# Optimal trading without optimal control111Bastien Baldacci gratefully
acknowledge the financial support of the ERC Grant 679836 Staqamof and would
like to thank Iuliia Manziuk (Ecole Polytechnique) for fruitful discussions.

Bastien Baldacci222École Polytechnique, CMAP, 91128, Palaiseau, France,
<EMAIL_ADDRESS>Jerome Benveniste333Courant Institute of
Mathematical Science, New York University, 251 Mercer St., New York, NY 10012
Gordon Ritter444Courant Institute of Mathematical Science, New York
University, 251 Mercer St., New York, NY 10012 (corresponding author),
<EMAIL_ADDRESS>
###### Abstract

A hypothetical risk-neutral agent who trades to maximize the expected profit
of the next trade will approximately exhibit long-term optimal behavior as
long as this agent uses the vector $p=\nabla V(t,x)$ as effective
microstructure alphas, where $V$ is the Bellman value function for a smooth
relaxation of the problem. Effective microstructure alphas are the steepest-
ascent direction of $V$, equal to the generalized momenta in a dual
Hamiltonian formulation. This simple heuristics has wide-ranging practical
implications; indeed, most utility-maximization problems that require
implementation via discrete limit-order-book markets can be treated by our
method.

##  1 Introduction

Consider an investor whose preferences are described by a utility function of
wealth, $u(w)$, as per Arrow (1963) and Pratt (1964). Let $w_{T}$ denote the
investor’s wealth at some known final time $T$. The investor attempts to
maximize the expectation of utility of final wealth, $\mathbb{E}[u(w_{T})]$,
by trading financial assets. The mechanism by which buyers meet sellers and
trades occur is known as the market microstructure. In this work, the
microstructure is assumed to be a continuous double auction electronic order
book with time priority, although our methods could be generalized to include
other kinds of market microstructure. In continuous limit-order-book
microstructure, trades are effected by submitting limit orders to an
exchange’s matching engine. For each security being traded, the investor must
determine the price levels at which to submit buy and sell orders and the
associated share quantities attached to those orders. In real markets, the
price levels are discrete; the minimum possible price increment is the quote
resolution allowed by the exchange, known as the tick size. Alternatively, the
investor may decide to refrain from placing any orders or cancel some existing
orders. Other decision variables include order type and venue. Considering all
of these details, we see that the instantaneous action space is an
inconveniently large discrete space; we discuss ways of simplifying it later
on.

######  Problem 1.

The investor seeks the optimal dynamic strategy for choosing an action
$a_{t}\in\mathcal{A}_{t}$ at each time $t$, where $\mathcal{A}_{t}$ is the set
of possible actions at time $t$, optimal in the sense of maximizing the
expected utility of final wealth, $\mathbb{E}[u(w_{T})]$.

Problem 1 is mathematically deep and perhaps intractable; it is essentially a
stochastic optimal control problem over high-dimensional discrete action and
state spaces. According to Cont and Kukanov (2017), “Although simultaneous
optimization of order timing, type, and routing decisions is an interesting
problem, it also appears to be intractable…” and even this is a special case
of Problem 1.

The purpose of the present paper is to give practically implementable methods
which execution desks could start using right away to solve Problem 1
approximately. Our approximation method breaks the problem up into two parts.
The first part is to construct a _smooth relaxation_ of the problem, which is
essentially the continuous-time and continuous-space limit; the second part is
to adjust our microstructure decisions to track the smooth relaxation
optimally. The key feature of all smooth relaxations is that they hide
microstructure details behind smooth cost functions meant to represent the
average cost of trading at a given rate; they provide no guidance on
microstructure-level decisions, effectively assuming all executions use market
orders. In particular, if we can predict the probability of a passive fill at
any given instant (e.g., based on order book imbalance), it is not clear how
to use this information in the context of a smooth relaxation, whereas our
model provides a very clear and obvious way for the implementor to take
advantage of predictions of passive fill completion.

Let $V(t,x)$ denote the Bellman value function for the smooth relaxation,
defined to be the remaining expected-utility gain from time $t$ obtained from
following the best policy when the current state at time $t$ is $x$. There is
also, in principle, a value function for Problem 1 defined in the same way,
but the latter appears to be intractable. The present paper’s key idea is to
exploit the value function of the smooth relaxation to provide effective
microstructure alphas that adjust the microstructure decisions toward long-
term optimality. The vector

$p:=\nabla V(t,x)\in\mathbb{R}^{d}$ (1.1)

plays a central role in our approach, where $d\geq 1$ is the number of traded
assets. As defined, $p$ is the direction of steepest ascent for the value
function. In our heuristics, (1.1) encodes all of the information about the
long-term utility function is needed to make the microstructure decision, so
it provides the key link between the trading schedule and the order routing
problem.

In order to describe our policy for selecting the best microstructure action,
we must first introduce some more notation. Write $R_{i,t}(v,a)$ for the
(random) profit (or loss, if negative) from an order of quantity $v$ on stock
$i$ using action $a$ over a short interval $[t,t+\delta t]$. The action $a$
includes the trader’s choice of whether to trade passively or aggressively. A
trader’s expected profit $\mathbb{E}[R_{i,t}(v,a)]$ depends on the trading
cost associated with the pair $(v,a)$, and also on the trader’s views
concerning the short-horizon midpoint price return

$r_{i}^{\text{mid}}=\text{mid}_{i}(t+\delta t)/\text{mid}_{i}(t)-1.$

With no subscript, $r^{\text{mid}}$ denotes the $d$-dimensional vector of all
midpoint returns for all assets.

######  Definition 1.1.

The term effective microstructure alphas, as used in this paper, will denote a
set of parameters given to a microstructure trader in the place of
$\mathbb{E}[r^{\text{mid}}]$, for the purpose of satisfying either a short-
term goal or a long-term goal.

Microstructure alphas, as defined, could be a simple prediction of
$r^{\text{mid}}$, or, more interestingly, they could be purposely skewed to
encourage trading to increase expected utility (ie. increase long-term alpha
and reduce risk) as we shall suggest in Equation (1.3). Our heuristics is, at
time $t$, for each security indexed by $i\in\\{1,\dots,d\\}$, choose an
instantaneous action $a_{i}^{\star}$ which solves the following maximization
(over the finite set $\mathcal{A}_{t,i}$ of possible actions on the asset $i$
at time $t$):

$\displaystyle{a_{i}}^{\star}$
$\displaystyle=\mathop{\text{argmax}}_{a\in\mathcal{A}_{t,i}}\,\mathbb{E}[R_{i,t}(v_{a},a)]\
\ \text{ for all }\ i\in\\{1,\dots,d\\},$ (1.2) where:
$\displaystyle\mathbb{E}[r^{\text{mid}}]=p\,:=\,\nabla V(t,x),$ (1.3)

and follow this action over the interval $[t,t+\delta t)$. Here $v_{a}$
denotes the quantity associated with action $a$; for example “$a=$ aggressive
buy 100 shares” means $v_{a}=100$. Note also that $r^{\text{mid}}$ and $p$ are
$n$-vectors, so the equation $\mathbb{E}[r^{\text{mid}}]=p$ expresses the
trader’s views in all $n$ assets.

There is a very good intuitive justification for (1.2)-(1.3). We show later
that, under certain conditions, the optimal instantaneous trading rate at any
time $t$ in the smooth relaxation is given by

$\mathop{\text{argmax}}_{v}\\{\langle p,v\rangle-c(v)\\},$ (1.4)

where $c(v)$ is the average cost of trading at rate $v$. The expression
$\langle p,v\rangle-c(v)$ is the instantaneous analogue of expected profit
minus cost, if your expected return is $p$.

The rest of the paper is organized as follows. In Section 2, we present an
example of a long-term trading schedule, the Almgren-Chriss case, and how to
compute the value function $V(t,x)$ and its gradient $p$. We further show how
$p$ is related to the generalized momenta of the Hamiltonian approach. In
Section 3, we derive the heuristics (1.2) by analogy with the smooth case and
show how the trader can choose its short-term alpha in order to minimize the
error with respect to the trading schedule. In Section 4, we present a general
microstructure trading framework on a portfolio of cross-listed assets, taking
into account long and short-term trading signals as well as many components of
market microstructure (spread, imbalance, probability of filling etc.). We
also show how this heuristics can be applied to the problem of multi-asset
market-making. Finally, Section 5 shows a detailed numerical example which
illustrates the dangers of separating portfolio construction from execution,
and in which our method generates an improvement that is both statistically
and economically significant.

##  2 The long-term trading curve

In this section, we recall the optimal portfolio liquidation framework of
Almgren and Chriss (2001) in continuous time and show how to solve the problem
using the Hamiltonian method. In the process, we introduce notation that is
used in the rest of the paper.

Variations of the Almgren-Chriss model are used in execution desks all over
the world. Indeed it is safe to say that the Almgren-Chriss model (usually
with some custom extensions) has been used to execute trillions of dollars’
worth of customer orders. We also believe that the Almgren-Chriss model, used
in conjunction with the presented heuristics for microstructure decisions, is
a reasonable choice if the trader has no alpha forecasts (with the possible
exception of microstructure forecasts). Moreover, the computations we do here
generalize in a straightforward way to extensions of the Almgren-Chriss model,
such as to include long-term alpha. One of the key pieces of intuition that
allows us to generalize the model, we feel, comes from Theorem 1 which makes
the connection between microstructure alpha, Hamilton’s generalized momentum,
and the steepest-ascent direction of the long-term value function. For all of
these reasons, we feel this is a useful example to do in detail.

We consider a trader in charge of a portfolio of $d\geq 1$ assets, of initial
positions $q_{0}=(q_{0}^{1},\dots,q_{0}^{d})^{\mathbf{T}}$ where
$q_{0}^{i}\in\mathbb{R}$ for all $i\in\\{1,\dots,d\\}$. The trader wants to
unwind this portfolio over the time horizon $[0,T]$, where $T>0$. Given a
control process $(v_{t}^{i})_{t\in[0,T]}$ representing the trading rate on
asset $i$, the inventory process of the $i$-th asset is given by

$\displaystyle q_{t}^{i}=q_{0}^{i}-\int_{0}^{t}v_{s}^{i}ds,\quad
i\in\\{1,\dots,d\\}.$ (2.1)

For each stock, we consider Gaussian price dynamics:

$\displaystyle dS_{t}^{i}=\sigma^{i}dW_{t}^{i},$

where the Brownian motions $(W_{t}^{i})_{t\in[0,T]}$ are such that
$(S^{1}_{t},\dots,S^{d}_{t})_{t\in[0,T]}$ has a nonsingular covariance matrix
$\Sigma$.

