Dealing with the Inventory Risk: A Solution to the Market Making Problem

Abstract

Market makers continuously set bid and ask quotes for the stocks they have under consideration. Hence they face a complex optimization problem in which their return, based on the bid-ask spread they quote and the frequency at which they indeed provide liquidity, is challenged by the price risk they bear due to their inventory. In this paper, we consider a stochastic control problem similar to the one introduced by Ho and Stoll (J Fin Econ 9(1): 47–73, 1981) and formalized mathematically by Avellaneda and Stoikov (Quant Fin 8(3):217–224, 2008). The market is modeled using a reference price S
t
following a Brownian motion with standard deviation σ, arrival rates of buy or sell liquidity-consuming orders depend on the distance to the reference price S
t
and a market maker maximizes the expected utility of its P&L over a finite time horizon. We show that the Hamilton–Jacobi–Bellman equations associated to the stochastic optimal control problem can be transformed into a system of linear ordinary differential equations and we solve the market making problem under inventory constraints. We also shed light on the asymptotic behavior of the optimal quotes and propose closed-form approximations based on a spectral characterization of the optimal quotes.

Full text

arXiv:1105.3115v1 [q-fin.TR] 16 May 2011
Dealing with the Inventory Risk
Olivier Gu´eant∗
, Charles-Albert Lehalle∗∗
, Joaquin Fernandez Tapia∗∗∗
2010
Abstract
Market makers have to continuously set bid and ask quotes for the stocks they have under
consideration. Hence they face a complex optimization problem in which their return, based
on the bid-ask spread they quote and the frequency they indeed provide liquidity, is chal-
lenged by the price risk they bear due to their inventory. In this paper, we provide optimal
bid and ask quotes and closed-form approximations are derived using spectral arguments.
Introduction
The optimization of the intra-day trading process on electronic markets was born with the
need to split large trades to make the balance between trading too fast (and possibly degrade
the obtained price via “market impact”) and trading too slow (and suffer from a too long
exposure to “market risk”). This “trade scheduling” viewpoint has been mainly formalized
in the late nineties by Bertsimas and Lo [8] and Almgren and Chriss [2]. More sophisticated
approaches involving the use of stochastic and impulse control have been proposed since
then (see for instance [9]). Another branch of proposals goes in the direction of modeling
the effect of the “aggressive” (i.e. liquidity consuming) orders at the finest level, for instance
via a martingale model of the behavior market depth and of its resilience (see [1]).
From a quantitative viewpoint, market microstructure is a sequence of auction games be-
tween market participants. It implements the balance between supply and demand, forming
an equilibrium traded price to be used as reference for valuation of the listed assets. The
rule of each auction game (fixing auction, continuous auction, etc), are fixed by the firm
operating each trading venue. Nevertheless, most of all trading mechanisms on electronic
markets rely on market participants sending orders to a “queuing system” where their open
interests are consolidated as “liquidity provision” or form transactions [3]. The efficiency of
such a process relies on an adequate timing between buyers and sellers, to avoid too many
non-informative oscillations of the transaction price (for more details and modeling, see for
example [18]).
To take profit of these oscillations, it is possible to provide liquidity to an impatient buyer
(respectively seller) and maintain an inventory until the arrival of the next impatient seller
(respectively buyer). Market participants focused on this kind of liquidity-providing activity
The authors wish to acknowledge the helpful conversations with Pierre-Louis Lions (Coll`ege de France), Jean-
Michel Lasry (Universit´e Paris-Dauphine), Yves Achdou (Universit´e Paris-Diderot), Vincent Millot (Universit´e
Paris-Diderot), Antoine Lemenant (Universit´e Paris-Diderot) and Vincent Fardeau (London School of Economics).
∗UFR de Math´ematiques, Laboratoire Jacques-Louis Lions, Universit´e Paris-Diderot. 175, rue du Chevaleret,
75013 Paris, France. olivier.gueant@ann.jussieu.fr
∗∗Head of Quantitative Research. Cr´edit Agricole Cheuvreux. 9, Quai du Pr´esident Paul Doumer, 92400
Courbevoie, France. clehalle@cheuvreux.com
∗∗∗PhD student, Universit´e Paris 6 - Pierre et Marie Curie. 4 place Jussieu, 75005 Paris, France.
1

are called “market makers”. On one hand they are buying at the bid price and selling at
the ask price they chose, taking profit of this “bid-ask spread”. On the other hand, their
inventory is exposed to price fluctuations mainly driven by the volatility of the market (see
[4, 7, 11, 13, 15, 21]).
The usual “market making problem” comes from the optimality of the quotes (i.e. the
bid and ask prices) that such agents provide to other market participants with respect to
the constraints on their inventory and their utility function as a proxy to their risk (see
[10, 16, 20, 23]).
The recent evolution of market microstructure and the financial crisis reshaped the nature
of the interactions of the market participants during electronic auctions, one consequence
being the emergence of “high-frequency market makers” who are said to be part of 70% of
the electronic trades and have a massively passive (i.e. liquidity providing) behavior. A
typical balance between passive and aggressive orders for such market participants is around
80% of passive interactions (see [19]).
Avellaneda and Sto¨ıkov proposed in [5] an innovative framework for “market making in
an order book” and studied it using different approximations. In such an approach, the “fair
price”Stis modeled via a Brownian motion with volatility σ, and the arrival of a buy or
sell liquidity-consuming order at a distance δof Stfollows a Poisson process with intensity
Aexp(−k δ). Our paper extends their proposal and provides results in two main directions:
•An explicit solution to the Hamilton-Jacobi-Bellman equation coming from the optimal
market making problem thanks to a non trivial change of variables and the resulting
expressions for the optimal quotes:
Main Result 1 (Theorems 1-2).The optimal quotes can be expressed as:
sb∗(t, q, s) = s−−1
kln vq+1(t)
vq(t)+1
γln 1 + γ
k
sa∗(t, q, s) = s+1
kln vq(t)
vq−1(t)+1
γln 1 + γ
k
where γis the risk aversion of the agent and where vis a family of strictly positive
functions (vq)q∈Zsolution of the linear system of ODEs (S)that follows:
∀q∈Z,˙vq(t) = αq2vq(t)−η(vq−1(t) + vq+1(t))
with vq(T) = 1, and α=k
2γσ2and η=A(1 + γ
k)−(1+ k
γ).
It means that to find an exact solution to the generic high-frequency market making
problem, it is enough to solve on the fly the companion ODEs in vq(t) provided by our
change of variables, and to plug the result in the upper equalities to obtain the optimal
quotes with respect to a given inventory and market state.
•Asymptotics of the solution that are numerically attained fast enough in most realistic
cases:
Main Result 2 (Theorem 3 (asymptotics) and the associated approximation equa-
tions).
lim
T→∞ s−sb∗(0, q, s) = δb∗
∞(q)≃1
γln 1 + γ
k+2q+ 1
2sσ2γ
2kA 1 + γ
k1+ k
γ
lim
T→∞ sa∗(0, q, s)−s=δa∗
∞(q)≃1
γln 1 + γ
k−2q−1
2sσ2γ
2kA 1 + γ
k1+ k
γ
2

