. This paper investigates the impact of anonymous trading on the agents’ strategy in an
optimal execution framework. It mainly explores the specificity of order attribution on the Toronto
Stock Exchange, where brokers can choose to either trade with their own identity or under a generic
anonymous code that is common to all the brokers. We formulate a stochastic differential game
for the optimal execution problem of a population of Nbrokers and incorporate permanent and
temporary price impacts for both the identity-revealed and anonymous trading processes. We then
formulate the limiting mean-field game of controls with common noise and obtain a solution in
closed-form via the probablistic approach for the Almgren-Chris price impact framework. Finally,
we perform a sensitivity analysis to explore the impact of the model parameters on the optimal
strategy.
1. Introduction, motivations and literature review
With the advent of algorithmic trading and electronic markets, automated and high frequency
trading have become an increasingly active area of research. As a result, a considerable amount of
effort has been devoted to understanding market microstructure. The existing literature covers a
wide variety of subjects, but three main categories can be broadly identified:
•Statistical arbitrage, or studying opportunities to make profits out of “predictable” returns
(short-term alpha) or benefit from short-term inefficiencies in the market (as with pair-
trading or futures-index arbitrage);
•Optimal execution, or determining the optimal schedule (for a given cost functional) in order
to sell and/or acquire a large position in one or multiple assets while mitigating several risks
(such as information leakage, price impact and adverse selection);
•Market making, or determining the optimal placement of limit orders to benefit by providing
liquidity to markets.
The study of market impact and the modelling of market frictions has been of prime interest in
designing efficient trading strategies. This paper analyzes a multi-agent optimal execution problem,
where many financial institutions and brokers must determine the strategy to liquidate or build a
position on a specific asset while maximizing an expected profit objective function.
Most of the early literature on optimal execution focuses on the single agent setting, where the
trader is facing a trade-off between choosing a fast trading rate (to reach their goal as soon as
possible to reduce the execution risk) and limiting their price impact (which pushes prices in an
E-mail address:{rcarmona, cszeng}[at] princeton.edu.
Date: October 11, 2022.
1
arXiv:2210.04167v1 [q-fin.MF] 9 Oct 2022
2 OPTIMAL EXECUTION WITH IDENTITY OPTIONALITY
unfavourable direction on average). The initial framework for the single trader case is attributed
to Almgren and Chriss, who consider both a permanent and an immediate price impact in [1]. [22]
considers the same problem but with a transient price impact that has exponential decay, while
[15] generalizes this approach by introducing a general decay kernel. We refer to [11] for a detailed
presentation on optimal execution with a more general objective function. [2] and [5] introduced the
two-player setting, where one agent has a liquidation target and the other is trying to benefit from
predatory trading by exploiting this information. Considering many agents is essential to model
the markets and is of particular interest to try to model some financial events. Indeed, there are
some liquidity events that may force some traders to liquidate a large position of an asset within
a relatively short time window. Changes in the membership in stock indexes such as the Russell
3000, is a case in point. ETFs are funds that track indexes; they try to minimize the tracking
error by replicating the index of interest in their portfolio. As a result, any major change in the
index composition causes the ETF to rebalance its portfolio, adding or dropping the same stocks as
the index. For instance, the Russell US indexes undergo an annual reconstitution process and the
benchmark composition is communicated in advance to the marketplace. This composition change
is essential to make sure that the indexes reflect accurately the US equity market. At the end of
May, the official modifications are announced, and will be effective at the end of June. June is
therefore a transition month, during which ETFs and other institutions tracking the index must
trade to rebalance their portfolio, so that it replicates the reconstitution portfolio by the end of
June.