We treat all temporary impact as instantaneous and permanent impact as linear,
hence irrelevant in the continuous-time case. In the single-asset case, if we
trade $\Delta q$ dollars in some small time interval of length $\Delta t$, and
this costs $\lambda\Delta q/\Delta t$ times traded notional for some
$\lambda>0$, then the total cost in dollars per unit time is

$\lambda(\delta q/\delta t)^{2}\equiv c(\delta q/\delta t),$

where $c(v)=\lambda v^{2}$. However, $c(\cdot)$ does not need to be quadratic,
merely convex. In the multi-asset case, we simply set
$c:\mathbb{R}^{d}\to\mathbb{R}_{+}$. As an aside, we note that these impact
assumptions are an approximation that is only ever assumed to be valid within
a certain regime. For example, if we repeatedly aggress with medium to large
order sizes within a short time-frame, it is unrealistic to assume that the
impact will revert instantly.

The trader attempts to maximize the expectation of utility of final wealth.
Due to the nonlinear nature of most utility functions, this is inconvenient to
work with. If time is discrete and if the multi-period asset return vector
follows an elliptical distribution, then there exists some constant $\kappa>0$
for which it is equivalent to maximize the mean-variance quadratic form:

$\mathbb{E}[w_{T}]-\frac{\kappa}{2}\mathbb{V}[w_{T}],$

where $w_{T}$ is the trader’s wealth at terminal time. This is essentially the
Markowitz prescription. Following Almgren and Chriss (2001), most authors and
practitioners replace the variance of final wealth $\mathbb{V}[w_{T}]$ with
the integrated instantaneous variance, leading to the standard continuous-time
approximation of the mean-variance form,

$\int_{0}^{T}\Big{[}-\frac{\kappa}{2}q_{t}^{T}\Sigma
q_{t}-c(\dot{q}_{t})\Big{]}\,dt.$

We shall now recast the maximization of the mean-variance form of the utility
function as a problem in the calculus of variations. As in classical
mechanics, it has both a Lagrangian and a Hamiltonian formulation, which are
convex duals to each other. The Hamiltonian is related to the Lagrangian by
the Legendre-Fenchel transform. In the following, we define what we mean by
the term “smooth relaxation” which is not standard terminology.

######  Definition 2.1.

Let $q_{T}\in\mathbb{R}^{d}$ be a desired final portfolio to be achieved at
time $T$. The smooth relaxation problem associated to $c(\cdot),\Sigma,q_{T}$
is defined to be:

$\displaystyle V(0,q_{0})$ $\displaystyle=\min_{q\in
C^{2}([0,T],\mathbb{R}^{d})}\int_{0}^{T}L(q_{s},\dot{q}_{s})\,ds\text{ subject
to }q_{0}=q_{0},q_{T}=q_{T},$ (2.2)

where the (autonomous) Lagrangian is given by:

$L(q,v)=c(v)+\frac{1}{2}\kappa(q-q_{T})^{\prime}\Sigma(q-q_{T}),$ (2.3)

and $\kappa>0$ is the risk-aversion constant.

The terminology of Definition 2.1 arises because Problem 1 is non-smooth and
perhaps intractable, given that the action space of Problem 1 is discrete and
quite large. In this sense (2.2) is a relaxation of the intractable problem to
the space of twice-differentiable paths $C^{2}([0,T],\mathbb{R}^{d})$. In this
paper, most of our results assume autonomous Lagrangians for simplicity.
However, non-autonomous Lagrangians also arise in trading problems. For
example, an alpha forecast which attenuates for large $t$ entails a time-
dependent linear term in (2.2).

An application of the Euler-Lagrange formula to (2.2) leads to a system of
second-order equations. By a standard trick, a first-order system can be
obtained if we introduce the so-called generalized momenta $p$, defined as

$p:=\partial_{v}L(q,\dot{q}).$ (2.4)

If the conditions of the implicit function theorem are satisfied, we could
solve (2.4) for $\dot{q}$, obtaining

$\dot{q}=\phi(q,p),$

for some function $\phi$ defined implicitly by (2.4). The Euler equation then
takes the form

$\dot{p}=\partial_{q}L(q,\dot{q})=\partial_{q}L(q,\phi(q,p))\equiv\psi(q,p),$

where this defines $\psi$.

As the functions $\phi,\psi$ are algebraic (not involving derivatives), we
have a system of $2d$ first-order ODEs given by

$\dot{q}=\phi(q,p),\quad\dot{p}=\psi(q,p)$ (2.5)

These equations can be expressed more symmetrically by introducing the
Hamiltonian

$H(q,p):=p\phi(q,p)-L(q,\phi(q,p)).$

Equations (2.5) are equivalently written in a form known as _Hamilton’s
equations_ :

$\dot{q}=H_{p}(q,p),\quad\dot{p}=-H_{q}(q,p).$ (2.6)

Suppose $c(v)=\frac{1}{2}v^{\mathbf{T}}\Lambda v$ where
$\Lambda=\text{diag}(\lambda^{1},\dots,\lambda^{d})$ is a diagonal matrix. We
assume that no trading is free of cost, so $\lambda^{i}>0$ for all $i$. From
(2.3) and (2.4), we see that the generalized momenta are $p=\Lambda\cdot q$
and hence algebraically solving, one has $\phi(q,p)=\Lambda^{-1}p$. The
Hamiltonian is then

$H(q,p)=\frac{1}{2}p\Lambda^{-1}p-\frac{1}{2}\kappa q^{\mathbf{T}}\Sigma q.$

Hamilton’s equations then become:

$\dot{q}=\Lambda^{-1}p,\qquad\dot{p}=\kappa\Sigma q,$ (2.7)

and some technical computations lead to the following proposition.

######  Lemma 2.2.

The solution to (2.2) is given by

$\displaystyle
q_{t}^{\star}=(C^{\mathbf{T}})^{-1}\Omega\Big{(}e^{D^{\frac{1}{2}}(T-t)}-e^{-D^{\frac{1}{2}}(T-t)}\Big{)}\Big{(}e^{D^{\frac{1}{2}}T}-e^{-D^{\frac{1}{2}}T}\Big{)}^{-1}\Omega^{\mathbf{T}}C^{\mathbf{T}}q_{0},$
(2.8)

where $\Sigma=CC^{\mathbf{T}}$ is the Cholesky decomposition of $C$ and
$\Omega D\Omega^{\mathbf{T}}$ is a spectral decomposition of the positive
definite matrix $\frac{\kappa}{2}C^{\mathbf{T}}\Lambda^{-1}C$.

The trading curve (2.8) can be computed in advance, and corresponds to the
order scheduling decision. We end this section by showing that the gradient of
the value function (2.2) is equal to the generalized momenta $p$.

######  Theorem 1.

Let $V:[0,T]\times\mathbb{R}^{d}\to\mathbb{R}$ be continuously differentiable
in time and space such that:

$\displaystyle V(t,q_{t})$ $\displaystyle=-\min_{q\in
C^{2}([t,T],\mathbb{R}^{d})}\int_{t}^{T}L(q_{s},v_{s})\,ds\text{ subject to
}q_{0}=q_{0},q_{T}=q_{T}$ (2.9)

where for all $s\in[0,T]$, $\dot{q}_{s}=v_{s}$ and $L(q,v)$ is
separable.555The value function (2.2) satisfies these hypothesis, as the
associated optimal control (2.8) is continuously differentiable with respect
to time and the Lagrangian of the problem is quadratic with respect to both of
its variables. The function $V$ defined in (2.9) satisfies the Hamilton-
Jacobi-Bellman differential equation:

$\partial_{t}V(t,q)+H(q,\nabla V)=0,$ (2.10)

where $H(q,p)=\sup_{v}\left\\{\langle p,v\rangle-L(q,v)\right\\}$, with the
singular final condition:

$V(T,q)=\begin{cases}0,\text{ if $q=q_{T}$}\\\ \infty,\text{ if $q\neq
q_{T}$}.\end{cases}$

###### Proof.

Let $q^{\star}$ be the path that solves (2.9) on $[t,T]$ with initial
condition $q^{\star}_{t}=q$. By the dynamic programming principle, we have for
$h>0$

$\displaystyle V(t,q)$
$\displaystyle=-\int_{t}^{t+h}L(q^{\star}_{s},\dot{q}^{\star}_{s})ds+V(t+h,q^{\star}_{t+h}).$
(2.11)

As
$V(t+h,q^{\star}_{t+h})=V(t,q)+\int_{t}^{t+h}\big{(}\partial_{t}V(s,q_{s})+\langle\nabla
V(s,q_{s})\cdot\dot{q}_{s}^{\star}\rangle\big{)}ds$, Equation (2.11) can be
rewritten as

$\displaystyle 0=\int_{t}^{t+h}\big{(}\partial_{t}V(s,q_{s})+\langle\nabla
V(s,q_{s})\cdot\dot{q}_{s}^{\star}-L(q^{\star}_{s},\dot{q}^{\star}_{s})\big{)}ds.$

The conclusion follows from an application of the Bellman’s optimality
principle, see Dreyfus (1960), which gives the desired Hamilton-Jacobi-Bellman
equation. ∎

The above theorem gives the desired interpretation of the generalized momenta
$p$ in terms of the value function. Indeed, along an optimal trajectory
$q^{\star}$, we have

$\left.\nabla
V\right|_{x=q^{\star}_{t}}=\partial_{v}L(q^{\star}_{t},\dot{q}^{\star}_{t})=p_{t}\
\forall\ t\in[0,T],$

where the second equality is just the definition of $p$ from (2.4).

##  3 From smooth relaxation to microstructure decision

In this section, we prove our main theorem, Theorem 2, which shows how a risk-
neutral instantaneous-profit maximizer (or “myopic agent”) can achieve long-
term optimality given a judicious choice of microstructure alpha model. In
other words, there is a specific microstructure alpha model related to
Hamilton’s generalized momenta, which, if used by a microstructure trader,
encourages the trader to take positions that are optimal at a much longer
horizon.