These results open doors to new directions of research involving the modeling and con-
trol of passive interactions with electronic order books. If some attempts have been made
that did not rely on stochastic control but on forward optimization (see for instance [22]
for a stochastic algorithmic approach for optimal split of passive orders across competing
electronic order books), they should be complemented by backward ones.
This paper goes from the description of the model choices that had to be made (section
1), through the main change of variables (section 2), exposes the asymptotics of the obtained
dynamics (section 3), its comparative statics (section 4), extends the framework to trends
in prices and constraints on the inventory (section 5), finally discusses the model choices
that had to be made (section 6) and ends with an application to real data (section 7).
Adaptations of our results are already in use at Cr´edit Agricole Cheuvreux to optimize the
brokerage trading flow.
In our framework, we follow Avellaneda and Sto¨ıkov in using a Poisson process model
pegged on a “fair price” diffusion (see section 1). As it is discussed in section 6, it is
an arguable choice since it does not capture “resistances” that can be built by huge passive
(i.e. liquidity-providing) orders preventing the market price to cross their prices. Our results
cannot be used as such for large orders, but are perfectly suited for high-frequency market
making as it is currently implemented in the market, using orders of small size (close to the
average trade size, see [19]).
Moreover, to our knowledge, no quantitative model of “implicit market impact” of such
small passive orders has never been proposed in the literature, despite very promising studies
linking updates of quantities in the order books to price changes (see [12]). Its combination
with recent applications of more general point processes to capture the process of arrival
of orders (like Hawkes models, see [6]) should give birth to such implicit market impact
models, specifying dependencies between the trend, the volatility and possible jumps in
the “fair price” semi-martingale process with the parameters of the multi-dimensional point
process of the market maker fill rate. At this stage, the explicit injection of such path-
dependent approach (once they will be proposed in the literature) into our equations are
too complex to be handled, but numerical explorations around our explicit formulas will be
feasible. The outcomes of applications of our results to real data (section 7) show that they
are realistic enough so that no more that small perturbations should be needed.
1 Setup of the model
1.1 Prices and Orders
We consider a market maker operating on a single stock and whose size is small enough to
consider price dynamics exogenous. For the sake of simplicity and since we will basically
only consider short horizon problems we suppose that the mid-price of the stock moves as a
brownian motion:
dSt=σdWt
The market maker under consideration will continuously propose bid and ask quotes denoted
respectively Sb
tand Sa
tand will hence buy and sell stocks according to the rate of arrival of
aggressive orders at the quoted prices.
His inventory q, that is the (signed) quantity of stocks he holds, is given by qt=Nb
t−Na
t
where Nband Naare the jump processes giving the number of stocks the market maker
respectively bought and sold. These jump processes are supposed to be Poisson processes
and to simplify the exposition (although this may be important, see the discussion part)
we consider that jumps are of size 1. Arrival rates obviously depend on the prices Sb
tand
Sa
tquoted by the market maker and we assume, in accordance with most datasets, that
3

intensities λband λaassociated to Nband Naare of the following form1:
λb(sb, s) = Aexp(−k(s−sb)) λa(sa, s) = Aexp(−k(sa−s))
This means that the closer to the mid-price an order is quoted, the faster it will be executed.
As a consequence of his trades, the market maker has an amount of cash whose dynamics
is given by:
dXt=Sa
tdNa
t−Sb
tdNb
t
1.2 The optimization problem
As we said above, the market maker has a time horizon Tand his goal is to optimize the
expected utility of his P&L at time T. In line with [5], we will focus on CARA utility
functions and we suppose that the market maker optimizes:
sup
Sa,Sb
E[−exp (−γ(XT+qTST))]
where γis the absolute risk aversion characterizing the market maker, where XTis the
amount of cash at time Tand where qTSTis the mid-price evaluation of the (signed) re-
maining quantity of stocks in the inventory at time T(liquidation at mid-price2).
2 Resolution
2.1 Hamilton-Jacobi-Bellman equation
The optimization problem set up in the preceding section can be solved using classical
Bellman tools. To this purpose, we introduce a Bellman function udefined as:
u(t, x, q, s) = sup
Sa,Sb
E[−exp (−γ(XT+qTST))|Xt=x, St=s, qt=q]
The Hamilton-Jacobi-Bellman equation associated to the optimization problem is then
given by the following proposition:
Proposition 1 (HJB).The Hamilton-Jacobi-Bellman equation for uis:
(HJB) 0 = ∂tu(t, x, q, s) + 1
2σ2∂2
ssu(t, x, q, s)
+ sup
sb
λb(sb, s)hu(t, x −sb, q + 1, s)−u(t, x, q, s)i
+ sup
saλa(sa, s) [u(t, x +sa, q −1, s)−u(t, x, q, s)]
with the final condition:
u(T, x, q, s) = −exp (−γ(x+qs))
This equation is not a simple 4-variable PDE. Rather, because the inventory is discrete,
it is an infinite system of 3-variable PDEs. To solve it, we will use a change of variables that
is different from the one used in [5] and transforms the system of PDEs in a system of linear
ODEs.
1Although this form is in accordance with real data, some authors used a linear form for the intensity functions
– see [17] for instance.
2We will discuss other hypotheses below.
4

2.2 Reduction to a system of linear ODEs
In [5], the authors proposed a change of variables to factor out wealth. Here we go further
and propose a rather non-intuitive change of variables that allows to write the problem in a
linear way.
Proposition 2 (A system of linear ODEs).Let’s consider a family of strictly positive func-
tions (vq)q∈Zsolution of the linear system of ODEs (S)that follows:
∀q∈Z,˙vq(t) = αq2vq(t)−η(vq−1(t) + vq+1(t))
with vq(T) = 1, where α=k
2γσ2and η=A(1 + γ
k)−(1+ k
γ).
Then u(t, x, q, s) = −exp(−γ(x+qs))vq(t)−γ
kis solution of (HJB).
Theorem 1 (Well-posedness of the system (S)).There exists a unique solution of (S)in
C∞([0, T ), ℓ2(Z)) and this solution consists in strictly positive functions.
2.3 Optimal quotes characterization
Theorem 2 (Optimal quotes and bid-ask spread).Let’s consider the solution vof the system
(S)as in Theorem 1. Then optimal quotes can be expressed as:
sb∗(t, q, s) = s−−1
kln vq+1(t)
vq(t)+1
γln 1 + γ
k
sa∗(t, q, s) = s+1
kln vq(t)
vq−1(t)+1
γln 1 + γ
k
Moreover, the bid-ask spread quoted by the market maker, that is ψ∗=sa∗(t, q, s)−sb∗(t, q, s),
is given by:
ψ∗(t, q) = −1
kln vq+1(t)vq−1(t)
vq(t)2+2
γln 1 + γ
k
We see that the difference between each quoted price and the mid-price has two compo-
nents. If we consider the case of the bid quote – the same analysis would be true in the case
of the ask quote –, we need to separate the term −1
kln vq+1(t)
vq(t)from the term 1
γln 1 + γ
k.
If σ= 0, then vq(t) = exp(2η(T−t)) defines a solution of the system (S). Hence, the
relations s−sb∗=sa∗−s=1
γln 1 + γ
kdefine the optimal quotes in the “no-volatility”
benchmark case3. Consequently, in the expression that defines the optimal quotes, the sec-
ond term corresponds to the “no-volatility” benchmark while the first one takes account of
the influence of volatility.
3 Examples and asymptotics
To motivate the asymptotic approximation we provide, and before discussing the way to
solve the problem numerically, let us present some graphs to understand the behavior in
time and inventory of both the optimal quotes and the bid-ask spread.
3Smaller quotes would lead to trade more often with less revenue per trade in a way that is not in favor of the
market maker. Symmetrically, larger quotes would lead to more revenue per trade but less trades and the welfare
of the market maker would also be reduced.
5

3.1 Numerical examples
−30 −20 −10 010 20 30
0
100
200
300
400
500
600
0.5
1
1.5
2
2.5
3
Inventory
Time [Sec]
s − sb [Tick]
−30 −20 −10 0 10 20 30
0.5
1
1.5
2
2.5
3
Inventory
s − sb [Tick]
−30 −20 −10 010 20 30 0100 200 300 400 500 600
0.5
1
1.5
2
2.5
3
Time [Sec]
Inventory
sa − s [Tick]
−30 −20 −10 0 10 20 30
0.5
1
1.5
2
2.5
3
Inventory
sa − s [Tick]
−30 −20 −10 010 20 30 0100 200 300 400 500 600
3.275
3.28
3.285
3.29
3.295
3.3
3.305
3.31
3.315
3.32
Time [Sec]
Inventory
ψ [Tick]
−30 −20 −10 0 10 20 30
3.3114
3.3116
3.3118
3.312
3.3122
3.3124
3.3126
3.3128
3.313
3.3132
Inventory
ψ [ Tick]
Figure 1: Left: Behavior of the optimal quotes and bid-ask spread with time and inventory.
Right: Behavior of the optimal quotes and bid-ask spread with inventory, at time t= 0. σ=
0.3 Tick ·s−1/2,A= 0.9 s−1,k= 0.3 Tick−1,γ= 0.01 Tick−1,T= 600 s.
3.2 Asymptotics
In [5], the authors propose a heuristic approximation for the bid-ask spread. Namely they
propose to approximate ψ∗(t, q) by γσ2(T−t) + 2
γln 1 + γ
k. However, as suggested by the
graphs exhibited above, the predominant feature is that both the bid-ask spread and the
distance between the quotes and the mid-price are rather constant, except near the time
horizon T(and in numerical examples, a few minutes are enough to be near the asymptotic
values), and certainly not linearly decreasing with time.
6