The first extensions to multi-agent settings modeled one large trader facing an exogenous order flow
such as in [10]. Modeling the interaction in an endogenous way, that is, when the order flow and
the price dynamics come from the interaction of the agents on the market, has been formulated
in finite-player stochastic differential games and in mean field games. Developed initially in [18],
[19], [20] and in parallel in [16], [3], mean field games have been applied to several problems in
economics and finance: for instance, [4] addresses the problem of crowd trading, where the traders
interact through the asset mid-price process. The use of mean field games requires the assumption of
symmetry among the agents, but heterogeneous preferences can be considered by introducing either
a Major-Minor framework as in [17] or several sub-populations. Several approaches are possible
when solving a mean field game problem. The probabilistic approach, formulated by [6], aims at
characterizing directly optimal controls. This method has been applied to the optimal execution
problem in several works starting with[7]. The variational approach, which relies on a Dynamic
Programming Principle, has been used in [4] and [17]: it characterizes value functions directly,
while incidentally characterizing optimal controls. Another approach based on applying convex
analysis techniques can also be used as in [14] and [21].
With the introduction of pre-trade anonymity in equity markets, many exchanges have shifted to-
wards a fully anonymous design; this has been further exacerbated by the increasing competition
from Alternative Trading Systems (ATS) and Electronic Communication Networks (ECN). Publi-
cations addressing the impact of anonymity on market quality, in particular on liquidity, are few
and far between. The consensus is that anonymity offers an additional opportunity for market par-
ticipants to enhance their trading strategies for a better execution. Previous studies compared the
effect of anonymity in different market designs. Some performed statistical analysis and comparison
between separate platforms dedicated to anonymous and non-anonymous trading ([24]) or before
and after a regulatory change in identity disclosure requirements. The Toronto Stock Exchange
(TSX) sets itself apart in that it has a hybrid system where anonymity and transparency co-exist
side-by-side. The TSX public trading tape (Level I data) is also helpful in overcoming the obsta-
cles mentioned above, both because it is one of the few markets where anonymity and identity are
both actively used (at least for the most liquid stocks) and because it is quite significant in size;
it represents the eleventh largest exchange world wide and is ranked third in North America in
OPTIMAL EXECUTION WITH IDENTITY OPTIONALITY 3
market capitalization. Voluntary identity disclosure is a key feature of the market design of the
TSX 1. Although all agents are identified by a unique identity code (which is publicly available),
every broker has the right to either disclose their identity or choose anonymity for each individual
trade. When anonymity is chosen, all the anonymous buyer/seller identities are reported under a
generic anonymous code 01 both pre-trade and post-trade; otherwise, the exchange discloses the
buyer and/or seller ID codes. Figure 1 below illustrates the inventory accumulated by the generic
broker (representing the anonymous orders of all the brokers trading anonymously) and on the right
the inventory of the specific broker with ID code #65 for the stock RCI.B (Rogers Communications
Inc. Class B) on 02/26/2022. The statistical analysis in [12] tries to identify the determinants of
the decision to trade anonymously. Based on a data set from the Market Regulation Services super-
vising the Canadian securities market, the authors find that reduced execution costs are associated
with anonymous orders from strategic traders, while the trades displaying the identity of special-
ists and dual capacity brokers have a higher price impact. Moreover, they infer that anonymity
is strategically selected, depending in particular on the market conditions but also on the order
source, size, aggressiveness, and expected execution costs.
Figure 1. Non-anonymous inventory for the anonymous broker #1 (left) and broker
#65 (right) for the stock RCI.B on 02/26/2021
This paper introduces an optimal execution stochastic differential game for a setting that takes
into account identity optionality and whose limiting problem is a mean-field game of controls with
common noise. Our analysis aims at determining if the voluntary disclosure of identity represents an
opportunity for the market participants and investigates its impact on the agents’ trading strategies,
rather than on the market quality. Section 2 presents the general N-player stochastic differential
game for the optimal execution problem with identity optionality. Section 3 describes the general
strategy to solve the mean field game limit. Section 4 specifies the model for an Almgren-Chriss-type
1An additional feature of the Toronto stock exchange lies in the difference between disclosing one’s identity or not.