Suppose now that $L$ is coercive of degree $r>1$. One may prove that $H$
coincides with the Fenchel conjugate of $L$:

$H(q,p)=\sup_{v\in\mathbb{R}^{d}}\left\\{\langle p,v\rangle-L(q,v)\right\\}.$
(3.1)

It follows that $H$ is convex in the $p$ variable. We now restrict attention
to autonomous and separable Lagrangians that take the form

$L(q,v)=c(v)+f(q),$ (3.2)

which includes the mean-variance example discussed before. Under the
assumption (3.2), duality between the Lagrangian and Hamiltonian implies that
the optimal instantaneous trade $\dot{q}^{*}(t)$ at each time $t$ is the
argument $v$ which solves the maximization problem in (3.1). That is,

${\dot{q}}^{\star}=\mathop{\text{argmax}}_{v}\big{\\{}\langle
p,v\rangle-c(v)\big{\\}}.$ (3.3)

where (3.2) implies we can safely omit the term $f(q)$.

One can interpret (3.3) as optimization in a risk-neutral world. Indeed,
suppose a risk-neutral agent had a vector of expected returns, which happened
to exactly equal the vector of generalized momenta, $p$, and sought only to
maximize net profit, irrespective of risk. In that case, (3.3) is the problem
faced by this agent. This agent can be considered “myopic” because any
information concerning more than one period ahead is available to the agent
only indirectly, insofar as $p$ depends on the rest of the trading path. This
intuition is related to the interpretation of the generalized momenta as the
gradient of the value function. Indeed, for a myopic risk-neutral investor who
does not face transaction costs, the “value function” of a position of size
$x$ is simply the expected profit over the next period, i.e., $V(t,q)=r\cdot
q$ where $r$ is the vector of expected returns, but then $\nabla V=r$. The
following definition characterizes what a myopic agent is. Moreover, we
emphasize that, throughout the paper, $V(\cdot,\cdot)$ corresponds to the
value function of a long-term optimization problem (whose gradient provide the
effective microstructure alphas), whereas $W(\cdot,\cdot)$ denotes the value
function of the myopic agent.

######  Definition 3.1.

A myopic agent with microstructure alphas $p$ is defined to be a risk-neutral
trader seeking to maximize instantaneous net profit by choosing trading rate
given by

$\displaystyle{\dot{q}}^{\star}=\mathop{\text{argmax}}_{v}\big{\\{}\langle
p,v\rangle-c(v)\big{\\}},$ (3.4)

where $p$ denotes a set of microstructure alphas, as in Definition 1.1. The
value function of the myopic trader at time $t\in[0,T]$ for an inventory
$q\in\mathbb{R}^{d}$ is defined as

$\displaystyle W(t,q)=\int_{t}^{T}\big{(}\langle
p_{s},v^{\star}_{s}\rangle-c(v^{\star}_{s})\big{)}ds,$ (3.5)

where $v^{\star}$ is defined as the solution of (3.4).

In other words, the value function of a myopic trader defined in (3.5) is
simply the sum of his instantaneous trading gains over time. The following
proposition shows that a myopic trader sending market orders only has to
choose $p=\nabla V$ in order to minimize the error between his value function
and the long-term objective function $V$.

######  Theorem 2.

Assume that $L$ takes the separable form (3.2). A myopic agent with
instantaneous cost function $c(\cdot)$ must choose microstructure alphas
$p=\nabla V$ in order to minimize the absolute error between his value
function and the long-term objective function $V$ defining the trading
schedule. More precisely, for all
$(t,q)\in[0,T]\times\prod_{i=1}^{d}\big{[}\min(q_{0}^{i},q_{T}^{i}),\max(q_{0}^{i},q_{T}^{i})\big{]}$,

$\displaystyle\big{|}W(t,q)-V(t,q)\big{|}\leq\kappa(T-t)\frac{|q-q_{T}|^{\mathbf{T}}\Sigma|q-q_{T}|}{2},$

and we have the uniform bound

$\displaystyle\sup_{(t,q)\in[0,T]\times\prod_{i=1}^{d}[\min(q_{0}^{i},q_{T}^{i}),\max(q_{0}^{i},q_{T}^{i})]}\big{|}W(t,q)-V(t,q)\big{|}\leq\kappa
T\frac{|q_{0}-q_{T}|^{\mathbf{T}}\Sigma|q_{0}-q_{T}|}{2},$

where
$|q_{0}-q_{T}|=\big{(}|q_{0}^{1}-q_{T}^{1}|,\dots|q_{0}^{d}-q_{T}^{d}|\big{)}$.

###### Proof.

The myopic trader aims at minimizing $c(v)-pv$, where for the moment $p$
remains undetermined, at each trading time. Over $[t,T]$ the trader’s problem
can be written:

$\displaystyle W(t,q)=\max_{v}\int_{t}^{T}[\langle
p_{s},v_{s}\rangle-c(v_{s})]\,ds.$

We choose quadratic costs $c(v)=\frac{\eta}{2}\|v\|_{2}^{2}$, where
$\|\cdot\|_{2}$ is the Euclidian norm and the first order condition with
respect to $v$ gives $W(t,q)=\int_{t}^{T}\frac{\|p_{s}\|_{2}^{2}}{2\eta}\,ds$.
On the other hand, the value function $V(t,q)$ becomes

$\displaystyle\int_{t}^{T}\left\\{\frac{1}{2\eta}\|\partial_{v}L(q^{\star}_{s},v^{\star}_{s})\|_{2}^{2}+\frac{1}{2}\kappa(q^{\star}_{s}-q_{T})^{\mathbf{T}}\Sigma(q^{\star}_{s}-q_{T})\right\\}\,ds,$

where $q^{\star}$ is defined by (2.8) and $v^{\star}$ is its derivative with
respect to time. Therefore,

$\displaystyle|V(t,q)-W(t,q)|=\Big{|}\int_{t}^{T}\Big{\\{}\frac{1}{2\eta}\|\partial_{v}L(q^{\star}_{s},v^{\star}_{s})\|_{2}^{2}+\frac{1}{2}\kappa(q^{\star}_{s}-q_{T})^{\mathbf{T}}\Sigma(q^{\star}_{s}-q_{T})-\frac{\|p_{s}\|_{2}^{2}}{2\eta}\Big{\\}}\,ds\Big{|},$

and the minimum with respect to $p$ is attained at
$p_{s}=\partial_{v}L(q^{\star}_{s},v^{\star}_{s})=\nabla V(s,q^{\star}_{s})$,
because $p$ must depend only of the instantaneous trading rate and $L$ is
separable and additive. The bounds are obtained easily by definition of the
space of inventories. ∎

This simple result has several important consequences. Suppose one wants to
avoid the use of optimal control and still wants to follow the Almgren-Chriss
trading curve. In that case, one can simply solve the static optimization
problem (3.3) at discrete times (the times of trading), using $p=\nabla V$.
Equation (3.3) does not give a full set of instructions for the trader with a
long-term trading schedule who has to interact with a continuous limit order
book market, but it can serve as a guide. Indeed, the order routing problem,
treated notably in Cont and Kukanov (2017), takes into account the possibility
to send limit, market, or cancel orders to several liquidity venues, depending
on their spread and imbalance. Stochastic control appears to be inefficient
for this problem, as one needs to solve a high-dimensional Hamilton-Jacobi-
Bellman equation. Methods involving deep reinforcement learning have been
developed for optimal trading, see for example Baldacci and Manziuk (2020),
but they lead to high computation time, especially if one wants to deal with a
portfolio of assets traded on several venues. The advantage of the methodology
presented in this paper is that one can avoid optimal control and solve a
simple static optimization problem to determine the optimal action at each
discrete trading time.

######  Remark 3.2.

The bounds on the absolute error between the value function of the myopic
agent and the long-term objective function enable to compute the accuracy of
the myopic trader. For example, take the liquidation over $T=1$ day of
$q_{0}^{1}=q_{0}^{2}=2000$ shares of $2$ assets, with correlation $\rho=0.6$
and daily volatilities $\sigma^{1}=0.015$ and $\sigma^{2}=0.02$. The absolute
error between the two value functions is uniformy bounded in time and
inventories by $2\times 10^{-2}$.

In Theorem 2, the myopic trader does not consider the properties of an order
book, such as the possibility to submit limit and market orders or to wait. In
this case, the optimal effective microstructure alpha $p$ (in the sense of
minimization of the error with respect to the Almgren-Chriss value function
$V$) should not be equal to the generalized momenta because we add
microstructure effects for the myopic trader that are not present in the
trading schedule represented by $\nabla V$.

With a sufficiently simple fill model, the myopic trader’s problem dealing
with microstructure effects can be solved in closed form. This closed-form
expression (see Example 3.3) illustrates the contrast between the two possible
decisions the trader must face, as mentioned above.

######  Example 3.3.

For the sake of readability, we assume $d=1$. Suppose that the myopic trader
can choose between submitting a limit order (with fill probability $0<f<1$) or
a market order (with the cost of crossing the spread equal to $s>0$). The
myopic trader’s optimization problem is:

$\displaystyle\sup_{v}\Big{\\{}(pv-\frac{\eta}{2}v^{2}-sv)\mathbf{1}_{\\{pv-\frac{\eta}{2}v^{2}-sv>f(pv-\frac{\eta}{2}v^{2})\\}}+f(pv-\frac{\eta}{2}v^{2})\mathbf{1}_{\\{pv-\frac{\eta}{2}v^{2}-sv<f(pv-\frac{\eta}{2}v^{2})\\}}\Big{\\}}.$

Computations lead to the following decisions:

$\displaystyle\text{ passive order if }p<\frac{2s}{1-f},\text{ aggressive
otherwise},$

and the optimal microstructure alpha is given by:

$\displaystyle p^{\star}_{t}=\frac{1}{\sqrt{f}}|\nabla
V(t,q^{\star}_{t})|\text{sgn}\big{(}\nabla V(t,q^{\star}_{t})\big{)}\text{ if
}p_{t}^{\star}<\frac{2s}{1-f},\quad p^{\star}_{t}=|\nabla
V(t,q^{\star}_{t})|\text{sgn}\big{(}\nabla V(t,q^{\star}_{t})\big{)}+s\text{
otherwise.}$

The use of the generalized momenta as effective microstructure alpha has a
wide range of practical implications. First, it offers a way to bridge the gap
between order placement decisions and scheduling decisions, usually decoupled
in practice. Second, the microstructure formulation helps to tackle classic
optimal control on limit order books. For example, a realistic optimal trading
framework dealing with a portfolio of assets on several liquidity venues is in
practice intractable due to the dimensionality of the problem. In the method
presented in this paper, the optimal controls of the trader (that is, the
volume sent on each venue for each asset by the mean of limit and market
orders) are derived through a simple static optimization problem, which can be
solved for a large number of assets on a large number of venues. The
convergence through the trading schedule is guaranteed by choice of the
effective microstructure alpha $p$.