In fact, we can prove the existence of an asymptotic behavior and provide semi-explicit
expressions for the asymptotic values of the bid-ask spread and the quotes:
Theorem 3 (Asymptotic quotes and bid-ask spread).
∀q∈Z,∃δb∗
∞(q), δa∗
∞(q), ψ∗
∞(q)∈R
lim
T→∞ s−sb∗(0, q, s) = δb∗
∞(q)
lim
T→∞ sa∗(0, q, s)−s=δa∗
∞(q)
lim
T→∞ ψ∗(0, q) = ψ∗
∞(q)
Moreover,
δb∗
∞(q) = 1
γln 1 + γ
k−1
kln f0
q+1
f0
q!δa∗
∞(q) = 1
γln 1 + γ
k+1
kln f0
q
f0
q−1!
and
ψ∗
∞(q) = −1
kln f0
q+1f0
q−1
f0
q2!+2
γln 1 + γ
k
where f0∈ℓ2(Z)is characterized by:
f0∈argmin
kfkℓ2(Z)=1 X
q∈Z
αq2f2
q+ηX
q∈Z
(fq+1 −fq)2
As we have seen in the above numerical examples, only these asymptotic values seem
to be relevant in practice. Consequently, we provide an approximation for f0that happens
to fit the actual figures. This approximation is based on the continuous counterpart of f0,
namely ˜
f0∈L2(R), a function that verifies:
˜
f0∈argmin
k˜
fkL2(R)=1 Z∞
−∞ αx2˜
f(x)2+η˜
f′(x)2dx
It can be proved4that such a function ˜
f0must be proportional to the probability distri-
bution function of a normal variable with mean 0 and variance qη
α. Hence, we expect f0
q
to behave as exp −1
2qα
ηq2.
This heuristic viewpoint induces an approximation for the optimal quotes and bid-ask
spread if we replace f0
qby exp −1
2qα
ηq2:
δb∗
∞(q)≃1
γln 1 + γ
k+1
2krα
η(2q+ 1)
≃1
γln 1 + γ
k+2q+ 1
2sσ2γ
2kA 1 + γ
k1+ k
γ
δa∗
∞(q)≃1
γln 1 + γ
k−1
2krα
η(2q−1)
≃1
γln 1 + γ
k−2q−1
2sσ2γ
2kA 1 + γ
k1+ k
γ
4To prove this we need to proceed as in the proof of Theorems 1 and 3. In a few words, we introduce the positive,
compact and self-adjoint operator Lcdefined for f∈L2(R) as the unique weak solution vof αx2v−ηv′′ =f
with R∞
−∞ αx2v(x)2+ηv′(x)2dx < +∞.Lccan be diagonalized and largest eigenvalue of Lccan be shown to
be associated to the eigenvector f(x) = exp −1
2qα
ηx2.
7

ψ∗
∞(q)≃2
γln 1 + γ
k+1
krα
η
≃2
γln 1 + γ
k+sσ2γ
2kA 1 + γ
k1+ k
γ
We exhibit below the values of the optimal quotes and the bid-ask spread, both with their
associated approximations. Empirically, these approximations for the quotes are satisfactory
in most cases and are always good for small values of the inventory q. The apparent difficulty
to approximate the bid-ask spread comes from the chosen scale (the bid-ask spread being
almost uniform across values of the inventory).
−30 −20 −10 0 10 20 30
0
0.5
1
1.5
2
2.5
3
3.5
Inventory
s − sb [Tick]
−30 −20 −10 0 10 20 30
−6
−4
−2
0
2
4
6
8
10
Inventory
s − sb [Tick]
−30 −20 −10 0 10 20 30
0
0.5
1
1.5
2
2.5
3
3.5
Inventory
sa − s [Tick]
−30 −20 −10 0 10 20 30
−6
−4
−2
0
2
4
6
8
10
Inventory
sa − s [Tick]
−30 −20 −10 0 10 20 30
3.321
3.3215
3.322
3.3225
3.323
3.3235
3.324
3.3245
Inventory
ψ [Tick]
−30 −20 −10 0 10 20 30
3.36
3.38
3.4
3.42
3.44
3.46
3.48
3.5
3.52
3.54
3.56
Inventory
ψ [Tick]
Figure 2: Asymptotic behavior of optimal quotes and the bid-ask spread (bold line). Approxima-
tion (dotted line). Left: σ= 0.4 Tick ·s−1/2,A= 0.9 s−1,k= 0.3 Tick−1,γ= 0.01 Tick−1,
T= 600 s. Right: σ= 1.0 Tick ·s−1/2,A= 0.2 s−1,k= 0.3 Tick−1,γ= 0.01 Tick−1,
T= 600 s.
8

4 Comparative statics
Before starting with the comparative statics, we rewrite the approximations done in the
previous section to be able to have some intuition about the behavior of the optimal quotes
and bid-ask spread with respect to the parameters:
δb∗
∞(q)≃1
γln 1 + γ
k+2q+ 1
2sσ2γ
2kA 1 + γ
k1+ k
γ
δa∗
∞(q)≃1
γln 1 + γ
k−2q−1
2sσ2γ
2kA 1 + γ
k1+ k
γ
ψ∗
∞(q)≃2
γln 1 + γ
k+sσ2γ
2kA 1 + γ
k1+ k
γ
Now, from these approximations, we can “deduce” the behavior of the optimal quotes
and the bid-ask spread with respect to price volatility, trading intensity and risk aversion.
4.1 Dependence on σ2
From the above approximations we expect the dependence of optimal quotes on σ2to be a
function of the inventory. More precisely, we expect:





∂δb∗
∞
∂σ2<0,∂ δa∗
∞
∂σ2>0,if q < 0
∂δb∗
∞
∂σ2>0,∂ δa∗
∞
∂σ2>0,if q= 0
∂δb∗
∞
∂σ2>0,∂ δa∗
∞
∂σ2<0,if q > 0
For the bid-ask spread we expect it to be increasing with respect to σ2:
∂ψ∗
∞
∂σ2>0
The rationale behind this is that a rise in σ2increases the inventory risk. Hence, to
reduce this risk, a market maker that has a long position will try to reduce his exposure and
hence ask less for his stocks (to get rid of some of them) and accept to buy at a cheaper
price (to avoid buying new stocks). Similarly, a market maker with a short position tries to
buy stocks, and hence increases its bid quote, while avoiding short selling new stocks, and
he increases its ask quote to that purpose. Overall, due to the increase in risk, the bid-ask
spread widens as it is well instanced in the case of a market maker with a flat position (this
one wants indeed to earn more per trade to compensate the increase in inventory risk.
These intuitions can be verified numerically on Figure 3.
4.2 Dependence on A
Because of the above approximations, and in accordance with the form of the system (S),
we expect the dependence on Ato be the exact opposite of the dependence on σ2, namely





∂δb∗
∞
∂A >0,∂δa∗
∞
∂A <0,if q < 0;
∂δb∗
∞
∂A <0,∂δa∗
∞
∂A <0,if q= 0
∂δb∗
∞
∂A <0,∂δa∗
∞
∂A >0,if q > 0
For the same reason, we expect the bid-ask spread to be decreasing with respect to A.
∂ψ∗
∞
∂A <0
9