Indeed, anonymous orders are excluded from what is called broker preferencing. This refers to the priority of the
order matching on TSX: attributed orders will follow the Price/Broker/Long Life/Time priority to be matched (Long
Life orders are committed to rest in the order book for a minimum period of time, during which they can be neither
modified nor canceled). At the best bid or offer price, attributed orders of one specific broker will be matched with
new offsetting attributed orders from the same broker, and this, ahead of the other brokers that arrived before them.
This enables attributed orders to “jump the queue” and happens only if the identity is disclosed on both sides. Once
these orders are matched, the Long Life/Time priority will be applied to the other orders. This broker preferencing
allows for lower transaction costs, since internal crosses are free of fees. Therefore, not disclosing the identity on
certain orders represents a potential cost, but this is not the object of our analysis.
4 OPTIMAL EXECUTION WITH IDENTITY OPTIONALITY
price impact model and develops the McKean-Vlasov Theory necessary to find an open-loop solution
of the limiting mean field game formulated in the strong formulation and presents a numerical
study. Finally, Section 5 provides numerical illustrations and interpretations of the equilibrium
characteristics for some realistic sets of parameters.
Notations:
We assume that we are given a complete probability space (Ω0,F0,P0) endowed with a complete
and right-continuous filtration F0= (F0
t)t∈[0,T ]generated by the Wiener process W0= (W0
t)t∈[0,T ]
and for each integer i≥1, a complete probability space (Ωi,Fi,Pi) endowed with a complete and
right-continuous filtration Fi= (Fi
t)t∈[0,T ]generated by a two-dimensional Wiener process Wi=
(Wi
t)t∈[0,T ]= ((Wi,1
t, W i,2
t)†)t∈[0,T ]. The independent Wiener processes Wi,1and Wi,2play the roles
of the idiosyncratic noises of player iassociated with the anonymous and non-anonymous trading
processes respectively. We assume independence of the Brownian motions W0,W1,...,WN, . . . .
2. Finite Player stochastic differential game
If N≥1 is an integer, when considering a model for Nplayers, we denote by (Ω0×Ω1×. . . ΩN,F0⊗
F1⊗ ··· ⊗ FN,P0⊗P1⊗ ··· ⊗ PN) endowed with a complete and right-continuous filtration F=
(F)t∈[0,T ]defined from the augmentation of the product filtration F0⊗F1⊗ ··· ⊗ FNso that it is
right-continuous and complete. So in such a model, the common noise W0is essentially constructed
on (Ω0,F0,P0) whiile the idiosyncratic noises (Wi)i≥1are essentially built on their respective
(Ωi,Fi,Pi).
2.1. Price Impact Model. For simplicity we assume that all players trade the same stock whose
price at time tis denoted by St. The inventory Qi
tat time tof broker iis the aggregation of two
separate inventories: an inventory Qi,a
taccumulated by trading anonymously at speed νi,a
tand an
inventory Qi,n
taccumulated through identity-revealed trading at speed νi,n
t. We ssume that these
inventories have non-trivial quadratic variations and we denote by σaand σn>0 their volatilities,
so:
dQi,a
t=νi,a
tdt +σadW i,1
t(1a)
dQi,n
t=νi,n
tdt +σndW i,2
t.(1b)
Remark 2.1. Note the presence of idiosyncratic Brownian motions in the dynamics of the inventory
processes, and as we shall see later on, in the associated wealth processes derived from the self-
financing conditions. The noise term σadW i,1
t(resp. σadW i,1
t) models a random stream of client
demands the broker faces. These demands affect the broker anonymous (resp. identity revealed)
inventory. This was first introduced and studied in [9], and later empirically supported in [8] where
statistical tests performed on the Toronto Stock Exchange data show that both the inventory and the
wealth dynamics should have non-zero quadratic variations. This assumption is also consistent with
the option for a client to specify if they want anonymity or the broker identity to appear when they
send their order.