######  Remark 3.4.

Note that, in the framework of Example (3.3), if the spread $s$ tends to zero
and the filling probability $f$ tends to one, we recover the framework of a
myopic sending market orders only, and the optimal effective microstructure
alpha is given by $\nabla V$.

This method can easily handle the increasing complexity coming from the
microstructure effects (short term alpha, imbalance, and spread of each venue,
etc.). In the next section, we present a general microstructure trading model
taking into account the main stylized facts combining order placement and
order routing of a portfolio of assets. We show that using the method proposed
in this paper can be applied to solve in practice two important problems in
systematic trading: the multi-asset, multi-venue optimal trading problem and
the multi-asset, multi-venue optimal market-making problem.

##  4 A general microstructure trading model with long-term trading schedule

###  4.1 Optimal trading

We first provide a definition of the problem.

######  Definition 4.1.

Consider an agent trading a portfolio of correlated assets, where each asset
is listed on one or more liquidity venues. The multi-asset, multi-venue
optimal trading problem consists in determining at a given time and for each
asset, the optimal quantity to buy or sell on each venue, for given market
conditions and a pre-computed trading schedule, as well as the optimal limit
at which such quantity should be posted.

The framework described here is inspired by Baldacci and Manziuk (2020).
Consider a trading schedule for $d\geq 1$ assets $q^{\star}\in\mathbb{R}^{d}$
(the Almgren-Chriss trading schedule described in Equation (2.8), for example)
with associated value function $V(t,q_{t}^{\star})$. For each asset
$i\in\\{1,\dots,d\\}$, the trader splits his limit and market orders between
$N^{i}\geq 1$ liquidity venues. We assume that he wants to unwind the
portfolio so that $q_{T}=0^{d}$. For all
$i\in\\{1,\dots,d\\},n\in\\{1,\dots,N^{i}\\}$, the order book of the asset $i$
on the venue $n$ is characterized by the following quantities:

  * •

the bid-ask spread process $(\psi_{t}^{i,n})_{t\in[0,T]}$ taking values in the
state space $\overline{\psi}^{i,n}=\\{\delta^{i,n},\dots,J\delta^{i,n}\\}$,

  * •

the imbalance process $(I^{i,n}_{t})_{t\in[0,T]}$ taking values in the state
space $\overline{I}^{i,n}=\\{I^{i,n}_{1},\dots,I^{i,n}_{K}\\}$,

where $J,K\in\mathbb{N}$ denote the number of possible spreads and imbalances
respectively and $\delta^{i,n}$ stands for the tick size of $i$-th asset on
the $n$-th venue. Note that the dynamics are unspecified, meaning that any
continuous-time stochastic process with discrete values can be considered for
the purpose of simulation.

######  Definition 4.2.

A market regime is, for asset $i\in\\{1,\dots,d\\}$, a set of spread and
imbalance values on the different venues $n\in\\{1,\dots,N^{i}\\}$.

We define the sets
$\Psi=\\{\Psi_{1},\dots,\Psi_{\\#\Psi}\\},\mathcal{I}=\\{\mathcal{I}_{1},\dots,\mathcal{I}_{\\#\mathcal{I}}\\}$
of disjoint intervals, representing different market regimes of interest in
terms of spreads and imbalances.

######  Example 4.3.

Assume $d=1$ and for all $n\in\\{1,\dots,N\\}$, $\delta^{n}=\delta$. The set
$\Psi=\big{\\{}\delta,\\{2\delta,3\delta\\},\\{4\delta,5\delta\\}\big{\\}}$
denotes three spread regimes: low (one tick), medium (two or three ticks), and
high (four or five ticks).

######  Example 4.4.

Assume $d=1$ and for all $n\in\\{1,\dots,N\\}$ and $k\in\\{1,\dots,K\\}$ that
$I^{n}_{k}=I_{k}$. In this case the set
$\mathcal{I}=\big{\\{}[-1,-0.66],(-0.66,-0.33],(-0.33,0.33],(0.33,0.66],(0.66,1]\big{\\}}$
denotes five regimes of imbalance: low ($-33\%$ to $33\%$), medium on the ask
(resp. bid) from $33\%$ to $66\%$ (resp. from $-66\%$ to $-33\%$) and high on
the ask (resp. bid) from $66\%$ to $100\%$ (resp. from $-100\%$ to $-66\%$).

The number of, possibly partially, filled ask orders on the asset $i$ in the
venue $n$ is modeled by a Cox process denoted by $N^{i,n}$ with intensities
$\lambda^{i,n}\big{(}\psi^{i}_{t},I^{i}_{t},p_{t}^{i,n},\ell^{i}_{t}\big{)}$
where $p_{t}^{i,n}\in Q_{\psi}^{i,n}$ represent the limit at which the trader
sends a limit order of size $\ell_{t}^{i,n}$, and

$\displaystyle Q_{\psi}^{i,n}=\\{0,1\\}\text{ if
}\psi^{i,n}=\delta^{i,n},\text{ and }\\{-1,0,1\\}\text{ otherwise.}$

Practically, on asset $i$, for $n\in\\{1,\dots,N^{i}\\}$, when the spread is
equal to the tick size, the trader can post at the first best limit
($p^{i,n}=0$) or the second best limit (if $p^{i,n}=1$). When the spread is
equal to two ticks or more, the trader can either create a new best limit
($p^{i,n}=-1$) or post at the best or the second best limit as previously. The
arrival intensity of a buy market order at time $t$ on the venue
$n\in\\{1,\dots,N^{i}\\}$ for asset $i$ at the limit $p\in Q_{\psi}^{i,n}$,
given a couple $(\psi^{i}_{t},I^{i}_{t})=\mathbf{m}$ of spread and imbalance
on each venue, is equal to $\lambda^{i,n,\mathbf{m},p}>0$. When the trader
posts limit orders of volume $\ell_{t}^{i,n}$ on the $n$-th venue for
$n\in\\{1,\dots,N^{i}\\}$, the probability that it is executed is equal to
$f^{\lambda}(\ell^{i}_{t})$, where $f^{\lambda}(\cdot)\in[0,1]$ is a
continuously differentiable function, decreasing with respect to each of its
coordinate. Therefore, the arrival intensity of an ask market order filling
the buy limit order of the trader for asset $i$ on the $n$-th venue at the
limit $p_{t}^{i,n}$, given spread and imbalance $(\psi^{i}_{t},I^{i}_{t})$ is
a multi-regime function defined by

$\displaystyle\lambda^{i,n}(\psi^{i}_{t},I^{i}_{t},p_{t}^{i,n},\ell^{i}_{t})=f^{\lambda}(\ell^{i}_{t})\sum_{\mathbf{m}\in\mathcal{M}^{i},p\in
Q_{\psi}^{i,n}}\lambda^{i,n,\mathbf{m},p}\mathbf{1}_{\\{(\psi^{i}_{t},I^{i}_{t})\in\mathbf{m},p_{t}^{i,n}=p\\}},$
(4.1)

where $\mathcal{M}^{i}=\Psi^{N^{i}}\times\mathcal{I}^{N^{i}}$. Moreover, we
allow for partial execution, the fact of which we represent by random
variables $\epsilon^{i,n}_{t}\in[0,1]$. The proportion of executed volume for
limit orders in each venue depends on the spread and the imbalance in all
$N^{i}$ venues for asset $i$, as well as the volume and the limit of the order
chosen by the trader. We assume a categorical distribution with $R>0$
different execution proportions $\omega^{r},r\in\\{1,\ldots,R\\}$ for each
venue with
$\mathbb{P}(\epsilon_{t}^{i,n}=\omega^{r})=\rho^{i,n,r}(\psi^{i}_{t},I^{i}_{t},p_{t}^{i,n},\ell^{i}_{t})$,
where

$\displaystyle\rho^{i,n,r}(\psi^{i}_{t},I^{i}_{t},p_{t}^{i,n},\ell^{i}_{t})=f^{\rho}(\ell^{i}_{t})\sum_{\mathbf{m}\in\mathcal{M}^{i},p\in
Q_{\psi}^{i,n}}\rho^{i,n,\mathbf{m},p,r}\mathbf{1}_{\\{(\psi^{i}_{t},I^{i}_{t})\in\mathbf{m},p_{t}^{i,n}=p\\}},$
(4.2)

where $f^{\rho}(\cdot)$ is a continuously differentiable function, decreasing
with respect to each of its coordinate.

We allow for the execution of market orders (denoted by a point process
$(J_{t}^{i,n})_{t\in[0,T]}$) on each venue of size
$(v_{t}^{i,n})_{t\in[0,T]}\in[0,\overline{v}]$ where $\overline{v}>0$ and
$J_{t}^{i,n}=J_{t^{-}}^{i,n}+1$. We assume that market orders are always fully
executed but this assumption can be relaxed easily. As each asset must be
bought or sold, we define $\Delta=(\Delta^{1},\dots,\Delta^{d})$ where for
$i\in\\{1,\dots,d\\}$, $\Delta^{i}=1$ if $q_{0}^{i}>0,-1$ otherwise. The
inventory process on each asset is defined by

$\displaystyle
q_{t}^{i}=q_{0}^{i}-\Delta^{i}\sum_{n=1}^{N^{i}}\Big{(}\int_{0}^{t}\ell_{s}^{i,n}\epsilon_{s}^{i,n}dN_{s}^{i,n}+\int_{0}^{t}v_{s}^{i,n}dJ_{s}^{i,n}\Big{)}.$

The myopic trader has an effective microstructure
$(p_{t}^{\text{eff},i})_{t\in[0,T]}$ in order to follow the pre-computed
execution curve $q_{t}^{\star i}$ on each asset, but also a short-term alpha
$(p_{t}^{\text{short},i})_{t\in[0,T]}$ which is a function of the current
spread and imbalance $\mathbf{m}^{i}$ of all the venues where asset $i$ is
listed.