The rationale behind these expectations is that an increase in Areduces the inventory
risk. An increase in Aindeed increases the frequency of trades and hence reduces the risk
of being stuck with a large inventory (either positive or negative). For this reason, a rise in
Ashould have the same effect as a decrease in σ2.
These intuitions can be verified numerically on Figure 4.
−30 −20 −10 010 20 30 0
0.1
0.2
0.3
0.4
0.5
0
0.5
1
1.5
2
2.5
3
3.5
σ[Tick/√Sec]
Inventory
s−sb[Tick]
−30 −20 −10 010 20 30 0
0.1
0.2
0.3
0.4
0.5
0
0.5
1
1.5
2
2.5
3
3.5
σ[Tick/√Sec]
Inventory
sa−s[Tick]
−30
−20
−10
0
10
20
30 0
0.1
0.2
0.3
0.4
0.5
3.28
3.29
3.3
3.31
3.32
3.33
3.34
3.35
σ[Tick/√Sec]
Inventory
Spread [Tick]
Figure 3: Asymptotic optimal quotes and bid-ask spread for different inventories and different
values for the volatility σ.A= 0.9 s−1,k= 0.3 Tick−1,γ= 0.01 Tick−1,T= 600 s.
−30
−20
−10
0
10
20
30
0
0.5
1
1.5
−4
−2
0
2
4
6
8
A[Sec−1]
Inventory
s−sb[Tick]
−30 −20 −10 010 20 30 0
0.5
1
1.5
−4
−2
0
2
4
6
8
A[Sec−1]
Inventory
sa−s[Tick]
10

−30 −20 −10 010 20 30 0
0.5
1
1.5
3.3
3.35
3.4
3.45
3.5
3.55
A[Sec−1]
Inventory
Spread [Tick]
Figure 4: Asymptotic optimal quotes and bid-ask spread for different inventories and different
values of A.σ= 0.3 Tick ·s−1/2,k= 0.3 Tick−1,γ= 0.01 Tick−1,T= 600 s.
4.3 Dependence on k
From the above approximations we expect δb∗
∞to be decreasing in kfor qgreater than some
negative threshold. Below this threshold we expect it to be increasing. Similarly we expect
δa∗
∞to be decreasing in kfor qsmaller than some positive threshold. Above this threshold
we expect it to be increasing.
Eventually, as far as the bid-ask spread is concerned, the above approximation indicates
that the bid-ask spread should be a decreasing function of k.
∂ψ∗
∞
∂k <0
In fact several effects are in interaction. On one hand, there is a “no-volatility” effect
that is completely orthogonal to any reasoning on the inventory risk: when kincreases,
trades occur closer to the mid price. For this reason, and in absence of inventory risk, the
optimal quotes have to get closer to the mid-price. However, an increase in kalso affects the
inventory risk since it decreases the probability to be executed (for δb, δa>0). Hence, an
increase in kis also, in some aspects, similar to a decrease in A. These two effects explain
the expected behavior.
Numerically, one of two effects dominates for the values of the inventory under consider-
ation:
−30
−20
−10
0
10
20
30
0
0.5
1
1.5
2
−1
0
1
2
3
4
5
Inventory
k[Tick−1]
s−sb[Tick]
−30
−20
−10
0
10
20
30 0
0.5
1
1.5
2
−1
0
1
2
3
4
5
k[Tick−1]
Inventory
sa−s[Tick]
11

−30
−20
−10
0
10
20
30 0
0.5
1
1.5
2
0
1
2
3
4
5
6
7
k[Tick−1]
Inventory
Spread [Tick]
Figure 5: Asymptotic optimal quotes and bid-ask spread for different inventories and different
values of k.σ= 0.3 Tick ·s−1/2,A= 0.9 s−1,γ= 0.01 Tick−1,T= 600 s.
4.4 Dependence on γ
Using the above approximations, we see that the dependence on γis ambiguous. The
market maker faces two different risks that contribute to the inventory risk: (i) trades occur
at random times and (ii) the mid price is stochastic. But if risk aversion increases, the
market maker will mitigate the two risks: (i) he may set his quotes closer to one another to
reduce the randomness in execution (as in the “no-volatility” benchmark) and (ii) he may
enlarge his spread to reduce price risk. The tension between these two roles played by γ
explains the different behaviors we may observe, as in the figures below:
−30
−20
−10
0
10
20
30
0
0.1
0.2
0.3
0.4
0.5
−6
−4
−2
0
2
4
6
8
γ
Inventory
s−sb[Tick]
−30
−20
−10
0
10
20
30
0
0.1
0.2
0.3
0.4
0.5
−3
−2
−1
0
1
2
3
4
γ
Inventory
s−sb[Tick]
−30 −20 −10 010 20 30 0
0.1
0.2
0.3
0.4
0.5
−6
−4
−2
0
2
4
6
8
γ
Inventory
sa−s[Tick]
−30 −20 −10 010 20 30 0
0.1
0.2
0.3
0.4
0.5
−3
−2
−1
0
1
2
3
4
γ
Inventory
sa−s[Tick]
12

−30
−20
−10
0
10
20
30
0
0.1
0.2
0.3
0.4
0.5
2
2.5
3
3.5
γ
Inventory
Spread [Tick]
−30
−20
−10
0
10
20
30
0
0.1
0.2
0.3
0.4
0.5
0.9
0.95
1
1.05
1.1
1.15
1.2
1.25
γ
Inventory
Spread [Tick]
Figure 6: Asymptotic optimal quotes and bid-ask spread for different inventories and different val-
ues for the risk aversion parameter γ. Left: σ= 0.3 Tick ·s−1/2,A= 0.9 s−1,k= 0.3 Tick−1,
T= 600 s. Right: σ= 0.6 Tick ·s−1/2,A= 0.9 s−1,k= 0.9 Tick−1,T= 600 s
5 Different settings
In what follows we provide the settings of several variants of the initial model. We will
alternatively consider a model with a trend in prices, a model with a penalization term for
not having cleared one’s inventory and a model with inventory constraints from which all
the figures have been drawn.
For each model, we enounce the associated results and some specific points are proved in
the appendix. However, the general proofs are not repeated since they can be derived from
adaptations of the proofs of the initial model.
5.1 Trend in prices
In the preceding setting, we supposed that the mid-price of the stock followed a brownian
motion. However, we can also build a model in presence of a trend:
dSt=µdt +σdWt
In that case we have the following proposition:
Proposition 3 (Resolution with drift).Let’s consider a family of functions (vq)q∈Zsolution
of the linear system of ODEs that follows:
∀q∈Z,˙vq(t) = (αq2−βq)vq(t)−η(vq−1(t) + vq+1(t))
with vq(T) = 1, where α=k
2γσ2,β=kµ and η=A(1 + γ
k)−(1+ k
γ).
Then, optimal quotes can be expressed as:
sb∗(t, q, s) = s−−1
kln vq+1(t)
vq(t)+1
γln 1 + γ
k
sa∗(t, q, s) = s+1
kln vq(t)
vq−1(t)+1
γln 1 + γ
k
and the bid-ask spread quoted by the market maker is :
ψ∗(t, q) = −1
kln vq+1(t)vq−1(t)
vq(t)2+2
γln 1 + γ
k
13

Moreover,
lim
T→∞ s−sb∗(0, q, s) = 1
γln 1 + γ
k−1
kln f0
q+1
f0
q!
lim
T→∞ sa∗(0, q, s)−s=1
γln 1 + γ
k+1
kln f0
q
f0
q−1!
lim
T→∞ ψ∗(0, q) = −1
kln f0
q+1f0
q−1
f0
q2!+2
γln 1 + γ
k
where f0∈ℓ2(Z)is characterized by:
f0∈argmin
kfkℓ2(Z)=1 X
q∈Z
αq−β
2α2
f2
q+ηX
q∈Z
(fq+1 −fq)2
Using the same continuous approximation as in the initial model we find the following
approximations for the optimal quotes and the bid-ask spread:
δb∗
∞(q)≃1
γln 1 + γ
k+−µ
γσ2+2q+ 1
2sσ2γ
2kA 1 + γ
k1+ k
γ
δa∗
∞(q)≃1
γln 1 + γ
k+µ
γσ2−2q−1
2sσ2γ
2kA 1 + γ
k1+ k
γ
ψ∗
∞(q)≃2
γln 1 + γ
k+sσ2γ
2kA 1 + γ
k1+ k
γ
5.2 Inventory liquidation below mid price
In the initial model we imposed a terminal condition based on the assumption that the
market maker liquidates his inventory at mid-price at time t=T. This hypothesis is ques-
tionable and we propose to introduce an additional term to model liquidation cost that can
also be interpreted as a penalization term for having a non-zero inventory at time T.
From a mathematical perspective it means that the control problem is now:
sup
Sa,Sb
E[−exp (−γ(XT+qTST−φ(|qT|)))]
where φ(·)≥0 is an increasing function with φ(0) = 0 that represents the penalization term
modeling the incurred cost at the end of the period for not having cleared the inventory5.
The analysis can then be done in the same way as in the initial model and we get the
following result:
Proposition 4 (Resolution with inventory liquidation cost).Let’s consider a family of
functions (vq)q∈Zsolution of the linear system of ODEs that follows:
∀q∈Z,˙vq(t) = αq2vq(t)−η(vq−1(t) + vq+1(t))
5For analytical reasons we supposed that this penalization term does not depend on ST. However, nothing
prevents this penalization term to depend on price through S0for instance, a rather acceptable hypothesis in a
short horizon perspective.
14

