Fast and Slow Optimal Trading with Exogenous Information Rama Cont1 Alessandro Micheli2 and Eyal Neuman3

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Fast and Slow Optimal Trading with Exogenous

Information

Rama Cont1, Alessandro Micheli∗2, and Eyal Neuman3

1Mathematical Institute, University of Oxford

2,3Department of Mathematics, Imperial College London

June 26, 2023

Abstract

We model the interaction between a slow institutional investor and a high-

frequency trader as a stochastic multiperiod Stackelberg game. The high-

frequency trader exploits price information more frequently and is subject to

periodic inventory constraints. We ﬁrst derive the optimal strategy of the

high-frequency trader given any admissible strategy of the institutional in-

vestor. Then, we solve the problem of the institutional investor given the

optimal strategy of the high-frequency trader, in terms of the resolvent of a

Fredholm integral equation, thus establishing the unique multi-period Stackel-

berg equilibrium of the game. Our results provide an explicit solution which

shows that the high-frequency trader can adopt either predatory or cooperative

strategies in each period, depending on the tradeoﬀ between the order-ﬂow and

the trading signal. We also show that the institutional investor’s strategy is

more proﬁtable when the order-ﬂow of the high-frequency trader is taken into

account.

Mathematics Subject Classiﬁcation (2010): 49N70, 49N90, 93E20, 60H30

JEL Classiﬁcation: C73, C02, C61, G11

∗AM is supported by the EPSRC Centre for Doctoral Training in Mathematics of Random

Systems: Analysis, Modelling and Simulation (EP/S023925/1)

arXiv:2210.01901v3 [q-fin.TR] 23 Jun 2023

Keywords: Optimal stochastic control, stochastic games, price impact, predictive

signals, Stackelberg equilibrium

1 Introduction

Modern ﬁnancial markets involve a range of participants who place buy and sell

orders across a wide spectrum of time scales: on one end, pension funds rebalance

their portfolio on an annual basis and mutual fund managers rebalance typically on a

monthly time scale while, on the other end of the spectrum, electronic market makers

and high frequency trading ﬁrms submit several thousands of orders per second (see

e.g Cont (2011)), while having strict inventory constraints (see p.4 of U.S. Securities

and Exchange Commission (2014)). Although this heterogeneity in time scales has

been always present, the development of computerized trading in electronic markets

has substantially widened the range of frequencies at which various market partici-

pants operate. The interaction between the ﬂow of buy and sell orders from these

diﬀerent participants results in an aggregate order ﬂow which is the superposition

of components across a wide range of frequencies. The consequences of this phe-

nomenon for market volatility, price dynamics and market stability have yet to be

systematically explored.

This heterogeneity of frequencies stands in contrast with mathematical models

of market microstructure and price dynamics which are often formulated in terms

of homogeneous agents operating at a single time scale as in (Gârleanu and Peder-

sen,2016;Evangelista and Thamsten,2020;Voß,2022;Neuman and Schied,2022;

Micheli et al.,2021;Drapeau et al.,2019;Casgrain and Jaimungal,2020;Fu et al.,

2020;Neuman and Voß,2021) among others. Yet, the repeated occurrence of ‘ﬂash

crashes’ (see e.g. Kirilenko et al. (2017)) demonstrates that components at diﬀer-

ent frequencies may strongly interact and possibly lead to market disruption, calling

for a modeling framework which incorporates the interaction of agents operating on

diﬀerent time scales.

As a ﬁrst step to investigate these phenomena, we propose a model for the dynam-

ics of prices and order ﬂow in a market where participants of two diﬀerent frequencies

submit buy and sell orders on a risky asset. Speciﬁcally, we consider a stochastic

game between an institutional investor and a high-frequency trader who are exploit-

ing an exogenous signal which interacts with the price process in the drift term. The

institutional investor and high-frequency trader, which will be referred to as major

agent and minor agent, respectively, interact through their aggregated order-ﬂow,

which is resulting by their own trades. The trades of both agents create temporary

and permanent price impact which aﬀect the asset price process. We model this sys-

tem by means of two coupled multi-period stochastic control problems over a ﬁxed

time horizon T, where the high-frequency trader exploits the exogenous information

continuously, but is also subject to periodic inventory constraints at the end of any

sub-period 0< t1< ... < tn=T, for some n≥1. On the other hand, the institu-

tional investor has a limited access to the signal but she is only subject to inventory

constraints at time T. Since in the setting that we wish to describe the minor agent

has a clear advantage in terms of information exploitation, it is natural to look for a

Stackelberg equilibrium in this game, where the minor agent takes advantage of the

signal and the order-ﬂow which is created by the major agent’s transactions.

Our ﬁrst result derives the unique optimal strategy of the high-frequency trader

given any admissible strategy of the major agent (see Theorem 3.2). The challenging

part in establishing a Stackelberg equilibrium is to derive the strategy of the player

who plays ﬁrst, namely the major agent. We develop a novel approach for this class

of Stackelberg games in order to derive the major agent’s optimal strategy given

the optimal signal-adaptive strategy of the minor agent using tools from the theory

of integral equations. Speciﬁcally, in Theorem 3.5 we describe the unique optimal

major agent’s strategy in terms of the resolvent of a Fredholm integral equation,

thus establishing the unique multi-period Stackelberg equilibrium of the game. In

Section 4we illustrate the solutions to the Stackelberg game and in Section 5we

derive the additional technical steps that are needed in order to obtain such explicit

results directly from Theorems 3.2 and 3.5.