######  Remark 4.5.

The microstructure alpha considered for each asset is the sum of a direct
microstructure alpha depending on the market regimes and an effective
microstructure alpha that gives a signal to follow the long-term objective
function. The sum of these two terms gives the magnitude of the buy or sell
signal. For example, suppose $p^{\text{eff},i}$ is small and positive,
indicating that filling a buy order would be a slight improvement to the value
function. Suppose with $p^{\text{eff},i}$ alone, the system would have
recommended a passive buy order. Now suppose a strongly-positive
microstructure alpha, denoted $p^{\text{short},i}$, is also present; then the
combination $p^{\text{eff},i}+p^{\text{short},i}$ in place of
$p^{\text{eff},i}$ should recommend a more aggressive action, such as a
spread-crossing buy order.

Finally, the cost function of a limit order of size $\ell$ at limit $p$ on
venue $n$ for asset $i$ is defined as $c^{i,n,\mathcal{L}}(\ell,p)$ and
$c^{i,n,\mathcal{M}}(v)$ for the cost function of a market order of size $v$
on venue $n$ for asset $i$.

The myopic trader acts at discrete times and at time $t\in[0,T]$ for
$(\psi^{i}_{t},I^{i}_{t})\in\mathbf{m}^{i},i\in\\{1,\dots,d\\}$, his
optimization problem is

$\displaystyle\begin{split}\max\bigg{\\{}&\sup_{\ell,p}\Big{\\{}\sum_{i=1}^{d}\sum_{n=1}^{N^{i}}\lambda^{i,n}(\mathbf{m}^{i},p^{i,n},\ell^{i})\mathbb{E}\Big{[}\big{(}p^{\text{eff},i}+p^{\text{short},i,n}(\mathbf{m}^{i})\big{)}\ell^{i,n}\epsilon^{i,n}-c^{i,n,\mathcal{L}}(\epsilon^{i,n}\ell^{i,n},p^{i,n})\Big{]}\Big{\\}},\\\
&\sup_{v}\Big{\\{}\sum_{i=1}^{d}\sum_{n=1}^{N^{i}}\big{(}p^{\text{eff},i}+p^{\text{short},i,n}(\mathbf{m}^{i})\big{)}v^{i,n}-c^{i,n,\mathcal{M}}(v^{i,n})\Big{\\}}\bigg{\\}},\end{split}$
(4.3)

where the expectation is taken with respect to the variables $\epsilon^{i,n}$
for all $i\in\\{1,\dots,d\\},n\in\\{1,\dots,N^{i}\\}$. This is a simple static
optimization which can be solved for a large number of assets and venues using
a multidimensional root-finding method. The output is, for each state
$\mathbf{m}^{i}$, the optimal volumes and limits $\ell^{\star
i,n}(\mathbf{m}^{i}),p^{\star i,n}(\mathbf{m}^{i})$ for each asset on each
liquidity venue. We define the value function of the myopic trader at time $t$
as

$\displaystyle W(t,\psi,I,q)=\int_{t}^{T}\max\bigg{\\{}$
$\displaystyle\sum_{i=1}^{d}\sum_{n=1}^{N^{i}}\lambda^{i,n}(\psi_{s}^{i},I_{s}^{i},p_{s}^{\star
i,n},\ell_{s}^{\star
i})\mathbb{E}\Big{[}\big{(}p_{s}^{\text{long},i}+p_{s}^{\text{short},i,n}(\psi_{s}^{i},I_{s}^{i})\big{)}\ell_{s}^{\star
i,n}\epsilon_{s}^{i,n}$
$\displaystyle-c^{i,n,\mathcal{L}}(\epsilon_{s}^{i,n}\ell_{s}^{\star
i,n},p_{s}^{\star
i,n})\Big{]},\sum_{i=1}^{d}\sum_{n=1}^{N^{i}}\big{(}p_{s}^{\text{long},i}+p_{s}^{\text{short},i,n}(\psi_{s}^{i},I_{s}^{i})\big{)}v_{s}^{\star
i,n}-c^{i,n,\mathcal{M}}(v^{\star i,n})\bigg{\\}}ds$

where $\psi_{t}=\psi,I_{t}=I,q_{t}=q$. As in Theorem 2, the myopic trader has
now to choose the long-term alpha $p^{\text{eff}}$ to match the trading
schedule $q^{\star}$. This leads to the following optimization setting for all
$\mathbf{m}^{i}\in\mathcal{M}^{i}$:

$\displaystyle\begin{split}\max\bigg{\\{}&\sup_{\ell,p}\Big{\\{}\sum_{i=1}^{d}\sum_{n=1}^{N^{i}}\lambda^{i,n}(\mathbf{m}^{i},p^{i,n},\ell^{i})\mathbb{E}\Big{[}\big{(}p^{\text{eff},i}+p^{\text{short},i,n}(\mathbf{m}^{i})\big{)}\ell^{i,n}\epsilon^{i,n}-c^{i,n,\mathcal{L}}(\epsilon^{i,n}\ell^{i,n},p^{i,n})\Big{]}\Big{\\}},\\\
&\sup_{v}\Big{\\{}\sum_{i=1}^{d}\sum_{n=1}^{N^{i}}\big{(}p^{\text{eff},i}+p^{\text{short},i,n}(\mathbf{m}^{i})\big{)}v^{i,n}-c^{i,n,\mathcal{M}}(v^{i,n})\Big{\\}}\bigg{\\}},\\\
&p^{\text{eff}}=\text{argmin}_{p^{\text{eff}}}\big{|}V(\cdot,q_{\cdot}^{\star})-W(\cdot,\psi_{\cdot},I_{\cdot},q_{\cdot})\big{|}.\end{split}$
(Opt-Trd)

In this general framework, order scheduling with a long-term target is easily
tractable even for a large portfolio of assets, as the trader has to solve a
static optimization problem at each trading time. For a parsimonious model of
filling probabilities, the effective microstructure alpha can be computed in
closed form. Note that each time a fill is received that changes the portfolio
holdings, and/or each time a significant amount of time passes, the effective
microstructure alpha $p^{\text{eff}}$ must be recomputed.

######  Remark 4.6.

The methodology presented in this paper leads to entirely tractable
optimization problems, even for a large number of assets. This is the case
when we have a closed-form solution for the long-term value function $V$,
which can be computed quickly. It also suggests an approximation of the
effective microstructure alpha, that is to take
$p_{t}^{\text{eff},i}=\nabla_{i}V(t,q_{\cdot}^{\star})$. This heuristics will
be used in the next section to solve a different control problem.

###  4.2 Market-making

The great advantage of the framework presented in this paper is that it avoids
the use of optimal control to tackle optimal trading problems. The trader
solves a simple static optimization problem, and the use of the generalized
momenta as a long-term alpha plays the role of the trading schedule. Similar
ideas can be applied to the market-making problem, with some minor changes.

######  Definition 4.7.

Consider an agent trading on a portfolio of correlated assets, where each of
them is listed on one or several liquidity venues. His goal is to earn the
difference between the bid and ask prices (the bid-ask spread) while keeping
his inventory close to zero to avoid an unwanted large exposure and be forced
to buy at a higher price or sell at a lower price in order to unwind this
position. The multi-asset, multi-venue optimal market-making problem consists
in deriving at a given time, for each asset, the optimal quantity to buy or
sell in each venue, for given market conditions, as well as the optimal limit
at which such quantity should be posted, with an inventory vector mean-
reverting around zero or some predetermined target.

The market-making problem has been introduced in the financial literature by
Ho and Stoll (1981), Glosten and Milgrom (1985). Ho and Stoll presented a
framework to tackle inventory management, while Grossman and Miller proposed a
$3$ periods model that encompassed both market-makers and final customers,
enabled them to understand what happens at equilibrium, and contributed to the
important literature on the price formation process. The seminal reference of
the recent literature on market-making is the work of Avellaneda and Stoikov
in Avellaneda and Stoikov (2008), who proposed a stochastic control framework
to tackle the quoting and inventory management problems. Since then, a vast
literature on optimal market-making has emerged, basically adding many
features to the Avellaneda and Stoikov framework, see for example Cartea et
al. (2014), Guéant et al. (2013) and the two textbooks Cartea et al. (2015),
Guéant (2016). These works deal with single asset market-making, and the
considered framework is more suitable for OTC markets rather than order-driven
markets. The problem of multi-asset market-making, dealing with the curse of
dimensionality, has been addressed via deep reinforcement learning methods,
see for example Guéant and Manziuk (2019). Models for optimal market-making in
limit order books have been developed for the single asset case, see Guilbaud
and Pham (2013), for example. All these models suffer from the same problem
when dealing with a portfolio of assets: solving a high-dimensional Hamilton-
Jacobi-Bellman equation makes the problem almost intractable in practice. In
this section, we propose an adaptation of the previously described heuristics
to tackle the multi-asset market-making problem in limit order books.

####  4.2.1 The long-term objective function

Our methodology to solve optimal control problems in high dimension relies on
the fact that the effective microstructure alphas come from a long-term
objective function computed analytically. This is the case of the Almgren-
Chriss trading curve, which hides the microstructure effects that are
incorporated in the myopic optimization problem. However, as stated
previously, the main constraint of the optimal market-making problem is that,
even for market-making on OTC markets, the value function’s computation is
very time-consuming. We propose to use the gradient of an approximation of the
value function of the optimal market-making problem on OTC markets as the
effective microstructure alphas for the optimal market-making problem in order
books. To this end, we borrow the OTC framework of Bergault et al. (2020) and
recall their modeling assumptions briefly.