with vq(T) = e−kφ(|q|), where α=k
2γσ2and η=A(1 + γ
k)−(1+ k
γ).
Then, optimal quotes can be expressed as:
sb∗(t, q, s) = s−−1
kln vq+1(t)
vq(t)+1
γln 1 + γ
k
sa∗(t, q, s) = s+1
kln vq(t)
vq−1(t)+1
γln 1 + γ
k
and the bid-ask spread quoted by the market maker is :
ψ∗(t, q) = −1
kln vq+1(t)vq−1(t)
vq(t)2+2
γln 1 + γ
k
Moreover, the asymptotic behavior of the optimal quotes and the bid-ask spread does not
depend on the liquidation cost term φ(|q|)and is the same as in the initial model.
5.3 Introduction of inventory constraints
Another possible setting is to consider explicitly in the model that the market maker cannot
have too large an inventory. This is interesting by itself but it also provides numerical
methods to solve the problem and all the graphs presented above have been made using this
model that approximates the general one when the inventory limits are large.
5.3.1 The model
In this model, we introduce limits on the inventory. This means that once the agent holds
a certain amount Qof shares, he does not propose an ask quote until he sells some of his
shares. Symmetrically, once the agent is short of Qshares, he does not short sell anymore
before he buys a share.
In modeling terms, it means that the Hamilton-Jacobi-Bellman equation of the problem
is the following:
∀q∈ {−(Q−1),...,0,...,Q−1},
0 = ∂tu(t, x, q, s) + 1
2σ2∂2
ssu(t, x, q, s)
+ sup
sb
λb(sb, s)hu(t, x −sb, q + 1, s)−u(t, x, q, s)i
+ sup
saλa(sa, s) [u(t, x +sa, q −1, s)−u(t, x, q, s)]
for q=Qwe have:
0 = ∂tu(t, x, Q, s) + 1
2σ2∂2
ssu(t, x, Q, s)
+ sup
saλa(sa, s) [u(t, x +sa, Q −1, s)−u(t, x, Q, s)]
and symmetrically, for q=−Qwe have:
0 = ∂tu(t, x, −Q, s) + 1
2σ2∂2
ssu(t, x, −Q, s)
+ sup
sb
λb(sb, s)hu(t, x −sb,−Q+ 1, s)−u(t, x, −Q, s)i
15

with the final condition:
∀q∈ {−Q, . . . , 0,...,Q}, u(T , x, q, s) = −exp (−γ(x+qs))
As in the initial model we can reduce it to a linear system of ODEs. However the linear
system associated to this model will be simpler since it involves 2Q+ 1 equations only.
Proposition 5 (Resolution with inventory limits).Let’s introduce the matrix Mdefined by:
M=















αQ2−η0··· ··· ··· 0
−η α(Q−1)2−η0.......
.
.
0................
.
.
.
.
.................
.
.
.
.
................0
.
.
.......0−η α(Q−1)2−η
0··· ··· ··· 0−η αQ2















where α=k
2γσ2and η=A(1 + γ
k)−(1+ k
γ).
Then, if
v(t) = (v−Q(t), v−Q+1(t),...,v0(t),...,vQ−1(t), vQ(t))′= exp(−M(T−t)) ×(1,...,1)′
the optimal quotes are:
∀q∈ {−Q, . . . , 0,...,Q−1}, sb∗(t, q, s) = s−−1
kln vq+1(t)
vq(t)+1
γln 1 + γ
k
∀q∈ {−(Q−1),...,0,...,Q}, sa∗(t, q, s) = s+1
kln vq(t)
vq−1(t)+1
γln 1 + γ
k
and the bid-ask spread quoted by the market maker is given by:
∀q∈ {−(Q−1),...,0,...,Q−1}, ψ∗(t, q) = −1
kln vq+1(t)vq−1(t)
vq(t)2+2
γln 1 + γ
k
Moreover, the asymptotic quotes and bid-ask spread can be expressed as:
δb∗
∞(q) = 1
γln 1 + γ
k−1
kln f0
q+1
f0
q!δa∗
∞(q) = 1
γln 1 + γ
k+1
kln f0
q
f0
q−1!
and
ψ∗
∞(q) = −1
kln f0
q+1f0
q−1
f0
q2!+2
γln 1 + γ
k
where f0∈R2Q+1 is an eigenvector corresponding to the smallest eigenvalue of M.
5.3.2 Application to numerical resolution
This model based on a slight modification of the initial one leads to a system of linear ODEs
whose associated matrix is solely tridiagonal. Hence, for all numerical resolutions we con-
sidered this modified problem with the inventory limit Qlarge enough and the numerical
resolution simply boiled down to exponentiate a tridiagonal matrix. The rationale behind
this is that the variant of the model under consideration imposes vq(t) = 0 for |q|> Q and
16

for all times. Since the solution of the initial problem is in ℓ2(Z) for t < T , this approxima-
tion will be valid when Qis large as long as tis far enough from the terminal time Tand q
not too close to Q.
Another possible method is to compute an eigenvector associated to the smallest eigen-
value of M. As we noticed before, we expect f0
qto behave as exp −1
2qα
ηq2.
Hence it’s a better idea to look for g0instead of f0where g0
q=f0
qexp 1
2qα
ηq2. To this
purpose, we replace the spectral analysis of Mby the spectral analysis of the tridiagonal
matrix DM D−1where Dis a diagonal matrix whose terms are exp 1
2qα
ηq2q∈{−Q,...,Q}
and g0will be an eigenvector associated to the smallest eigenvalue of DM D−1.
Now, once g0has been calculated, the asymptotic values of the optimal quotes and the
bid-ask spread are:
δb∗
∞(q) = 1
γln 1 + γ
k−1
kln g0
q+1
g0
q!+1
2krα
η(2q+ 1)
δa∗
∞(q) = 1
γln 1 + γ
k+1
kln g0
q
g0
q−1!−1
2krα
η(2q−1)
and
ψ∗
∞(q) = 2
γln 1 + γ
k−1
kln g0
q+1g0
q−1
g0
q2!+1
krα
η
6 Discussion on the model
6.1 Exogenous nature of prices
In our model, the mid-price is modeled by a brownian motion independent of the behavior of
the agent. Since we are modeling a single market maker who operates through limit orders,
it seems natural to consider the price process exogenous in the medium run. However, even
if we neglect the impact of our market maker on the market, the very notion of mid-price
must be clarified. Indeed, one may consider that, although it has little impact on the mar-
ket, the market maker can put an order inside the bid-ask spread of the market order book
and hence change the mid-price. This would be a misunderstanding of the model since the
mid-price is to be considered in the model before any insertion of an order. Hence, the
mid-price in this model must be understood as the mid-price of an order book in which our
market maker’s orders would be removed. More generally, it may be viewed as a generalized
mid-price calculated across trading facilities or any reference price for which the hypothesis
on orders arrival is a good approximation of reality.
6.2 Dependence on price
What may be counterintuitive at first sight is that the bid-ask spread or any of the two
spreads between quoted prices and market mid-price seems not to depend on the mid-price
itself. In fact, in our model, the bid-ask spread and the gap between quoted prices and
the market mid-price depends on price, though indirectly, through parameters. Prices are
indeed hidden in the trading intensity λ, and more specifically into the parameter k. We
indeed considered a trading intensity depending on the distance between the quoted prices
and the mid-price. Thus, kmust depend indirectly on prices to normalize prices differences.
17