Our method for solving this Stackelberg game introduces concepts from the theory

of integral equations, which according to our knowledge were not used before in order

to solve such equilibrium problems. These tools allow us to derive explicitly the

equilibrium of this multi-period Stackelberg model, which is quite a surprising as

typically these models are highly intractable and can be solved only in some special

cases. See the comparison with related papers by Carlin et al. (2007), Schöneborn

and Schied (2009) and Roşu (2019) which is discussed later in this section. For this

reason we put an eﬀort to create a self-contained text, which introduces the required

background for the application of our method. In particular in Sections 5and 10 and

in Appendix Awe describe in detail the numerical scheme which is used in order to

plot the optimal strategy of the major agent in Theorem 3.5, and we provide the proof

of its convergence. These details are given in order demonstrate how the theoretical

solution in Theorem 3.5, which is given in terms of a spectral decomposition for a

resolvent of an operator in (3.13), can be approximated and plotted. The discussed

operator Garises from the solution to the minor agent’s problem (see (3.10)).

From our main theoretical results we derive explicit expressions for both agents

equilibrium strategies which have fascinating economic interpretation regarding the

trading behaviour of high-frequency traders and on the best practices for institutional

investors who are executing large meta-orders. We summarise these insights in the

following list and refer the reader for the comprehensive discussion in Section 4:

(i) Our results suggest that the high-frequency trader can adopt either predatory

or cooperative strategy with respect to the major agent in each period, de-

pending on the tradeoﬀ between the order-ﬂow of the major agent and the

trading signal during the period. See Figure 1for speciﬁc realisations of such

strategies.

(ii) We compare the revenues of the major agent’s optimal order execution with a

benchmark optimal strategy in which the agent is not taking into account of

the minor agent’s trading activity. In Figure 5we show that the major agent’s

optimal strategy on average considerably outperforms the benchmark strategy.

This contrasts with the common belief that high-frequency traders order-ﬂow

can be regarded as noise.

(iii) We show that the major agent’s and minor agent’s optimal trading strategies

induce the well-known U-shaped pattern of intraday trading volume, where the

traded volume peaks at the beginning and at the end of the day (see Figure 6).

Our model is related to a class of predatory trading models which was introduced

by Carlin et al. (2007) for a single period and further developed by Schöneborn and

Schied (2009) for two-periods. In Carlin et al. (2007) a single-period multi-agent

game was introduced where traders are liquidating simultaneously where while cre-

ating both temporary and permanent price impact which aﬀects the price process. In

their model there are two types of agents: sellers which start with a positive amount

of assets and competitors who have zero initial positions. All agents are seeking

to maximise simultaneously similar revenue functionals, using strictly deterministic

strategies. Their main results derive a Nash equilibrium for the game. In the single

period case it is shown that, under some assumptions on the model parameters, if the

seller is liquidating then the competitor is ﬁrst selling and later buying her position

back due to inventory constraints (see Figure 1 in Schöneborn and Schied (2009)).

In the two period model the seller can liquidate only in the ﬁrst period, while the

competitor can execute her strategy over two periods. Depending on the price im-

pact parameters, there are two possible scenarios: either the competitor is buying in

the ﬁrst period and then selling in the second period, i.e. introducing cooperative

strategies in the game (see Figure 8 therein), or doing a round trip of selling ﬁrst

and then closing the position all in the ﬁrst period.

Our model is diﬀerent from the Schöneborn and Schied (2009) in a few critical

points. First, we assume that minor agent (resp. competitor) is trading at a higher

frequency than the major agent (resp. seller). This is reﬂected in the model as

periodic inventory constraints in the minor agent’s revenue functional. This term

do not appear in the major agent’s objective, who has a fuel constraint only at the

end to the trading time horizon. The minor agent is also reacting continuously to

exogenous information while the major agent has access to the information only at

the beginning of the trade. This means also that the minor agent’s optimal strat-

egy is stochastic, unlike the deterministic game which was studied in Schöneborn

and Schied (2009). Another major diﬀerence between these models is in the type

of equilibrium which is derived. In Schöneborn and Schied (2009) an open-loop

Nash equilibrium was derived, which means that all traders optimise simultaneously.

From market microstructure setting with various frequencies, it is essential to con-

sider a Stackelberg equilibrium as the minor agent is indeed reacting to the major

agent’s selling strategy. As stated before, neither Carlin et al. (2007) nor Schöneborn

and Schied (2009) take into account exogenous information, therefore, their optimal

strategies are always found to be deterministic. One of the main conclusions of our

analysis is that this aspect has a prominent eﬀect on the behaviour of the major agent

and the minor agent, which is not captured in Carlin et al. (2007) and Schöneborn

and Schied (2009). Finally, despite the clear asymmetry in our model between the

agents in the access to information, type of equilibrium and inventory constraints,

which make the problem quite involved and required us to introduce new methods for

Stackelberg games, we are able to derive explicit solutions for any number of time

periods, in contrast to Schöneborn and Schied (2009), where only the two period

model is tractable.

We brieﬂy mention in this context that Roşu (2019) studied a discrete-time model

where fast traders, whose decisions depend on a market signal, trade simultaneously

with slow traders, who can only observe a lagged version of that same signal. However

besides this diﬀerence in the access to information, the fast agents do not have

diﬀerent objective functionals nor inventory constraints which diﬀer them from the

slow agents, which are some of the main ingredients in our model.

Structure of the paper. The rest of the paper is organised as follows. In Section

2we deﬁne the two player model. Our main results regarding the explicit solution to

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FastandSlowOptimalTradingwithExogenousInformationRamaCont1,AlessandroMicheli∗2,andEyalNeuman31MathematicalInstitute,UniversityofOxford2,3DepartmentofMathematics,ImperialCollegeLondonJune26,2023AbstractWemodeltheinteractionbetweenaslowinstitutionalinvestorandahigh-frequencytraderasastochasticmultiper...

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Fast and Slow Optimal Trading with Exogenous Information Rama Cont1 Alessandro Micheli2 and Eyal Neuman3

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