For $i\in\\{1,\dots,d\\}$, the reference price of asset $i$ is modeled by a
process $S_{t}^{i}$ with dynamics

$\displaystyle dS_{t}^{i}=\sigma^{i}dW_{t}^{i},$

where $(W_{t}^{i},\dots,W_{t}^{d})$ is a $d$-dimensional Brownian motion with
variance-covariance matrix $\Sigma$. At each $t\in[0,T]$, the market-maker
chooses the prices $P_{t}^{i,b},P_{t}^{i,a}$ at which she is ready to buy/sell
each asset $i$. These prices are given by

$\displaystyle P_{t}^{i,b}=S_{t}^{i}-\delta_{t}^{i,b},\quad
P_{t}^{i,a}=S_{t}^{i}+\delta_{t}^{i,a},$

where
$\delta_{t}=(\delta_{t}^{1,b},\delta_{t}^{1,a},\dots,\delta_{t}^{d,b},\delta_{t}^{d,a})$
are the control processes of the market-maker corresponding to the bid and ask
spreads set on each asset $i$. For $i\in\\{1,\dots,d\\}$, the point processes
$N_{t}^{i,b},N_{t}^{i,a}$ denote the total number of bid and ask transactions
between $0$ and $t$ on asset $i$. Their intensities are given by
$\Lambda^{i,b}(\delta_{t}^{i,b}),\Lambda^{i,a}(\delta_{t}^{i,a})$ where the
functions $\Lambda^{i,b},\Lambda^{i,a}$ satisfy some technical conditions, see
Bergault et al. (2020) for details. These conditions are sufficiently general
to allow the use of several form of intensity such as exponential, logistic,
SU Johnson etc.

The transaction size for asset $i$ is constant and denoted by $z^{i}$, and the
inventory process of the market-maker for asset $i$ is

$\displaystyle dq_{t}^{i}=z^{i}\big{(}dN_{t}^{i,b}-dN_{t}^{i,a}\big{)},\quad
q_{t}=(q_{t}^{1},\dots,q_{t}^{d})^{\mathbf{T}}.$

The cash process of the market-maker has the following dynamics:

$\displaystyle
dX_{t}=\sum_{i=1}^{d}\big{(}P_{t}^{i,a}dN_{t}^{i,a}-P_{t}^{i,b}dN_{t}^{i,b}\big{)}.$

The optimization problem of the market-maker is defined by

$\displaystyle\sup_{\delta}\mathbb{E}\Big{[}X_{T}+\sum_{i=1}^{d}q_{T}^{i}S_{T}^{i}-\frac{\gamma}{2}\int_{0}^{T}q_{s}^{\mathbf{T}}\Sigma
q_{s}dt\Big{]},$

and simple computations666It can be shown by simple change of variables that
the value function of this control problem is only a functional of the time
and the inventories. give the associated value function at time $t$ for a
given inventory vector $q_{t}=q$:

$\displaystyle
V(t,q)=\sup_{\delta}\mathbb{E}_{t}\bigg{[}\sum_{i=1}^{d}\int_{t}^{T}\Big{(}\delta_{s}^{i,a}\Lambda^{i,a}(\delta_{s}^{i,a})+\delta_{s}^{i,b}\Lambda^{i,b}(\delta_{s}^{i,b})-\frac{\gamma}{2}q_{s}^{\mathbf{T}}\Sigma
q_{s}\Big{)}dt\bigg{]},$ (4.4)

where $\mathbb{E}_{t}$ denote the conditional expectation with respect to the
canonical filtration at time $t$ and $\gamma>0$ is the risk-aversion of the
market-maker. He wishes to maximize the sum of his cash process and the mark-
to-market value of his inventory. The running penalty forces him to mean-
revert his inventories to zero. We now state the main proposition of Bergault
et al. (2020) that provides a closed form approximation of $V(t,q)$, and refer
to this article for the proof.

######  Proposition 4.8.

Define the functions

$\displaystyle
H^{i,b}(p)=\sup_{\delta}\\{\Lambda^{i,b}(\delta)(\delta-p)\\},\quad
H^{i,a}(p)=\sup_{\delta}\\{\Lambda^{i,a}(\delta)(\delta-p)\\},$

and the constants
$\alpha_{j}^{i,b}=(H^{i,b})^{j}(0),\alpha_{j}^{i,a}=(H^{i,a})^{j}(0)$, where
the superscript $j\in\\{0,1,2\\}$ denote the derivative of order $j$. Define
also for $k\in\mathbb{N}$

$\displaystyle\Delta_{j,k}^{i,b}=\alpha_{j}^{i,b}(z^{i})^{k},\quad\Delta_{j,k}^{i,a}=\alpha_{j}^{i,a}(z^{i})^{k},$
$\displaystyle
V_{j,k}^{b}=\Big{(}\Delta_{j,k}^{1,b},\dots,\Delta_{j,k}^{d,b}\Big{)},\quad
V_{j,k}^{a}=\Big{(}\Delta_{j,k}^{1,a},\dots,\Delta_{j,k}^{d,a}\Big{)},$
$\displaystyle
D_{j,k}^{b}=\text{diag}\Big{(}\Delta_{j,k}^{1,b},\dots,\Delta_{j,k}^{d,b}\Big{)},\quad
D_{j,k}^{a}=\text{diag}\Big{(}\Delta_{j,k}^{1,a},\dots,\Delta_{j,k}^{d,a}\Big{)}.$

Then if $\alpha^{i,b}_{2}+\alpha^{i,a}_{2}>0$, the value function of the
optimal control problem (4.4) can be approximated by the function

$\displaystyle\tilde{V}(t,q)=-q^{\mathbf{T}}A(t)q-q^{\mathbf{T}}B(t)-C(t),$
(4.5)

where $A:[0,T]\to\mathcal{S}_{d}^{++},B:[0,T]\to\mathbb{R}^{d}$ and
$C:[0,T]\to\mathbb{R}$ are deterministic functions given by

$\displaystyle
A(t)=\frac{1}{2}D_{+}^{-\frac{1}{2}}\hat{A}\big{(}e^{\hat{A}(T-t)}-e^{-\hat{A}(T-t)}\big{)}\big{(}e^{\hat{A}(T-t)}+e^{-\hat{A}(T-t)}\big{)}^{-1}D_{+}^{-\frac{1}{2}},$
$\displaystyle
B(t)=-2e^{-2\int_{t}^{T}A(u)D_{+}du}\int_{t}^{T}e^{2\int_{s}^{T}A(u)D_{+}du}A(s)\Big{(}V_{-}+D_{-}\mathcal{D}\big{(}A(s)\big{)}\Big{)}ds$
$\displaystyle
C(t)=-\text{Tr}(D_{0,1}^{b}+_{0,1}^{a})(T-t)-\text{Tr}\Big{(}(D_{1,2}^{b}+D_{1,2}^{a})\int_{t}^{T}A(s)ds\Big{)}-V_{-}^{\mathbf{T}}\int_{t}^{T}B(s)ds$
$\displaystyle-\frac{1}{2}\int_{t}^{T}\mathcal{D}\big{(}A(s)\big{)}^{\mathbf{T}}(D_{2,3}^{b}+D_{2,3}^{a})\mathcal{D}\big{(}A(s)\big{)}ds-\frac{1}{2}\int_{t}^{T}B(s)^{\mathbf{T}}D_{+}B(s)ds$
$\displaystyle-\int_{t}^{T}B(s)^{\mathbf{T}}D_{-}B(s)ds,$

with

$\displaystyle
D_{+}=D_{2,1}^{b}+D_{2,1}^{a},D_{-}=D_{2,2}^{b}-D_{2,2}^{a},V_{-}=V_{1,1}^{b}-V_{1,1}^{a},\hat{A}=\sqrt{\gamma}\big{(}D_{+}^{\frac{1}{2}}\Sigma
D_{+}^{\frac{1}{2}}\big{)}^{\frac{1}{2}},$

and $\mathcal{D}$ is the linear operator mapping a matrix onto the vector of
its diagonal and $\mathcal{S}_{d}^{++}$ is the set of $d\times d$ definite
positive matrix.

The approximated value function (4.5) is quadratic, therefore, sub-
differentiable with respect to the vector of inventories $q$ and the
deterministic functions $A(t),B(t),C(t)$ can be computed in closed form. It
takes into account the main property of a “high-level” multi-asset market-
making problem, that is, the correlation structure between the assets. By
analogy with Section 4.1, its sub-gradient can be chosen as an effective
microstructure alpha to mean revert toward a flat inventory.

######  Remark 4.9.

In order to compute efficiently the value function in (4.5), note that the
matrix $A$ can be diagonalized and therefore approximated with a principal
component analysis. The expressions of $A,B,C$ do not provide intuition about
the long-term behavior of the market-maker. However, in an asymptotic
framework, that is when $T\to+\infty$, we obtain

$\displaystyle A\to_{T\to+\infty}\frac{1}{2}\sqrt{\gamma}\Gamma,$
$\displaystyle
B\to_{T\to+\infty}-D_{+}^{-\frac{1}{2}}\hat{A}\hat{A}^{+}D_{+}^{-\frac{1}{2}}\big{(}V_{-}+\frac{1}{2}\sqrt{\gamma}D_{-}\mathcal{D}(\Gamma)\big{)},$

where $\Gamma=D_{+}^{-\frac{1}{2}}\big{(}D_{+}^{\frac{1}{2}}\Sigma
D_{+}^{\frac{1}{2}}\big{)}^{\frac{1}{2}}D_{+}^{-\frac{1}{2}}$ and
$\hat{A}^{+}$ is the Moore-Penrose generalized inverse of $\hat{A}$. If we
perform a principal component analysis on the variance-covariance matrix
$\Sigma$, we observe that the buy or sell signal (depending on the sign of the
inventories) coming from the sub-gradient of $V(t,q)$ is an increasing
function of the eigenvalues of $\Sigma$ and the risk-aversion parameter
$\gamma$. Thus, choosing the sub-gradient of $V(t,q)$ as an effective
microstructure alpha should provide a mean-reverting signal for a myopic
agent, taking into account the correlation between the assets.