6.3 Constant size of orders
Another apparent issue of the model is that market makers set orders of size 1 at all times.
A first remark is that, if this is an issue, it is only limited to the fact that orders are of
constant size since we can consider that the unitary orders stand for orders of constant size
δq or equivalently, though more abstractly, orders of size 1 on a bunch of δq stocks.
If all the orders are of size δq then the stochastic process representing cash is:
d˜
Xt=δq(Sa
tdNa
t−Sb
tdNb
t) = δq ×dXt
where the jump processes model the event of being hit by an aggressive order (of size δq).
Then, if we consider that an order of size δq is a unitary order on a bunch of δq stocks, we
can write ˜qt=δq ×qtand the optimization criterion becomes:
sup
Sa,Sb
Eh−exp −γ(˜
XT+ ˜qTST)i= sup
Sa,Sb
E[−exp (−γδq(XT+qTST))]
Hence, we can solve the problem for orders of size δq using a modified risk aversion,
namely solving the problem for unitary orders, with γmultiplied by δq.
However, if we can transform the problem with orders of constant size δq into the ini-
tial problem where δq = 1, the parameters must be adjusted in accordance with the fact
that orders are of size δq. As such, it must be noticed that Ahas to be estimated to take
account of the expected proportion of an order of size δq filled by a single trade and approx-
imations must be made to take account of market making with orders of constant size. In
fact, we can consider that, after each trade that partially filled the order, the market maker
sends a new order so that the total size of his orders is δq, using a convex combination of the
model recommendations for the price6, since in that case the inventory is not a multiple of δq.
In our view, this issue is important but it should not be considered a problem to describe
qualitative market maker’s behavior and appropriate approximation on Aallows us to believe
that the error made, as far as quantitative modeling results are concerned, is relatively small.
6.4 Constant parameters
Another issue is that the parameters σ,Aand kare constant. While models can be devel-
oped to take account of deterministic or stochastic variations of the parameters, the most
important point is to take account of the links between the different parameters. σ,Aand
kshould not indeed be considered independent of one another since, for instance, an in-
crease in Ashould induce an increase in the number of trades and hence an increase in price
volatility.
Some attempts have been made in this direction to model the link between volatility and
trades intensity. Hawkes processes (see [6]) for instance may provide good modeling per-
spectives to link the parameters but this has been left aside for future work.
7 Applications
In spite of the limitations discussed above we used this model to backtest the strategy on
real data. We rapidly discuss the change that have to be made to the model and the way
backtests have been carried out. Then we present the result on the French stock AXA.
6If the optimal quote changed, the remaining part of the order is canceled before a new order is inserted.
Otherwise, the market maker may just insert a new order so that the cumulated size of his orders is δq.
18

7.1 Empirical use
Before using the above model in reality, we need to discuss some features of the model that
need to be adapted before any backtest is possible.
First of all, the model is continuous in both time and space while the real control problem
under scrutiny is intrinsically discrete in space, because of the tick size, and in time, because
orders have a certain priority and changing position too often reduces the actual chance to
be reached by a market order. Hence, the model has to be reinterpreted in a discrete way.
In terms of prices, quotes must not be between two ticks and we decided to ceil or floor the
optimal quotes with probabilities that depend on the respective proximity to the neighboring
quotes. In terms of time, an order is sent to the market and is not canceled nor modified
for a given period ∆t(say 20 or 60 seconds), unless a trade occurs and, though perhaps
partially, fills one of the market maker’s orders. Now, when a trade occurs and changes the
inventory or when an order stayed in the order book for longer than ∆t, then the optimal
quotes on both sides are updated and, if necessary, new orders are inserted.
Now, concerning the parameters, σ,Aand kcan be calibrated easily on trade-by-trade limit
order book data while γhas to be chosen. However, it is well known by practitioners that
Aand khave to depend at least on the actual market bid-ask spread. Since we do not
explicitly take into account the underlying market, there is no market bid-ask spread in
the model. Thus, we simply chose to calibrate kand Aas functions of the market bid-ask
spread, making then an off-model hypothesis.
Turning to the backtests, they were carried out with trade-by-trade data and we assumed
that our orders were entirely filled when a trade occurred above (resp. below) the ask (resp.
bid) price quoted by the market maker.
7.2 Results
To present the results, we chose to illustrate the case of the French stock AXA on November
2nd 2010.
We first show the evolution of the inventory and we see that this inventory mean-reverts
around 0.
10:00 10:30 11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30 15:00 15:30 16:00
−4
−3
−2
−1
0
1
2
3
4
5
Figure 7: Inventory when the strategy is used on AXA (02/11/2010) from 10:00 to 16:00 with
γ= 0.05 Tick−1
19

Now, to better understand the very nature of the strategy, we focused on a subperiod
of 20 minutes and we plotted the state of the market both with the quotes of the market
maker. Trades occurrences involving the market maker are signalled and we can see on the
following plot the corresponding evolution of the inventory.
12:30 12:32 12:34 12:36 12:38 12:40 12:42 12:44 12:46 12:48 12:50
13.2
13.21
13.22
13.23
13.24
13.25
13.26
13.27
13.28
Figure 8: Details for the quotes and trades when the strategy is used on AXA (02/11/2010) with
γ= 0.05 Tick−1. Thin lines represent the market while bold lines represent the quotes of the
market maker. Dotted lines are associated to the bid side while plain lines are associated to the
ask side. Black points represent trades in which the market maker is involved.
12:30 12:32 12:34 12:36 12:38 12:40 12:42 12:44 12:46 12:48 12:50
−4
−3
−2
−1
0
1
2
3
Figure 9: Details for the inventory when the strategy is used on AXA (02/11/2010) with γ=
0.05 Tick−1
20

Conclusion
In this paper we presented a model for the optimal quotes of a market maker. Starting with
the model by Avellaneda ans Stoikov [5] we introduced a change of variables7that allows
to find semi-explicit expressions for the quotes. Then, we exhibited the asymptotic value of
the optimal quotes and argued that the asymptotic values were very good approximations
for the quotes even for rather small times. Closed-form approximations were then obtained
using spectral arguments. The model is finally backtested on real data and the results are
promising.
7In a companion paper (see [14]) we used a change of variables similar to the one introduced above to solve
the Hamilton-Jacobi-Bellman equation associated to an optimal execution problem with passive orders.
21

Appendix
Proof of Proposition 1:
This is the classical PDE representation of a stochastic control problem with jump pro-
cesses.
Proof of Proposition 2 and Theorem 2:
Let’s consider a solution (vq)qof (S) and introduce u(t, x, q, s) = −exp (−γ(x+qs)) vq(t)−γ
k.
Then:
∂tu+1
2σ2∂2
ssu=−γ
k
˙vq(t)
vq(t)u+γ2σ2
2q2u
Now, concerning the bid quote, we have:
sup
sb
λb(sb, s)hu(t, x −sb, q + 1, s)−u(t, x, q, s)i
= sup
sb
Ae−k(s−sb)u(t, x, q, s)"exp γ(sb−s)vq+1(t)
vq(t)−γ
k
−1#
The first order condition of this problem corresponds to a maximum (because uis nega-
tive) and writes:
(k+γ) exp γ(sb∗−s)vq+1(t)
vq(t)−γ
k
=k
Hence:
s−sb∗=−1
kln vq+1(t)
vq(t)+1
γln 1 + γ
k
and
sup
sb
λb(sb, s)hu(t, x −sb, q + 1, s)−u(t, x, q, s)i
=−γ
k+γAexp(−k(s−sb∗))u(t, x, q, s)
=−γA
k+γ1 + γ
k−k
γvq+1(t)
vq(t)u(t, x, q, s)
Similarly,
sa∗−s=1
kln vq(t)
vq−1(t)+1
γln 1 + γ
k
and
sup
saλa(sa, s) [u(t, x +sa, q −1, s)−u(t, x, q, s)]
=−γ
k+γAexp(−k(sa∗−s))u(t, x, q, s)
=−γA
k+γ1 + γ
k−k
γvq−1(t)
vq(t)u(t, x, q, s)
22