Here, we use the modeling assumptions introduced in [7] and developed in [9] with a nonlinear
order book. At each time t, every agent faces a cost structure given by two transaction cost curves
ca, cn:R7→ [0,∞], which are convex and satisfy ca(0) = cn(0) = 0. The order book incorporates
each trade and reconstructs itself instantly around a new mid-price St, impacted in the following
way by the transactions. If a single agent iplaces an anonymous (resp. identity-revealed) market
order of νi,a
t(resp. νi,n
t) when the mid-price is St, the transaction will cost them νi,a
tSt+ca(νi,a
t)
OPTIMAL EXECUTION WITH IDENTITY OPTIONALITY 5
(resp. νi,n
tSt+cn(νi,n
t)). Hence an anonymous and an identity-revealed trades will trigger changes
in cash given by:
dKi,a
t=−(νi,a
tSt+ca(νi,a
t))dt (2a)
dKi,n
t=−(νi,n
tSt+cn(νi,n
t))dt (2b)
respectively . Let us make the assumption that ca(·) = κac(·) and cn=κnc(·) for some positive
constants κaand κn, and a function c:R7→ Rwhich is convex and satisfies c(0) = 0.
Once a market orders are executed, the order book relocates around a price incorporating the
permanent price impact composed of two terms :
•the impact from the anonymous market orders : γa
NPN
j=1 κac0(νj,a
t)
•the impact from the identity-revealed market orders : γn
NPN
j=1 κnc0(νj,n
t)
Hence the mid-price process can be modeled as a martingale plus a drift representing this permanent
impact.
dSt=γa
N
N
X
j=1
κac0(νj,a
t) + γn
N
N
X
j=1
κnc0(νj,n
t)dt +σ0dW 0
t(3)
The wealth Vi,a
t(resp. Vi,n
t) accumulated by agent ithrough anonymous (resp. attributed) trading
at time twill therefore be the sum of the initial value, the mark-to-market value of the anonymous
(resp. attributed) inventory and the anonymous (resp. attributed) cash process:
Vi,a
t=Vi,a
0+Qi,a
tSt+Ki,a
t
Vi,n
t=Vi,n
0+Qi,n
tSt+Ki,n
t,
and since the Brownian motions W0
tis independent of Wi,1
tand Wi,2
t, the wealth processes have
the following dynamics:
dV i,a
t=Qi,a
tdSt+StdQi,a
t+dKi,a
t(4a)
dV i,n
t=Qi,n
tdSt+StdQi,n
t+dKi,n
t(4b)
We assume that the agents are risk-neutral and seek to maximize the expectation of their terminal
wealth, a running cost fand a terminal cost gfunctions of their inventories. Agent itherefore
wants to maximize the following objective functional:
Ji(νi,ν−i) = EhVi,a
T+Vi,n
T+g(Qi,a
T, Qi,n
T) + ZT
0
f(Qi,a
t, Qi,n
t)dti(5)
By using the expression of the wealth processes and discarding the constant terms, the objective
functional to maximizes becomes:
Ji(νi,ν−i) = Ehg(Qi,a
T, Qi,n
T)+(Qi,a
T+Qi,n
T)ST+ZT
0d(Ki,a
t+Ki,n
t) + f(Qi,a
t, Qi,n
t)dti (6)
By using the expression of Kt, the objective functional becomes :
Ji(νi,ν−i) = Ehg(Qi,a
T, Qi,n
T)+(Qi,a
T+Qi,n
T)ST−ZT
0(νi,a
t+νi,n
t)St+c(νi,a
t)+c(νi,n
t)−f(Qi,a
t, Qi,n
t)dti
(7)
and we are using the various rates of trading as controls which we assume to be progressively
measurable for the filtrations generated by the idiosyncratic noise and the common noise.
摘要：

OPTIMALEXECUTIONWITHIDENTITYOPTIONALITYRENECARMONA*,CLAIREZENG**DepartmentofOperationsResearchandFinancialEngineeringPrincetonUniversityPrinceton,NJ08544Abstract.Thispaperinvestigatestheimpactofanonymoustradingontheagents'strategyinanoptimalexecutionframework.Itmainlyexploresthespeci cityoforderatt...

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