####  4.2.2 Multi-asset multi-venue optimal market-making in limit order book

We now derive the solution to the multi-asset multi-venue optimal market-
making problem using our heuristics. We take the same modeling notation as in
4.1. Assuming bid and ask symmetry for sake of simplicity, we introduce the
processes $N^{i,n,b},N^{i,n,a}$ to model the number of (possibly partially-
filled) bid and ask orders on the asset $i$ in the venue $n$ of intensity
$\lambda^{i,n}(\psi^{i},I^{i},p^{i,n,b},\ell^{i,b})$ for the bid side and
$\lambda^{i,n}(\psi^{i},I^{i},p^{i,n,a},\ell^{i,a})$ for the ask side, where
the function $\lambda$ is defined by (4.1). The quantity $\ell^{i,n,b}$ (resp.
$\ell^{i,n,a}$) is the volume sent on the $n$-th venue of the $i$-th asset on
the bid (resp. ask) side. The quantity $p^{i,n,b}$ (resp. $p^{i,n,a}$) is the
limit chosen by the market-maker on the $n$-th venue of the $i$-th asset on
the bid (resp. ask) side to send a limit order. The distribution of the random
variables $\epsilon_{t}^{i,n,b}$ and $\epsilon_{t}^{i,n,a}$ are defined as in
Equation (4.2). The market-maker can also send market orders on the bid and
ask sides (denoted by point processes $J_{t}^{i,n,b},J_{t}^{i,n,b}$) on each
venue of size $v_{t}^{i,n,b},v_{t}^{i,n,a}$. Its inventory process on each
asset is defined by

$\displaystyle
q_{t}^{i}=\Big{(}\int_{0}^{t}\ell_{s}^{i,n,b}\epsilon_{s}^{i,n,b}dN_{s}^{i,n,b}+\int_{0}^{t}v_{s}^{i,n,b}dJ_{s}^{i,n,b}\Big{)}-\Big{(}\int_{0}^{t}\ell_{s}^{i,n,b}\epsilon_{s}^{i,n,b}dN_{s}^{i,n,b}+\int_{0}^{t}v_{s}^{i,n,b}dJ_{s}^{i,n,b}\Big{)}\in\mathbb{Z}.$

The problem faced by a market-maker is slightly different compared to a
classic trader. While the trader must follow a predetermined target, the
market-maker’s inventory must revert toward zero. Therefore, we seek a long-
term alpha that gives a signal to our myopic market-maker of the form “sell
for high inventory, buy for low inventory” with different type of
aggressiveness (limit or market order) depending on the level of inventory.
Contrary to optimal execution, there is no optimal inventory in market-making
problems at a given time $t\in[0,T]$, which explains the dependence of the
long-term alpha on the current inventory.

As stated previously, the effective microstructure alphas should be the
gradient of a value function corresponding to a “high-level” multi-asset
market-making problem (which hides the microstructure effects). This value
function should be in closed form to recompute the gradient quickly when the
market-maker trades and too much time passed. Thus, we propose to use the sub-
gradient of the value function (4.5) corresponding to an approximation of the
multi-asset market-making value function in OTC markets as a proxy for
effective microstructure alphas used by a market-maker acting on a portfolio
of assets listed on several order book platforms. By analogy with Section 4.1,
at each time step $t$, for an inventory vector $q\in\mathbb{R}^{d}$, the
market-maker solves the following optimization problem:

$\displaystyle\max\bigg{\\{}\\!$
$\displaystyle\sup_{\ell,p}\\!\Big{\\{}\\!\sum_{i=1}^{d}\sum_{n=1}^{N^{i}}\\!\sum_{j\in\\{b,a\\}}\\!\\!\\!\\!\lambda^{i,n}(\mathbf{m}^{i},\\!p^{i,n,j},\\!\ell^{i,j})\mathbb{E}\Big{[}\big{(}p^{\text{eff},i}\\!+\\!p^{\text{short},i,n}(\mathbf{m}^{i})\big{)}\ell^{i,n,j}\epsilon^{i,n,j}\\!-\\!c^{i,\\!n,\\!\mathcal{L}}(\epsilon^{i,n,j}\ell^{i,n,j},\\!p^{i,n,j})\Big{]}\Big{\\}},$
$\displaystyle\sup_{v^{b}}\Big{\\{}\sum_{i=1}^{d}\sum_{n=1}^{N^{i}}\big{(}p^{\text{eff},i}+p^{\text{short},i,n}(\mathbf{m}^{i})\big{)}v^{i,n,b}-c^{i,n,\mathcal{M}}(v^{i,n,b})\Big{\\}},$
$\displaystyle\sup_{v^{a}}\Big{\\{}\sum_{i=1}^{d}\sum_{n=1}^{N^{i}}\big{(}p^{\text{eff},i}+p^{\text{short},i,n}(\mathbf{m}^{i})\big{)}v^{i,n,a}-c^{i,n,\mathcal{M}}(v^{i,n,a})\Big{\\}}\bigg{\\}},$
$\displaystyle p^{\text{eff},i}=\nabla^{i}\tilde{V}(t,q).$

The control problem is essentially a choice between sending limit orders or
market orders in each venue for each asset. The effective microstructure alpha
helps the market-maker to mean revert his inventory toward zero. For example,
assume that the market-maker received a large buy passive filling in asset
$i$. The effective microstructure alpha, that is the $i$-th component of the
gradient of the long-term utility function $\tilde{V}$, will point down which
is a strong sell signal. Therefore, the market-maker will send a sell market
order to reduce his long position. Note that the effective microstructure
alpha takes into account the correlation structure between the assets, meaning
that the market-maker can hedge a long position in an asset with a short
position in another positively correlated asset.

##  5 Numerical results

Mathematical elegance and simplicity are to be prized, of course, but an
execution model cannot pass the test of practicality until it helps us execute
portfolio transitions.

One of the most important features of our framework, as compared with a plain-
vanilla, Almgren-Chriss executor, it allows the executor to consider market
microstructure and use passive orders, hence avoiding certain types of market
impact and spread costs. The main point we wish to make in this example is
that our method potentially avoids the pitfalls of a purely-passive execution
model because it can consider the utility gradient (and its multi-period
analog, the gradient of the value function) in the formation of aggression
levels. With this specific aim in mind, we consider the liquidation of a
market-neutral portfolio with our method and contrast this with comparable
results for a purely-passive method.

The specific example we choose is the liquidation of a market-neutral
portfolio on October $15,2008$. The portfolio to be liquidated is long IBM and
short AAPL. We choose the long position in IBM arbitrarily to be 1000 shares.
We estimate the CAPM beta of each security, denoted $\hat{\beta}_{i}$ where
$i=1,2$, using three years of daily data, and size the short position so that
the beta exposure of the portfolio $\sum_{i}h_{i}\hat{\beta}_{i}$ is near
zero.

###  5.1 Transaction cost model and microstructure simulation

We take $\kappa=10^{-3}$ and we assume $c(v)=v^{\mathbf{T}}\Lambda v$ for some
diagonal matrix $\Lambda=\text{diag}(\lambda^{1},\ldots,\lambda^{d})$. This
reduces the transaction cost modeling problem to one of estimating appropriate
values for each $\lambda_{i}$. Let $\text{advp}_{i}$ denote our prediction of
the daily dollar volume in the $i$-th security. The notation “advp” comes from
the fact that it is computed as the average daily volume “adv” in shares,
times the price “p”. For simplicity we assume trading one percent of
$\text{advp}_{i}$ will cause 20 basis points of market impact, with extension
by linearity, meaning that

$\lambda^{i}=20\times 10^{-4}\times\frac{1}{0.01\text{advp}_{i}}.$ (5.1)

For very large trades (say, more than 0.05 $\text{advp}_{i}$), simple models
such as (5.1) break down. For this reason, we restrict our attention in this
example to trades that are relatively small with respect to the anticipated
volume.

One of the most challenging aspects of this study is simulating passive
execution, which we defined previously as a process of continually joining the
queue on the near side of the limit order book until the order is filled, but
never crossing the spread.

We are limited to the academic data sets available via the Wharton Research
Data Services (WRDS). For this exercise, we used the New York Stock Exchange
Trade and Quote (TAQ) database, which contains intraday transactions data on
trades and quotes for all securities listed on the New York Stock Exchange
(NYSE) and American Stock Exchange (AMEX), as well as Nasdaq National Market
System (NMS) and SmallCap issues.

The TAQ database represents the aggregate inside quote for each exchange.
Therefore, it includes both specialists and the public limit order book. Only
having access to the consolidated feed, we construct a conservative simulation
of when passive fills occur. Specifically, if we have a “buy” limit order (the
entire process is similar for limit “sell” orders with “bid” replaced by
“ask”) which is simulated as existing in the queue on the bid side of the
order book, when can we assume such an order was filled? Conservatively, if
the order book changes and the new ask price is less or equal to the existing
limit order price, we assume that markets would have cleared in the process of
this change, and our limit order would have been filled, at least partially.
We limit the amount of fill to the posted quantity at the new ask price. If
this quantity is simulated to have been taken out, then no further fills are
allowed to occur in the simulation until the price level changes. We assume
that when the price level of the NBBO has changed, the liquidity is also
replenished to the reported value at the new price level. This is a fairly
conservative set of conventions; in reality, a larger number of passive fills
could occur than merely the ones we simulate. This is because if there are
multiple limit orders in the queue, one limit order can, of course, be filled
without either bid or ask price levels changing.

Predicting the probability of a passive fill, denoted $f_{i}$ above, is
equivalent to predicting the next transition of the limit order book and hence
requires a model of limit order book dynamics. Indeed, such fill probabilities
are one of the possible outputs of the very detailed model of Cont et al.
(2010) or the microstructure trading model presented in the previous section.
As our data set is only the consolidated feed, we simply take $f_{i}=0.1$ as
the passive fill probability.

###  5.2 Results

As indicated above, we construct a market-neutral portfolio of $d=2$
securities in which the long side is initially 1000 shares of IBM. Security
$i=1$ is IBM and $i=2$ is AAPL. We estimate the security betas to the S&P 500
(via regression on several years of daily data) as

$\hat{\beta}_{1}=0.705,\quad\hat{\beta}_{2}=1.276\,.$ (5.2)

We begin the simulation at 10:00 am on October 15, 2008, rather than
immediately at the open since there are often outlier quotes, wide spreads and
other effects around the open. The most recent midpoint price of IBM at 10:00
am was $p_{1}=93.06$ and for AAPL, $p_{2}=105.985$.