Hence, putting the three terms together we get:
∂tu(t, x, q, s) + 1
2σ2∂2
ssu(t, x, q, s)
+ sup
sb
λb(sb, s)hu(t, x −sb, q + 1, s)−u(t, x, q, s)i
+ sup
saλa(sa, s) [u(t, x +sa, q −1, s)−u(t, x, q, s)]
=−γ
k
˙vq(t)
vq(t)u+γ2σ2
2q2u−γA
k+γ1 + γ
kk
γvq+1(t)
vq(t)+vq−1(t)
vq(t)u
=−γ
k
u
vq(t)"˙vq(t)−kγσ2
2q2vq(t) + A1 + γ
k−1+ k
γ(vq+1(t) + vq−1(t))#= 0
Now, noticing that the terminal condition for vqis consistent with the terminal condition
for u, we get that uverifies (HJB).
Proof of Theorem 1 and Theorem 3:
Before starting the very proof, let’s introduce the necessary functional framework.
Let’s introduce H=nu∈ℓ2(Z),Pq∈Zαq2u2
q+η(uq+1 −uq)2<+∞o.His a Hilbert
space equipped with the scalar product:
hv, wiH=X
q∈Z
αq2vqwq+η(vq+1 −vq)(wq+1 −wq),∀v, w ∈H
The first preliminary lemma indicates that the ℓ2(Z)-norm can be controlled by the H-
norm.
Lemma 1.
∃C > 0,∀w∈H, kwkℓ2(Z)≤CkwkH
Proof:
∀w∈H, kwk2
ℓ2(Z)=|w0|2+X
q∈Z∗
|wq|2≤ |w0|2+1
αkwk2
H
≤(|w1|+|w0−w1|)2+1
αkwk2
H≤
 r1
α+r1
η!2
+1
α
kwk2
H
A second result that is central in the proof of our results is that His compactly embedded
in ℓ2(Z).
Lemma 2. His compactly embedded in ℓ2(Z)
23

Proof:
To prove the result we consider a bounded sequence (sk)k∈Nin HN. For each k∈Nwe
introduce ˆskdefined by:
ˆsk
0=sk
0,∀q∈Z∗,ˆsk
q=qsk
q
For (sk)k∈Nis a bounded sequence in HN, (ˆsk)k∈Nis a bounded sequence in ℓ2(Z)N. Hence,
we can find ˆs∞∈ℓ2(Z) so that there exists a subsequence indexed by (kj)j∈Nsuch that
(ˆskj)j∈Nweakly converges toward ˆs∞in ℓ2(Z).
Now, we can define s∞∈Hby the inverse transformation:
s∞
0= ˆs∞
0,∀q∈Z∗, s∞
q=1
qˆs∞
q
and it is easy to check that (skj)Nconverges in the ℓ2(Z) sense toward s∞. Indeed,
kskj−s∞k2
ℓ2(Z)≤ |skj
0−s∞
0|2+X
q∈Z∗
1
q2|ˆskj
q−ˆs∞
q|2
=|ˆskj
0−ˆs∞
0|2+X
q6=0,|q|≤N
1
q2|ˆskj
q−ˆs∞
q|2+X
|q|>N
1
q2|ˆskj
q−ˆs∞
q|2
The first two terms tend to 0 because (ˆskj)Nweakly converges in ℓ2(Z) towards ˆs∞. The
last one can be made smaller than any ǫ > 0 as Nbecomes large because P|q|>N 1
q2tends
to zero as Ntends to infinity and (kˆskj−ˆs∞kℓ2(N))j∈Nis a bounded sequence.
Now, we are going to consider a linear operator Lthat is linked to the system (S). Lis
defined by
L:f∈ℓ2(Z)7→ v∈H⊂ℓ2(Z)
where
∀q∈Z, αq2vq−η(vq+1 −2vq+vq−1) = fq
We need to prove that Lis well-defined and we use the weak formulation of the equation.
Lemma 3. Lis a well-defined linear (continuous) operator.
Moreover ∀f∈ℓ2(Z),∀w∈H, hLf , wiH=hf, wiℓ2(Z)
Proof:
Let’s consider f∈ℓ2(Z).
Because of Lemma 1, w∈H7→ hf, wiℓ2(Z)is continuous. Hence, by Riesz representation
Theorem there exists a unique v∈Hsuch that ∀w∈H, hv, wiH=hf , wiℓ2(Z).
This equation writes
∀w∈H, X
q∈Z
fqwq=X
q∈Z
αq2vqwq+η(vq+1 −vq)(wq+1 −wq)
=αX
q∈Z
q2vqwq+ηX
q∈Z
(vq+1 −vq)wq+1 −ηX
q∈Z
(vq+1 −vq)wq
=αX
q∈Z
q2vqwq+ηX
q∈Z
(vq−vq−1)wq−ηX
q∈Z
(vq+1 −vq)wq
24

This proves ∀q∈Z, αq2vq−η(vq+1 −2vq+vq−1) = fq.
Conversely, if ∀q∈Z, αq2vq−η(vq+1 −2vq+vq−1) = fq, then we have by the same
manipulations as before that:
∀w∈H, hv, wiH=hf, wiℓ2(Z)
Hence Lis well-defined, obviously linear and Lf ∈H.
Now, if we take w=Lf we get
hLf, Lf iH=hf, Lf iℓ2(Z)≤ kfkℓ2(Z)kLfkℓ2(Z)≤Ckfkℓ2(Z)kLf kH
so that kLfkH≤Ckfkℓ2(Z)and Lis hence continuous.
Now, we are able to prove important properties about L.
Lemma 4. Lis a positive, compact, self-adjoint operator
Proof:
As far as the positiveness of the operator is concerned we just need to notice that, by
definition:
∀f∈ℓ2(Z),hLf, f iℓ2(Z)=kLfk2
H≥0
For compactness, we know that kLf kH≤Ckfkℓ2(Z)and Lemma 2 allows to conclude.
Now, we prove that the operator Lis self-adjoint.
Consider f1, f 2∈ℓ2(Z), we have:
hf1, Lf 2iℓ2(Z)=hLf1, Lf 2iH=hLf 2, Lf 1iH=hf2, Lf 1iℓ2(Z)=hLf1, f 2iℓ2(Z)
Now, we can go to the very proof of Theorem 1 and Theorem 3.
Step 1: Spectral decomposition and building of a solution when the terminal condition
is in ℓ2(Z).
We know that there exists an orthogonal basis (fk)k∈Nof ℓ2(Z) made of eigenvectors of
L(that in fact belongs to Hand we can take for instance ||fk||H= 1) and we denote λk>0
the eigenvalue8associated to fk(we suppose that the eigenvalues are ordered, λ0being the
largest one). We have:
αq2fk
q−η(fk
q+1 −2fk
q+fk
q−1) = 1
λkfk
q
Hence, if we want to solve (S′) that is similar to (S) but with a terminal condition
v(T)∈ℓ2(Z) instead of v(T) = 1 (where 1 stands for the sequence equal to 1 for all indices),
classical argument shows that we can search for a solution of the form v(t) = Pk∈Nµk(t)fk.
80 cannot be an eigenvalue. If indeed λk= 0 then ∀w∈H, hfk, wiℓ2(Z)=hLf k, wiH= 0. But because His
dense in ℓ2(Z), fk= 0.
25