For convenience, we keep track of a cash balance for each position. The
$n_{1}=1000$ shares of IBM are financed by borrowing $n_{1}p_{1}=$ USD 93,060
in cash and purchasing a position initially worth USD 93,060, so the net value
(cash plus stock) of that position is initially zero. Similarly, the short
position in AAPL is obtained by borrowing $n_{2}=-485$ shares and immediately
selling them for USD 51,403, and this position also initially has a net (cash
+ stock) value of zero. Note that with these holdings, (5.2) implies that the
portfolio’s beta is

$n_{1}p_{1}\hat{\beta}_{1}+n_{2}p_{2}\hat{\beta}_{2}\approx 0\,.$

Any cash generated from further stock sales or cash used for further purchases
of the same security is considered part of the separate cash balance allocated
to that position. As prices change and as orders are filled, the values of
each position will fluctuate.

Let $n_{i,t}$ denote the number of shares held in the $i$-th security at time
$t$, and $p_{i,t}$ the latest midpoint price as of time $t$. Also, let
$c_{i,t}$ denote the amount of cash (which can be positive or negative)
attributed to the $i$-th security at time $t$, according to the accounting
conventions outlined above. These variables change throughout the lifetime of
the execution.

The value of a position is the number of shares held times the most recent
midpoint price, plus the total amount of cash associated to the position, i.e.
$n_{i,t}p_{i,t}+c_{i,t}$. The value of a portfolio is the sum of the values of
all its positions, i.e.

$\text{value}_{t}:=\sum_{i=1}^{d}(n_{i,t}p_{i,t}+c_{i,t}).$ (5.3)

The value process (5.3), and especially its drift, is one measure of the
execution’s quality. If the value tends to drift downward, as in the
AlwaysPassive model detailed below, then the execution desk is losing money
due to slippage. This is perhaps the typical situation – one expects execution
to have associated costs. A particularly pleasant situation arises when the
drift of the portfolio value process (5.3) is zero, as in Figure 2, and it is
possible that with very good microstructure alphas added to the generalized
momenta, the drift could even become positive. All monetary values are
reported in USD. The predicted daily volumes are estimated to

$\text{advp}_{1}\approx 1.16\times 10^{9},\quad\text{advp}_{2}\approx
6.13\times 10^{9}\,.$

The covariance matrix is

$\Sigma=10^{-4}\times\left(\begin{array}[]{cc}15.5728&17.7558\\\
17.7558&28.6519\\\ \end{array}\right),$

which implies a correlation of 0.84 among the two assets and daily
volatilities of approximately 3.9% and 5.4%.

The output of our algorithm is the instantaneous aggression level: aggressive,
passive, or wait. There is not a unique benchmark to gauge such an algorithm’s
performance, but it is sensible to compare a complicated method of choosing
the aggression level to a simple method for choosing the aggression level, to
see if the additional complexity is justified. Hence one could compare it to a
constant aggression level – always passive.

Figure 1 reveals that, as the market was falling, the passive “buy” orders in
AAPL were all filled very quickly, while unsurprisingly the “sell” orders in
IBM were filled very slowly, and indeed were not even finished by the end of
the trading day. This drove the Gross Market Value (gmv) down while pushing
the net and beta higher, where we define

$\beta_{t}:=\sum_{i}n_{i,t}p_{i,t}\hat{\beta}_{i},\ \
\text{net}_{t}:=\sum_{i}n_{i,t}p_{i,t},\ \
\text{gmv}_{t}:=\sum_{i}|n_{i,t}p_{i,t}|,$ (5.4)

with $\hat{\beta}_{i}$ given by (5.2). Thus the portfolio had $\beta_{t}>0$ in
a falling market. Note that the losses incurred in this manner do not become
gains if the sign of the market move is reversed; they remain losses
irrespective of the market’s direction. In a rising market, the “always
passive” model would have the same problem: the “sell” orders would be filled
quickly, the “buy” orders would linger, and the portfolio would build up
negative beta in a rising market.

Figure 1: Portfolio holdings in the “always passive” model, and portfolio
characteristics: gross market value (gmv), net, and $\beta_{t}$ given by
(5.4).

We now show the analogous graphs for the simplest version of our execution
model developed in the previous section. Note that the model retains a fairly
small beta exposure throughout the lifetime of the execution. This is because
CAPM beta is also a factor in the APT risk model, and the generalized momenta
point along the gradient of the Hamilton-Jacobi-Bellman value function and
hence drive trading towards the optimal value of multiperiod utility
(including the risk term). This is the key advantage of our model over simpler
execution algorithms.

Figure 2: Portfolio holdings in the sophisticated model, and portfolio
characteristics: gross market value ($\mathrm{gmv}_{t}$), $\mathrm{net}_{t}$,
and $\beta_{t}$ given by (5.4).

Finally, we consider the portfolio value over the lifetime of the execution.
Note that in our model, the value process (5.3) is approximately driftless,
which as explained above is a desirable property, and outperforms the “always
passive” value process realization. In particular, in our model
$\text{value}_{t}$ is able to avoid negative drift in a falling market
precisely because the portfolio remains approximately beta-neutral. In a
portfolio with many assets (large $n$), our method would allow it to remain
approximately neutral to all factors in the APT model.

Figure 3: Portfolio value (5.3) over the lifetime of the execution, for both
execution methods.

The difference in Figure 3 is both statistically and economically significant.
The t-statistic for the difference is about 78, hence significant at the
99.999% level. Moreover, the dollar value of the difference between the two
methods is about 1.5% of the initial gross market value to be liquidated.

##  6 Conclusion

In this paper, we present a framework to perform optimal trading, taking into
account market microstructure and a long-term trading schedule without the use
of optimal control. This approach relies on the use of the generalized momenta
$p=\nabla V(t,q)$ as the effective microstructure alpha. We show that a myopic
agent sending only market orders with such alpha will minimize the error with
respect to the long-term trading schedule. Moreover, when we add the
possibility of passive execution, the long-term alpha can be chosen as a
transformation of the generalized momenta $p$. We also present a general
microstructure trading framework for the multi-asset multi-venue optimal
trading problem. For a parsimonious model of fill probabilities, the effective
microstructure alpha can be computed in closed form. We apply the same
heuristics to derive an optimal market-making model that is tractable for a
large number of assets and venues.

Based on the dual formulation of the classic Almgren-Chriss optimization
problem, this simple heuristics has wide-ranging practical implications. In
addition to bridging the gap between order placement and scheduling, it
simplifies optimal trading problems that are usually intractable using optimal
control due to the high-dimensional Hamilton-Jacobi-Bellman equation resulting
from the control problem. This is of particular importance for a quantitative
execution desk wishing to trade a high number of cross-listed assets. It opens
up many avenues for future exploration. One set of projects is to consider
trading problems beyond the typical buy-side utility-maximization, which can
still be viewed within the unifying framework of a myopic risk-neutral wealth-
maximizer, whose microstructure alphas are aligned with the value function
gradient.

## References

  * Almgren and Chriss (2001) R. Almgren and N. Chriss.  Optimal execution of portfolio transactions.  _Journal of Risk_ , 3:5–40, 2001.
  * Arrow (1963) K. J. Arrow.  Liquidity preference, lecture vi in “lecture notes for economics 285, the economics of uncertainty”, pp 33-53.  1963\.
  * Avellaneda and Stoikov (2008) M. Avellaneda and S. Stoikov.  High-frequency trading in a limit order book.  _Quantitative Finance_ , 8(3):217–224, 2008.
  * Baldacci and Manziuk (2020) B. Baldacci and I. Manziuk.  Adaptive trading strategies across liquidity pools.  _arXiv preprint arXiv:2008.07807_ , 2020.
  * Bergault et al. (2020) P. Bergault, D. Evangelista, O. Guéant, and D. Vieira.  Closed-form approximations in multi-asset market making, 2020.
  * Cartea et al. (2014) Á. Cartea, S. Jaimungal, and J. Ricci.  Buy low, sell high: A high frequency trading perspective.  _SIAM Journal on Financial Mathematics_ , 5(1):415–444, 2014.
  * Cartea et al. (2015) Á. Cartea, S. Jaimungal, and J. Penalva.  _Algorithmic and high-frequency trading_.  Cambridge University Press, 2015.
  * Cont and Kukanov (2017) R. Cont and A. Kukanov.  Optimal order placement in limit order markets.  _Quantitative Finance_ , 17(1):21–39, 2017.
  * Cont et al. (2010) R. Cont, S. Stoikov, and R. Talreja.  A stochastic model for order book dynamics.  _Operations Research_ , 58(3):549–563, 2010.
  * Dreyfus (1960) S. E. Dreyfus.  Dynamic programming and the calculus of variations.  _Journal of Mathematical Analysis and Applications_ , 1(2):228–239, 1960.
  * Glosten and Milgrom (1985) L. R. Glosten and P. R. Milgrom.  Bid, ask and transaction prices in a specialist market with heterogeneously informed traders.  _Journal of financial economics_ , 14(1):71–100, 1985.
  * Guéant (2016) O. Guéant.  _The Financial Mathematics of Market Liquidity: From optimal execution to market making_ , volume 33.  CRC Press, 2016.
  * Guéant and Manziuk (2019) O. Guéant and I. Manziuk.  Deep reinforcement learning for market making in corporate bonds: beating the curse of dimensionality.  _Applied Mathematical Finance_ , 26(5):387–452, 2019.
  * Guéant et al. (2013) O. Guéant, C.-A. Lehalle, and J. Fernandez-Tapia.  Dealing with the inventory risk: a solution to the market making problem.  _Mathematics and financial economics_ , 7(4):477–507, 2013.
  * Guilbaud and Pham (2013) F. Guilbaud and H. Pham.  Optimal high-frequency trading with limit and market orders.  _Quantitative Finance_ , 13(1):79–94, 2013.
  * Ho and Stoll (1981) T. Ho and H. R. Stoll.  Optimal dealer pricing under transactions and return uncertainty.  _Journal of Financial economics_ , 9(1):47–73, 1981.
  * Pratt (1964) J. W. Pratt.  Risk aversion in the small and in the large.  _Econometrica: Journal of the Econometric Society_ , pages 122–136, 1964.