Since ∀q∈Z
˙vq(t) = αq2vq(t)−η(vq−1(t) + vq+1(t))
=αq2vq(t)−η(vq+1(t)−2vq(t) + vq−1(t)) −2ηvq(t)
We must have dµk
dt (t) = ( 1
λk−2η)µk(t) and hence, since λk→0 we can easily define a
solution of (S′) by:
v(t) = X
k∈N
hv(T), f kiℓ2(Z)exp 2η−1
λk(T−t)fk
and the solution is in fact in C∞([0, T ], ℓ2(Z))
Step 2: Building of a solution when v(T) = 1
The first thing to notice is that H⊂ℓ1(Z) (indeed, w∈H⇒(qwq)q∈ℓ2(Z)⇒(wq)q∈
ℓ1(Z) by Cauchy-Schwarz inequality). Hence, the sequence v(T) that equals 1 at each index
is in H′, the dual of H. As a consequence, to build a solution of (S), we can consider a
similar formula:
v(t) = X
k∈N
h1, f kiH′,H exp 2η−1
λk(T−t)fk
Step 3: Uniqueness
Uniqueness follows easily from the ℓ2(Z) analysis. If indeed v(T) = 0 we see that we
must have that
∀k∈Z,hfk,˙v(t)iℓ2(Z)=dhfk, v(t)iℓ2(Z)
dt = ( 1
λk−2η)hfk, v(t)iℓ2(Z)
Hence, ∀k∈Z,hfk, v(T)iℓ2(Z)= 0 =⇒ ∀k∈Z,∀t∈[0, T ],hfk, v(t)iℓ2(Z)= 0 and v= 0.
Step 4: Asymptotics
To prove Theorem 3, we will show that the largest eigenvalue λ0of Lis simple and that
the associated eigenvector f0can be chosen so as to be a strictly positive sequence.
If this is true then we have ∀q∈Z, vq(0) ∼
T→∞ h1, f 0iH′,Hf0
qexp (2η−1
λ0)Tso that the
result is proved with
δb∗
∞(q) = 1
γln 1 + γ
k−1
kln f0
q+1
f0
q!δa∗
∞(q) = 1
γln 1 + γ
k+1
kln f0
q
f0
q−1!
and
ψ∗
∞(q) = −1
kln f0
q+1f0
q−1
f0
q2!+2
γln 1 + γ
k
Hence we just need to prove the following lemma:
Lemma 5. The eigenvalue λ0is simple and any associated eigenvector is of constant sign
(in a strict sense).
26

Proof:
Let’s consider the following characterization of λ0and ofthe associated eigenvectors (this
characterization follows from the spectral decomposition):
1
λ0= inf
f∈ℓ2(Z)
kfk2
H
kfk2
ℓ2(Z)
= inf
kfkℓ2(Z)=1 kfk2
H= inf
kfkℓ2(Z)=1 X
q∈Z
αq2f2
q+η(fq+1 −fq)2
Let’s consider fan eigenvector associated to λ0. We have that:
X
q∈Z
αq2|fq|2+η(|fq+1| − |fq|)2≤X
q∈Z
αq2f2
q+η(fq+1 −fq)2
Hence, since |f|has the same ℓ2(Z)-norm as f, we know that |f|is an eigenvector asso-
ciated to λ0.
Now, since αq2|fq| − η(|fq+1| − 2|fq|+|fq−1|) = 1
λ0|fq|, if |fq|= 0 at some point q, we
have −η(|fq+1|+|fq−1|) = 0 and this induces |fq+1|=|fq−1|= 0, and |f|= 0 by immediate
induction. Since f6= 0, we must have therefore |f|>0.
This proves that there exists a strictly positive eigenvector associated to λ0.
Now, if the eigenvalue λ0were not simple, there would exist an eigenvector gassociated
to λ0with h|f|, giℓ2(Z)= 0. Hence, there would exist both positive and negative values in
the sequence g. But, in that case, since |g|must also be an eigenvector associated to λ0, we
must have equality in the following inequality:
X
q∈Z
αq2|gq|2+η(|gq+1| − |gq|)2≤X
q∈Z
αq2g2
q+η(gq+1 −gq)2
In particular, we must have that ||gq+1| − |gq|| =|gq+1 −gq|,∀q. This implies that ∀q,
either gqand gq+1 are of the same sign or at least one of the two terms is equal to 0. Thus,
since gcannot be of constant sign, there must exist ˆqso that gˆq= 0. But then, because
|g|is also an eigenvector associated to λ0we have by immediate induction, as above, that
g= 0.
This proves that there is no such gand that the eigenvalue is simple.
Step 5: Positiveness
So far, we did not prove that v, the solution of (S) was strictly positive. A rapid way to
prove that point is to use a Feynman-Kac-like representation of v. If (qs)sis a continuous
Markov chain on Zwith intensities ηto jump to immediate neighbors, then we have the
following representation for v:
vq(t) = Eexp −ZT
t
(αq2
s−2η)ds
qt=q
This representation guarantees that v > 0.
This ends the proof of Theorems 1 and 3
Proof of Proposition 3:
If dSt=µdt +σdWtthen the (HJB) equation becomes:
27

(HJB) 0 = ∂tu(t, x, q, s) + µ∂su(t, x, q, s) + 1
2σ2∂2
ssu(t, x, q, s)
+ sup
sb
λb(sb, s)hu(t, x −sb, q + 1, s)−u(t, x, q, s)i
+ sup
sa
λa(sa, s) [u(t, x +sa, q −1, s)−u(t, x, q, s)]
Using the same change of variables as in the proof of Proposition 2, we can consider a
family of strictly positive functions (vq)q∈Zis solution of the linear system of ODEs that
follows:
∀q∈Z,˙vq(t) = (αq2−βq)vq(t)−η(vq−1(t) + vq+1(t))
with vq(T) = 1, α=k
2γσ2,β=kµ and η=A(1 + γ
k)−(1+ k
γ).
Then u(t, x, q, s) = −exp (−γ(x+qs)) vq(t)−γ
kis a solution of (HJB) and the final condition
is satisfied.
Also, as in the proof of Proposition 2, the optimal quotes are given by:
sb∗(t, q, s) = s−−1
kln vq+1(t)
vq(t)+1
γln 1 + γ
k
sa∗(t, q, s) = s+1
kln vq(t)
vq−1(t)+1
γln 1 + γ
k
and the bid-ask spread follows straightforwardly.
To prove the counterparts of Theorem 1 and Theorem 3, we have to write:
˙vq(t) = (αq2−βq)vq(t)−η(vq−1(t) + vq+1(t))
="αq−β
2α2
vq(t)−η(vq−1(t)−2vq(t) + vq+1(t))#−2η+β2
4αvq(t)
Hence the operator Land the Hilbert space Hare modified but the results are the same
mutatis mutandis.
Proof of Proposition 4:
In this setting, the Bellman function is defined by:
u(t, x, q, s) = sup
Sa,Sb
E[−exp (−γ(XT+qTST−φ(qT)))|Xt=x, qt=q, St=s]
The (HJB) equation is:
(HJB) 0 = ∂tu(t, x, q, s) + 1
2σ2∂2
ssu(t, x, q, s)
+ sup
sb
λb(sb, s)hu(t, x −sb, q + 1, s)−u(t, x, q, s)i
+ sup
saλa(sa, s) [u(t, x +sa, q −1, s)−u(t, x, q, s)]
with u(T, x, q, s) = −exp (−γ(x+qs −φ(q))).
28

Using the same change of variables as in the proof of Proposition 2, we can consider a
family of strictly positive functions (vq)q∈Zis solution of the linear system of ODEs that
follows:
∀q∈Z,˙vq(t) = αq2vq(t)−η(vq−1(t) + vq+1(t))
with vq(T) = e−kφ(q),α=k
2γσ2and η=A(1 + γ
k)−(1+ k
γ).
Then u(t, x, q, s) = −exp (−γ(x+qs)) vq(t)−γ
kis a solution of (HJB) and the terminal con-
dition is satisfied.
Also, as in the proof of Proposition 2, the optimal quotes are given by:
sb∗(t, q, s) = s−−1
kln vq+1(t)
vq(t)+1
γln 1 + γ
k
sa∗(t, q, s) = s+1
kln vq(t)
vq−1(t)+1
γln 1 + γ
k
and the bid-ask spread follows straightforwardly.
Proof of Proposition 5:
The demonstration is the same as for Proposition 2, with the same change of variables.
The only difference here is that the system of ODEs for (vq)qcan be solved explicitly using
a matrix exponentiation. The characterization of f0is the same as in Proposition 2 once it
is noticed that the counterpart of Lis now (M+ 2ηI )−1.
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